ts Righ - Georgetown University

Nonlinear Response Of Strongly Intera ting Quantum Systems in

Nonequilibrium

A Thesis

submitted to the Fa ulty of the

Graduate S hool of Arts and S ien es

of Georgetown University

in partial ful�llment of the requirements for the

degree of

Do tor of Philosophy

in Physi s

By

Khadijeh Naja�, M.S .

Washington, DC

April 5, 2018

Copyright

© 2018 by Khadijeh Naja�

All Rights Reserved

ii

Nonlinear Response Of Strongly Intera ting Quantum Systems

in Nonequilibrium

Khadijeh Naja�, M.S .

Thesis Advisor: James K. Freeri ks, Ph.D. in Physi s

Abstra t

The re ent developments in experimental te hniques su h as pump-probe spe -

tros opy, new te hnologies in building two dimensional materials, and the advent of

highly tunable systems in the form of ultra old opti al latti es, trapped ions and

array of atoms have opened a new path to investigate the dynami s of strongly or-

related system out of equilibrium, whi h would not be possible with the onventional

methods. These new developments not only provided a powerful tool to study some of

the most fundamental questions in the nonequilibrium regime of quantum systems but

also made it possible to explore new phenomena whi h has not been observed before.

Motivated by some of these experiments, in this thesis, we study a wide range of

strongly orrelated systems out of equilibrium. We will use a ombination of theoret-

i al and omputational methods su h as sum rules, nonequilibrium Green's fun tion,

nonequilibrium dynami al mean-�eld theory (DMFT), and an exa t solution of the

non-intera ting model to address a wide spe trum of systems out of equilibrium. First,

we have generalized a formalism for the nth derivative of a time-dependent operator

in the Heisenberg representation and employ it to the spe tral sum rules in whi h we

obtain a qualitative understanding of the pump e�e t on ele tron-phonon oupling in

high Tc super ondu tors. Then, by using the nonequilibrium DMFT for the Fali ov-

Kimball model, we obtain the urrent-voltage pro�le of a multilayer devi e whi h

onsists of a single barrier region (usually insulator plane) onne ted to a number

of metalli leads in both sides. To improve the onservation of urrent and �lling

iii

a ross the barrier region, we further develop an optimization method. From a dif-

ferent aspe t, we use the DMFT solution of the Fali ov-Kimball model to enhan e

the riti al temperature of quantum ordering, whi h is a hallenging problem as the

riti al temperature lies below urrently a essible temperatures. We further propose

a few mixtures su h as Yb− Cs and Sr− Cs as a possible andidate for dete ting the

riti al temperature enhan ement e�e t. Finally, in the last hapter, we study some

of the most interesting dynami al quantities su h as the probability of revivals, the

light one velo ity, formation probabilities and Shannon information in the XY hain.

Although XY hain is a free fermioni system, it has been onsidered as one of the

most interesting models from both theoreti al and experimental views. Be ause this

model, not only manifests a quantum phase transition but it has been the subje t of

the array of trapped ions and neutral atoms whi h are the most promising andidates

for a quantum simulator. We show that the formation probabilities, revival probabili-

ties, and observed propagation velo ity are a tually state-dependent and non-trivially

predi table in the XY hain.

Index words: Nonequilbrium dynami al mean-�eld theory, Fali ov-Kimball

model, ele tron-phonon intera tion, high Tc super ondu tor,

Sum rules, Multilayer devi e, Current-voltage pro�le,

Antiferromagneti ordering, quantum ordering enhan ement,

Quantum spin hain, Formation probabilities, Shannon

information, Light- one velo ities, thesis

iv

Dedi ation

To my parents, Akbar and Batool.

v

A knowledgments

First and foremost I would like to thank my advisor, Professor. Jim Freeri ks,

who has supported me throughout my Ph.D. with his patien e and guidan e at key

moments while allowing me to work independently. It has been a true honor to work

with him. In parti ular, I appre iate all of his e�orts, en ouragements, and time to

make my Ph.D. experien e very produ tive and rewarding. Thanks to him, I had a

spe ta ular opportunity to work on multiple subje ts whi h have helped me tremen-

dously to deepen my intuition and knowledge in the �eld of strongly orrelated sys-

tems out of equilibrium.

Next, I would like to o�er my sin erest gratitude to my olleague and the best

ollaborator I ould ask for, Mohammad Ali Rajabpour, whom not only has been a

great person but also has thauht me how to be ome a persistent resear her. I am

truly grateful for all of his endless support in my professional areer and personal life

sin e 2010.

I would like also to take the opportunity to thank my dissertation ommittee:

Prof. Amy Liu, Edward Van Kueren and Hans Engler for their time and e�ort for

helping me through my dissertation. In parti ular, Prof. Amy Liu, who has been a

great support throughout my Ph.D. and an ex ellent example of a su essful female

physi ist.

I am also grateful to my other ollaborators: Prof. Ma iej Maska, Paul. S. Julienne,

Thomas Devereaux, Alexander F. Kemper, and Ja opo Viti who I had the honor to

work with on di�erent proje ts. Additionally, I would like to o�er my spe ial thanks

vi

to Woonki Chung, the systems administrator who has helped me with o asional

omputational issues.

During my graduate study, I have met many post-do s whose advi e and friendship

I have bene�ted from in my professional areer. I am espe ially grateful to Ehsan

Khatami, Oleg Matveev, Herbert Fotso, Karlis Mikelsons, Juan Carrasquilla, Rubem

Mondiani, Mohammad Maghrebi, Miles Stoudenmire, Andreas Dirks, Greg Boyd,

Louk Rademaker, and Valentine Stanev.

In my daily work, I have been blessed with a friendly and supportive group of

fellow students and post-do s. In parti ular, I would like to thank Jesús Cruz-Rojas,

Bry e Yoshimura, Oliver Albertini, Je� Cohn, Claudia Dessi, Mona Kaltho�, and

Manuel Weber, with whom I have enjoyed to share many s ienti� dis ussions and

unforgettable memories.

Prof. Freeri ks, provided me a fantasti opportunity to mentor multiple under-

graduate students. Parti ularly, I would like to thank Kahlil Dixon, Forest Yang, and

Alex Ja obi who ontributed to my Ph.D. resear h with their talent and enthusiasm.

I also would like to express my gratitude to Physi s department sta�; Amy Hi ks,

Thomas Lewis, Jennifer Liang, Jannet Gibson, and Mary Rashid who have been a

great support in administrative work.

Beyond physi s, my time at Georgetown was made memorable due to the many

friends that be ame a part of my life. More spe i� ally, I am grateful to the friend-

ship of Eri Urano, Maryam Shojaie, Azadeh Ghayomi, Afsaneh Ranghiani, Miriam

Bolanes, Azadeh Bagheri, Neghar Ghahremani, Samyeh Mahmoodian, Sara Shahabi,

and Behnaz Bagheri from whom I have bene�ted tremendously of their emotional

support.

Last but not least, I would like to express my sin ere gratitude to my parents

and family members for their endless support and love throughout my life. Thanks

vii

to them and espe ially my oldest brother Dariush Naja�, who made it possible for

me to pursue my dream of be oming a physi ist.

viii

Table of Contents

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Nonequilibrium dynami al mean �eld theory . . . . . . . . . . . . . . . 7

2.1 Nonequilibrium Green's fun tion formalism . . . . . . . . . . . . 8

2.2 Nonequilibrium dynami al mean �eld theory for Fali ov-Kimball

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Numeri al implementation of nonequilibrium DMFT . . . . . . . 25

3 Nonequilibrium spe tral moment sum rules for systems with ele tron-

phonon intera tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Spe tral sum rules for the retarded Green's fun tion . . . . . . 36

3.2 Nonequilibrium sum rules for the Holstein model . . . . . . . . . 44

3.3 Nonequilibrium sum rules for the Hubbard-Holstein model . . . 54

4 Current-voltage pro�le of a strongly orrelated materials heterostru ture

using non-equilibrium dynami al mean �eld theory . . . . . . . . . . . 73

4.1 Non equilibrium DMFT formalism for a multilayer devi e . . . . 75

4.2 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Fixing the boundary ondition . . . . . . . . . . . . . . . . . . . 81

4.4 Impurity solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 Cal ulating the urrent in multilayer devi e . . . . . . . . . . . . 88

4.6 Optimizing of the DMFT-Zip algorithm . . . . . . . . . . . . . . 99

4.7 Con lusion: Current-Voltage (I-V) pro�le . . . . . . . . . . . . . 102

5 Designing mixtures of ultra old atoms to boost Tc by using dynami al

mean �eld theory solution . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1 Intera tion of light and matter . . . . . . . . . . . . . . . . . . . 107

5.2 DMFT solution for mixture of heavy and light parti les in opti al

latti e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3 Enhan ing quantum order with fermions by in reasing spe ies

degenera y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Nonequilbrium dynami s of XY hain . . . . . . . . . . . . . . . . . . . 124

6.1 XY spin hain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2 Dynami s of observables in XY spin hain . . . . . . . . . . . . 130

ix

6.3 On the possibility of omplete revivals after quantum quen hes

to a riti al point . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.4 Light- one velo ities after a global quen h in a non-intera ting

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Formation probabilities and Shannon information and their time

evolution after quantum quen h in the transverse-�eld XY hain 173

7 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8 Publi ation List of Khadijeh Naja� . . . . . . . . . . . . . . . . . . . . 212

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

x

List of �gures

2.1 Kadano�-Baym-Keldysh ontour . . . . . . . . . . . . . . . . . . . . 12

2.2 Mapping a latti e model into a single impurity problem . . . . . . . 18

2.3 S hemati of dis retization on the Keldysh-Baym ontour . . . . . . . 26

2.4 Boundary ondition on the Keldysh-Baym ontour . . . . . . . . . . . 28

3.1 The s hemati pi ture of tr-Arpes experiment . . . . . . . . . . . . . 33

3.2 Time-resolved spe tra of high Tc uprate super ondu tor . . . . . . . 35

4.1 S hemati pi ture of a multilayer devi e . . . . . . . . . . . . . . . . 75

4.2 Retarded Green's fun tion in bulk and barrier . . . . . . . . . . . . . 91

4.3 Retarded Green's fun tion at the �rst plane . . . . . . . . . . . . . . 92

4.4 Retarded Green's fun tion at one and two adja ent planes to the barrier 93

4.5 Lo al lesser Green's fun tion in the bulk at low temperature T = 0.01 94

4.6 Current through the multilayer devi e. . . . . . . . . . . . . . . . . . 95

4.7 Current of multilayer devi e in the presen e of ele tri �eld A = π/20 97

4.8 Current and �lling through the multilayer devi e for A = 6π/20 . . . 98

4.9 Current and �lling through the multilayer devi e in the presen e of

ele tri �eld for A = 2π/20 . . . . . . . . . . . . . . . . . . . . . . . 98

4.10 Optimization of urrent for di�erent values of ve tor potential with

single layer nonzero ele tri �eld . . . . . . . . . . . . . . . . . . . . . 101

4.11 Comparing the urrent obtained from the optimization algorithm . . 102

4.12 Current-Voltage pro�le for a multilayer devi e . . . . . . . . . . . . . 103

5.1 Enhan ement of quantum ordering riti al temperature . . . . . . . . 105

5.2 Comparison of the 2D DMFT and MC . . . . . . . . . . . . . . . . . 118

5.3 The maximal riti al temperature plotted as a fun tion of mobile

fermion degenera y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1 Di�erent riti al regions in the quantum XY hain . . . . . . . . . . 128

6.2 S hemati set up of quen h of trapped ion experiment . . . . . . . . . 132

6.3 Distribution of domain size of a one dimensional spin hain with 53 spins133

6.4 Di�erent regions in the phase diagram of the quantum XY hain . . 137

6.5 Logarithmi �delity for the periodi riti al Ising hain starting . . . 140

6.6 Logarithmi �delity for the open riti al Ising hain . . . . . . . . . . 141

6.7 Group velo ity vφ with respe t to φ . . . . . . . . . . . . . . . . . . . 142

6.8 Maximum group velo ity . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.9 Los hmidt e ho on the Ising line for periodi and open hains . . . . 144

6.10 Los hmidt e ho on the XY riti al line for periodi and open hains . 144

6.11 Los hmidt e ho on the line h = 0.8 for periodi and open boundary

ondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

xi

6.12 ∆ln(t) for the XX hain with di�erent initial states . . . . . . . . . . 157

6.13 ∆ln(t) for the XY hain with various γ and h parameters . . . . . . . 159

6.14 |Cln|2, |Fln|2, and ∆ln(t) for the XY hain . . . . . . . . . . . . . . . 160

6.15 The ontinuous blue urve is the fun tion −Ai2(−X) . . . . . . . . . 163

6.16 The evolution of the entanglement entropy in the XY hain . . . . . . 166

6.17 ve�max(2) and ve�

max(1) for di�erent values of a and h . . . . . . . . . . . 172

6.18 Π(l, L)− αl for periodi system with total length L = 2000 . . . . . . 187

6.19 Π(l, L)− αl for open system with total length L = 2000 . . . . . . . . 188

6.20 The oe� ient of the logarithmi term for two on�gurations . . . . . 190

6.21 Π(l, L)−αl for periodi system with total length L = 300 with respe t

to l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.22 Values of f(x), αmin and αmax with respe t to x . . . . . . . . . . . . 196

6.23 The ontributions of di�erent ranks k in the Shannon information for

two sizes l = 14 and 26 . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.24 The error E(xm, l) in the evaluation of Shannon information oming

from the trun ation at the rank k = xml . . . . . . . . . . . . . . . . 198

6.25 The ontributions of di�erent ranks k in the Shannon information . . 199

6.26 The evolution of logarithmi formation probability of di�erent on�g-

urations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.27 The evolution of Shannon information of a subsystem with di�erent sizes201

6.28 The mutual information between a pair of dimers . . . . . . . . . . . 202

6.29 The evolution of mutual information between two adja ent subsystems

in two di�erent ases . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

xii

List of tables

6.1 Properties of the four L × L blo ks of the matrix T for a quadrati

Hamiltonian (6.37). The notation is obvious and time dependen e is

omitted here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2 Fitting parameters for the logarithmi formation probabilities . . . . 187

6.3 Fitting parameters for the logarithmi formation probability of anti-

ferromagneti on�gurations . . . . . . . . . . . . . . . . . . . . . . . 191

6.4 Fitting parameters for di�erent on�gurations . . . . . . . . . . . . . 192

A1 Shannon information al ulated for sizes l = 1, 2, ..., 39 . . . . . . . . 207

xiii

1

Introdu tion

� More often seems like it should be more of the same, but more is di�erent.�

� P.W. Anderson

The �eld of nonequilibrium phenomena of strongly orrelated systems has be ome

one of the most a tive and hallenging parts of ondensed matter physi s. The strong

intera tion between ele tron, phonon, and spin degrees of freedom in strongly or-

related systems has reated a variety of interesting phases of matter su h as Mott

insulators (metal to insulator transition), super ondu tivity, harge density waves,

and antiferromagneti ordering to name a few. The number of exoti phenomena

with emerging phases of the matter even in reases as we drive the system out of equi-

librium [1℄. In general, theoreti al study of these systems be omes hallenging be ause

the strongly orrelated systems have a non-perturbative nature, and most of the usual

te hniques whi h are based on a perturbative approa h annot be applied. Moreover,

in the nonequilibrium ase, one needs to study the time evolution of the many-body

system whi h be omes problemati as the Hilbert spa e grows exponentially. The

exponential growth of the Hilbert spa e also remains a problem even in the equilib-

rium ase. However, there has been a number of omputational methods developed

for studying strongly orrelated systems with di�erent hara teristi s, ea h one with

its own advantages and limitations. In addition, various developments in experiments

with higher ontrol and a ura y have provided a great opportunity to study the

1

dynami s of su h systems. Here, we provide a list of su h experimental developments

on whi h we will be fo using in this thesis:

1. Developments in ultrafast pump-probe spe tros opy whi h have provided a

powerful tool to study the ex itation and relaxation dynami s of orrelated ele troni

and phononi degrees of freedom.

2. Designing and building two-dimensional materials that manifest di�erent ele -

troni properties from their bulk omponents, whi h leads to new emerging phe-

nomena.

3. Development of opti al latti es and trapped ion hain that provides a highly

tunable and ontrollable system.

The nonlinear behavior of a wide range of strongly orrelated systems whi h are

driven into nonequilibrium states will be the subje t of this thesis. For this purpose,

we have used various theoreti al and omputational methods of whi h we will provide

a brief des ription as follows. As we know, one of the simplest models to des ribe the

strongly orrelated systems is the Hubbard model whi h is de�ned as,

HHub = −∑

ijσ

tij(c†iσcjσ + c†jσciσ) + U

∑

i

ni↑ni↓ (1.1)

where throughout this thesis, c†i and cj indi ate the fermioni reation and annihilator

operator on site i and j, respe tively, and U is the Coulomb repulsion. However, ea h

hapter of this thesis will be devoted to slightly di�erent model in order to investigate

the various intera tions between harge, spin, and phononi degrees of freedom. Below,

�rst, we will des ribe some of the related experiments as a motivation to our work,

and subsequently, des ribe the models and methods that we have used to study ea h

parti ular system in nonequilibrium regime.

In hapter 2, we will fo us on the simpli�ed version of the Hubbard model, the

so- alled Fali ov-Kimbal model where only one of the spe ies of ele trons hop on the

2

latti e ( ele tron) and the other spe ies is lo alized (f ele tron) [2℄:

HFK = −∑

ij

tij(c†icj + c†jci) + Ucf

∑

i

c†icif†i fi + Ef

∑

i

f †i fi, (1.2)

where, Ucf is the intera tion between itinerant and lo alized ele trons, and Ef is the

energy of the lo alized ele trons. The advantages of the Fali ov-Kimball model is that

the dynami al mean �eld theory(DMFT) an be solved exa tly as nf = f †i fi ommutes

with Hamiltonian and onsequently, one an obtain the Green's fun tion with respe t

to c ele tron [3℄. The DMFT is onsidered as one of the most powerful methods to

study strongly orrelated systems out of equilibrium. In hapter 2, �rst, we provide

an introdu tion to nonequilibrium Green's fun tion whi h is essential to des ribe the

DMFT. Then, we explain the nonequilibriumDMFT whi h was su essfully developed

by Freeri ks, Turkowski, and Zlati¢ (2006) where they employed the Kadano�-Baym

formalism to des ribe the real-time evolution of a latti e system deriven by an external

ele tri �eld [4℄. Then, we provide a brief des ription of how one an implement

the nonequilibrium DMFT whi h has been proposed by Freeri ks et al in Ref.[5℄.

Furthermore, we will use the DMFT algorithm in hapters 4 and 5.

As we mentioned, the pump-probe spe tros opy provides a powerful tool to ex ite

individual ele tron or phonon degrees of freedom [6, 7℄. Subsequently, in hapter 3,

we will fo us on a re ent study where the role of the pump on the weakening of the

ele tron-phonon oupling in a high Tc Cuprate based super ondu tor[8, 9℄. In order

to investigate the ele tron-phonon intera tion and its interplay with the ele tron-

ele tron intera tion, we will onsider the Hubbard-Holstein model whi h is des ribed

3

as

HHH(t) = −∑

ijσ

tijc†iσcjσ +

∑

i

Uini↑ni↓

+∑

iσ

[gixi − µi]niσ

+∑

i

1

2Mip2i +

∑

i

1

2κix

2i

, (1.3)

where in addition to the hopping and on-site Coulomb intera tion, we have to add

the Hamiltonian of the harmoni os illator and it's oupling with harges. In the

above Hamiltonian, p and x indi ate the momentum and position of the phonon

degrees of freedom, and µ indi ates the hemi al potential. Furthermore, g is the

oupling strength between ele tron and phonon degrees of freedom. First, we derive

a general pro edure for evaluating the nth derivative of a time-dependent operator

in the Heisenberg representation and employ this approa h to al ulate the spe tral

fun tion of retarded Green's fun tion. The sum rules are an exa t formalism whi h an

be applied both in equilibrium and nonequilibrium to study the nonlinear behavior of

strongly orrelated systems. We will losely follow the generalization of the sum rule

in nonequilibrium whi h has been previously obtained by Freeri ks et al for various

models[10, 11, 12, 13℄ and report the results of the spe tral moment sum rules both for

the Holstein model whi h has been published in Ref. [14℄ and, the Hubbard-Holstein

model whi h is near submission.

In addition, from a di�erent aspe t, strongly orrelated materials entered a new

era with the development of thin-�lm growth te hnology to reate two-dimensional

materials whi h have revolutionized our daily life. One ategory of su h materials is

transition metal oxides (TMO) whi h is an ideal andidate for the study of ele tron

orrelation as ele trons in transition metals lie in narrow d orbitals whi h ause strong

4

intera tions. This be omes even more fas inating as new phenomena an emerge in

the interfa e of TMOs [15℄. For example, in a re ent study, two layers of intrinsi

insulators LaAlO3 and SrTiO3 grown on top of ea h other forms a highly ondu tive

layer at the interfa e [16℄. In fa t. one of the most important hara teristi s of these

systems is the urrent-voltage pro�le. In hapter 4, �rst, we introdu e a multilayer

devi e onsisting of an array of metal and insulator planes, and then, we provide

a summary of urrent biasing DMFT method for the Fali ov-Kimball model whi h

has been developed by Freeri ks [17℄. In order to maintain the urrent and �lling

onservation, we develop an optimization method. We report the results before and

after the optimization in whi h manifest an improvement in both urrent and �lling

results. Finally, we present a urrent-voltage pro�le for the di�erent regime of barrier

planes.

Furthermore, the DMFT solution of Fali ov-Kimball model provides a path to

enhan e the quantum ordering in the mixture of heavy and light mixture [18℄. In

hapter 5, we fo us on the hallenges of dete ting a quantum ordering at low temper-

ature. Although, there has been an intensive e�ort to dete t the antiferromagneti

phase of Hubbard model, a hieving low temperature below the Neel ordering tem-

perature still remains hallenging[19, 20℄. Instead, we use the results of al ulation of

spin and harge sus eptibility of Fali ov-Kimball model obtained by Freeri ks et al in

Ref. [18℄ whi h suggest that in reasing the degenera y of light(fermioni ) spe ies will

lead to an in rease in the riti al temperature of quantum ordering. First, we brie�y

explain the intera tion of light and matter whi h plays a ru ial role both in trapping

and ooling of atoms in ultra old atoms. Then, we losely follow the derivation of

harge sus eptibility of the Fali ov-Kimball model from Ref. [18℄ for a pedagogi al

reason. Finally, we report the published results where we have used the DMFT to al-

ulate the enhan ement of riti al temperature of the light-heavy mixture for di�erent

5

degenera ies of light spe ies and proposed a few di�erent mixtures whi h satisfy the

required riteria for su h enhan ement whi h an be onsidered as a good andidates

to manifest su h e�e ts[21℄.

Finally, we have devoted the last hapter of this thesis to study the exa t dynami s

of the free fermioni system. There are multiple reasons to study su h a system.

First, it turns out that one an map the free fermioni system into the XY quantum

hain[22℄ whi h provides a solid laboratory to study the quantum phase transition[23℄

and has importan e from a fundamental point of view. Furthermore, theXY quantum

hain plays an important role in quantum simulators su h as trapped ion experi-

ment [25, 26, 27, 28, 29℄ and old atoms [30℄. In this hapter, we �rst, introdu e

the XY model and we explain how one an map it onto the fermioni degrees of

freedom. Then, we report our published results about the dynami al properties of

the XY hain su h as time evolution of orrelation and entanglement, revival proba-

bility and �nally, the formation probability, and Shannon information and their time

evolution [31, 32, 33℄. Furthermore, we have al ulated the post-measurement entan-

glement entropy and full ounting statisti s of the quantum spin hain, whi h are

beyond the s ope of this thesis and we refer the interested reader to Refs.[34, 35℄.

6

2

Nonequilibrium dynami al mean field theory

�The simpli ities of natural laws arise through the omplexities of the lan-

guage we use for their expression.�

� Eugene Wigner

There exist a variety of methods to study systems in equilibrium su h as mean

�eld theory and the renormalization group. These methods allow us to understand

the experimental data of a omplex systems at low temperature by des ribing their

intera tions in terms of simple e�e tive models with relatively few degrees of freedom.

The problem be omes less lear for systems driven into the nonequilibrium regime,

as it is not obvious that the dynami s of a omplex system an be explained by the

dynami al des ription of simpli�ed models. The e�ort to �nd an analyti al approa h

to understand the time evolution of generi many body quantum systems started in

early 1960's, when S hwinger, Kadano�, Baym and, Keldysh developed the nonequi-

librium Green's fun tion formalism [36, 37, 38℄. This formalism is an extension of

the imaginary time equilibrium formulation [39℄. Later, the quantum kineti approa h

was developed by Rammer [40℄, whi h is an alternative method based on the nonequi-

librium Green's fun tion. The kineti approa h des ribes the evolution of a system

towards its thermal state on a long time s ale and it an not apture the dynami s of

the system at short times. Other methods su h as the density matrix renormalization

group (DMRG) [41℄, are dire tly involved in the evolution of wavefun tion and are

7

more suitable for one-dimensional systems [42, 43℄. In ontrast, the nonequilibrium

dynami al mean-�eld theory (DMFT) works e�e tively in the thermodynami limit

and it an be applied to both short and long times ales. Although the main restri tion

of DMFT lies in the lo al approximation of the self-energy, whi h may not be appro-

priate for systems with spatial nonlo al orrelations; these orrelations an be taken

into a ount by additional diagrammati and luster expansions. In this hapter, we

will explain the nonequilibrium dynami al mean-�eld theory. First, in se tion 2.1,

we will explain the nonequilibrium Green's fun tion formalism and will des ribe the

Kadano�-Baym formalism for the ontour ordered Green's fun tion and the Keldysh

formalism for the nonequilibrium steady state. In se tion 2.2, we will explain the

nonequilibrium dynami al mean-�eld theory and in parti ular, we will dis uss an

exa t formalism for the the Fali ov-Kimbal model in whi h it was su essfully gen-

eralized by Freeri ks and et al in Ref. [4℄. Then, in se tion 2.3 we will explain how

one an imply the omputational for the DMFT formalism in nonequilbrium whi h

is proposed by Freeri ks et al in Ref. [5℄.

2.1 Nonequilibrium Green's fun tion formalism

The main goal of many body theory is to al ulate orrelation fun tions. In this se -

tion, we will des ribe the nonequilibrium Green's fun tion approa h whi h an be

appli able to arbitrary time evolution of orrelated systems and ontains informa-

tion regarding the orrelation of the strongly orrelated system. First, we will review

the Heisenberg representation as we will use it frequently in this thesis. Then, we

explain the Kadano�-Baym formalism as a general formulation of a Green's fun tion

in nonequilbrium, whi h an immediately be des ribed by the Heisenberg representa-

tion of the operators. The nonequilibrium Green's fun tion is a general formalism in

8

whi h does not involve any assumption about the distribution of strongly orrelated

system out of equilibrium as the time evolution of the distribution fun tion an be

obtained from the initial distribution. So in this sense the nonequilibrium Green's

fun tion formalism is an initial value problem while in a di�erent formalism needed

for open system whi h is based on boundary ondition problem. For open systems

with driving for es and dissipation, the system is expe ted to rea h a nonequilibrium

steady state, and onsequently, the Green's fun tion does not hange with time and

they an be solved by the boundary ondition �xed by the bath. This spe i� ase,

is known as the Keldysh formalism, in whi h we will explain afterwards. Finally, in

the last part of this se tion, we will review the results for the equilibrium ase in the

eigenstate basis whi h is known as Lehmann representation.

2.1.1 Heisenberg representation

Before explaining the nonequilibrium formalism for the Green's fun tion, we �rst

review the Heisenberg representation as a powerful tool to study the time evolution

of physi al system in nonequilibrium. For simpli ity, we start with a general time-

dependent impurity Hamiltonian denoted by Hs(t) in the S hrödinger representation.

We know that, in the Heisenberg representation, the time dependen e is en oded in

the operator AH(t), whi h is related to the S hrödinger representation operator As(t)

by

AH(t) = U †(t, t0) As(t) U(t, t0), (2.1)

where t0 is the referen e time (whi h in our ase is equal to −∞) and U(t, t0) is the

evolution operator whi h is de�ned as

U(t, t0) = Te−i

∫ tt0

dtHs(t), (2.2)

9

and

U †(t, t0) = T ei∫ tt0

dtHs(t), (2.3)

Here, T is the time ordering operator whi h orders the operator with later time to

the left. A ordingly T is the anti time ordering operator whi h a ts in the opposite

way. Moreover, the evolution operator satis�es the equation of motion

i∂tU(t, t0) = Hs(t) U(t, t0). (2.4)

Now, let us derive a useful identity for the derivative of c(t). Starting from the de�-

nition of the time dependent operator

c(t) = T ei∫ tt0

dtHs(t) c Te−i

∫ tt0

dtHs(t), (2.5)

we see immediately that

dc(t)

dt= i e

i∫ tt0

dtHs(t)Hs(t) c e−i

∫ tt0

dtHs(t)

− i e−i

∫ tt0

dtHs(t) c Hs(t) e−i

∫ tt0

dtHs(t), (2.6)

whi h be omes

dc(t)

dt= iU †(t, t0) [Hs(t), c] U(t, t0). (2.7)

Now we onsider the following time-independent Hamiltonian

H0s (t) = H − µN = −µc†c, (2.8)

where H0s is the full Hamiltonian in the S hrödinger representation and, for simpli ity,

we absorb the hemi al potential into the de�nition of H0s (t). For this Hamiltonian,

we have

[H0s (t), c] = −µ[c†c, c] = µc, (2.9)

10

so that

dc(t)

dt= iµc(t), (2.10)

whi h is solved by

c(t) = eiµtc. (2.11)

In a similar way, using the ommutation relation [H0s , c

†] = −µ[c†c, c†] = −µc† we

�nd

c†(t′) = e−iµt′c†. (2.12)

2.1.2 Kadanoff-Baym-Keldysh ontour

As we mentioned, orrelation fun tions are one of the important quantities to des ribe

strongly orrelated systems. In this se tion, we fo us on single parti le Green's fun -

tions whi h provide information about the single ex itation and distribution of par-

ti les. We start with the ontour-ordered Green's fun tion whi h is de�ned as:

GCij(t, t

′) = − i〈TC ci(t)c†j(t

′)〉

= − iθC(t, t′)Tre−βH(−∞)ci(t)c

†j(t

′)/Z

+ iθC(t′, t)Tre−βH(−∞)c†j(t

′)ci(t)/Z. (2.13)

whi h is de�ned on the ontour C that runs from tmin to tmax, and ba k from tmax

to tmin and then, goes to tmin − iβ. In addition, β = 1/kBT is de�ned as the inverse

temperature and Z = Tre−βH(−∞). In this approa h, one onsiders the initial orrela-

tion by introdu ing it via time evolution on the imaginary axis, see �gure 2.1. The Tc

is time ordering on the ontour with ordering along tmin → tmax → tmin → tmin − iβ.

One an determine di�erent types of Green's fun tion by hoosing the time variables

11

Figure 2.1: Kadano�-Baym-Keldysh ontour with thermal state at t = tmin.

to lie on di�erent bran hes. For example, the time-ordered Green's fun tion is when

t and t′ both lie on the upper bran h,

GTij(t, t

′) = −iTr TT e−βH(−∞)ci(t)c

†j(t

′)/Z, (2.14)

while, the anti-time-ordered Green's fun tion is for the ase when both times lies on

the lower bran h,

GTij(t, t

′) = −iTr TT e−βH(−∞)ci(t)c

†j(t

′)/Z, (2.15)

where TT denotes anti-time ordering and a ts in opposite way of time ordering. Sim-

ilarly, if t lies on the upper bran h and t′ lies on the lower bran h, we get the lesser

Green's fun tion de�ned as

G<ij(t, t

′) = iTr e−βH(−∞)c†j(t′)ci(t)/Z, (2.16)

and greater Green's fun tion where t and t′ lies on the lower and upper bran h respe -

tively,

G>ij(t, t

′) = −iTr e−βH(−∞)ci(t)c†j(t

′)/Z. (2.17)

Subsequently, there is a ase where both terms lie on the imaginary axis,

GMij (τ, τ

′) = −〈T ci(τ)c†j(τ ′)〉. (2.18)

12

this ase has spe ial properties whi h we will explain in the end of this se tion. In

addition, there are Green's fun tions whi h are an be obtained from the mix of real

and imaginary bran hes whi h normally are useful when one studies the transient

behavior of the systems. In general, the Green's fun tion an be expressed as 3 × 3

matrix, where, for simpli ity, we drop the position dependen e,

G =

G++ G+− G+I

G−+ G−− G−I

GI+ GI− GII

, (2.19)

where the indi es +,−, and I orresponds to the time lying on the upper, lower and

imaginary bran hes, respe tively. In fa t, it turns out that in addition to the above

Green's fun tion that an be dire tly obtained from di�erent ordering on the ontour,

one may de�ne a few other important Green's fun tion known as retarded, advan ed,

and Keldysh Green's fun tion,

GRij(t, t

′) = −iθ(t− t′)〈{ci(t), c†j(t′)}〉, (2.20)

GAij(t, t

′) = iθ(t′ − t)〈{ci(t), c†j(t′)}〉, (2.21)

GKij (t, t

′) = −i〈[ci(t), c†j(t′)]〉, (2.22)

where the simple and urely bra ket denotes the ommuting and anti ommuting rela-

tion between fermioni operators, respe tively. In fa t, the lesser and greater Green's

fun tion an be written in terms of retarded, advan ed and Keldysh Green's fun tion:

G<ij(t, t

′) =1

2[GK

ij (t, t′)−GR

ij(t, t′) +GA

ij(t, t′)]

G>(t, t′) =1

2[GK

ij (t, t′) +GR

ij(t, t′)−GA

ij(t, t′)], (2.23)

13

in whi h one may get GKij (t, t

′) = G<ij(t, t

′) + G>(t, t′). From the de�nition of the

Green's fun tion it is straightforward to �nd a relation between the Green's fun tion

and their Hermitian onjugates:

GRij(t, t

′)∗ = GAij(t

′, t),

G<,>,Kij (t, t′)∗ = −G<,>,K

ij (t′, t), (2.24)

Now, we fo us on the Matsubara Green's fun tion as we mentioned that has a few

advantages: it is always translationally invariant GMij (τ, τ

′) = GMij (τ − τ ′), it is Her-

mitian GMij (τ)

∗ = GMij (τ), and it is anti-periodi for fermions GM

ij (τ) = −GMij (τ + β).

Furthermore, one an perform a Fourier transform into the frequen y domain by

GMij (τ) = T

∑

n

e−iωn(τ)GMij (iωn), (2.25)

where

GMij (iωn) =

∫ β

0

dτeiωn(τ)GMij (τ), (2.26)

where ωn = (2n+1)πT for fermioni ase. Finally, in the Green's fun tion formalism,

the intera tion an be taken into a ount by introdu ing the self-energy, whi h is

de�ned on the ontour and it satis�es the same symmetry and boundary ondition

as the Green's fun tion. The self-energy may be introdu ed by the Dyson equation :

G(t, t′) = G0(t, t′) +

∫

Cdt

∫

Cdt′G0(t, t)Σ(t, t′)G(t′, t

′). (2.27)

In the next se tion, we will explain how one may employ the dynami al mean-�eld

theory to obtain the self-energy of systems with lo al intera tions.

2.1.3 Keldysh formalism for nonequilibrium steady state

It turns out that, it is possible to simplify the nonequilibrium formalism for open

systems. This is possible, be ause in this ase, the energy pumped into system by

14

external driving an dissipate through the onne tion with a heat bath, and the

system an rea h a steady state. Furthermore, one may also ignore the imaginary

bran h, as there is no orrelation between the initial state and time evolving states.

The reasoning for ignoring the initial orrelation arises from the dissipation in the

system and onne tion to the heat bath whi h provides large number of degrees of

freedom that in�uen es the long time dynami of the system by stripping out the

information from the initial states. So, this makes it possible to use the Keldysh

formalism. By this methodology, the ontour ordered Green's fun tion be omes a

2× 2 matrix,

G =

G++ G+−

G−+ G−−

. (2.28)

One an rewrite the above matrix with respe t to retarded, advan ed and, Keldysh

Green's fun tions by performing the Larkin-Ov hinkov transformation [45℄

G ≡ Lτ3GL† =

GR GK

0 GA

, (2.29)

where L and τ3 are de�ned as

L =1√2

1 −1

1 1

, (2.30)

and

τ3 =

1 0

0 −1

. (2.31)

We will return to the Keldysh formalism in hapter 4 and provide more information

of how it an be imply to obtain the urrent-voltage pro�le of a multilayer devi e.

15

2.1.4 Lehmann representation

Before ending this se tion, we also brie�y omment on the so alled Lehmann repre-

sentation in whi h one uses a set of eigenstates {|n〉} of the full Hamiltonian. In equi-

librium, the Hamiltonian is independent of time, and the Green's fun tion be omes a

fun tion of time di�eren e, trel = t′ − t. Now, if the energy eigenstates of the Hamil-

tonian are de�ned as, H|n〉 = En|n〉, then by onsidering the diagonal form of the

greater Green's fun tion in equation 2.17, we have

G>(k, t− t′) = −i 1Z 〈ck(t)c†k(t′)〉 = −i 1Z∑

n

〈n|e−βHck(t)c†k(t

′)|n〉

= −i 1Z∑

nn′

e−βEn〈n|ck|n′〉〈n′|c†k|n〉ei(En−En′ )(t−t′). (2.32)

where in the se ond line, we have inserted 1 =∑

n |n〉〈n| and we have absorbed

the hemi al potential inside the Hamiltonian. In addition, by rewriting the ck(t)

in Heisenberg representation, we have used the following identity: 〈n|ck(t)|n′〉 =

〈n|eiHt ck e−iHt|n′〉 = ei(En−En′)(t−t′)〈n|ck|n′〉. By performing the Fourier transform

with respe t to trel = t− t′ into the frequen y domain, we get:

G>(k, ω) =−2πi

Z∑

nn′

e−βEn〈n|ck|n′〉〈n′|c†k|n〉δ(En −En′ + ω), (2.33)

similarly, for the lesser Green's fun tion we get:

G<(k, ω) =2πi

Z∑

nn′

e−βEn〈n|c†k|n′〉〈n′|ck|n〉δ(En − En′ − ω)

=2πi

Z∑

nn′

e−βEn′ 〈n′|c†k|n〉〈n|ck|n′〉δ(En′ − En − ω)

. =2πi

Z∑

nn′

e−β(En+ω)〈n′|c†k|n〉〈n|ck|n′〉δ(En′ −En − ω)

= −G>(k, ω)e−βω. (2.34)

where we have ex hanged the En and En′on the se ond line. In the ase of retarded

Green's fun tion, when we take the Fourier transform,we have to remind ourself about

16

the in�nitesimal onvergen e fa tor iη → i0+

GR(k, ω) = − i

Z

∫ ∞

0

dt

ei(ω+iη)t∑

nn′

e−βEn

(

〈n|ck|n′〉〈n′|c†k|n〉ei(En−En′ )t + 〈n|c†k|n′〉〈n′|ck|n〉e−i(En−En′ )t)

=1

Z∑

nn′

e−βEn

( 〈n|ck|n′〉〈n′|c†k|n〉ω + En − En′ + iη

+〈n|c†k|n′〉〈n′|ck|n〉ω −En + En′ + iη

)

=1

Z∑

nn′

〈n|ck|n′〉〈n′|c†k|n〉ω + En − En′ + iη

(e−βEn + e−βEn′ ). (2.35)

The imaginary part of of the retarded Green's fun tion is de�ned as the single parti le

spe tral fun tion,

A(k, ω) =−1

πIm[GR(k, ω)] =

1

Z∑

nn′

(e−βEn + e−βEn′ )|〈n|ci|n′〉|2δ(ω + En −En′)

=−1

πIm[GR(k, ω)] =

1

Z∑

nn′

e−βEn(1 + e−βω)|〈n|ci|n′〉|2δ(ω + En −En′)

= −i(1 + e−βω)G>(k, ω). (2.36)

where we have used the Dira identity

1ω+iη

= P 1ω− iπδ(ω). Consequently, we may

derive the following relations known as �u tuation-dissipation relations:

G<ij(ω) = 2πiAij(ω)f(ω),

G>ij(ω) = −2πiAij(ω)[1− f(ω)], (2.37)

where f(ω) = 11+eβω is the Fermi-Dira distribution.

2.2 Nonequilibrium dynami al mean field theory for Fali ov-Kimball

model

A stati mean �eld theory, su h as a Weiss mean-�eld theory (whi h maps a spin

system onto an e�e tive single site in the average �eld) has been known for a long time.

A similar stati mean-�eld theory for intera ting ele trons is the Hartree approa h

17

whi h takes an average, time-independent potential as an approximation for the

Coulombi intera tion. However, the orrelation of ele trons are time dependent,

as they move in the latti e and one needs to onsider their dynami features to

apture the physi al behavior of strongly orrelated system. The DMFT approa h

approximates a latti e problem with many degrees of freedom by means of a single

site e�e tive problem with fewer degrees of freedom. The underlying physi al idea is

to map the many-body latti e problem into a many-body lo al problem (an impu-

rity problem), intera ting with an e�e tive bath reated by all the other degrees of

freedom of other sites. The ru ial di�eren e between stati and dynami mean-�eld

arises from the fa t that the DMFT has been built upon the e�e tive theory for the

lo al frequen y-dependent Green's fun tion and the impurity site is onne ted with

the bath via time dependent oupling λ(t, t′) whi h mimi s the dynami s of the hop-

ping of ele trons in the latti e and it is determined in a self- onsistent formalism, see

�gure 2.2.

λ(t, t′)

Lattice model Single impurity

Figure 2.2: Mapping a latti e model into a single impurity problem with a time

dependent hybridization fun tion λ(t, t′). The urved arrow in the �gure demonstrates

the hopping of an ele tron from a site onto a nearest neighbor site in a latti e model

and hopping from a single impurity into the bath in the impurity model.

In this se tion, we will explain the DMFT for the Fali ov-Kimball model. As we

mentioned in introdu tion, the Fali ov-Kimball model des ribes itinerant ele trons

hopping in the ba kground of lo alized ele trons, whi h are denoted by c and f

18

ele trons, respe tively. For a reminder, we rewrite the Hamiltonian in Eq. 2.38:

HFK(t) = −∑

ij

tij(c†icj + c†jci) + Ucf (t)

∑

i

c†icif†i fi + Ef

∑

i

f †i fi, (2.38)

Noti e that, in this ase, the Hamiltonian is time-dependent. The DMFT approa h

has been su essful in des ribing the so alled Mott-Hubbard transition and it pro-

vides insight about the dynami s of the quantum system. For the half-�lled ase

(µ = Ucf (t → −∞)/2), half of the sites are �xed as they are o upied with lo al-

ized f ele trons. For large enough intera tions, the repulsion between lo alized and

ondu tion ele trons prevents the double o upan y of sites, and, sin e there are no

other empty sites, the system will be frozen and be ome an insulator. In addition,

the Fali ov-Kimball model has an advantage sin e the impurity model in DMFT an

be solved exa tly [3℄. In this se tion, we will demonstrate how to use DMFT to study

the time-dependent Fali ov-Kimball model. First, we will explain how one an �nd

the impurity solver for the Fali ov-Kimball model, whi h is normally the hardest part

of the DMFT algorithm and then, we will des ribe the DMFT algorithm.

2.2.1 Impurity problem solver

As we explained earlier, in the DMFT approa h, one maps the many body latti e

problem onto an impurity problem, that is the lo al many-body problem in the pres-

en e of an external dynami al mean-�eld. The idea in DMFT is to onsider the evolu-

tion operator with an additional time dependent �eld, the so- alled dynami al mean

�eld that mimi s the dynami s of the latti e many body problem. In general, any

time dependen e of the many-body problem, su h as hopping of ele trons on latti e

sites an be onsidered inside the dynami al mean-�eld parameter. In this se tion, we

will des ribe the impurity problem solver losely following the derivation proposed

by Freeri ks in Ref. [17℄. For simpli ity, we will solve the impurity problem in the

19

presen e of an evolution operator for H0s (t) and H1

s (t) separately, and we will use

those results to �nd the impurity Green's fun tion for the generalized Hamiltonian

HFKs (t) = −µc†c+ Eff

†f + Ucf(t)c†cf †f .

First, we onsider the impurity Hamiltonian in equation (2.8). Now, in the presen e

of the �eld, the time ordered Green's fun tion is de�ned as

GC(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉

= − iθC(t, t′)Tr{e−βH0

s (−∞)S(λC)c(t)c†(t′)}/Z0

+ iθC(t′, t)Tr{e−βH0

s (−∞)S(λC)c†(t′)c(t)}/Z0, (2.39)

where S(λC) is the evolution operator on the ontour whi h satis�es

S(λC) = TC exp[−i∫

Cdt

∫

Cdt′c†(t′)λC(τ, τ

′)c(t)], (2.40)

and the operators are in the Heisenberg representation. Moreover, the impurity par-

tition fun tion is de�ned as Z0(λ, µ) = TrC{TCe−βH0s (−∞)S(λC)}. Due to the presen e

of the evolution operator it is not easy to al ulate the tra e, so we use a fun tional

derivative to obtain the hange of the partition fun tion due to a small hange of λc

δZ0(λC, µ) = TrC{TCe−βH0s (−∞)δS(λC)}. (2.41)

Using the al ulus of variations, we obtain

δS(λC) = −iTC [S(λC)∫

Cdt

∫

Cdt′δλC(t, t

′)c†(t)c(t′)], (2.42)

Re alling the de�nition of the Green's fun tion in Eq 2.13, we obtain

δZ0(λc, µ) = −Z0(λ, µ)

∫

Cdt

∫

Cdt′δλC(t, t

′)GC,0(t′, t), (2.43)

whi h an be solved by

GC,0(t, t′) = −δ lnZ0(λC, µ)

δλC(t′, t). (2.44)

20

The next step is to solve the EOM to �nd the impurity Green's fun tion. Due

to the presen e of the time ordered produ t with respe t to two times in the double

integral, al ulating the derivative of S(λc) is ompli ated. Fortunately, one may split

the time evolution operator into two parts to simplify the time derivative

TC[S(λC) c(t)c†(t′)] = TC[S1(λC)]c(t)[S2(λc)]c

†(t′), (2.45)

with

S1(λc) = exp[−i∫ ∞

t

dt′′∫

Cdt′′′λc(t

′′, t′′′)c†(t′′)c(t′′′)], (2.46)

and

S2(λc) = exp[−i∫ t

−∞dt′′∫

Cdt′′′λc(t

′′, t′′′)c†(t′′)c(t′′′)]. (2.47)

Evaluating the time derivative of the Green's fun tion in equation(2.13), we obtain

∂GC,0(t, t′)

∂t′= −iδC(t, t′) + iµG0

C(t, t′)− i

∫

Cdt′′λC(t, t

′′)GC,0(t′′, t′), (2.48)

where the �rst term −iδC(t, t′) omes from the derivative of the step fun tion along

the ontour, while the se ond term is the expli it derivative of c(t) and the last term

omes from the time derivative of the evolution operator. It is straightforward to show

that the above equation an be solved by

GC,0(t, t′) = [(i∂ + µ)δC(t, t′)− λC(t, t

′)]−1. (2.49)

Using the same pro edure, we al ulate the impurity Green's fun tion for the following

Hamiltonian.

H1s (t) = −µc†c+ Ucf(t)c

†c, (2.50)

where the Green's fun tion is de�ned as

GC,1(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉

= − iθC(t, t′)Tr{e−βH1

s (−∞)S(λC)c(t)c†(t′)}/Z1

+ iθC(t′, t)Tr{e−βH1

s (−∞)S(λC)c†(t′)c(t)}/Z1. (2.51)

21

The impurity partition fun tion be omes Z1(λ, µ−Ucf(−∞)) = TrC{TCe−βH1s (−∞)S(λC)},

while S(λC) is de�ned by equation (2.40) and it is straightforward to show that the

Green's fun tion satis�es a similar equation to equation (2.44),

GC,1(t, t′) = −δ lnZ1(λc, µ)

δλc(t′, t). (2.52)

To �nd the impurity Green's fun tion, we have to take the time derivative of the

Green's fun tion; the only hange is the derivative of c(t), whi h involves the om-

mutation [H1s (t), c(t)], and �nally we obtain

GC,1(t, t′) = [(i∂ + µ− Ucf(t))δC(t, t′)− λC(t, t

′)]−1. (2.53)

Now, we may use the impurity Green's fun tion that we obtained for H0s and H1

s

to al ulate the impurity Green's fun tion for the time-dependent Fali ov-Kimball

model:

HFKs (t) = −µc†c+ Eff

†f + Ucf(t)c†cf †f, (2.54)

where wi = f †f is the number of lo alized ele trons, and the se ond term is the

energy of the lo alized ele trons on the latti e, and the last term denotes the repul-

sive intera tion between ondu tion and lo alized ele trons, whi h is expli itly time

dependent. The fa t that wi = f †f ommutes with the Hamiltonian and the evolution

operator, separates the Hilbert spa e into two subspa es with wi = 0 and wi = 1,

making the al ulation of the tra e simpler. The partition fun tion an be written as

ZFK = Z0(λ, µ) + e−βEfZ0(λ, µ− Ucf(t)), (2.55)

and the Green's fun tion be omes

GC,FK(t, t′) = − i〈TC S(λC) c(t)c†(t′)〉

= − iθC(t, t′)Tr{e−βHFK

s (−∞)S(λC)c(t)c†(t′)}/ZFK

+ iθC(t′, t)Tr{e−βHFK

s (−∞)S(λC)c†(t′)c(t)}/ZFK. (2.56)

22

Due to the presen e of the time-ordered produ t and the tra e operator, whi h a ts

over ondu tion and lo alized ele trons, �nding the equation of motion is ompli ated.

However, employing the al ulus of variations, it is straightforward to show that the

Green's fun tion satis�es

GFKC (t, t′) = −δ lnZ

FK(λC, µ)

δλC(t′, t), (2.57)

whi h simpli�es to

GC,FK(t, t′) = − Z0(λC, µ)

ZFK

δ lnZ0(λC, µ)

δC(t′, t)

+e−βEfZ1(λC, µ)

ZFK

δ lnZ0(λC, µ− Ucf(−∞))

δλC(t′, t). (2.58)

Using equation(2.44) and equation(2.52), we an rewrite it as

GC,FK(t, t′) = w0GC,0(t, t′) + w1G

C,1(t, t′), (2.59)

where w0 =Z0(λC ,µ)ZFK , w1 =

e−βEfZ0(λC ,µ−Ucf (−∞))

ZFK and it is straightforward to show that

w0 + w1 = 1. Finally, using equations 2.49 and 2.53, we obtain

GC,FK(t, t′) = (1− w1)[(i∂t + µ)δC(t, t′)− λC(t, t

′)]−1

+ w1[(i∂t + µ− Ucf (t))δC(t, t′)− λC(t, t

′)]−1. (2.60)

2.2.2 Self- onsistent Dynami al mean field theory loop

In order to solve the many-body problem, we need to determine the ele troni Green's

fun tion. In large spatial dimensions d → ∞, the ele tron self-energy be omes lo al

and makes the many-body problem simpler. This approximation is alled dynam-

i al mean-�eld theory [3℄. We start with the non-intera ting ontour-ordered Green's

fun tion in momentum spa e, whi h is de�ned as

GC(eq,non)(t, t′) = − iT r TC e−βHeq,nonc

k

(t)c†k

(t′)/Zeq,non, (2.61)

23

where ck

(t) and c†k

(t′) are reation and annihilator operator in momentum spa e:

ck

=1√L

∑

j

e−ik.Rjcj

c†k

=1√L

∑

j

eik.Rjc†j . (2.62)

Furthermore, Heq,non is the equilibrium non-intera ting Hamiltonian de�ned as

Heq,non =∑

k

(ǫk

− µ)c†k

ck

, (2.63)

with ǫR

= limd→∞− t∗∑di=1 cos(ki)/

√d to be the band stru ture for hyper- ubi lat-

ti e. Metzner and Vollhardt showed that hoosing the nearest-neighbor hopping s ales

as t = t∗

2√don a simple hyper- ubi latti e leads to �nite average kineti energy. [46℄

. In this ase, the nonintera ting density of states be ome Gaussian with an in�nite

bandwidth de�ned as

ρ(ǫ) =1√πe−ǫ2 , (2.64)

noti e that in our al ulation we have used t∗ as our energy unit and the latti e

onstant is set equal to 1.

Here, we explain the DMFT algorithm for nonequilibrium in whi h losely follows

the algorithm des ribed in Ref. [17℄. First, we need to generalize the Hilbert transform

for the nonequilibrium ase. Primarily, we onstru t the lo al ordered Green's fun tion

as GCloc =

∑

kGCk(t, t

′). Starting with lo al self-energy

∑C(t, t′) (whi h is usually

hosen as equilibrium self-energy) and by using Dyson's equation, we an rewrite the

lo al Green's fun tion as

GCloc(t, t

′) =

∫

dǫρ(ǫ)[(I −GC,nonΣC)−1GC,non](t, t′), (2.65)

24

noti e that within the nonequilibrium formalism, all the Green's fun tions and the

self-energies are ontinuous matrix operators with two time variables running over

the ontour. The next step is to extra t an e�e tive dynami al mean-�eld λC(t, t′).

First, we need to �nd the e�e tive medium, using the Dyson equation, we have

GC0(t, t

′) = [(GCloc)

−1 + ΣC ]−1(t, t′), (2.66)

then the dynami al mean �eld be omes

λC(t, t′) = (i∂Ct + µ)δC(t, t′)− (GC

0)−1(t, t′)

= (i∂Ct + µ)δC(t, t′)− (GC

loc)−1(t, t′) + ΣC(t, t′). (2.67)

On e the dynami al mean �eld has been determined, we need to �nd the impurity

Green's fun tion whi h evolves in the presen e of the dynami al mean �eld. We have

already explained this step in great detail for the Fali ov-Kimball model when the

intera tion parameter is time dependent, see equation(2.60). To lose the DMFT loop,

we use the Dyson equation to extra t the impurity self-energy from the impurity

Green's fun tion and the e�e tive medium. Then we need to iterate the loop until it

onverges.

2.3 Numeri al implementation of nonequilibrium DMFT

To implement the algorithm numeri ally, we need to dis retize the ontour, as there is

no way to al ulate ontinuous matrix operators. We will de�ne the ontinuous matrix

operator as the limit where the dis retization size goes to zero using an extrapolation

pro edure. For pedagalogi al reason, in this se tion, we explain the dis retization

pro edure whi h has been proposed in Ref. [5℄ by Freeri ks et al. First, we will split

the upper bran h ontour with Nt points starting from tmin to tmax −∆t, Nt points

25

in the lower bran h starting from tmax to tmin +∆t and Nτ points on the imaginary

axis starting from tmin to tmin − iβ +∆τ , see �gure 2.3

ti = tmin + (i− 1)∆t 1 ≤ i ≤ Nt

= tmax − (i−Nt − 1)∆t Nt + 1 ≤ i ≤ 2Nt

= tmin − (i− 2Nt − 1)∆τ 2Nt + 1 ≤ i ≤ 2Nt + 100, (2.68)

where ∆t = tmax−tmin

Ntand we �xed the Nτ = 100 points on the imaginary axis.

−tmin

−tmin − iβ

tmax

∆t

∆τ Upper branch

Lower branch

Imaginary branch

Figure 2.3: S hemati of dis retization on the Keldysh-Baym ontour.

The ontour ordered Green's fun tion satis�es the boundary ondition similar to

the antiperiodi ity property of thermal Green's fun tion

GCii(tmin, t

′) = −GCii(tmin − iβ, t′). (2.69)

Our numeri al dis retization varies from ∆t = 0.1 to 0.02 on the real axis and it

is �xed to be ∆τ = 0.1i on the imaginary axis. We will use the so- alled Wigner

oordinates to transform to new oordinates de�ned as [47℄

trel = t− t′ , T =t+ t′

2. (2.70)

Now we explain the te hni al details for performing the DMFT algorithm. Starting

from equation(2.65), one performs the Hilbert transform to �nd the lo al Green's

26

fun tion; it is ne essary to perform the matrix multipli ation and matrix inverse. The

matrix multipli ation in a dis ritized form is de�ned as

∫

Cdt′′A(t, t′′)B(t′′, t′) =

∑

k

A(ti, tk)wkB(tk, tj). (2.71)

Using the identity

∫

C dt′′A(t, t′′)A−1(t′′, t′) = δC(t, t′), we an de�ne the inverse for the

ontinuous matrix

∑

k

A(ti, tk)wkA−1(tk, tj) =

1

wiδij . (2.72)

Hen e, to �nd the inverse of any matrix, we need to multiply the rows and olumns

by quadrature weights. The standard linear algebra pa kage (LAPACK and BLAS)

is used to manipulate the general, omplex operators. Additionally, we will use the

left point re tangular integration rule, de�ned as

∫

Cdtf(t) =

2Nt+Nτ∑

i=1

wif(ti), (2.73)

where

wi = ∆t 1 ≤ i ≤ Nt

= −∆t Nt + 1 ≤ i ≤ 2Nt

= −0.1i 2Nt + 1 ≤ i ≤ 2Nt + 100. (2.74)

However, in our ase, the omputations will be less intensive, sin e the Hilbert trans-

form is redu ed to integration over one band stru ture ǫ, simplifying the equation

and making possible the exploration of longer times in the nonequilibrium formalism.

We will dis uss the details in next se tion, when we explain about the ben hmarking

of nonequilibrium formalism. In this ase, the numeri s works better by using the

so alled point splitting for delta fun tion, we will losely follow the omputational

27

pro edure as des ribed in Ref.[5℄. Re alling that the delta fun tion is the derivative

of the theta fun tion, the dis retized delta fun tion an be de�ned as

δC(ti, tj) =1

wiδij+1 for integration over j

=1

wi−1

δij+1 for integration over i (2.75)

where wi are quadrature weight as de�ned in Eq. 2.74. Noti e that the above formula

works only for i 6= 1, when i = 1 the only nonzero elements is 1, 2Nt +Nτ , whi h has

a sign hange due to a boundary ondition, see �gure 2.4 For al ulating the impurity

i = 1

i = 2Nt + 100

i i+ 1

Nt

Figure 2.4: Boundary ondition on the Keldysh-Baym ontour. The points tmin(i =1) and tmin − iβ are identi al, ex ept the sign hange due to boundary ondition.

problem, one needs to evaluate (i∂t +µ)δC(t, t′) and (i∂t +µ−Ucf(t))δC(t, t′). We are

looking for proper matrix Mjk as following

[i∂t + µ]δC(tj, tk) = i1

wjMjk

1

wk. (2.76)

28

Using the de�nition of derivative in dis retized form, we an write the �rst operator

as

[i∂t + µ]δC(tj , tk) =[

i(δjk − δjk+1

∆t) + µδjk+1

] 1

wj. (2.77)

It is obvious that the matrix will have many zero elements and will have nonzero

elements on diagonal and sub diagonal,

[

i(δjk − δjk+1

∆t) + µδjk+1

] 1

wj

= i1

wj

1

wk

k = j

= [−1− iµ∆t]1

wj

1

wk

k = j + 1. (2.78)

Comparing equations(2.76), and (2.78), the matrix Mjk be omes

Mjk =

1 0 0 · · · 1 + i∆τµ

−1 − i∆tµ 1 0 · · · 0

0 −1 − i∆tµ 1 0

.

.

.

0 −1 + i∆tµ 1 0

0 −1 + i∆tµ 1

.

.

.

−1 − ∆τµ 1

−1 − ∆τµ 1

,

where the top blo k orresponds to upper bran h, the middle blo k orresponds to

lower bran h and the right bottom blo k orresponds to imaginary bran h. The upper

right hand matrix element omes from the orre t boundary ondition. One ould

dire tly al ulate the Green's fun tion of a spinless Green's fun tion with a hemi al

potential, and the inverse should equal to Mjk. In our ase, the determinant of the

29

inverse of the matrix should be equal the partition fun tion.

det Mjk = 1 +(−1)2Nt+Nτ−1(1 + i∆tµ)(−1 − i∆tµ)Nt−1

× (−1 + i∆tµ)Nt(−1− i∆tµ)Nτ

≃ 1 + (1 + ∆τµ)Nτ +O(∆t2). (2.79)

Re alling that ∆τ = βNτ(sin e the imaginary bran h has length β), we lead to 1 +

exp(βµ) in the limit ∆t,∆τ → 0. This shows the importan e of point splitting and

veri�es that our al ulation is orre t. In similar way

[i∂t + µ− Ucf(t)]δC(tj , tk) = i1

wj

Ljk1

wk

, (2.80)

where

Ljk =

1 0 0 · · · 1 + i∆tu1

−1 − i∆tu2 1 0 · · · 0

0 −1 − i∆tu3 1 0

.

.

.

0 −1 + i∆tuNt+1 1 0

0 −1 + i∆tuNt+2 1

.

.

.

−1 − ∆τu2Nt+1 1

.

.

.

−1 − ∆τu2Nt+100 1

and the ui = µ − Ucf(ti) oe� ients are time-dependent and evaluated at ea h dis-

ritized point using the left point rule, as we des ribed in equation(2.68). For example,

the oe� ient hanges from u1 = µ−Ucf (t1) to uNt = µ−Ucf(tmax−∆t) in the upper

bran h, and from uNt+1 = µ − Ucf(tmax) to u2Nt = µ − Ucf (tmin + ∆t) in the lower

bran h, and from u2Nt+1 = µ−Ucf(tmin) to u2Nt+100 = µ−Ucf(tmin− iβ+0.1i) in the

imaginary bran h. The time-dependent parameter is hanging from some initial value

30

at tmin to rea h the �nal value at tmax in upper bran h, and then needs to evolve ba k

to it's initial value toward tmin in the lower bran h. This way of time evolution, on

the time ordered ontour, is responsible for orresponden e in time dependent oe�-

ients; for example the oe� ients uNt+1,...,u2Nt in the lower bran h be ome identi al

to uNt,...,u1 in upper bran h. Moreover, the imaginary oe� ient are the same and

we will denote them by uβ = µ− Ucf(−∞). So the matrix Lij simpli�es to

Ljk =

1 0 0 · · · 1 + i∆tu1

−1 − i∆tu2 1 0 · · · 0

0 −1 − i∆tu3 1 0

.

.

.

0 −1 + i∆tuNt1 0

0 −1 + i∆tuNt−1 1

.

.

.

.

.

.

−1 − ∆τuβ 1

. Consequently, the determinant be omes

det Ljk = 1 + (−1)2Nt+Nτ−1 (1 + i∆tu1)(−1 − i∆tu2) · · · (−1 − i∆tuNt)

× (−1 + i∆tuNt)(−1 + i∆tuNt−1) · · · (−1 + i∆tu1)

× (−1 −∆τuβ)(−1−∆τuβ) · · · (−1 −∆τuβ) ,(2.81)

simplifying the above equation we obtain

det Ljk = 1 + [1 + ∆t2u21] · · · [1 + ∆t2u2Nt]× (1 + ∆τuβ)

Nτ , (2.82)

and the determinant equals the partition fun tion when ∆t,∆τ → 0:

det Ljk = 1 + exp[β(µ− Ucf(−∞))]. (2.83)

31

3

Nonequilibrium spe tral moment sum rules for systems with

ele tron-phonon intera tion

�S ien e is spe tral analysis. Art is light synthesis.�

� Karl Kraus

Most of the interesting strongly orrelated systems su h as strongly orrelated

oxide multilayers, harge- and spin-density wave materials and high Tc uprates

exhibit strong ele tron-ele tron or ele tron-phonon ouplings while understanding

the non equilibrium dynami s of ele tron-phonon intera tion still remains as one of

the most intriguing topi s both from experimental and theoreti al points of view.

Re ent development in pump-probe spe tros opy has provided a powerful tool to

study the non-equilibrium properties of a large variety of strongly orrelated sys-

tems with oupled ele tron, phonon and spin degrees of freedom within the relevant

time-s ale for the ele tron-phonon dynami s. The original angle-resolved photoemis-

sion spe tros opy (Arpes) is used to obtain the energy and momentum of ele troni

band stru ture with high resolution, but an not provide information regarding the

band stru ture in ex ited states. Instead, the time and angle-resolved photoemission

spe tros opy (trARPES) adds the femtose ond time-resolution to the onventional

ARPES whi h provides a powerful tool to study the elementary s attering pro ess

in the ele troni stru tures. The set up for the pump-probe spe tros opy is shown in

�gure 3.1. The pump is a strong ultrashort laser pulse whi h an be used to ex ite

32

either the ele troni or phononi state generating a non-equilibrium state, followed

by the UV pulse whi h probes the pump-indu ed hanges in the system after a time

delay∆t. Moreover, in a re ent experiment [8℄, the ultrafast response of the self-energy

Figure 3.1: The s hemati pi ture of tr-Arpes experiment. The probe whi h is a

femtose ond infrared pulse reates an ex itation in sample, while the se ond pulse

whi h is a UV pulse, is used to probe the transient behavior of ele troni stru ture

after the time ∆t. Pi ture taken from Shen Laboratory, Stanford.

(a fundamental quantity des ribing many-body intera tions) of a high-temperature

super ondu tor has been investigated both in the normal and super ondu ting state.

The most dire t eviden e of ele tron-phonon oupling in uprate high Tc super on-

du tors is known as a universal ele tron self-energy renormalization whi h manifests

itself as a kink in photoemission spe tra [9℄, although whether it is related to super-

ondu tivity still remains un lear. In �gure 3.2, we show the results of the ele troni

dispersion of a high Tc uprate super ondu tor [8℄. The data is taken at 17K < Tc

at equilibrium, t = −1ps and in nonequlibrium at t = 1ps and t = 10ps. In part

(a), we an observe the kink around 70mev at equilibrium, while after applying the

pump, there is a signi� ant loss in the spe tral weight, whi h is happening at t = 1ps

33

and �nally, at a later time of t = 10ps the transient spe tra re overs the equilibrium

state. In part (b), the ele troni dispersion is shown before and after applying the

laser pump. By omparing the momentum dispersion urve (MDC) after applying the

pump shown in red and the equilibrium dispersion urves shown in bla k, we observe

that the signi� ant hanges in the MDC are happening around the kink energy. Fur-

ther results from the same experiment [8℄, show that the pump indu ed hanges in

the self-energy in the super ondu ting state are di�erent from the normal state as it

is more on�ned in the vi inity of the kink and an not be explained by a temper-

ature broadening e�e t whi h hanges over a broad range of energies. These results

raise interesting questions regarding the origin of su h behavior and it an provide

an insight to understand the role of ele tron-phonon ouplings in high Tc uprate

super ondu tors.

In this hapter, we will explain how one an use sum rules in a form of exa t

analyti al results to investigate the response of strongly orrelated system out of

equilibrium. In addition to the fa t that sum rules have been used to develop a

di�erent approximation to investigate the feature of systems, they provide powerful

tools to be used in the ben hmarking of omputational work to he k the pre ision

of numeri al solutions. This approa h has been developed by White by applying the

exa t sum rules for the zeroth and the �rst two moments of the spe tral fun tion

to estimate the a ura y of Monte-Carlo solutions of the Hubbard model in two-

dimensions [54℄. Moreover, higher moments an reveal useful information about the

spontaneous magneti order in orrelated systems [55℄. The appli ation of the sum

rules for the self-energy has attra ted mu h interest from angle-resolved experiments

on the high-temperature super ondu tors [56, 57, 58℄. However, in nonequilibrium the

Green's fun tion and self-energies depend on two times, nevertheless, the exa t sum

rules have been developed by Freeri ks et al [10, 11, 12, 13℄. In the next se tion, we

34

Figure 3.2: Time-resolved spe tra of high Tc uprate super ondu tor,

Bi2Sr2Ca2Cu2O8+δ. a) Photoemission intensity as a fun tion of energy and momentum

measured at equilibrium (t = −1ps) and nonequilibrium (t = 1ps and t = 10ps) afterapplying the ultra fast pump. The intensity shown in false olor, indi ates a lear

loss of spe tral weight after applying the pump at t = 1ps. The arrow indi ates

the position of kink in the momentum distribution urve whi h happens at 70mev.b) Comparing the momentum distribution urve (MDC) at di�erent time shown by

bla k for t = −1ps, red for t = 1ps and gray for t = 10ps. In the insets, omparison

of (MDC) is done at di�erent binding energies [8℄.

35

will show how one an al ulate the nth moment sum rules for the spe tral fun tion

of a generi time-dependent Hamiltonian. Then, in se tion 2, we will fo us on the

Holstein model as a basi model to des ribe the ele tron-phonon intera tion. We will

derive the moments of the retarded Green's fun tion and self-energy up to se ond and

zeroth order respe tively. These results are reported from a paper where I al ulated

the atomi limit of the Holstein model [14℄. In se tion 3, we go one step further and

we add the ele tron-ele tron intera tion. We have al ulated the sum rules for the

retarded Green's fun tion up to third order whi h allows us to al ulate the moment

of the self-energy to �rst order.

3.1 Spe tral sum rules for the retarded Green's fun tion

The nonequilibrium retarded Greens fun tion an be de�ned as,

GRij(t1, t2) = −iθ(t1, t2)〈{ci(t1), c†j(t2)}〉, (3.1)

where 〈O〉 = Tr [exp(−βHeq)O]/Z. Moreover the fermioni operators are written in

the Heisenberg representation ci(t) = U †(t, t′) ci U(t, t′), where the evolution operator

satis�es the S hrödinger equation, idU(t,t′)dt

= H(t)U(t, t′). The two times lie on the

Kadano�-Baym-Keldysh ontour, where one starts from tmin and runs in the positive

dire tion until tmax and ba k to tmin in opposite dire tion and �nally goes to tmin− iβ

parallel to the imaginary axis with β = 1/T . It is onvenient to onvert to so alled

Wigner oordinates: the average time T = t+t′

2and the relative time t = t1 − t2. By

using a Fourier transform with respe t to relative time, we an �nd the frequen y

dependent retarded Green's fun tion for ea h average time

GRij(T, ω) =

∫ ∞

0

dt eiωtGRij(T +

t

2, T − t

2). (3.2)

36

The nth spe tral moment in real spa e is de�ned as

µRnij (T ) = −1

π

∫ ∞

−∞dω ωnImGR

ij(T, ω). (3.3)

It is straightforward to show that one an rewrite the moments as derivatives in time

µRnij (T ) = Im 〈in+1 d

n

dtn{ci(T +

t

2), c†j(T − t

2)} |t=o+〉. (3.4)

So pra ti ally the problem of �nding the nth moment of the spe tral fun tion redu es

to al ulating the nth derivative of {ci(T + t2), c†j(T − t

2)} with respe t to t. Below

we show how one an al ulate this quantity in the Heisenberg representation. Let

us onsider a physi al system with a generi time-dependent Hamiltonian denoted by

Hs(t) in the S hrödinger representation. We know that in the Heisenberg represen-

tation the time dependen e is en oded in the operator AH(t) whi h is related to the

S hrödinger representation operator As(t) by

AH(t) = U †(t, t0) As(t) U(t, t0), (3.5)

where t0 is the referen e time and U(t, t0) is the evolution operator whi h is de�ned

as

U(t, t0) = T e−i∫ tt0

dtHs(t), (3.6)

and the onjugate operator is de�ned as

U †(t, t0) = T ei∫ tt0

dtHs(t), (3.7)

here T is the time ordering operator whi h orders the operator with later time to the

left. A ordingly T is the anti time ordering operator and a ts in the opposite way.

Moreover, the evolution operator satis�es the equation of motion

i∂tU(t, t0) = Hs(t) U(t, t0). (3.8)

37

Then it is easy to show that the Heisenberg equation of motion implies,

dAHdt

= iU †(t, t0)[Hs(t), A]U(t, t0) + U †(t, t0)∂A

∂tU(t, t0) (3.9)

for simpli ity hereafter we will drop the index for the Hamiltonian and we get

dAHdt

= i[H(t), AH] +∂AH∂t

, (3.10)

where the Hamiltonian appearing in the equation is in the Heisenberg represen-

tation and the symbol AH = U †(t, t0)AU(t, t0). Using the de�nition LnAH =

[...[[AH, H(t)], H(t)]...H(t)] and DnAH =∂An

H

∂tn, we an rewrite above equation as,

idAHdt

= L1AH + iD1AH, (3.11)

and onsequently we an al ulate the higher order derivatives as follows,

i2dA2

Hdt2

= L1L1AH + iD1L1AH + iL1D1AH + i2D1D1AH

= L2AH + iD1L1AH + iL1D1AH + i2D2AH, (3.12)

i3dA3

Hdt3

= L3AH + iL1D1L1AH + iL2D1AH + i2L1D2AH +

iD1L2AH + i2D2L1AH + i2D1L1D1AH + i3D3AH. (3.13)

Noti e that we have used a ontra tion rule, where we have ombined the two oper-

ators in the ase that identi al operators lie beside ea h other, for example L1L1 an

be written as L2. In general, when the order of the derivative is smaller than the

number of the nested ommutator, one has to onsider all possible ommutators, we

will show it with a further example later. As we an see there are 2n distin t terms

for the nth derivative. So n = 4 has 16 di�erent terms and the number of terms grows

exponentially. However most of the time, the operator has no expli it time depen-

den e in the S hrödinger representation and the derivative terms with respe t to the

38

operator itself vanishes, here we use a tilde notation to indi ate that the operator

does not have expli t time dependen e. So we lead to,

idAHdt

= L1AH, (3.14)

i2dA2

Hdt2

= L2AH + iD1L1AH, (3.15)

i3dA3

Hdt3

= L3AH + iL1D1L1AH + iD1L2AH + i2D2L1AH. (3.16)

One ould see that for higher derivatives the equation involves a di�erent order of

derivative of the Hamiltonian and keeping tra k of all terms be omes a hard task.

Here we derive a simple identity that helps to tra k all possible terms.

indnAHdtn

=2n∑

sequence=1

(i)m Tuple[{D1, L1}, n]AH, (3.17)

where Tuple[list, n] is n-tuple, whi h is de�ned as a sequen e of elements with length

n. The sum runs over all the sequen es of n-tuple and as we mentioned before, we

have 2n possible sequen es as the list only ontains two elements, L1 and D1. The

index m indi ates the order of derivative of ea h sequen e and it an be obtained by

summing over the number of times that operator D1 appears in that sequen e.

To larify the notation, let us al ulate the derivative for n = 1 and n = 2 by using

the de�nition of n-tuple. For n = 1, the 1-tuple list ontains only two sequen es with

one element, Tuple[{D1, L1}, 1] = {{L1}, {D1}}, while for n = 2, we have 4 sequen es

with two elements, Tuple[{D1, L1}, 2] = {{L1, L1}, {L1, D1}, {D1, L1}, {D1, D1}}.

Now using the identity (3.17), we an al ulate the derivative of the operator AH as

follows,

39

idAHdt

= L1AH + iD1AH, (3.18)

and

i2dA2

Hdt2

= L1L1AH + iL1D1AH + iD1L1AH + i2D1D1AH, (3.19)

Noti e that we an use similar ontra tion rule to ombine the index of identi al

operators that lie beside ea h other to rea h equations (3.11) and (3.12). Again for

operators whi h do not have expli t time dependen e, half of the terms be ome zero.

Now that we �nd a generalized way to al ulate the nth derivative of the operator

in the Heisenberg representation, we ome ba k to equation (3.4). Here we expli itly

derive the derivatives of {ci(T + t2), c†j(T − t

2)} up to third order. The zeroth order

is trivial sin e it simply be omes the anti ommutator of {ci, c†j} whi h is equal to δij

for the fermioni operators. For simpli ity we drop the time arguments of operators,

but we keep in mind that ea h time we take the derivative with respe t to ci(T + t2)

and c†j(T − t2), we get a fa tor of 1/2 and −1/2 respe tively. Now we show how one

an al ulate the nth moment independent of the lower order derivatives. Sin e both

operators ci(T + t2) and c†j(T − t

2) are time dependent, �rst we use Leibniz rule to

al ulate the proper nth derivative,

indn

dtn{ci(T +

t

2), c†j(T − t

2)} =

1

2n

n∑

k=0

(−1k)

(n

k

) {

[d

dt]n−kci(T +

t

2) , [

d

dt]kc†j(T − t

2)}

,(3.20)

where the derivative with respe t to t will bring down a 1/2n fa tor for ci and (−1/2)n

for c†j. After generating the orre t orders of derivatives and using equation 3.17 one

an generate the proper terms for individual derivatives. For n = 0, we simply get

40

the anti ommutator {ci, c†j}. Below we show the results from n = 1 to n = 3,

id

dt{ci, c†j} =

1

2

({ d

dtci , c

†j

}

−{

ci ,d

dtc†j

})

=1

2

(

{L1ci, c†j} − {ci, L1c

†j})

. (3.21)

i2d2

dt2{ci, c†j} =

1

4

({ d2

dt2ci , c

†j

}

− 2{ d

dtci ,

d

dtc†j

}

+{

ci ,d2

dt2c†j

})

=1

4

({

(L2 + iD1L1)ci , c†j

}

− 2{

L1ci , L1c†j

}

+{

ci , (L2 + iD1L1)c†j

})

,

separating the di�erent order of derivatives we get,

i2d2

dt2{ci, c†j} =

1

4

(

{L2ci, c†j} − 2{L1ci, L1c

†j}+ {ci, L2c

†j})

+i

4

(

{D1L1ci, c†j}+ {ci, D1L1c

†j})

. (3.22)

Similarly, for n = 3 we get,

i3d3

dt3{ci, c†j} =

1

8

({ d3

dt3ci , c

†j

}

− 3{ d2

dt2ci ,

d

dtc†j

}

+ 3{ d

dtci ,

d2

dt2c†j

}

−{

ci ,d3

dt3c†j

})

=1

8

(

{L3ci, c†j} − 3{L2ci, L1c

†j}+ 3{L1ci, L2c

†j} − {ci, L3c

†j})

+i

8

(

{L1D1L1ci, c†j}+ 2{D1L2ci, c

†j} − 3{D1L1ci, L1c

†j}

+ 3{L1ci, D1L1c†j} − {ci, L1D1L1c

†j} − 2{ci, D1L2c

†j})

− 1

8

(

{D2L1ci, c†j} − {ci, D2L1c

†j})

, (3.23)

Now we write the results for the spe tral moments,

µR0ij = Re {ci, c†j}, (3.24)

41

µR1ij = Re

1

2

({

[ci, H ] , c†j

}

−{

ci , [c†j , H ]})

, (3.25)

µR2ij = Re

1

4

({

[[ci, H ], H ] , c†j

}

− 2{

[ci, H ] , [c†j , H ]}

+{

ci , [[c†j, H ], H ]

})

+ Rei

4

({

[ci,∂H

∂t] , c†j

}

+{

ci , [c†j ,∂H

∂t]})

, (3.26)

µR3ij = Re

1

8

({

[[[ci, H ], H ], H ] , c†j

}

− 3{

[[ci, H ], H ] , [c†j, H ]}

+ 3{

[ci, H ] , [c†j, H ], H ]}

−{

ci , [[[c†j , H ], H ], H ]

})

+ Rei

8

(

2{

[[ci,∂H

∂t], H ] , c†j

}

+{

[[ci, H ],∂H

∂t] , c†j

}

− 3{

[ci,∂H

∂t, [c†j , H ]

}

+ 3{

[ci, H ] , [c†j,∂H

∂t]}

− 2{

ci , [[c†j ,∂H

∂t], H ]

}

−{

ci , [[c†j , H ],

∂H

∂t]})

− Re1

8

({

[ci,∂2H

∂t2] , c†j

}

−{

ci , [c†j,∂2H

∂t2]})

, (3.27)

As n in reases, the number of terms in reases, but we go one step further and we

show that we an sum up all terms with m = 0 whi h are the zero derivative. Let us

start with {L0ci, Lnc†j}, whi h an be written as

{L0ci, Lnc†j} = {L0ci, L1Ln−1c

†j}} = {ci, [[...[c†j , H ]...], H ]}. (3.28)

where, the urly bra kets indi ate the anti- ommutation, so we an use the following

Ja obian identity [X, {Y, Z}] − {Z, [X, Y ]} + {Y, [Z,X ]} ≡ 0. Now, we ompare the

42

�rst and the last term of nth derivative with zero derivative order,

{ci, [[...[c†j , H ]...], H ]} = −{[ci, H ], [[...[c†j , H ]...]]} − [H, {ci, [[...[c†j , H ]...]}], (3.29)

and

{[[...[ci, H ]...], H ], c†j} = −{[[[...[ci, H ]...]], [c†j, H ]} − [H, {[[...[ci, H ]...], c†j}], (3.30)

the se ond term is zero sin e the internal anti ommutator always is a number and

ommutes with Hamiltonian. So we end up with

{L0ci, Lnc†j} = −{L1ci, Ln−1c

†j}. (3.31)

We repeat the same pro edure one more time,

{L1ci, Ln−1c†j} = −{L2ci, Ln−2c

†j} (3.32)

ombining Eqs. 3.31 and 3.32, we get

{L0ci, Lnc†j} = {L2ci, Ln−2c

†j} (3.33)

repeating the pro edure s times we observe that

{L0ci, Lnc†j} = (−1)s{Lsci, Ln−sc

†j} (3.34)

where the sign of term will be positive or negative depending on whether the degree

of L shifts even or odd times respe tively. Similarly, using the Ja obi identity the

se ond term

{

ci , [c†j ,∂H∂t]}

an be obtained by swapping

∂H∂t

over the ommutator

from the �rst term

{

[ci,∂H∂t] , c†j

}

. However, as we des ribed above, there is a (−1)s

oe� ient whi h makes this two term to an el ea h other. One has to be areful

for the derivatives in higher moments as there are terms that are not identi al due

to the fa t that H and

∂H∂t

do not ommute. However, one an simplify some of the

43

terms from the Ja obi identity. Below, we rewrite the �nal answers for the spe tral

moments after the simpli� ation,

µR0ij = Re {ci, c†j}, (3.35)

µR1ij = Re

{

[ci, H ] , c†j

}

, (3.36)

µR2ij = Re

{

[[ci, H ], H ] , c†j

}

(3.37)

µR3ij = Re

{

[[[ci, H ], H ], H ] , c†j

}

+ Rei

2

({

[[ci,∂H

∂t], H ] , c†j

}

−{

[[ci, H ],∂H

∂t] , c†j

}

− Re1

8

({

[ci,∂2H

∂t2] , c†j

}

−{

ci , [c†j,∂2H

∂t2]})

, (3.38)

3.2 Nonequilibrium sum rules for the Holstein model

In re ent years, we have seen signi� ant advan es in time-resolved experiments on

systems that have strong ele tron-phonon intera tions [59, 60℄. These experiments

study how energy is transferred between the ele troni and phononi parts of the

system. One of the interesting e�e ts that have been seen in these experiments is

the so- alled phonon-window e�e t [61℄, where ele trons with energies farther than

the phonon frequen y from the Fermi level relax qui kly ba k to equilibrium after

the pulsed �eld is applied, but those lose to the Fermi level relax on a mu h longer

time s ale, be ause their relaxation involves multiparti le pro esses due to a restri ted

phase spa e. It is lear that this experimental and theoreti al work is just starting

This hapter is reprinted from J. K. Freeri ks, K. Naja�, A. F. Kemper, and T. P. Dev-

ereaux, FEIS 2013, Copyright(2013) Key West, FL, USA, 2013O.

44

to analyze ele tron-phonon intera ting systems in the time domain. Hen e, any exa t

results that an be brought to bear on this problem will be important.

In this work, we derive sum rules for the zeroth and �rst two moments of the

retarded ele troni Green's fun tion and for the zeroth moment of the retarded self-

energy. The moment sum rules have already been derived in equilibrium [56, 58℄, but

they a tually hold true, un hanged, in nonequilibrium as well [10, 11, 12, 13℄. With

these sum rules, one an understand how the ele tron-phonon intera tion responds

to nonequilibrium driving, and how di�erent response fun tions will behave.

We start with the so- alled Holstein model [62, 63℄, given by the following Hamil-

tonian in the S hroedinger representation:

HH(t) = −∑

ijσ

tij(t)c†iσcjσ +

∑

iσ

[g(t)xi − µ]c†iσciσ +∑

i

p2i2m

+1

2κ∑

i

x2i (3.39)

where c†iσ (ciσ) are the fermioni reation (annihilation) operators for an ele tron at

latti e site i with spin σ (with anti ommutator {ciσ, c†jσ′}+ = δijδσσ′), and xi and

pi are the phonon oordinate and momentum (with ommutator [xi, pj]− = i~δij),

respe tively. The hopping −tij(t) between latti e sites i and j an be time dependent

[for example, an applied ele tri �eld orresponds to the Peierls' substitution [64℄, µ

is the hemi al potential for the ele trons, g(t) is the time-dependent ele tron-phonon

intera tion, m is the mass of the opti al (Einstein) phonon and κ is the orresponding

spring onstant. The frequen y of the phonon is ω =√

κ/m. It is often onvenient

to also express the phonon degree of freedom in terms of the raising and lowering

operators a†i and ai (with ommutator [ai, a†j]− = δij) with xi = (a†i + ai)

√

~/(2mω)

and pi = (−a†i + ai)√

~mω/2/i. This Hamiltonian involves ele trons that an hop

between di�erent sites on a latti e and intera t with harmoni Einstein phonons that

have the same phonon frequen y for every latti e site. The hopping and the ele tron-

45

phonon oupling are taken to be time dependent for the nonequilibrium ase. We set

~ = 1 and kB = 1 for the remainder of this work.

3.2.1 The ele troni sum rules

These moments an now be evaluated straightforwardly, although the higher the

moment is the more work it takes. We �nd the well-known result

µR0ijσ(T ) = δij (3.40)

for the zeroth moment. The �rst moment satis�es

µR1ijσ(T ) = −tij(T )− µδij + g(T )〈xi(T )〉δij (3.41)

and the se ond moment be omes

µR2ijσ(T ) =

∑

k

tik(T )tkj(T ) + 2µtij(T ) + µ2δij (3.42)

− tij(T )g(T )〈xi(T ) + xj(T )〉 − 2µg(T )〈xi(T )〉δij

+ g2(T )〈x2i (T )〉δij.

Unlike in the ase of the Hubbard or Fali ov-Kimball model, where the sum rules

relate to onstants or simple expe tation values [11, 12, 13℄, one an see here that

one needs to know things like the average phonon oordinate and its �u tuations in

order to �nd the moments. We will dis uss this further below.

Our next step is to al ulate the self-energy moments in the normal state, whi h

are de�ned via

CRnijσ(T ) = −1

π

∫

dω ωnImΣRijσ(T, ω). (3.43)

Note that the self-energy is de�ned via the Dyson equation

GRijσ(t, t

′) = GR0ijσ(t, t

′) +∑

kl

∫

dt

∫

dt′GR0ikσ(t, t)Σ

Rklσ(t, t

′)GRljσ(t

′, t′), (3.44)

46

where GR0is the nonintera ting Green's fun tion and the time integrals run from −∞

to ∞. The strategy for evaluating the self-energy moments is rather simple. First,

one writes the Green's fun tion and self-energy in terms of the respe tive spe tral

fun tions

GRijσ(T, ω) = −1

π

∫ImGR

ijσ(T, ω′)

ω − ω′ + i0+dω′

(3.45)

and

ΣRijσ(T, ω) = ΣR

ijσ(T,∞)− 1

π

∫ImΣR

ijσ(T, ω′)

ω − ω′ + i0+dω′. (3.46)

Next, one substitutes those spe tral representations into the Dyson equation that

relates the Green's fun tion and self-energy to the nonintera ting Green's fun tion.

By expanding all fun tions in a series in 1/ω for large ω, one �nds the spe tral formulas

involve summations over the moments. By employing the exa t values for the Green's

fun tion moments, one an extra t the moments for the self-energy. Details for the

formulas appear elsewhere [12℄. The end result is

ΣRijσ(T,∞) = g(T )〈xi(T )〉δij (3.47)

and

CR0ijσ(T ) = g2(T )[〈x2i (T )〉 − 〈xi(T )〉2]. (3.48)

So, the total strength (integrated weight) of the self-energy depends on the �u tua-

tions of the phonon �eld.

3.2.2 Formalism for the phononi sum rules

The retarded phonon Green's fun tion is de�ned in a similar way, via

DRij(t, t

′) = −iθ(t − t′)Tre−βH(tmin)[xi(t), xj(t′)]−/Z, (3.49)

with the operators in the Heisenberg representation. The moments are de�ned in the

same way as before. First one onverts to the average and relative time oordinates

47

and Fourier transforms with respe t to the relative oordinate

DRij(T, ω) =

∫

dtreleiωtrelDR

ij(T +1

2trel, T − 1

2trel), (3.50)

and then one omputes the moments via

mRnij (T ) = −1

π

∫

dω ωnImDRij(T, ω). (3.51)

The zeroth moment vanishes be ause xi ommutes with itself at equal times. For the

higher moments, we also derive a formula similar to what was used for the ele troni

Green's fun tions. In parti ular, we have

mR1ij (T ) = −1

2Im{

〈[x′i(T ), xj(T )]−〉 − 〈[xi(T ), x′j(T )]−〉}

(3.52)

for the �rst moment. But x′i(T ) = −i[xi(ttave),HH(T )]− = pi(T )/m, so we �nd

mR1ij (T ) =

1

mδij . (3.53)

Similarly,

mR2ij (T ) = −1

4Imi{

〈[x′′i (T ), xj(T )]−〉 − 2〈[x′j(T ), x′j(T )]−〉

+ 〈[xi(T ), x′′j (T )]−〉}

. (3.54)

Using the fa t that x′′i (T ) = −i[pi(T ),HH(T )]− = −g(T )(ni↑(T ) + ni↓(T ))− κxi(T ),

then we an show that mR2ij (T ) = 0, sin e all ommutators vanish. We don't analyze

the phonon self-energy here. Unlike the ele troni moments, the phononi moments

are mu h simpler, and do not require any expe tation values to evaluate them.

We end this se tion by showing that the imaginary part of the retarded phonon

Green's fun tion is an odd fun tion of ω, whi h explains why all the even moments

vanish. If one evaluates the omplex onjugate of the retarded phonon Green's fun -

tion, one �nds

DRij(t, t

′)∗ = iθ(t− t′)Tr[xj(t′), xi(t)]−e

−βH(tmin)/Z = DRij(t, t

′) (3.55)

48

where the last identity follows by swit hing the order of the operators in the ommu-

tator and the invarian e of the tra e under a y li permutation. Hen e, the phonon

propagator in the time representation is real. Evaluating the frequen y-dependent

propagator, then it shows that DR∗ij (T, ω) = DR

ij(T,−ω) by taking the omplex onju-

gate of equation (3.50). Hen e the real part of the retarded phonon propagator in the

frequen y representation is an even fun tion of frequen y while the imaginary part is

an odd fun tion of frequen y, and therefore all even moments vanish.

3.2.3 Atomi limit of the Holstein model

To get an idea of the phonon expe tation values and the �u tuations, we solve expli -

itly for the expe tation values for the Holstein model in the atomi limit, where

tij(t) = 0 and we an drop the site index from all operators. In this limit, one an

exa tly determine the Heisenberg representation operator x(t) by solving the equation

of motion for the Heisenberg representation operators a(t) and a†(t). This yields

x(t) =ae−iωt + a†eiωt√

2mω− 2Re

{

ie−ωt

∫ t

0

dt′eiωt′

g(t′)

}n↑ + n↓2mω

(3.56)

where the ele troni number operators ommute with H now, so they have no time

dependen e. Sin e the atomi sites are de oupled from one another, we an fo us on

just a single site. The partition fun tion for a single site an be evaluated dire tly by

employing standard raising and lowering operator identities. To begin, we note that

the Hilbert spa e is omposed of a dire t produ t of the harmoni os illator states

|n〉 = 1√n!

(a†)n |0〉 (3.57)

and the fermioni states

|0〉, | ↑〉 = c†↑|0〉, | ↓〉 = c†↓|0〉, | ↑↓〉 = c†↑c†↓|0〉. (3.58)

49

The partition fun tion satis�es

Zat =∑

0,↑,↓,↑↓

∞∑

nb=0

〈nb, nf | exp[−β{(gx− µ)(nf↑ + nf

↓) + ω(nb +1

2)}]|nb, nf〉, (3.59)

where nfdenotes the Fermi number operator and nb

the Boson number operator

(we will drop the exp[βω/2] term whi h provides just a onstant). Sin e the produ t

states are not eigenstates ofH, we annot immediately evaluate the partition fun tion.

Instead, we need to �rst go to the intera tion representation with respe t to the

bosoni Hamiltonian in imaginary time (and we drop the onstant term from the

Hamiltonian), to �nd that

Zat = TrfTrbe−βωnbTτ exp

[

−∫ β

0

dτ ′{e−ωτ ′a+ eωτ

′a†√

2mωg(tmin)− µ

}

nf

]

,

= TrfTrbeβωnb

UI(β) (3.60)

where the time-ordering operator is with respe t to imaginary time and the time-

ordered produ t is the evolution operator in the intera tion representation and

denoted by UI(τ). Be ause the only operators that don't ommute in the evolution

operator are a and a†, and their ommutator is a -number, one an get an exa t

representation for the evolution operator via the Magnus expansion [65℄, as worked

out in the Landau and Lifshitz [66℄ or Gottfried [67℄ texts. The end result for the

time-ordered produ t in equation (3.60) be omes

UI(β) = exp

[

−g(tmin)nf

√2mω3

(1− e−βω

)a

]

exp

[

−g(tmin)nf

√2mω3

(eβω − 1

)a†]

× exp

[

−g2(tmin)n

f2

√2mω3

(eβω − 1− βω + µnf

)]

, (3.61)

whi h used the Campbell-Baker-Hausdor� theorem

eA+B = eBeAe12[A,B]−

(3.62)

for the ase when the ommutator [A,B]− is a number, not an operator, to get the

�nal expression. We substitute this result for the evolution operator into the tra e

50

over the bosoni states, expand the exponentials of the a and a† operators in a power

series, and evaluate the bosoni expe tation value to �nd

Zat = Trf

∞∑

m=0

∞∑

n=0

(n+m)!

n!m!m!

(g2(tmin)n

f2

2mω3

(eβω − 1 + e−βω − 1

))m

× exp

[

−βωn+ βµnf − g2(tmin)nf2

2mω3

(eβω − 1− βω

)]

. (3.63)

Next, we use Newton's generalized binomial theorem

∞∑

n=0

(n +m)!

n!m!zn =

1

(1− z)m+1, (3.64)

to simplify the expression for the partition fun tion to

Zat = Trf exp

[βg2(tmin)n

f2

2mω2+ βµnf

]1

1− e−βω. (3.65)

Performing the tra e over the fermioni states then yields

Zat =1

1− e−βω

{

1 + 2eβµ exp

[βg2(tmin)

2mω2

]

+ e2βµ exp

[2βg2(tmin)

mω2

]}

. (3.66)

We al ulate expe tation values following the same pro edure, but inserting the

relevant operators in the Heisenberg representation at the appropriate pla e, and

arrying out the remainder of the derivation as done for the partition fun tion. For

example, sin e the Fermi number operators ommute with the atomi Hamiltonian,

they are the same operator in the Heisenberg and S hroedinger representations, and

we immediately �nd that the ele tron density is a onstant in time and is given by

〈n↑ + n↓〉 =2eβµ exp

[βg2(tmin)

2mω2

]

+ 2e2βµ exp[2βg2(tmin)

mω2

]

1 + 2eβµ exp[βg2(tmin)

2mω2

]

+ e2βµ exp[2βg2(tmin)

mω2

] . (3.67)

We next want to al ulate 〈x(t)〉 and 〈x2(t)〉 − 〈x(t)〉2, where the operator is in

the Heisenberg representation, and given in equation (3.94). It is straightforward but

tedious to al ulate the averages. After mu h algebra, we �nd

〈x(t)〉 = 〈n↑ + n↓〉(

−g(tmin)

mω2cosωt− Re

{

ie−iωt

∫ t

0

dt′eiωt′ g(t′)

mω

})

. (3.68)

51

(Note that if we are in equilibrium, g(t) = g is a onstant, then one �nds 〈x(t)〉 =

−〈n↑ + n↓〉g/mω2, whi h has no time dependen e, as expe ted.) The �u tuation sat-

is�es

〈x2(t)〉 − 〈x(t)〉2 = [〈n↑〉(1− 〈n↑〉) + 〈n↓〉(1− 〈n↓〉) + 2〈n↑n↓〉 − 2〈n↑〉〈n↓〉]

×[g(tmin)

mω2cosωt+ Re

{

ie−iωt

∫ t

0


mω

}]2

+1

2mωcoth

(βω

2

)

, (3.69)

whi h onsists of two terms: a time-dependent pie e (whi h be omes a onstant when

g is a onstant) that represents the quantum �u tuations due to the ele tron-phonon

intera tion and a phonon pie e that varies with temperature (and is independent of

g). The latter pie e be omes large when T → ∞ (being proportional to T at high T ),

whi h tells us that �u tuations generi ally grow with in reasing the temperature of

the system, so that one expe ts the zeroth moment of the self energy to in rease as

the temperature in reases, or if the system is heated up by being driven by a large

ele tri �eld. Note that if one expands the self-energy perturbatively, as in Migdal-

Eliashberg theory, then only the term independent of g survives, as the other term is

higher order in g and lies outside of the Migdal-Eliashberg result [68℄.

3.2.4 Dis ussion and appli ations of the sum rules

One of the most important re ent experiments in ele tron-phonon intera ting sys-

tems involves time-resolved angle-resolved photoemission (tr-ARPES), whi h an be

analyzed in su h a way that one an extra t information about the ele troni self-

energy [61, 68℄. If one assumes that the phonons form an in�nite heat apa ity bath,

then they are not hanged by the ex itation of the ele trons, and the �u tuations of

the phonon �eld remain a onstant as a fun tion of time. This leads to a self-energy

that an transiently hange shape as a fun tion of time, but does not hange its

52

spe tral weight. Re ent al ulations show pre isely this behavior [61, 68℄. One an

also understand it from the perturbation theory expansion, where a dire t evalua-

tion of the diagrams for the self-energy, and the sum rules for the ele troni Green's

fun tions, establish that the zeroth moment of the retarded ele troni self-energy is

a onstant [68℄. What is perhaps more interesting, is when one treats a fully self-

onsistent system where the ele trons and phonons both an ex hange energy with

one another, and the phonon bath properties hange transiently. In this ase, one

has to examine the self- onsisten y for both the ele trons and the phonons within

the perturbation theory, and the general form of the sum-rules hold. The al ulations

shown above in the atomi limit indi ate that it is likely that adding energy into

the phonon system in reases the phonon �u tuations and thereby reates a stronger

ele troni self-energy. One would expe t there to be os illations of the spe tral weight

as well. It is also likely that these ideas an be in orporated into the quantitative

analysis of experiments that we expe t to see o ur over the next few years.

3.2.5 Con lusions and future work

In this work, we have shown the simplest sum rules for ele trons intera ting with

phonons. These sum rules have been established in equilibrium for some time now, but

our work shows that they dire tly extend to nonequilibrium. We also established new

sum rules for the phonon propagator. In general, these sum rules are ompli ated to

use, be ause they require one to determine both the average phonon expe tation value

and its �u tuations, so they might �nd their most important appli ation to numeri s

as ben hmarking, assuming one an al ulate the relevant expe tation values with

the numeri al te hniques employed to solve the problem. But they also allow us to

examine the physi al behavior we expe t to see if we look at how the moments might

hange in time due to the e�e t of a transient light pump applied to the system.

53

For example, we expe t that as energy is ex hanged from ele trons to phonons, the

ele tron self-energy should in rease its spe tral weight, with the opposite o uring as

the phonons transfer energy ba k to the ele trons. This result is not one that ould

have been easily predi ted without the sum rules.

In the future, there are a number of ways these sum rules an be extended. One

an examine more realisti models, like the Hubbard-Holstein model and �nd those

sum rules. One an look into the e�e ts of anharmoni ity on the sum rules, and �nally,

one an arry out the al ulations to higher order, to examine more moments. We

plan to work on a number of these problems in the future.

3.3 Nonequilibrium sum rules for the Hubbard-Holstein model

Re ent developments in pump-probe spe tros opy have provided a powerful tool to

study nonequilibrium properties of a large variety of strongly orrelated systems with

oupled ele trons, phonons, and spin degrees of freedom within femtose ond time

s ales [72, 73, 74, 75, 76, 77, 78℄. This te hnique has been applied to study high Tc

uprates, whi h exhibit strong ele tron-ele tron and ele tron-phonon ouplings [59,

79, 80℄. Hen e, understanding the nonequilibrium dynami s of the ele tron-phonon

intera tion and it's interplay with the ele tron-ele tron intera tion still remains one

of the most intriguing topi s in ondensed matter physi s.

The pump is an ultra-strong and ultrashort ele tri �eld pulse, whi h an be used

to dire tly ex ite either the ele trons or the phonons. The resulting nonequilibrium

state an subsequently be explored by a probe, whi h is a weaker pulse that mea-

sures the temporal response of the system. Amongst di�erent materials that have

been studied by pump-probe spe tros opy, high temperature super ondu tors have

attra ted signi� ant attention, in part, be ause the role played by the ele tron-phonon

54

intera tion in these material's properties is still not well-understood. For example,

Zhang et al. [8℄ re ently investigated the ultrafast response of the self-energy of a

high-temperature super ondu tor in both the normal and super ondu ting states.

The most dire t eviden e of an ele tron-phonon oupling in the uprates (or, more

generally, an ele tron-boson oupling) is the universal ele tron self-energy renormal-

ization, whi h manifests itself as a kink in the photoemission spe tra that o urs below

the Fermi energy pre isely at the oupled phonon energy [9℄. The strength of the kink

is dire tly related to the strength of the ele tron-phonon oupling. Whether this

phenomenon is dire tly related to high temperature super ondu tivity still remains

un lear. One intriguing result from the pump/probe experiments [8℄ is that the kink

softens when in the super ondu ting state, even with a relatively weak pump. This

raises the question, is the pump dynami ally redu ing the ele tron-phonon oupling

in the super ondu ting state? It turns out sum rules an be employed to examine

this question. A numeri al study [68℄, shows that the kink softens when the system

is pumped, even if there is no dynami redu tion of the ele tron-phonon oupling, as

determined by the zeroth moment of the retarded self-energy as a fun tion of time.

Hen e, kink softening alone is insu� ient to tell whether there is a dynami redu tion

of the ele tron-phonon oupling.

We explain how one an use exa t sum rules to investigate the e�e t of a pump

on a system with both ele tron-ele tron and ele tron-phonon intera tions. We use the

Holstein-Hubbard model, whi h is one the simplest models to des ribe the interplay

between ele tron-ele tron and ele tron-phonon intera tions [81, 82, 83, 84, 85℄. The

sum rules also provide a powerful tool in the ben hmarking of omputational work to

he k the pre ision of numeri al solutions. The approa h was developed to al ulate

the �rst two moments of the spe tral fun tion in order to estimate the a ura y of

Monte-Carlo solutions of the Hubbard model in two-dimensions [54℄. Sin e then, the

55

appli ation of sum rules has extended to a variety of strongly orrelated systems

in equilibrium and nonequilibrium both for homogeneous and inhomogeneous ases

[10, 11, 12, 13℄. Sum rules for the retarded Green's fun tion through se ond order for

the Holstein model [86℄ and the zeroth-order self-energy sum rule for the Holstein-

Hubbard model [87℄ have also appeared in equilibrium. Preliminary work already

found the lowest-order sum rules in the nonequilibrium Holstein model [14℄. Here,

we fo us on the full Holstein-Hubbard model and derive the nonequilibrium spe tral

moments to one higher order.

The remainder of the paper is organized as follows. In Se . 3.3.1, we introdu e the

Holstein-Hubbard model and we derive the exa t sum rules for the spe tral fun tion

of retarded Green's fun tion up to the third moment. Periodi ity of the system is

not needed for these al ulations. In Se . 3.3.2, we derive the orresponding spe tral

moments for the retarded self-energy. For translationally invariant systems, we obtain

the moments in momentum spa e in Se . 3.3.3. To verify the expe tation values and

the �u tuations, we al ulate the atomi limit of the Holstein-Hubbard model in Se .

3.3.4 and ompare those exa t results to the moments. A summary and on lusions

are provided in Se . 3.3.5.

3.3.1 Formalism for the sum rules of the spe tral fun tion for the

Holstein-Hubbard model

The Holstein-Hubbard model has been widely used to des ribe the systems with both

ele tron-phonon and ele tron-ele tron intera tions [81, 82, 83, 84, 85℄. The Hamilto-

56

nian for the (inhomogeneous) Holstein-Hubbard model is given by

HH−H(t) = −∑

ijσ

tij(t)c†iσcjσ +

∑

i

Ui(t)ni↓ni↑

+∑

i

[gi(t)xi − µi(t)](ni↑ + ni↓)

+∑

i

1

2Mip2i +

∑

i

1

2κix

2i

, (3.70)

where niσ = c†iσciσ is the ele tron number at site i. The phonon oordinate and

momentum are denoted by xi and pi, respe tively. The ele tron hopping matrix tij(t)

is a (possibly time-dependent) Hermitian matrix and Ui(t) is the (possibly time-

dependent) on-site Hubbard repulsion. The ele trons are oupled to phonons by ou-

pling strength gi(t) whi h is parametrized by an energy per unit length (and may be

time-dependent). A lo al site energy µi is also in luded (it is the hemi al potential

if it is independent of i). This model aptures the features of variety of interesting

phenomena su h as the Mott transition, polaron and bipolaron formation. It also

has ordered phases to super ondu tivity, harge-density-wave order, and spin-density-

wave order. The dynami al mean-�eld theory (DMFT) has been applied to investigate

the model exa tly [81, 88, 89, 90, 91℄. We next use the formulas in Eqs. (3.24- 3.27), to

determine the sum rules for the nonequilibrium and inhomogeneous Holstein-Hubbard

model. To simplify the formulas, we introdu e the notation [O = O(Tave)℄ to indi ate

the operator is evaluated at the average time Tave, after taking the limit trel → 0. In

addition, we de�ne νiσ = µi(Tave) − Ui(Tave)〈niσ(Tave)〉 − gi(Tave)〈xi(Tave)〉 to make

the expressions more readable (we also use the notation σ = −σ). The zeroth moment

is trivial,

µR0ijσ(Tave) = δij , (3.71)

57

and higher moments are shown below, where we employ the fermioni operator iden-

tity n2iσ = niσ

µR1ijσ(T ) = − tij − νiσδij (3.72)

µR2ijσ(Tave) =

∑

k

tik tkj + tij νiσ + tij νjσ + ν2iσδij + U2i [〈niσ〉 − 〈niσ〉2]δij

+ g2i [〈x2i 〉 − 〈xi〉2]δij + 2Uigi[〈niσxi〉 − 〈niσ〉〈xi〉]δij, (3.73)

µR3ijσ(Tave) = −

∑

kl

tik tkltlj − νiσ∑

k

tik tkj −∑

k

tikνkσ tkj −∑

k

tik tkj νjσ

−ν2iσ tij − U2i [〈niσ〉 − 〈niσ〉2]tij − g2i [〈x2i 〉 − 〈xi〉2]tij − 2Uigi[〈niσxi〉 − 〈niσ〉〈xi〉]tij

−ν2jσ tij − U2j [〈njσ〉 − 〈njσ〉2]tij − g2j [〈x2j〉 − 〈xj〉2]tij − 2Uj gj[〈njσxj〉 − 〈njσ〉〈xj〉]tij

−νiσ tij νjσ − UiUj[〈c†iσ ciσ c†jσcjσ〉 − 〈niσ〉〈njσ〉]tij − gigj[〈xixj〉 − 〈xi〉〈xj〉]tij

−giUj [〈njσxi〉 − 〈njσ〉〈xi〉]tij − gjUi[〈niσxj〉 − 〈niσ〉〈xj〉]tij − ν3iσδij +giκiMi

〈xi〉

+U3i [〈niσ〉 − 〈niσ〉3]δij + g3i [〈x3i 〉 − 〈xi〉3]δij − 3µiU

2i [〈niσ〉 − 〈niσ〉2]δij

−3µig2i [〈x2i 〉 − 〈xi〉2]δij + 3U2

i gi[〈niσxi〉 − 〈niσ〉〈xi〉]δij + 3Uig2i [〈niσx

2i 〉 − 〈niσ〉〈x2i 〉]δij

−6µigiUi[〈niσxi〉 − 〈niσ〉〈xi〉] + Uiδij∑

kl

[tik tkl〈c†iσ clσ〉+ tkltli〈c†lσciσ〉 − 2tik tli〈c†lσ ckσ〉]

+Uiδij∑

k

[µk − µi][tki〈c†kσciσ〉+ tik〈c†iσ ckσ〉] + Uiδij∑

k

[tkigi〈c†kσciσxi〉+ tikgk〈c†iσ ckσxk〉]

+δijUi

∑

k

Uk[tki〈c†kσciσ c†kσckσ〉+ tik〈c†iσ ckσc†kσckσ〉] + UiUj [tji〈c†jσciσ c†jσciσ〉+ tij〈c†iσ cjσc†jσ ciσ〉]

+1

2Re i

∑

k

[dtikdTave

tkj − tikdtkjdTave

]− 1

2Re i

dtijdTave

[µi − µj] +1

2Re i[

dµi

dTave− dµj

dTave]tij

+1

2Re i

dtijdTave

[gi〈xi〉 − gj〈xj〉]−1

2Re i[

dgidTave

〈xi〉 −dgjdTave

〈xj〉]tij

+1

2Re i

dtijdTave

[Ui〈niσ〉 − Uj〈njσ〉]−1

2Re i[

dUi

dTave〈niσ〉 −

dUj

dTave〈njσ〉]tij

+1

4Re

d2tijdT 2

ave

+1

4Re δij [

dµ2i

dT 2ave

− dU2i

dT 2ave

〈niσ〉 −dg2idT 2

ave

〈xi〉]. (3.74)

These are the main results of this work.

58

Note that in the third moment, we have a term

giκi

Mi〈xi〉 whi h arises from the

phononi part of the Hamiltonian. The reason of absen e of su h terms in lower order

is as follows. The �rst moment is obvious be ause the ommutator of the fermioni

operator with phonon part of Hamiltonian is zero. Although in the se ond moment,

there are multiple terms whi h in lude ommutator of x operator with momentum

operator p, these terms be ome imaginary and onsequently do not ontribute in

the se ond moment and they only ontribute in the third moment. Furthermore, we

noti e that the sum rules depend on a number of di�erent expe tation values. Hen e,

they are not just fun tions of the parameters of the model, but also they depend on

the expe tation values of a number of di�erent operators. One expe tation value an

be immediately determined, namely 〈xi〉. This an be done by shifting the phonon

oordinate by xi → xi+ gi(t)(ni↑+ni↓)/κi = xi. The Hamiltonian is an even fun tion

of xi, so we must have 〈xi〉 = 0. This implies that 〈xi〉 = −gi(t)〈ni↑ + ni↓〉/κi.

Unfortunately, similar arguments will not allow us to determine higher power law

expe tation values of the oordinate xi.

3.3.2 Formalism for the sum rules for retarded ele troni self-

energy

Next, we derive the retarded self-energy moments. The self-energy does not vanish at

high frequen y, but approa hes a onstant value, whi h we denote as ΣRijσ(Tave, ω =

∞) and is real. The moments are de�ned from integrals over the imaginary part of

the self-energy via

CRnijσ = −1

π

∫

dωωnImΣijσ(ω) (3.75)

The zeroth moment gives the overall strength of the self-energy. These moments an be

obtained from the Dyson equation whi h onne ts the self-energy with the Green's

fun tion. For the nonequilibrium ase, it's useful to work in the Larkin-Ov hinkov

59

representation where the Green's fun tion and the self-energy ea h be ome 2 × 2

matri es [45℄. The omplete derivation of this Dyson equation for the nonequilibrium

self-energy has already been derived in Ref. [12℄ so we only report the �nal results.

These equations are used, in turn, to determine the moments of the self-energies, as

shown below. Note that the tilde here means the moments for the nonintera ting ase,

when all intera tions vanish [Ui(t) = 0 and gi(t) = 0℄:

µR0ijσ(Tave) = µR0

ijσ(Tave), (3.76)


ijσ(Tave)∑

kl

µR0ikσ(Tave)Σ

Rklσ(Tave, ω = ∞)µR0

ljσ(Tave), (3.77)


ijσ(Tave) +∑

kl

µR0ikσ(Tave)Σ


ljσ(Tave)

+∑

kl

µR0ikσ(Tave)C

R0klσ(Tave)µ

R0ljσ(Tave) +

∑

kl

µR1ikσ(Tave)Σ


ljσ(Tave)

, (3.78)


ijσ(Tave) +∑

kl

µR0ikσ(Tave)Σ


ljσ(Tave)

+∑

kl

µR0ikσ(Tave)C

R0klσ(Tave)µ

R1ljσ(Tave) +

∑

kl

µR0ikσ(Tave)C

R1klσ(Tave)µ

R0ljσ(Tave)

+∑

kl

µR1ikσ(Tave)Σ


ljσ(Tave) +∑

kl

µR1ikσ(Tave)C

R0klσ(Tave)µ

R0ljσ(Tave)

+∑

kl

µR2ikσ(Tave)Σ

Rkl(Tave, ω = ∞)µR0

ljσ(Tave), (3.79)

where ΣRij(ω = ∞) is the high-frequen y limit of the self-energy, i. e., the real onstant

term of the self-energy. Using the fa t that


ijσ(Tave) = δij , (3.80)

the self-energy moment sum rules an be expli itly determined after some algebra.

We �nd

ΣRijσ(Tave, ω = ∞) = [Ui〈niσ〉+ gi〈xi〉]δij, (3.81)

60

CR0ijσ(Tave) = U2

i [〈niσ〉 − 〈niσ〉2]δij + g2i [〈x2i 〉 − 〈xi〉2]δij + 2giUi[〈xiniσ〉 − 〈xi〉〈niσ〉]δij,

(3.82)

CR1ijσ(Tave) = −µig

2i [〈x2i 〉 − 〈xi〉2]δij − µiU

2i [〈niσ〉 − 〈niσ〉2]δij

+ g3i [〈x3i 〉 − 2〈x2i 〉〈xi〉+ 〈xi〉3]δij − giU2i [4〈xiniσ〉〈niσ〉+ 3〈niσ〉2〈xi〉+ 2〈niσ〉〈xi〉 − 3〈xiniσ〉]δij

+ U3i [〈niσ〉 − 2〈niσ〉2 + 〈niσ〉3]δij − Uig

2i [4〈xiniσ〉〈xi〉+ 5〈xi〉2〈niσ〉 − 3〈x2i niσ〉]δij +

giκiMi

〈xi〉

+ Uiδij∑

kl

[tik tkl〈c†iσclσ〉+ tkltli〈c†lσ ciσ〉 − 2tik tli〈c†lσckσ〉] + Uiδij∑

k

[µk − µi][tki〈c†kσciσ〉+ tik〈c†iσ ckσ〉]

+ Uiδij∑

k

[tkigi〈c†kσciσxi〉+ tikgk〈c†iσckσxk〉] + Uiδij∑

k

Uk[tki〈c†kσciσ c†kσckσ〉+ tik〈c†iσckσ c†kσckσ〉]

+ UiUj [tji〈c†jσciσ c†jσ ciσ〉+ tij〈c†iσ cjσc†jσciσ〉]

+1

2Re i

dtijdTave

[gi〈xi〉 − gj〈xj〉]−1

2Re i[

dgidTave

〈xi〉 −dgjdTave

〈xj〉]tij

+1

2Re i

dtijdTave

[Ui〈niσ〉 − Uj〈njσ〉]−1

2Re i[

dUi

dTave〈niσ〉 −

dUj

dTave〈njσ〉]tij

+1

4Re

d2tijdT 2

ave

+1

4Re δij [

dµ2i

dT 2ave

− dU2i

dT 2ave

〈niσ〉 −dg2idT 2

ave

〈xi〉]. (3.83)

Note that the zeroth moment is lo al (diagonal) even if the self-energy has

momentum dependen e, while the �rst moment an be nonzero only for lo al terms

(i = j) and for terms where the hopping is nonvanishing (tij 6= 0). In parti ular, if

we use the zeroth moment to determine the strength of the e�e tive ele tron-phonon

intera tion, then for a pure Holstein model, the only way the ele tron-phonon inter-

a tion is dynami ally hanged is if the orrelation fun tion of the phonon oordinate

hanges as a fun tion of time. This an happen, for example, if energy �ows into the

phonon bath, but is likely to be delayed due to the bottlene k for energy �ow from

ele trons to phonons. S reening e�e ts, whi h an hange the net ele tron-phonon

oupling, are not in the Holstein-Hubbard model, and require a more omplex model

to be properly des ribed.

61

3.3.3 Spe tral sum rules in momentum spa e

When the system is translationally invariant, it is more onvenient to work in

momentum spa e instead of real spa e. So, we next examine the situation where

tij is a periodi hopping matrix and the lo al hemi al potential, ele tron-phonon

oupling, and Hubbard intera tion are all uniform throughout the latti e. This al-

ulation requires us to make an appropriate Fourier transformation. The al ulations

are tedious, but straightforward. To begin, we start with the de�nition of the retarded

Green's fun tion in the momentum spa e,

GRkσ(t, t

′) = −iθ(t, t′)〈{ckσ(t), ckσ(t′)}〉, (3.84)

where k denotes the momentum. The orresponding reation and annihilation oper-

ators in momentum spa e an be obtained by performing a Fourier transform, ckσ =∑

i eik·Riciσ/N , and c†

kσ =∑

i e−ik·Ric†iσ/N . Here, N is the number of latti e sites.

Substituting the inverse Fourier transformation into the formula for the real-spa e

moments, then yields

µRnkσ (Tave) =

1

N

∑

ij

e−ik·(Ri−Rj)µRnijσ(Tave). (3.85)

The momentum-based sum rules then be ome

µR0kσ(Tave) = 1, (3.86)

µR1kσ(Tave) = ǫ

k

− νσ, (3.87)

where ǫk

= −∑{δ} ti i+δeik·δ

, {δ} is the set of all of the translation ve tors for whi h

the hopping matrix is nonzero (the index i+ δ s hemati ally denotes the latti e site

orresponding to site Ri+δ), and νσ = µ(Tave)−U(Tave)〈nσ(Tave)〉−g(Tave)〈x(Tave)〉.

62

Note, that in a paramagneti solution, the �lling will be independent of the spin σ.

The higher moments be ome the following:

µR2kσ(Tave) = ǫ2

k

− 2ǫk

νσ + ν2σ + U2[〈nσ〉 − 〈nσ〉2] + g2[〈x2〉 − 〈x〉2] + 2U g[〈nσx〉 − 〈nσ〉〈x〉],

(3.88)

µR3kσ(Tave) = ǫ3

k

− 3ǫ2k

νσ + ǫk

ν2σ + 3ǫk

U2[〈nσ〉+ 3〈nσ〉2] + ǫk

g2[〈x2〉 − 〈x〉2]

+ 6ǫk

U g[〈nσx〉 − 〈nσ〉〈x〉]− ν3σ + U3[〈nσ〉 − 〈nσ〉3] + g3[〈x3〉 − 〈x〉3]− 3µU2[〈nσ〉 − 〈nσ〉2]

− 3µg2[〈x2〉 − 〈x〉2] + gκ

M〈x〉+ 3U2g[〈nσx〉 − 〈nσ〉〈x〉] + 3U g2[〈nσx

2〉 − 〈nσ〉〈x2〉]

− 6µgU [〈nσx〉 − 〈nσ〉〈x〉] + 2U gǫ2k

〈nσx〉+ U2∑

q,p′,q′

(ǫq+q′−p + ǫ

p

′−q−q′)〈c†q

′+q′−p′cq

c†p

′cq

′〉

+ U2∑

q,p′,q′

(ǫp−q′ + ǫ

q−q′)〈c†k+q′cqc

†q−kcq′〉+

1

4Re

d2ǫ2k

dT 2ave

+1

4Re [

dµ2

dT 2ave

− dU2

dT 2ave

〈nσ〉 −dg2

dT 2ave

〈x〉].

(3.89)

Similarly, we an obtain the sum rules for the retarded self-energy,

ΣRkσ(Tave, ω = ∞) = U〈nσ〉+ g〈x〉, (3.90)

CR0kσ (Tave) = U2[〈n〉 − 〈n〉2] + g2[〈x2〉 − 〈x〉2] + 2gU [〈xn〉 − 〈x〉〈n〉], (3.91)

CR1kσ (Tave) = −µg2[〈x2〉 − 〈x〉2]− µU2[〈nσ〉 − 〈nσ〉2] + g3[〈x3〉 − 2〈x2〉〈x〉+ 〈x〉3]

− gU2[4〈xnσ〉〈nσ〉+ 3〈nσ〉2〈x〉+ 2〈nσ〉〈x〉 − 3〈xnσ〉] + U3[〈nσ〉 − 2〈nσ〉2 + 〈nσ〉3]

− U g2[4〈xn〉〈x〉+ 5〈x〉2〈nσ〉 − 3〈x2nσ〉] + 2U gǫ2k

〈nσx〉+giκ

M〈x〉

+ U2∑

q,p′,q′

(ǫq+q′−p + ǫ

p

′−q−q′)〈c†q

′+q′−p′cq

c†p

′cq

′〉+ U2∑

q,p′,q′

(ǫp−q′ + ǫ

q−q′)〈c†k+q′cqc

†q−kcq′〉

+1

4Re

d2ǫ2k

dT 2ave

+1

4Re [

dµ2

dT 2ave

− dU2

dT 2ave

〈nσ〉 −dg2

dT 2ave

〈x〉]. (3.92)

These forms of the di�erent sum rules may be more useful for most al ula-

tions, whi h work with translationally invariant systems. Note that, as expe ted, the

63

moments either have no momentum dependen e, or inherit a momentum dependen e

from the bandstru ture, be ause the o�-diagonal moments always had a dependen e

on the hopping matrix element. As noted before, the higher moments require many

di�erent expe tation values to be known in order to properly employ them. If one is

using methods like quantum Monte Carlo simulation, where one an measure su h

expe tation values in addition to determining the Green's fun tion and self-energy,

then one an employ these results as a he k on the a ura y of the al ulations.

Similarly, if one has an approximation method that is employed for the Holstein-

Hubbard model, then by al ulating these di�erent expe tation values within the

approximation, one an test the overall self- onsisten y of the approximation to see if

it satis�es these exa t relations. Of ourse, if everything is evaluated with an approx-

imate solution, there is no guarantee that the approximation is a urate even if it

self- onsistently satis�es these sum rules. But if it does not satisfy them, then you

immediately know they are in error.

We also want to emphasize that these results hold in a wide range of di�erent

nonequilibrium situations and are quite general. This makes them quite valuable

be ause there often are few exa t results known about nonequilibrium solutions. We

hope the ommunity will regularly use these sum rules to he k the a ura y of

di�erent al ulations, espe ially those in nonequilibrium.

Unfortunately, we have not been able to �nd su� iently a urate numeri al al u-

lations on the full Holstein-Hubbard model, along with the al ulation of the required

expe tation values, to readily he k the results of these sum rules against state-of-the-

art al ulations. So, as a substitute, we examine the atomi limit next, whi h allows

us to he k the pie es of the sum rule that do not depend on the hopping.

64

3.3.4 Atomi limit of the Holstein-Hubbard model

We an solve the atomi limit (tij → 0) exa tly, and thereby he k all terms in the

moments that survive when the hopping vanishes. The atomi Hamiltonian be omes

HatH−H(t) = U(t)n↑n↓ + [g(t)x− µ](n↑ + n↓) +

p2

2M+

1

2κx2, (3.93)

where, for simpli ity, we assume the hemi al potential is independent of time, be ause

time dependen e with respe t to the hemi al potential an be trivially handled due

to the fa t that the total ele tron number operator ommutes with the Hamiltonian.

Using the equation of motion for the raising and lowering operators, in the Heisenberg

representation, we �nd the phonon oordinate operator be omes

xH(t) =ae−iω(t−tmin) + a†eiω(t−tmin)

√2mω

− Re

{

ie−iωt

∫ t

tmin

dt′eiωt′

g(t′)

}n↑ + n↓mω

. (3.94)

Note that be ause the ele tron number operator now ommutes with the Hamiltonian,

it has no time dependen e. The Hilbert spa e is a dire t produ t of the harmoni

os illator number states given by the number operator representation

|n〉 = 1√n!

(a†)n |0〉, (3.95)

and the four fermioni states

|0〉, | ↑〉 = c†↑|0〉, | ↓〉 = c†↓|0〉, | ↑↓〉 = c†↑c†↓|0〉. (3.96)

The partition fun tion is

Zat =∑

0,↑,↓,↑↓

∞∑

nb=0

〈nb, nf | exp[−β{(gx− µ)(nf↑ + nf

↓) + Unf↑ n

f↓ + ω(nb +

1

2)}]|nb, nf〉.

(3.97)

where nb and nf are the bosoni and fermioni number operators, respe tively. Note

that the partition fun tion is al ulated in the initial equilibrium state, so g = g(tmin)

65

and U = U(tmin) have no time dependen e. To al ulate the partition fun tion, we

need to go to the intera tion representation with respe t to the bosoni Hamiltonian

in imaginary time. The steps of the al ulation are similar to what we have already

done for the Holstein model in Ref. [11℄ with the modi� ation to in lude the Hubbard

intera tion. Here, we will only report the �nal results and we refer the interested

reader to Ref. [11℄ for the details. The �nal results for the partition fun tion and the

other observables be omes,

Zat =1

1− e−βω

{

1 + 2eβµ exp

[βg2(tmin)

2mω2

]

+ eβ(2µ−U(tmin)) exp

[2βg2(tmin)

mω2

]}

,(3.98)

and

〈n↑ + n↓〉 =2eβµ exp

[βg2(tmin)

2mω2

]

+ 2eβ(2µ−U(tmin)) exp[2βg2(tmin)

mω2

]

1 + 2eβµ exp[βg2(tmin)

2mω2

]

+ eβ(2µ−U(tmin)) exp[2βg2(tmin)

mω2

] , (3.99)

〈x(t)〉 = 〈n↑ + n↓〉(

−g(tmin)

mω2cosω(t− tmin)− Re

{

ie−iωt

∫ t

tmin


mω

})

.

(3.100)

We noti e that, in equilibrium, g(t) = g be omes a onstant and the average

phonon oordinate be omes time-independent, 〈x(t)〉 = −〈n↑ +n↓〉g/mω2, as we saw

previously. Additionally, the �u tuation satis�es,

〈x2(t)〉 − 〈x(t)〉2 = [〈n↑〉(1− 〈n↑〉) + 〈n↓〉(1− 〈n↓〉) + 2〈n↑n↓〉 − 2〈n↑〉〈n↓〉]

×[g(tmin)

mω2cosω(t− tmin) + Re

{

ie−iωt

∫ t

tmin


mω

}]2

+1

2mωcoth

(βω

2

)

. (3.101)

One an see that the �u tuation is onsists of both quantum and thermal parts.

The �rst term denotes the quantum �u tuation due to the interation in the system.

The role of ele tron-phonon intera tion in the �u tuation has manifested itself by

66

a lear time dependent of oupling fa tor. However, we noti e that the ele tron-

ele tron intera tion is somewhat hidden in the oe� ient of the square term whi h

is a fun tion of ele tron density given by equation 3.99. One noti es that by setting

U = 0, we re over the results for the Holstein model in Ref. [14℄ as we expe ted. Then,

the se ond term denotes the thermal �u tuation whi h is temperature dependent and

it grows as temperature in reases. To omplete the al ulation of the atomi limit,

we also need to al ulate the following expe tation value,

〈x(t)n↓(t)〉 = 〈(n↓(t) + n↑(t))2〉(

−g(tmin)


{

ie−iωt

∫ t

tmin


mω

})

.

(3.102)

Re all that the Pauli ex lusion prin iple implies that 〈nσ(t)2〉 = 〈nσ(t)〉 and onse-

quently,

〈x(t)n↓(t)〉 = (〈n↓(t)〉+ 〈n↑(t)〉+ 2〈n↓(t)n↑(t)〉)

×(

− g(tmin)


{

ie−iωt

∫ t

tmin


mω

})

.

(3.103)

By substituting the expe tation value of position from equation 3.100, we get

〈x(t)n↓(t)〉 = 〈x(t)〉+ 2〈n↓(t)n↑(t)〉(

− g(tmin)


{

ie−iωt

∫ t

tmin


mω

})

,

(3.104)

whi h expli itly shows that di�erent orrelations are dependent to ea h other be ause

of the ele tron-phonon oupling. This identity be omes more valuable as the ele tron

density orrelation for up and down spin an be evaluated easily for ases su h as

half-�lling whi h implies a simple to evaluate ele tron harge and phonon oordinate

orrelation. Before ending this se tion, we omment about the zero moment of the

self-energy in the atomi limit. Putting together all three terms in equation (3.82) that

67

we have evaluated in atomi limit, we observe the following behaviors: First, the zero

moment of the self-energy in nonequilibrium is time dependent through dependen e

to original time-dependent parameters. Se ond, the zero moment of the self-energy

hanges with temperature whi h omes from the �u tuation term of phonons. This

temperature dependen e an be reated by an external driving �eld, for example , it

an be the ele tri �eld of the pump in the pump-probe tr-Arpes experiment. Our

result is onsistent with a re ent study performed by Kemper and his ollaborates

where they have performed numeri al simulation to study the e�e t of the pump

on the kink softening in the Holstein-Hubbard model[68℄. Their result indi ates that,

although the self-energy an hange with time, the spe tral weight of self-energy stays

onstant in all times.

We end this se tion with proposing a slightly di�erent method to verify the sum

rule result in the atomi limit. First, we al ulate the c(t1) and c†(t2) in the Heisenberg

representation in the atomi limit:

c(t1) = T e−i∫ t1t0

[−U(t)n(t)+µ(t)−g(t)x(t))]dtc,

c†(t2) = T ei∫ t2t0

[−U(t)n(t)+µ(t)−g(t)x(t))]dtc†. (3.105)

Although x(t) does not ommute at di�erent times, the time ordered produ t an be

evaluated as this ommutator be omes a simple fun tion of time and by transforming

in the intera tion representation we an obtain the following produ t after performing

long algebra, see Ref. [11℄ for more details,

c(t1) = e−i∫ t1t0

[−U(t)n(t)+µ(t)]dteig2(t1)

2mω3 (ωt1−sinωt1) ×−g(t1)√2mw3

[(eiωt1−1)a†+(1−e−iωt1)a]c,(3.106)

c†(t2) = ei∫ t2t0

[−U(t)n(t)+µ(t)]dte−ig2(t2)

2mω3 (ωt2−sinωt2) ×−g(t2)√2mw3

[(e−iωt2−1)a+(1−eiωt2 )a†]c†.(3.107)

68

Then, by using the equation 3.4, we an rewrite the spe tral moment in the atomi

limit as,

µRnat = (∂t1 − ∂t2)

nGRat(t1, t2)|(t2−t1)→0 (3.108)

where

GRat(t1, t2) = −iθ(t1, t2)〈{c(t1), c†(t2)}〉. (3.109)

By substituting the fermioni operators we get:

GRat(t1, t2) = −iθ(t1, t2)e−i

∫ t1t2

[−U(t)n(t)+µ(t)]dt e−i

2mw2 [g2(t1)t1−g2(t2)t2]e−

i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)]

×(

〈e[−g(t1)√2mw3

(eiωt1−1)a†+(1−e−iωt1)a]e[− g(t2)√

2mw3(e−iωt2−1)a+(1−eiωt2 )a†]

c c†〉

+〈e[−g(t2)√2mw3

(e−iωt2−1)a+(1−eiωt2 )a†]e[− g(t1)√

2mw3(eiωt1−1)a†+(1−e−iωt1 )a]

c† c〉)

. (3.110)

In the next step, we an evaluate the expe tation value, similar to what we did for

the other ones:

GRat(t1, t2) = −iθ(t1, t2)

1

Z∑

0,↑,↓,↑↓

∞∑

nb=0

e−βωn+βµnf− g2(tmin)nf2

2mω3 (eβω−1−βω)

e−i∫ t1t2


2mw2 [g2(t1)t1−g2(t2)t2]e−

i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)]

×(

〈nb, nf |eA1a eA2a†eA3a+A4a†eA5a+A6a†c c† + eA1a eA2a†eA4a+A6a†eA3a+A5a†c† c|nb, nf 〉,(3.111)

where, the oe� ients are as follows

A1 = −g(τmin)nf

√2mw3

(1− e−ωβ),

A2 = −g(τmin)nf

√2mw3

(eωβ − 1),

A3 = − g(t1)√2mw3

(eiωt1 − 1),

A4 = − g(t1)√2mw3

(1− e−iωt1),

A5 = − g(t2)√2mw3

(eiωt2 − 1),

A6 = − g(t2)√2mw3

(1− e−iωt2). (3.112)

69

After rearranging di�erent terms by using the identity eαA+βB = eαAeβBe−12αβ[A,B]

,

and evaluating taking the tra e with respe t to both fermioni and bosoni operator,

we have

GRat(t1, t2) = −iθ(t1, t2)

1

(1− e−βω)ZF1F2[e−A4A5 + 2eβµ+

g2β

2mω2 eA6A3

+e2(βµ−U(τmin))+g2β

mω2 (2e−A6A3 − e−A4A5)]. (3.113)

where

F1 = e−i∫ t1t2


2mw2 [g2(t1)t1−g2(t2)t2]e−

i2mw3 [g(t1) sin(ωt1)−g(t2) sin(ωt2)],

F2 = e−A2A3−A4A5− 12A3A4− 1

2A5A6 e

1

1−e−βω [A1A4+A1A6+A3A2+A3A6+A5A2+A5A4+A5A6].(3.114)

At this stage, we need to evaluate the time derivative of the above expression. For

example, the zero moment is straightforward and it an be obtained by taking the

limit (t2 − t1) → 0 whi h be omes

µR0at = GR

at(t1, t2)|(t2−t1)→0 = 1. (3.115)

3.3.5 Dis ussion and on lusion

In this paper, we have derived a general formalism whi h enables us to evaluate the

nth derivative of a time dependent operator in the Heisenberg representation. We have

noti ed that, this identity an be useful in some of other studies su h as full ounting

statisti s problem [92℄ or al ulating the dynami al algebra of the bosons [93℄ where

one needs to evaluate the derivative of di�erent operators. These are beyond the s ope

of the urrent study and we will postpone the deep studies for future works.

Then, we have used this identity to evaluate a sequen e of spe tral moment sum

rules for retarded Green's fun tion and orresponding self-energy in the normal state.

70

These spe tral fun tions have been obtained for the nonequilibrium and generi non-

homogenous ase. The spe tral fun tions provide an exa t formalism in nonequilib-

rium whi h an be used to he k the self- onsisten y he k of both experimental

and omputational results. We have al ulated the spe tral moment for the retarded

Green's fun tion whi h indi ate that both ele tron-phonon and ele tron-ele tron inter-

a tion has ontribution in the spe tral fun tion and only will hange with time if the

original parameter su h as tij , U , and g are fun tion of time.

Although we an not on lude whether the ele tron-ele tron or ele tron-phonon

intera tion has main ontribution in spe tral fun tion of retarded Green's fun tion,

the results for the spe tral moment of the self-energy are more interesting. In fa t,

our results in the atomi regime has provided an intuitive pi ture for studying the

behavior of the spe tral moment of the zero moment of the self-energy in the presen e

of external deriving. This result learly shows a temperature dependen e through

phonon �u tuation whi h arises from both quantum and thermal �u tuation and

provides an strong eviden e that the zero moment of the spe tral fun tion in reases

with in reasing temperature. Furthermore, this result agrees with a re ent numeri al

study in Ref. [68℄, and suggest that the softening of the kink in tr-Arpes of pump-probe

experiment an be indu ed from the applied pump without any hange in ele tron-

phonon oupling. This an be understand as following: although the self-energy itself

hanges with time whi h indu es a spe tral redistribution, however this redistribution

will ompensate ea h other at di�erent interval of frequen y and the integral over the

frequen y will remain onstant as we expe t it from the sum rule.

All the results presented in this paper have been al ulated for the normal state.

So, one important question is how the sum rules are hanged in the super ondu ting

state. To answer this question one needs to arefully extend the de�nition of the

spe tral fun tion into the super ondu ting state. One possible pro edure would be

71

using the Nambu-Gorkov formalism whi h ontains both the normal and anoma-

lous Green's fun tion. Therefore, for the self-energy one needs also determine the

proper Dyson equation in the super ondu ting state. In addition to expanding the

sum rules into super ondu ting state, a re ent study has indi ated an enhan ed tran-

sient ele tron-phonon oupling in a driven bilayered graphene whi h attra ted a lot of

attention[94, 95℄. However, the me hanism for su h enhan ement of ele tron-phonon

oupling remain un lear. One of the possible me hanism is the possibility of non-

linear ele tron-phonon oupling. So it would be interesting to investigate the role

of the non-linear ele tron-phonon oupling in the presen e of the external �eld by

using the spe tral moment sum rules. We hope that we will be able to address these

issues and expand the sum rules results to both super ondu ting state and the ase

of Hubbard-Holstein model with nonlinear oupling in the presen e of the external

�eld.

72

4

Current-voltage profile of a strongly orrelated materials

heterostru ture using non-equilibrium dynami al mean field theory

�The approximate omputations are a fundamental part of physi al s ien e.�

� Steven Weinberg

The multilayer devi e of our interest has onsisted of a number of planes built on

top of ea h other. Depending on the physi al system, the layers an be insulators or

metals. These planes have a small thi kness in one dire tion (here we onsider the z

dire tion) while they are in�nite in the other two dimensions. A �nite number of layers

an be onsidered as the barrier region where the interesting physi al phenomena

o ur and it's onne ted to the bulk via a �nite number of metalli planes, see Figure

4.1, where the bulk will determine the boundary ondition of the system. We will use

the Fali ov-Kimball [2℄ model to des ribe our multilayer devi e, the Hamiltonian of

the system in the general ase in the presen e of an ele tri �eld an be written as

HFK(t) = −∑

αl

t⊥αα+1[eieAα(t)c†αlcα+1l + e−ieAα(t)c†α+1lcαl]

−∑

αlδ

t‖αc†αlcαl+δ +

∑

αl

(−µα + Uαwαl)c†αlcαl (4.1)

73

where t⊥αα+1 is the hopping from plane α to α+1 and it equals to t⊥α+1α for a Hermitian

Hamiltonian. t‖α is the hopping within ea h plane between nearest neighbors, j = i+δ

with δ being a latti e onstant. µα = µ+Vα and Uα denotes the hemi al potential and

the intera tion at ea h plane, where they an hange for di�erent planes depending

on the intera tion and the voltage a ross the plane. Moreover wαl = f †αlfαl whi h an

be 0 or 1 for the heavy parti le. The Aα(t) and Vα are the time dependent ve tor

potential and s alar potential to des ribe the ele tri �eld,

Eα(t) = −∇Vα − ∂Aα(t)

∂t(4.2)

In the next se tion, we will explain the gauge that we will hoose for the ve tor

potential and s alar potential. As we know regardless of the hoi e of the gauge one

should obtain the same results for physi al quantities.

74

L

Bulk

Bulk

R

αα− 1 α+ 1

tαα+1

E

Metal

Insulator

Figure 4.1: S hemati pi ture of a multilayer devi e. The devi e onsists of a di�erent

number of metalli and insulator layers. In this paper, we have a single insulator layer

in the enter of the devi e, whi h is atta hed to the bulk via metalli leads at both

sides. The di�erent planes are indi ated by a Greek index α and tαα+1 denotes the

hopping from plane α to α+ 1 in z dire tion. An extra ele tri �eld is applied in the

barrier region to ompensate for the s attering.

4.1 Non equilibrium DMFT formalism for a multilayer devi e

In this se tion, we explain the main formalism for studying the steady-state transport

through a multilayer devi e. The most ommon method to obtain the urrent-voltage

pro�le is based on a voltage biased method where the left and right leads are separated

from the barrier and their hemi al potential is shifted by an opposite voltage on

opposite sides µl,r = ±V/2 [69, 70℄. Then, after turning on the hopping between

the leads and the barrier, the urrent drives into the system and after some time

eventually rea hes the steady state. But here, we will use a di�erent formalism whi h

has been proposed by Freeri ks et al, the so- alled urrent biased approa h [17℄. In

this formalism, the left and right leads are always onne ted to the barrier. We assume

75

that the system is in equilibrium at temperature T in the in�nite past. Then we apply

an ele tri �eld to generate urrent in the system and after that we turn it o� and

we let the system evolve so that the ve tor potential be omes Aα(t) = −A0. Sin e

the metalli planes are ballisti , the urrent will ontinue to �ow in the system even

after the ele tri �eld has been turned o�. Sin e the system rea hes a steady state,

and we an do the Fourier transform to the frequen y domain, whi h makes a huge

simpli� ation in our al ulation. Both urrent and �lling on ea h plane are onstant,

but due to the s attering in the insulator, the urrent will drop unless we add an

external ele tri �eld at the barrier to ompensate the s attering in those planes. In

pra ti e, this an be done in two ways, one with the time-dependent ve tor potential,

whi h leads to an ele tri �eld Eα = −∂Aα(t)∂t

. The se ond method is to apply a voltage

bias whi h gives rise to a s alar potential, and we get an ele tri �eld on the planes

whi h experien e the hange in the s alar potential E = −∇Vα. Here, we will work

with ve tor potential gauge, but one an show that the result is identi al to the se ond

gauge.

4.2 Equation of motion

In this se tion, we will derive the equation of motion whi h leads to a set of iterative

equations in the multilayer devi e whi h has been previously obtained in Ref. [71℄.

To solve the iterative equations one needs to know the boundary ondition that we

will explain in the next se tion. We start with the ontour ordered Green's whi h is

generalized to the multilayer system,

gcαβ ij(t, t′) = − i

ZTr[Tce−βH(−∞)cαi(t)c

†βj(t

′)], (4.3)

where Z = Tr[e−βH(−∞)] is a partition fun tion with H(−∞) being the Hamiltonian

in the in�nite past at equilibrium. As usual, cα i and c†β j are reation and annihilation

76

operators with indi es α and β denoting di�erent planes, while i and j denotes the

di�erent sites at ea h plane. Tc is the time ordering and it depends on where two

times t, t′ lie on the Keldysh ontour, see �gure 2.1. Sin e the ele tri �eld a ts in

the z dire tion, we an transfer to the mixed basis, where we an perform a Fourier

transform for the xy degrees of freedom into momentum spa e, while keeping the z

dire tion in the real spa e basis, so the ontour ordered Greens fun tion be omes,

gcαβ(k‖; t, t′) = − i

ZTr[Tce−βH(−∞)cαk‖(t)c

†βk‖

(t′)]. (4.4)

To obtain the equation of motion, we take the time derivative of the Green's fun tion

with respe t to t,

∫

c

dt{[(−i∂t + µα − ǫ‖α(k‖))δc(t, t)− Σc

α(t, t)]gcαβ(k

‖; t, t′)

+ [t⊥αα+1eiAα(t)gcα+1β(k

‖; t, t′) + t⊥α−1αe−iAα(t)gcα−1β(k

‖; t, t)]δc(t, t′)}

= δαβδc(t, t′), (4.5)

where we have introdu ed the planar band stru ture de�ned as ǫ‖α(k‖) = −t‖α

∑

δ eik‖.δ

with δ being the nearest neighbor translation ve tor and the ontour ordered delta

fun tion is de�ned as

∫

cdt′δc(t, t′)f(t′) = f(t). Moreover, Σc

α(t, t′) denotes the ontour

ordered self-energy at the α's plane. As we an see, due to the shift of indi es of the

Green's fun tion appearing in equation 4.5, we will get two sets of re ursive equations.

Before deriving the re ursive equations, we assume that we have �xed the α and β

indi es and the inverse will be done with respe t to time. To derive these re ursive

equations, we �rts multiply equation 4.5 by gc −1αβ (k‖; t, t′) from the right side and we

substitute β → α and α→ α− n

77

0 = [−i∂t + µα−n − ǫ‖(k‖)]δc(t, t′)− Σc

α−n(t, t′)

+ t⊥α−nα−n+1eiAα−n(t)

∫

c

dtgcα−n+1α(k‖; t, t)gc −1

α−nα(k‖; t, t′)

+ t⊥α−n−1α−neiAα−n−1(t)

∫

c

dtgcα−n−1α(k‖; t, t)gc −1

α−nα(k‖; t, t′). (4.6)

Now we introdu e the ontour-ordered left fun tion Lcα−n de�ned as

Lcα−n(k

‖;µα−n; t, t′) = −t⊥α−nα−n+1e

iAα−n(t)

∫

c

dtgcα−n−1α(k‖; t, t)gc −1

α−nα(k‖; t, t′). (4.7)

Substituting equation 4.7 in 4.6, we an rewrite it as

Lcα−n(k

‖;µα−n; t, t′) = [−i∂t + µα−n − ǫ

‖α−n(k

‖)]δc(t, t′)− Σα−n(t, t

′)− t⊥2α−n−1α−n

×e−iAα−n−1(t)Lc −1α−n−1(k

‖;µα−n−1; t, t′)eiAα−n−1(t′). (4.8)

Next, we an perform similar pro edure to obtain another re ursive relation, we let

α→ α+ n and β → α then we get

Rcα+n(k

‖;µα+n; t, t′) = −t⊥α+n−1α+ne

−iAα+n−1(t)

∫

c

dtgcα+n−1α(k‖; t, t′)gc −1

α+nα(k‖; t, t′),(4.9)

and the re ursion relation be omes

Rcα+n(k

‖;µα+n; t, t′) = [−i∂t + µα+n − ǫ

‖α+n(k

‖)]δc(t, t′)− Σc

α+n(t, t′)− t⊥2

α+n+1α+n

×eiAα+n(t)Rc −1α+n+1(k

‖;µα+n+1; t, t′)e−iAα+n(t′). (4.10)

Using the de�nition of the left and right fun tions and setting α = β, the equation of

motion an be written as

gc −1αα (k‖; t, t′) = [−i∂t + µα − ǫ‖α(k

‖)]δc(t, t′)− Σc

α(t, t′)

− t⊥2α−1αe

−iAα−1(t)Lc −1α−1 (k

‖;µα−1; t, t′)eiAα−1(t′)

− t⊥2αα+1e

iAα(t)Rc −1α+1 (k

‖;µα+1; t, t′)e−iAα(t′). (4.11)

78

Noti e that the sum of the inverse of left and right part an be onsidered as a

ontribution to the self-energy of the leads. Using the left and right iterative equations,

we an rewrite the above equation in the following form,

gc −1αα (k‖; t, t′) = −[−i∂t + µα − ǫ‖α(k

‖)]δc(t, t′) + Σc

α(t, t′)

+ Lcα(k

‖;µα; t, t′) +Rc

α(k‖;µα; t, t

′). (4.12)

After rea hing the steady state, the system re overs the time translational invarian e

and the Green's fun tion, self-energy, left and right fun tion, all will depend only

on the relative time in the presen e of onstant urrent �ow, so that we an Fourier

transform into the frequen y domain. But prior to that, we take an extra step to

simplify the above equation as all are ontour ordered obje ts. So we transfer into

the Larkin-Ov hinkov representation [45℄. By solving the matrix equation, we get 3

equations for the retarded, advan ed and Keldysh fun tions. Now lets look at their

respe tive de�nitions

gRαβ(k‖; t, t′) = −iθ(t− t′)〈{cαk‖(t), c

†β k‖

(t′)}+〉, (4.13)

gAαβ(k‖; t, t′) = iθ(t′ − t)〈{cαk‖(t), c

†β k‖

(t′)}+〉, (4.14)

and

gKαβ(k‖; t, t′) = −i〈[cα k‖(t), c

†β k‖

(t′)]−〉, (4.15)

then we an obtain the following identities,

gR∗αβ(k

‖; t, t′) = gAβ α(k‖; t′, t), (4.16)

and

gK∗αβ(k

‖; t, t′) = −gKβ α(k‖; t′, t). (4.17)

79

By performing a Fourier transform into the frequen y domain, we get,

gR∗αβ(k

‖;ω) = gAβ α(k‖;ω), (4.18)

and

gK∗αβ(k

‖; t, t′) = −gKβ α(k‖;ω), (4.19)

whi h shows that the Keldysh Green's fun tion is purely imaginary in the frequen y

domain. Using the above identities, we only need to al ulate retarded and Keldysh

Green's fun tion. To do so, we �rst start with the left and right fun tion in the

frequen y domain,

LRα−n(k

‖;µα−n;ω) = ω + µα−n − ǫ‖(k‖)− ΣRα−n(ω)

− t⊥2α−nα−n−1

LRα−n−1(k

‖;µα−n−1;ω + Eα−n−1), (4.20)

where we have introud ed the ele tri �eld by hoosing non-zero ve tor potential

whi h leads to Aα(t) = −A0 − Eαt. Similarly, we obtain the Keldysh left fun tion as

well,

LKα−n(k

‖;ω) = −ΣKα−n(ω) +

t⊥2α−nα−n−1L

Kα−n−1(k

‖;ω + Eα−n−1)

| LRα−n−1(k

‖;ω + Eα−n−1 + Eα−n−1) |2. (4.21)

We an write similar equations for the right fun tions,

RRα+n(k

‖;ω) = ω + µα+n − ǫ‖α+n(k

‖)− ΣRα+n(ω)

− t⊥2α+n+1α+n

RRα+n+1(k

‖;ω −Eα+n), (4.22)

and

RKα+n(k

‖;ω) = −ΣKα+n(ω) +

t⊥2α+n+1α+nR

Kα+n+1(k

‖;ω − Eα+n)

| LRα+n+1(k

‖;ω − Eα+n) |2. (4.23)

In the next se tion, we will show how one an �x the boundary ondition in the bulk

to use the re ursive equation for obtaining the retarded and Keldysh Green's fun tion.

80

4.3 Fixing the boundary ondition

As we an see, the retarded and advan ed fun tions an be al ulated independently

from the Keldysh part. First, we obtain L−∞ and R∞. Using the fa t that the self-

energy is zero for ballisti metals, ΣR±∞(ω) = 0 and also, t⊥αα−1 = t⊥ for the limit

α→ ±∞, we get

LR2−∞(k‖;ω)− [ω + µα − ǫ‖(k‖)]LR

−∞(k‖;ω) + t2⊥ = 0 (4.24)

whi h leads to

LR2−∞(k‖;ω) =

ω + µα − ǫ‖(k‖)

2± 1

2

√

[ω + µα − ǫ‖(k‖)]2 − 4t2⊥. (4.25)

The sign needs to be hosen in su h a way that the imaginary part of LRis greater than

zero. In a similar way, we �nd the solution for R∞ and we observe that, L−∞ = R∞.

These results for the left and right fun tions are the same as for equilibrium. As we

expe t, in the absen e of an ele tri �eld, even in the urrent arrying state, the states

are un hanged and only the momentum hanges whi h leads to hanges in o upan y

of states. Within the formalism, we �x a number of planes at both side of the barrier.

And here we onsider 30 planes. As we expe t, the planes far from the barrier will

not be a�e ted and they equal the bulk values, so we onsider L1 = L−∞ on the

left side and R61 = R∞. In the se ond step, we an use equations 4.20 and 4.22

to evaluate the left and right fun tion at all planes, then we an al ulate the lo al

Greens fun tion from equation 4.12 whi h we need to integrate over two-dimensional

density of states,

gR,Aαα (ω) =

∫

dǫ‖αρ2Dα (ǫ‖α)g

R,Aαα (ǫ‖α, ω), (4.26)

where

81

gRαα(k‖, ω) =

1

−[ω + µα − ǫα(k‖)− ΣRα (ω)] + LR

α (k‖;ω) +RR

α (k‖;ω)

. (4.27)

The advan ed fun tion an be found by taking the omplex onjugate of the retarded,

as we derived in equation 4.17. The retarded Greens fun tion provides information

about the density of states. To obtain information about the �lling and ultimately to

al ulate the urrent, we need to evaluate the Keldysh Green's fun tion. But we an't

al ulate the Keldysh part in the same way that we did for the retarded. The reason

being the di�eren e between the re ursive equations got the retarded and Keldysh

Green's fun tions. As we an see, for zero self-energy, the Keldysh re ursive equation

4.21 be omes homogeneous and it an not be de�ned uniquely, as we an multiply

both sides by any value, and we rea h the trivial solution of 0 = 0. However, we an

take a di�erent path. Starting from the de�nition of the left and right fun tions 4.7

- 4.10 , we an derive the following identities for retarded

LRα−1(k

‖;ω) = −t⊥α−1αeiAα−1(t)

gRαα(k‖;ω)

gRα−1α(k‖;ω)

, (4.28)

RRα+1(k

‖;ω) = −t⊥αα+1e−iAα(t)

gRαα(k‖;ω)

gRα+1α(k‖;ω)

, (4.29)

and Keldysh fun tions,

LKα−1(k

‖;ω) = −LRα−1(k

‖;ω)gKα−1α(k

‖;ω)

gAα−1α(k‖;ω)

− t⊥α−1αeieAα−1(t)

gKαα(k‖;ω)

gAα−1α(k‖;ω)

, (4.30)

RKα+1(k

‖;ω) = −RRα+1(k

‖;ω)gKα+1α(k

‖;ω)

gAα+1α(k‖;ω)

− t⊥α+1αe−ieAα+1(t)

gKαα(k‖;ω)

gAα+1α(k‖;ω)

. (4.31)

This means that to evaluate the left and right Keldysh fun tion we need to al ulate

the lo al and nearest neighbour Green's fun tion in bulk. In the following, we will

82

derive the retarded and Keldysh Green's fun tion in the presen e of a onstant ve tor

potential Aα(t) = −A0 + Eα(t) . This has been al ulated in [11℄ and here we only

write down the results,

gR(k; t, t′) = −iθ(t− t′)e−i[ǫ‖(k‖)−2t cos(kz+A0)−µbulk ](t−t′), (4.32)

where kz is the momentum in the z dire tion and both t and t′ are mu h larger

ompare to −∞. By performing the Fourier transform we get,

gR(k;ω) =1

ω + µbulk − ǫ‖(k‖) + 2t cos(kz + A0) + i0+. (4.33)

To make sure this gives the same results as we found for the self-energy of the leads,

we need to transform into the mixed basis by performing just a Fourier transform on

kz into real spa e, where we get

gRαβ(k‖;ω) =

1

2π

∫ π

−π

dkze−i(zα−zβ)kzgR(k;ω). (4.34)

By hanging the variables ω+µbulk− ǫ‖(k‖)− i0+ = 2tγ and k′z = k+A0, the integral

an be al ulated by the residue theorem. As the details have been shown in [12℄, we

only report the results here for the lo al Green's fun tion,

gRαβ(k‖;ω) =

1

±√

[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2⊥, (4.35)

the sign of the retarded Green's fun tion an be hosen from the ausality posterior,

whi h implies that the Green's fun tion has negative imaginary part when it is om-

plex. This result is onsistent with what we have obtained in equation 4.27. Similarly,

we an al ulate the nearest neighbor Green's fun tion,

gRαα+1(k‖;ω) = e−iA0

−ω+µbulk−ǫ‖(k‖)2t

±√

[ω+µbulk−ǫ‖(k‖)]2

4t2− 1

±√

[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2, (4.36)

83

where the sign needs to be hosen a ording to the same argument as above. By

substituting k′z → −k′z we an also evaluate the other integral whi h be omes,

gRαα−1(k‖;ω) = eiA0

−ω+µbulk−ǫ‖(k‖)2t

±√

[ω+µbulk−ǫ‖(k‖)]2

4t2− 1

±√

[ω + µbulk − ǫ‖(k‖)− i0+]2 − 4t2. (4.37)

We an he k that by substituting equation 4.35 into 4.33, we re over equation 4.25.

Next we need to al ulate the Keldysh Green's fun tion. In the bulk, it be omes [12℄,

gK(k; t, t′) = ifK [ǫ‖(k‖)− 2t cos(kz)− µbulk]e−i[ǫ‖(k‖)−2t cos(kz+A0)−µbulk ](t−t′). (4.38)

In the steady state, we an perform the Fourier transform into frequen y spa e,

gK(k;ω) = 2πifK [ǫ‖(k‖)− 2t cos(kz)− µbulk]

× δ[ω + µbulk + ǫ‖(k‖) + 2t cos(kz + A0)]. (4.39)

To go ba k into mixed basis, we perform another Fourier transform from kz into the

mixed-basis Green's fun tion,

gKαβ(k‖;ω) =

1

2π

∫ π

−π

dkze−i(zα−zβ)kzgK(k;ω). (4.40)

This integral an be done, as gK(k;ω) is proportional to a delta fun tion, but one needs

to divide the integrand by the absolute value of the derivative of the argument inside

the delta fun tion, as delta fun tion be omes zero for two di�erent roots. Moreover,

the Keldysh fun tion is non-zero only inside the band,

| ω + µbulk − ǫ‖(k‖)

2t|≤ 1. (4.41)

84

Similar to the retarded ase, we an do this integral with th residue theorem and the

�nal result be omes,

gKαβ(k‖;ω) = i

{

fK [ω+(k‖;A0)][

− ω + µbulk − ǫ‖(k‖)

2t− i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]|zα−zβ |

+fK [ω−(k‖;A0)][


2t+ i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]|zα−zβ |}

× 1√

4t2 − (ω + µbulk − ǫ‖(k‖))2ei(zα−zβ)A0 , (4.42)

where ω±is de�ned as follows:

ω+(k‖;A0) = ω cos(A0)− (µbulk − ǫ‖(k‖))(1− cos(A0)

±√

4t2 − [ω + µbulk − ǫ‖(k‖)] sin(A0). (4.43)

Then the lo al Keldysh Green's fun tion be omes

gKαα(k‖;ω) = i

fK [ω+(k‖;A0)] + fK [ω−(k‖;A0)]√

4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.44)

where fK(ω) = 11+eβω is the Fermi-Dira distribution. Similarly, we an �nd the

nearest neighbor Keldysh Green's fun tion,

85

gKαα+1(k‖;ω) = i

{

fK [ω+(k‖;A0)][


2t− i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]

+fK [ω−(k‖;A0)][


2t+ i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]}

× e−iA0

√

4t2 − (ω + µbulk − ǫ‖(k‖))2, (4.45)

and

gKαα+1(k‖;ω) = i

{

fK [ω+(k‖;A0)][


2t+ i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]

+fK [ω−(k‖;A0)][


2t− i

√

1−(ω + µbulk − ǫ‖(k‖)

2t

)2]}

× eiA0

√

4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.46)

Now, we verify that when A0 = 0, we re over the equilibrium results,

gKαα(k‖;ω) =

2ifK(ω)√

4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.47)

Now we an go ba k to equations 4.30 and 4.31 to al ulate the left and right Keldysh

fun tions. After doing some long algebra we obtain,

LK−∞(k‖;ω) = −ifK [ω+(k‖;A0)]

√

4t2 − (ω + µbulk − ǫ‖(k‖))2 (4.48)

86

and

RK∞(k‖;ω) = ifK [ω−(k‖;A0)]

√

4t2 − (ω + µbulk − ǫ‖(k‖))2. (4.49)

After al ulating the left fun tion at the �rst plane and the right fun tion at last

plane, we an use equations 4.21 and 4.23 to al ulate the left and right Keldysh

fun tions at other planes. Then we integrate over the two dimensional density of

states to get the lo al Keldysh fun tion. At the �nal step, we need to al ulate the

impurity Green's fun tion, whi h we will explain in next se tion.

4.4 Impurity solver

One of the most hallenging parts of nonequlibrium dynami al mean-�eld theory is

to �nd the proper impurity solver. There have been few di�erent methods to �nd the

impurity solver for various models. Most of the impurity solvers, su h as the ontin-

uous time quantum Monte Carlo, weak oupling and strong oupling perturbation

theory are based on diagrammati expansions. However, the Fali ov-Kimball model

has spe ial advantages, as the impurity solver is exa tly solvable,

gcimp = (1− w1)gc0 + w1(g

c−1

0 − Uδc)−1. (4.50)

Similar to what we did before, using the Larkin-Ov hinkov transform, we an get the

Keldysh and retarded (or advan ed) parts separately,

gRimp = (1− w1)gR0 + w1(g

R−1

0 − U)−1, (4.51)

and

gKimp = (1− w1)gK0 + w1(g

R−1

0 − U)−1gR−1

0 gK0 gA−1

0 (gA−1

0 − U)−1. (4.52)

87

4.5 Cal ulating the urrent in multilayer devi e

We start with explaining how one an determine the urrent operator for a parti -

ular model. Let's onsider a system with an external ele tri �eld E(r, t). Then, the

equation of ontinuity, whi h onne ts number urrent to harge density is

∂ρ(r, t)

∂t+∇. j(r, t) = 0. (4.53)

Introdu ing the ele tri polarization P(r, t) = rρ(r, t), we take the partial derivative

with respe t to time to get, ∂P(r, t)/∂t = r∂ρ(r, t)/∂t. Substituting ∂ρ(r, t)/∂t with

−∇. j(r, t) from equation 4.53 and integrating over spa e, we get

j(r, t) =∂P(r, t)

∂t. (4.54)

Using the Heisenberg equation of motion that we showed in hapter 2, the time

derivative of operators an be repla ed by ommutators with Hamiltonian, so we an

write

j(r) = i[H,P(r)], (4.55)

where on the latti e, the polarization operator an be written as Pj = Rjc†jcj . Now

we try to evaluate the urrent operator for the multilayers. Sin e the potential term

depends on the number operator, we just need to al ulate the ommutator of the

polarization operator with the hopping terms. Noti e that the polarization operator

in this ase will depend on the site index j and the plane index α. We have listed the

ommutators that need to be al ulated below:

[c†γlcγ+1l,Rαjc†αjcαj] = Rαj(c

†γlcαjδγ+1αlj − c†αjcγlδγαlj)

(4.56)

88

[c†γ+1lcγl,Rαjc†αjcαj ] = Rαj(c

†γ+1lcαjδγαlj − c†αjcγlcγlδγ+1αlj)

(4.57)

[c†γlcγl+δl,Rαjc†αjcαj ] = Rαj(c

†γlcαjδγαjl+δ − c†αjcγlcγlδγαlj)

. (4.58)

So the �nal answer be omes

j = i{−t⊥∑

γl

(eieAγ(t)Rγ+1lc

†γlcγ+1l − eieAγ(t)

Rγlc†γlcγ+1l

+ e−ieAγ(t)Rγlc

†γ+1lcγl − e−ieAγ(t)

Rγ+1lc†γ+1lcγl)

− t‖∑

γlδ

(Rγl+δc†γlcγl+δ −Rγlc

†γlcγl+δ). (4.59)

Using the nearest neighbors ve tors δ = Rγl+δ −Rγl and δ′ = Rγl −Rγ+1l, we get

j = i{−t⊥∑

γl

δ′(eieAγ(t)c†γlcγ+1l + e−ieAγ(t)c†γ+1lcγl)− t‖∑

γlδ

δc†γlcγl+δ} (4.60)

where we will al ulate the perpendi ular part whi h is the harge transport through

the multilayer. Using the de�nition of the lesser Green's fun tion, we get

j = −it⊥∑

γl

δ′[eieA0g<γ+1 γ + e−ieA0g<γ γ+1]. (4.61)

So in order to al ulate the urrent, one has to al ulate the nearest neighbor lesser

Green's fun tion. We start from the formula derived in 4.28- 4.31 for the left and

right fun tions.

LRα−1 = −te−iA0

gRααgRα−1α

, (4.62)

89

RRα+1 = −teiA0

gRαα

gRα+1α

, (4.63)

where we get,

gRαα+1 = −te−iA0gRα+1α+1

LRα

, (4.64)

gRα+1α = −teiA0gR,Aα,α

RRα+1

. (4.65)

Similarly starting from the left and right fun tions for the Keldysh fun tion, we �nd

LKα−1 = −LK

α−1

gKα−1α

gAα−1α

− te−ieA0gKααgAα−1α

, (4.66)

RKα−1 = −RK

α+1

gKα+1α

gAα+1α

− teieA0gKααgAα+1α

. (4.67)

After performing some simple algebra and substituting gAαα+1 and gAα+1α, we get,

gKαα+1 = t[eiA0LK

α gAα+1α+1 − e−iA0LA

αgKα+1α+1

LRαL

Aα

], (4.68)

and

gKα+1α = t[e−iA0RK

α+1gAαα − eiA0RA

α+1gKαα

LRα+1L

Aα+1

]. (4.69)

Then the �nal equation for the urrent be omes,

j = −it2⊥∑

k‖

∫ ∞

−∞dω[ RK

α+1gAαα

LRα+1L

Aα+1

+LAαg

Kα+1α+1

LRαL

Aα

− e−2iA0gKα+1α+1

LRα

− e2iA0gKαα

LRα+1

]

(4.70)

Now that we have derived the formula for the urrent a ross the multilayer devi e,

in the rest of this se tion, we will present the results whi h we have obtained from

the zipper algorithm for the lo al density of states and urrent both in bulk and at

di�erent layers.

90

4.5.1 Results of the lo al density of states

In this se tion, we present some results that have been already reported in Ref. [71℄.

We have onsidered total number of planes Nplane = 61 with a single insulator in the

middle N31 as the barrier plane. First, we start with the lo al density of states, whi h

is − Im[GR]π

. In Fig. 4.2, We present the results for di�erent values of the intera tion

as U = 1 for the dirty metal, U = 4 for the near Mott insulator and, U = 16 deep

in the insulating phase. These results have been already reported in Ref. [17℄ and we

only report it for the pedagogi al purposes.

-15 -10 -5 0 5 10 15ω

0

0.05

0.1

0.15

0.2

-Im

[GR]/

π U=1

U=4

U=16

(a)

-15 -10 -5 0 5 10 15ω

0

0.05

0.1

0.15

0.2

-Im

[GR]/

π

U=1

U=4

U=16

(b)

Figure 4.2: Retarded Green's fun tion for di�erent values of the intera tion in the

(a) bulk, ompared to (b) barrier plane for di�erent intera tions at A0 = π/20 and

T = 0.1. In both �gures, the red, blue, and green urves indi ate U = 1, U = 4 and

U = 16, respe tively.

As we an see, for a small value of intera tion U = 1, the density of the states is

smooth and there is no gap, while with in reasing the intera tion, for example, lose to

the Mott insulator U = 4, we see that the density of states develops a dip and �nally

for a large value of intera tion at U = 16 a wide gap appears. Now, we look at the

density of states of the barrier plane. We observe that the general hara teristi s are

similar to the bulk ase, while the features be ome sharper. Moreover, an interesting

91

phenomenon is happening in the insulating regime, where we an dete t nonzero

density of states entered around zero frequen y. This an be explained by a proximity

e�e ts of metalli planes, where some ele trons an tunnel through the barrier plane

be ause it is so thin. The lo al density of state in the �rst plane is shown in Fig. 4.3

-15 -10 -5 0 5 10 15ω

0

0.05

0.1

0.15

0.2

-Im

[GR]/

π

U=1U=4U=16

(a)

-2 -1 0 1 2ω

0.135

0.14

0.145

0.15

-Im

[GR]/

π

U=1U=4U=16

(b)

Figure 4.3: Retarded Green's fun tion for di�erent values of the intera tion at the

�rst plane for A0 = π/20 and T = 0.1. In (b) we have shown the blow out, in order

to show the Friedel os illations whi h are the e�e t of the barrier plane.

whi h may relax ba k to the bulk values as one in reases the number of planes.

Although, the �rst plane is far from the barrier, however one an see the e�e t of

the barrier by small os illations whi h is known as Friedel os illation. In Fig. 4.4, we

have shown the result of the lo al density of states for planes lose to barrier. The

presen e of an insulating plane auses a broadening of the lo al density of states in

the adja ent plane to it, however, it gets weaker on the se ond adja ent plane and

one would expe t to ompletely disappear in the next planes.

92

-15 -10 -5 0 5 10 15ω

0

0.05

0.1

0.15

0.2-I

m[G

R]π

U=1U=4U=16

(a)

-15 -10 -5 0 5 10 15ω

0

0.05

0.1

0.15

0.2

-Im

[GR]/

π

U=1U=4U=4

(b)

Figure 4.4: Retarded Green's fun tion for di�erent values of the intera tion at one

and two adja ent planes to the barrier for A0 = π/20 and T = 0.1. In (a), one an

learly observe the broadening in the density of states due to the e�e t of barrier,

while the e�e ts gets weaker for the next adja ent planes as it is shown in (b).

4.5.2 Results of the urrent in bulk

Again, we start with reprodu ing the results of urrent reported in Ref. [17, 71℄. After

we al ulate the retarded Green's fun tion, we al ulate the lesser Green's fun tion,

whi h enables us to al ulate the urrent in the multilayer devi e. Here, we show the

results for the urrent in bulk. We an al ulate the urrent in bulk in the presen e

of a ve tor potential whi h shifts the momentum by −A0z,

Jz(t, A0) = −i∑

k

vk+A0g<(k; t, t′) (4.71)

where vk = dǫ(k)/dz = 2t⊥ sin(k) is the band velo ity and using g<(k; t, t′) =

if<(ǫ(k)− µ) leads us,

Jz(t, A0) =∑

k

2t⊥ sin(kz + A0)f<(ǫ(k)− µ). (4.72)

93

-6 -4 -2 0 2 4 6ω/t

0

0.1

0.2

0.3

0.4

0.5Im

glo

c<(ω

)/2π

Bulk DOS

A0=4π/20

A0=π/20

A0=0

(a)

(a)

-6 -4 -2 0 2 4 6ω/t

0

0.1

0.2

0.3

0.4

0.5

Im g

loc<

(ω)/

2π

0 π/2 π0

0.2

0.4

A0=πA

0=17π/20

A0=13π/20

A0=π/2

Current

A0

Bulk DOS

(b)

T=0.01

(b)

Figure 4.5: Lo al lesser Green's fun tion for various urrent arrying states in the

bulk at low temperature T = 0.01. Inside inset at (b), we plotted the urrent as a

fun tion of the ve tor potential, whi h shows sinusoidal behavior. For omparison, we

also plotted the minus of imaginary part of lo al retarded Green's fun tion divided

by π/2.

Jz(t, A0) = −1

3

∑

k

ǫ(k)f<(ǫ(k)− µ) sin(A0). (4.73)

This shows that the urrent has sinusoidal dependen e to the ve tor potential and it

is also proportional to the average of the kineti energy in equilibrium. In Fig. 4.5,

we plotted the urrent for di�erent values of A0 at a temperature T = 0.01. As we

an see for A = 0, the lesser Green's fun tion simpli�es into the Fermi distribution,

as we expe ted. We also plotted the retarded Green's fun tion divided by π/2 for

omparing to the data. Then, as we in rease the ve tor potential, the higher states

start to get o upied so that to keep the �lling onserved, the �lling of lower energy

state should hange as well.

4.5.3 Results of the urrent a ross the multilayer devi e

As our goal is to evaluate the urrent-voltage pro�le of a multilayer devi e, here, we

present the results of urrent a ross the devi e as it is been al ulated at ea h layer.

We have used Eq. 4.70, whi h relates the urrent to nearest neighbor lesser Green's

94

fun tion. In Fig. 4.6, we present the results for dirty metal regime U = 1, for di�erent

initial ve tor potential. As it is shown in Fig. 4.6, the urrent drops as it passes

5 10 15 20 25 30 35 40 45 50 55 60Plane

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cur

rent

A=0, π/2

A=π/20

A=3π/20

A=2π/20

A=4π/20

A=5π/20

A=6π/20

A=7π/20A=8π/20A=9π/20

(a)

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60Plane

0.6

0.615

0.63

2.24

2.32

2.4

Cur

rent

3.6

3.8

4

4.2

A=π/20

A=4π/20

A=9π/20

(b)

Figure 4.6: Current through the multilayer devi e for di�erent values of A0 for U = 1and T = 0.1. The urrent drops as it passes through the barrier region, and the loss

of urrent be omes bigger as one in reases the initial ve tor potential. (b) To larify

the amount of urrent drop a ross the barrier, we have shown the urrents for three

di�erent values of the ve tor potentials A = π/20,A = 4π/20, and A = 9π/20 from

bottom to top, respe tively. One an learly see that, the amount of drop in urrent

for the di�erent ve tor potentials are approximately 0.02,0.12 and, 0.35 respe tively

from the bottom plot to top plot.

through the barrier. This is the result of s attering in the insulator plane. For the

bigger values of ve tor potential, whi h drives a bigger amount of urrent through the

devi e, the loss of urrents be omes bigger. We noti e that for A = 0 and, A = π/2

the urrent a ross the devi e is zero whi h is onsistent with our formula. To keep

the onservation of urrent and �lling, we will apply the lo al ele tri �eld a ross the

barrier.

The most simple pro edure is applying the ele tri �eld only on the barrier region.

We have examined this idea for a few di�erent values of ve tor potential and the

results are shown below. As it is shown Fig. 4.7, the urrent starts to re over as one

95

applies the ele tri �eld. However, the e�e t of the ele tri bias on di�erent planes are

di�erent and they ompete with ea h other. For example, for a single ele tri �eld,

although the urrent at the barrier improves, however, the plane before the barrier

shows the opposite e�e t and the urrent starts to drop dramati ally as one in reases

the ele tri �eld. Similar behavior also happens for some other ases, where one applies

ele tri �eld on the other planes before and after barrier. Of ourse, sin e the other

two layers or not insulating, we need a mu h smaller ele tri �eld ompared to the

barrier region. Furthermore, one noti es that there is a threshold in whi h in reasing

the ele tri �eld will make the results worse. In Fig. 4.11 and 4.9, we present the

results for both urrent and �lling for (U = 1, A = 6π/20) and U = 4, A = 2π/20. As

one an see, for the larger intera tion value, the loss of urrent be omes magni� ently

bigger, whi h indi ate higher s attering probability in the barrier plane.

Some of other approa hes that one may try is to hange the right ve tor potential

or in rease the numbers of planes with nonzero ele tri �eld. In fa t, we have examined

all of this pro edures and, as one might have guess from the above results, this

pro edure be omes a daunting task and even after performing al ulation, there is

no guarantee that one might have found the best value of parameters for the urrent

and �lling onservation. For this reason, we have taken di�erent approa h by using

the Newton method as a optimization pro edure whi h provides a systemati way of

obtaining the best values for the ele tri �eld and other possible parameters. This

will be the subje t of the next se tion.

96

0 5 10 15 20 25 30 35 40 45 50 55 60Plane

1.17

1.18

1.19

1.2

1.21

1.22

1.23

1.24

1.25

1.26

Cur

rent

E=0E

31=-0.04

E31

=-0.05

E31

=-0.06

E31

=-0.07

E31

=-0.08

E31

=-0.09

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60Plane

1.18

1.2

1.22

1.24

1.26

Cur

rent

E30

=E31

=0.0

E30

=-0.02, E31

=-0.04

E30

=-0.02, E31

=-0.05

E30

=-0.02, E31

=-0.06

E30

=-0.02,E31

=-0.07

E30

=-0.2,E31

=-0.08

E30

=-0.02,E31

=-0.09

(b)

0 10 20 30 40 50 60Plane

1.18

1.2

1.22

1.24

1.26

Cur

rent

E31

=E32

=0.0

E31

=-0.04,E3=-0.02

E31

=-0.05,E32

=-0.04

E31

=-0.07,E32

=-0.04

E31

=-0.08,E30

=-0.04

E31

=-0.09, E32

=-0.04

( )

0 10 20 30 40 50 60Plane

1.18

1.19

1.2

1.21

1.22

1.23

1.24

1.25

Cur

rent

E30=E31=E32=0.0

E30=-0.02, E31=E32=0.0

E32=-0.02, E31=E32=0.0

E30=-0.02, E31=-0.04, E32=0.0E31=-0.04, E30=E32=0.0

E30=E32=-0.2, E31=0.0

E30=0.0, E31=-0.04, E32=-0.02

E30=E32=-0.02, E31=-0.04

(d)

Figure 4.7: Current of multilayer devi e in the presen e of ele tri �eld lose to

barrier region for U = 1, A = π/10, and T = 0.1. For all di�erent ases, the urrentstarts to re over as one applies the ele tri �eld. (a) Shows the results of urrent

for di�erent values of ele tri �eld for single plane(E31 6= 0).(b) Shows the results of urrent for di�erent value of ele tri �eld at the barrier plane and one plane before the

barrier (E30 6= 0 and E31 6= 0 ). ( )Shows the results of urrent for di�erent value of

ele tri �eld at the barrier plane and a plane after barrier(E31 6= 0 and E32 6= 0 ). (d)Shows the results of urrent for di�erent values of ele tri �eld for triple ase.(E30 6= 0,E31 6= 0, E32 6= 0).

97

5 10 15 20 25 30 35 40 45 50 55 60Plane

3

3.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

Cur

rent

E30

=E31

=E32

=0.0

E30

=-0.04,E31

=-0.06,E32

=0.0

E30

=-0.10, E31

=-0.16,E32

=-0.08

E30

=-0.11,E31

=-0.17,E32

=0.0

E30

=-0.11,E31

=-0.18,E32

=0.0

E30

=-0.12,E31

=-0.19,E32

=0.0

(a)

0 10 20 30 40 50 60Plane

0.485

0.49

0.495

0.5

0.505

0.51

0.515

Filli

ng

E30

=E31

=E32

=0.0

E30

=-0.04,E31

=-0.06,E32

=0.0

E30

=-0.10,E31

=-0.16,E32

=-0.08

E30

=-0.11,E31

=-0.17,E32

=0.0

E30

=-0.11,E31

=-0.18,E32

=0.0

E30

=-0.12,E31

=-0.19,E32

=0.0

(b)

Figure 4.8: Current and �lling through the multilayer devi e in the presen e of

ele tri �eld lose to barrier region for U = 1, A = 6π/20, and T = 0.1.

0 5 10 15 20 25 30 35 40 45 50 55 60Plane

0.6

0.7

0.8

0.9

1

1.1

Cur

rent

E30

=E31

=E32

=0.0

E30

=-0.04,E31

=-0.06,E32

=0.0

E30

=-0.04,E31

=-0.10,E32

=0.0

E30

=-0.04,E31

=-1.10,E32

=0.0

E30

=-0.10,E31

=-1.30,E32

=0.0

E30

=-0.10,E31

=-1.50,E32

=0.0

E30

=-0.10,E31

=-1.60,E32

=0.0

E30

=-0.10,E31

=-1.70,E32

=0.0

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60Plane

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

Filli

ng

E30

=E31

=E32

=0.0

E30

=-0.04,E31

=-0.06,E32

=0.0

E30

=-0.10,E31

=-1.5,E32

=0.0

E30

=-0.10,E31

=-1.6,E32

=0.0

E30

=-0.10,E31

=-1.7,E32

=0.0

(b)

Figure 4.9: Current and �lling through the multilayer devi e in the presen e of

ele tri �eld lose to barrier region for U = 4, A = 2π/20, and T = 0.1. (b) For the�lling, we have only plotted some of the ases as they get very lose to ea h other.

98

4.6 Optimizing of the DMFT-Zip algorithm

As we mentioned, in order to ompensate the urrent and �lling loss a ross the barrier,

one needs to apply an external lo al ele tri �eld lose to barrier region. The idea is

to �nd the best values for an external ele tri �eld on a number of planes lose to the

barrier plane whi h leads to values that improve simultaneously the urrent and �lling

onservations. This method is based on the Newton root �nding method in whi h one

�nds the root for the matrix equation. Below, we explain the optimization pro edure

with arbitrary number of parameters and we explain the details of the al ulation for

the ase with three ele tri parameters.

1. Set the initial value for parameters for ea h parameters.

2. Cal ulate urrent and �lling at ea h plane as Ji, and ρi respe tively.

3. Obtain the di�eren e of the urrent and �lling from the target value at ea h

plane:

∆Ji = Jtarget − Ji

∆ρi = ρtarget − ρi (4.74)

4. Add small shift to ea h values of the parameters and evaluate the hanges in urrent

and �lling at ea h plane,

δJi(δJ)Parameterj

=J

′

i − Ji(δJ)Parameterj

δρi(δρ)Parameterj

=ρ

′

i − ρi(δρ)Parameterj

. (4.75)

5. Solve the following matrix equation,

∑

j

[δJi

(δJParameter)j

,δρi

(δρ)Parameterj

]δEj = [∆Ji,∆ρi], (4.76)

The above matrix equation an be solved by singular value de omposition(SVD)

whi h is required to �nd the pseudo-inverse of a re tangular matrix. Then, one repeats

99

the pro edure to rea h onvergen e. For the ase with three ele tri �elds, the vari-

ational matrix is 122 × 3, as there are 61 values for urrent and 61 values for �lling

and there are 3 olumns, regarding the applied ele tri �eld at 3 di�erent planes. The

target ve tor in right hand side is 122 and onsequently, by using the SVD, we �nd

the best values for the ele tri �eld at ea h plane. Sin e this method, is based on the

Newton root �nding method, we expe t that the iterations to onverge fast. Before

presenting our results for di�erent ases, we omment on adding more parameters.

For example, one an add more ele tri �elds or even varying the ve tor potential

or hemi al potential. In any of these ases, for ea h parameter, one needs to al u-

late both urrent and �lling for ea h parameter whi h adds a olumn to variational

matrix in left hand side, and the solution of the SVD will have orresponding number

of values.

In Fig. 4.10, we present the results of the optimization for di�erent ve tor poten-

tials for single ele tri �eld on the barrier. We have set the total riteria of the

onvergen e by introdu ing a new variable, whi h is the sum over all ∆J for the new

set of variables found by SVD. We have also repeated the same pro edure for double

and triple ele tri �eld, in whi h, we have obtained the optimized value for ea h ase.

Below in Fig. 4.11, we ompare the results with the original results in Fig. 4.6 whi h

makes lear how the loss in the �lling gets ompensated with the external ele tri �eld

a ross barrier. Of ourse, the results are mu h better for a smaller ve tor potential,

as one may guess that one needs to add further parameters in order to fully re over

the onservation. We have also, reported the best values obtained for ele tri �eld on

the barrier plane and the plane before it.

100

10 20 30 40 50 60Plane

0.6

0.605

0.61

0.615

0.62

0.625

0.63

Cur

rent

E30

=0.0,E31

=0.0,E32

=0.0

E30

=0.0,E31

=-0.01,E32

=0.0

E30

=0.0,E31

=-0.02,E32

=0.0

E30

=0.0,E31

=-0.03,E32

=0.0

E30

=0.0,E31

=-0.04,E32

=0.0

E30

=0.0,E31

=-0.05,E32

=0.0

E30

=0.0,E31

=-0.06,E32

=0.0

A=π/20

10 20 30 40 50 60Plane

1.65

1.7

1.75

1.8

1.85

Cur

rent E

30=0.0,E

31=0.0,E

32=0.0

E30

=0.0,E31

=-0.025,E32

=0.0

E30

=0.0,E31

=-0.050,E32

=0.0

E30

=0.0,E31

=-0.075,E32

=0.0

E30

=0.0,E31

=-0.100,E32

=0.0

E30

=0.0,E31

=-0.125,E32

=0.0

E30

=0.0,E31

=-0.150,E32

=0.0

A=3π/20

10 20 30 40 50 60Plane

2.2

2.25

2.3

2.35

Cur

rent

E30

=0.0,E31

=0.0,E32

=0.0

E30

=0.0,E31

=-0.030,E32

=0.0

E30

=0.0,E31

=-0.060,E32

=0.0

E30

=0.0,E31

=-0.090,E32

=0.0

E30

=0.0,E31

=-0.120,E32

=0.0

E30

=0.0,E31

=-0.150,E32

=0.0

E30

=0.0,E31

=-0.180,E32

=0.0

A=4π/20

10 20 30 40 50 60Plane

3

3.05

3.1

3.15

3.2

3.25

3.3

Cur

rent

E30

=0.0,E31

=0.0,E32

=0.0

E30

=0.0,E31

=-0.04,E32

=0.0

E30

=0.0,E31

=-0.08,E32

=0.0

E30

=0.0,E31

=-0.12,E32

=0.0

E30

=0.0,E31

=-0.16,E32

=0.0

E30

=0.0,E31

=-0.20,E32

=0.0

E30

=0.0,E31

=-0.24,E32

=0.0

A=6π/20

10 20 30 40 50 60Plane

3.2

3.3

3.4

3.5

3.6

3.7

Cur

rent

E30

=0.0,E31

=0.0,E32

=0.0

E30

=0.0,E31

=-0.045,E32

=0.0

E30

=0.0,E31

=-0.090,E32

=0.0

E30

=0.0,E31

=-0.135,E32

=0.0

E30

=0.0,E31

=-0.170,E32

=0.0

E30

=0.0,E31

=-0.215,E32

=0.0

E30

=0.0,E31

=-0.260,E32

=0.0

A=7π/20

10 20 30 40 50 60Plane

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

Cur

rent

E30

=0.0,E31

=0.0,E32

=0.0

E30

=0.0,E31

=-0.065,E32

=0.0

E30

=0.0,E31

=-0.130,E32

=0.0

E30

=0.0,E31

=-0.185,E32

=0.0

E30

=0.0,E31

=-0.245,E32

=0.0

E30

=0.0,E31

=-0.305,E32

=0.0

E30

=0.0,E31

=-0.370,E32

=0.0

A=9π/20

Figure 4.10: Optimization of urrent for di�erent values of ve tor potential with

single layer nonzero ele tri �eld for U = 1 and T = 0.1. As the ve tor potential

in reases, the solution of the optimization pro edure, leads to bigger value for ele tri

�eld as one would have expe ted.

101

5 10 15 20 25 30 35 40 45 50 55 60Plane

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cur

rent

A=0,π/2

A=π/20

A=2π/20

A=3π/20

A=4π/20

A=5π/20

A=6π/20

A=7π/20A=8π/20A=9π/20

[E30=-0.01052 , E31=-0.03419]

[E30=-0.02036 , E31=-0.04348]

[E30=-0.01858 , E31=-0.08108]

[E30=-0.03662 , E31=-0.13436]

[E30=-0.05914 , E31=-0.15200]

[E30=-0.01149 , E31=-0.19980]

[E30=-0.07885 , E31=-0.22037][E30=-0.06787 , E31=-0.20016]

[E30=-0.068571 , E31=-0.2237]

(a)

Figure 4.11: Comparing the urrent obtained from the optimization algorithm with

the one at zero ele tri �eld. The results for ele tri �eld at ea h plane has been been

reported for ea h ve tor potential for U = 1 and T = 0.1.

4.7 Con lusion: Current-Voltage (I-V) profile

As we mentioned earlier, one of the most intriguing problem is to des ribe di�erent

properties of a multilayer devi e atta hed to metalli leads. One of the most popular

methods is to onsider left and right reservoir at both side of the metalli leads whi h

are set a voltage ±V/2, respe tively. Then, the tunneling matrix between the metalli

leads and the reservoir are hanged slowly in whi h the system eventually rea hes the

steady state. Consequently, the urrent through the leads an be measured whi h

allows one to obtain the urrent-voltage pro�le. Instead, in this hapter, we have

presented a di�erent method based on urrent biasing the multilayer devi e, as we

hange the ve tor potential a ross the devi e to drive the urrent in devi e. In the

previous se tion, we have obtained the optimized value for the ele tri �eld a ross the

barrier region. In Fig. 4.12 , we present the IV pro�le for the single and double layers

with the external ele tri �eld for U = 1. As it is shown in above, the I-V pro�le

102

0 0.05 0.1 0.15 0.2 0.25 0.3Voltage

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cur

rent

Single

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3Voltage

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Cur

rent

Double

(b)

Figure 4.12: Current-Voltage pro�le for a multilayer devi e. (a) it shows the pro�le

whi h is obtained where we have only applied ele tri �eld on the barrier plane while

in (b) we have obtained the pro�le for the double ele tri �eld applied on barrier

and a plane before the barrier. We noti e that for larger values of ve tor potential

the result starts deviating from the linear behavior with slope passing the zero as is

expe ted for the dirty metal ase with U = 1 and at T = 0.1.

starts deviating from the linear behavior for the larger values of ve tor potential, this

is expe ted for the dirty metal regime. The next step would be al ulating the I-V

pro�le for insulating regime. This step is in progress and we hoping to get results

soon.

103

5

Designing mixtures of ultra old atoms to boost Tc by using

dynami al mean field theory solution

�One of the joys of physi s is the startlingly wide lasses of situations we

an treat with essentially identi al analyses.�

� Ri hard Feynman

Re ent development in ultra old atoms in opti al latti es has reated a great

potential to probe di�erent exoti phenomenas with various degrees of ontrol whi h

are not dire tly a essible in onventional strongly orrelated systems. As we men-

tioned in hapter 1, one of the simplest models to study strongly orrelated systems

is the Hubbard model whi h des ribes the hopping of ele trons in a latti e and with

Coulomb repulsion as two parti les sit at the same site. For reminder, we rewrite the

Hubbard Hamiltonian from Eq. whi h is 5.1: is de�ned as follows:

HHub = −∑

ijσ

tij(c†iσcjσ + c†jσciσ) + U

∑

i

ni↑ni↓ (5.1)

Even, this simple model has a ri h phase diagram. Here, we will only fo us on the

Antiferromagneti (AFM) part of the phase diagram, see �gure 5.1. Although, the

ooling and trapping of fermioni gases are more hallenging, as the Pauli ex lusion

prin iple prevents the s-wave s attering of fermioni spe ies, the Mott insulator of

fermioni atoms has been observed in

40K atoms in ultra old opti al latti es [19℄,

see �gure 5.1. However, the temperature required to a ess the long-range AFM is

104

lower than urrent available te hniques. However, the antiferromagneti orrelation

of quantum degenerate of

6Li fermi gas has been observed in 3 dimensional Hubbard

model, were the temperature lowered to 1.4 Tc, where Tc is the riti al temperature of

Neel order [20℄. In addition, re ent development in site-resolved imaging for fermion

quantum gases has provided a dire t tool to dete t the orrelations in fermioni

gasses [96℄. However, the experiment performed on 2 dimensional Hubbard model

of quantum degenerate of

6Li fermioni gases demonstrates an in rease of orrelation

fun tion in lower temperature and the ability to measure AFM orrelation at any

distan e [96℄, while a ording to Mermin-Wagner theorem there is no phase transition

in 2D. In this hapter, we dis uss the possibility of boosting the riti al temperature

Figure 5.1: Enhan ement of quantum ordering riti al temperature for di�erent

values of degenera y of trapped fermioni spe ies.

by in reasing the degenera y of the light spe ies (fermion) in the mixture of heavy

and light mixture of atoms in ultra old opti al latti es. The heavy-light mixture an

be des ribed by the Fali ov-Kimball model where one needs to onsider the heavy

parti le as a lo alized parti le [2℄. As we have shown in hapter 2, the DMFT solution

of the Fali ov-Kimball model is exa t in in�nite dimension. The idea of enhan ement

of the riti al temperature omes from the relation between the riti al temperature

105

and degenera y of spe ies [18℄. The riti al temperature of the phase transition an

be found by al ulating the di�erent sus eptibilities as they diverge at a riti al

temperature. It turns out that the harge and spin sus eptibilities for heavy-light

parti les an be des ribed as [18℄

χcf =1

1− ξ.dT

=1

1− Tc

T

, (5.2)

whi h indi ates that the riti al temperature should be in reasing as the degenera y

of fermioni spe ies is in reases. Consequently, the main riteria to a hieve a higher

phase transition temperature would be �nding light(fermioni ) atoms with higher

degenera y. In �gure 5.1, we have depi ted the enhan ement of the riti al tempera-

ture for N = 3, and N = 4. A ording to our al ulation, for a fermioni spe ies with

N = 4 degenera y, the enhan ement of riti al temperature happens and one an

dete t the AFM in higher temperature, whi h is a essible with urrent ooling te h-

niques. The idea of riti al temperature enhan ement and required riteria for �nding

an appropriate mixture will be the topi of this hapter. The rest of this hapter is

organized as follows. In se tion 5.1, we will review the basi s of the intera tion of

light with atoms, whi h is ru ial to understand the trapping and ooling of atoms

in opti al latti es. In se tion 5.2, we will brie�y explain the DMFT solution for the

mixture of heavy and light parti les in opti al latti e that has been al ulated in

[18℄. Then, we list the riteria for �nding the right mixture of ultra old atoms whi h

present this e�e t and �nally, in the last se tion, we present our published results on

numeri al al ulation and we dis uss the possibility of di�erent mixture whi h satisfy

the riteria required for riti al temperature enhan ement [21℄.

106

5.1 Intera tion of light and matter

Understanding the intera tion between light and atoms is the most important tool for

building the opti al latti es and trapping the atoms. The intera tion between laser

and atom onsists of two parts, the so alled absorptive and dispersive for e. The

absorptive part of intera tion arises from the momentum transfer during a photon

absorption and emission into another mode, whi h is widely used as ooling method if

the frequen y of light is lose to an atomi resonan e. First, we explain the dispersive

part of intera tion. We all the detuning parameter δ = ω − ω0, where ω is the

frequen y of light and ω0 is the transition frequen y of atom. The dispersive part

arises from the dipole for e. When an atom is pla ed into laser light, the ele tri �eld

of light E indu es an atomi dipole moment in the atom, whi h leads to intera tion

of the atom with light whi h is proportional to the intensity of light. By detuning

far away from the transition frequen y we an enter the regime where we an ignore

the s attering and the dipole for e be omes dominant. In this way we an build an

arti� ial latti e where we an ontrol the shape and the depth of the latti es.

First, we explain the me hanism of intera tion of laser light and atom with a

lassi al model. As we know when we pla e a lassi al dipole inside ele tri �eld, it

tends to align with the ele tri �eld, where it stores the potential energy as

Vdip = −p.E, (5.3)

where p is the dipole moment. In the following we use lassi al os illator model to

al ulate the dipole for e. Let's onsider that the atom is onsists of two ele troni

levels with the frequen y transition des ribed by ω0 and the ele tri �eld and the

107

dipole moment an be des ribed as,

E = E e−iωtk + c.c,

p = p e−iωtk+ c.c.

The dipole moment for atom inside ele tri �eld is [97℄

p = α(ω)E, (5.4)

where α(ω) is the wavelength dependent omplex polarizability of the atom. The

e�e tive potential is given by

Vdip = −1

2〈p.E〉. (5.5)

Sin e the light is rapidly os illating we take the average over one os illation period 〈.〉

and the 1/2 omes from the fa t that dipole moment is indu ed and is not permanent.

Vdip = − 1

2ǫ0cRe(α)I(r), (5.6)

where intensity I(r) = 2ǫ0c|E|2 and ǫ0 is the ele tri permeability. We also are inter-

ested to al ulate the s attering rate whi h is proportional to the imaginary part,

ΓSc =1

~ǫ0cIm(α)I(r), (5.7)

using the lassi al os illator model, we an al ulate the polarizability [97℄:

α(ω) = 6πǫ0c3 Γ/ω3

0

ω20 − ω2 − i ω3

ω20Γ

, (5.8)

where Γ denotes the linewidth of the atomi transition. If we assume ∆ = ω−ω0 > Γ,

then we get,

Vdip = −3πc2

2ω30

( Γ

ω0 − ω+

Γ

ω0 + ω

)

I(r), (5.9)

108

Γsc =3πc2

2ω30

( ω

ω0

)3( Γ

ω0 − ω+

Γ

ω0 + ω

)2

I(r). (5.10)

Within the rotating wave approximation ∆ << ω, we an simplify the equations

Vdip(r) = −3πc2

2ω30

Γ

∆I(r), (5.11)

and

Γsc(r) =3πc2

2~2ω30

Γ2

∆2I(r). (5.12)

This is the basi equation whi h des ribes the hara teristi of trapping of atoms

in opti al latti es. If the frequen y of light is smaller than the transition frequen y

∆ < 0, the dipole for e F = −∇Vdip will attra t atoms toward the high intensity, the

so alled the red detuned, while for ∆ > 0 the atoms feel for e toward the intensity

minima. It is easy to generalize the the above equation for the ase of multi-level

atoms. In this ase we need to sum the dipole potential and s attering rate for ea h

individual transitions multiplied by their line strength fa tors f , for example for a

J → J ′�ne stru ture transition, the line width γ be omes [97, 98, 99℄,

Γ =e2ω2

0

2πǫ0mec32J + 1

2J ′ + 1f. (5.13)

Opti al latti es is onsists of laser beams whi h a t like periodi dipole potential

to trap ultra old atoms in a similar way to the Coulomb potential a ting on atoms

in real latti e. A ommon way of reating opti al latti es is using linear polarized

mono hromati laser with wavelength λ beams whi h is retro-re�e ted to reate the

standing waves,

Vlat(x) = −V0sin2(kx), (5.14)

where the waveve tor is k = 2πλand the latti e depth V0 is typi ally given in the units

of re oil energy Er = ~2k2

2m. Moreover in the simple ase of retro-re�e ted, the sites

109

are separated by a latti e spa ing d = λ/2. The above potential resembles the latti e

stru ture in a rystal.

Now we explain the absorptive intera tion that an be used in ooling pro edure.

When an atom absorbs a photon from a single resonant laser, it gains a moment ~k,

where k is the wavelength of laser. Then the atom will will emit a photon so that the

net for e after many absorption-emission y les be omes,

Fsc = σΓ.~k, (5.15)

where Γ is the natural line width and σ(∆, I) is the o upation probability of the

ex ited state. Noti e that the for e depends on detuning ∆ = ω−ω0 and the intensity

of laser light.

5.2 DMFT solution for mixture of heavy and light parti les in

opti al latti e

The se ond order phase transition in physi al systems manifest itself by diver-

gen e of physi al observable su h as spe i� heat and magneti sus eptibility at

riti al temperature Tc. The Fali ov-Kimball model also undergoes di�erent phase

transitions. Here, we will fo us on the so alled he kerbord phase transition. The

he kerboard-like ele troni modulation has been dete ted in di�erent physi al sys-

tems whi h involves the instability of low dimensional harge leading to the harge

density wave(CDW) [100℄. As we mentioned before, the DMFT solution for the

Fali ov-Kimball model predi ted that the riti al temperature is proportional with

the degenera y of spe ies. Di�erent harge and spin sus eptibility have been al-

ulated from the DMFT solution for the Fali ov-Kimball model by Freeri ks and

Zlati¢(1998) [18℄ whi h expli itly shows the degenera y dependen e of the riti al

110

temperature. Here, we brie�y review the derivation of a stati sus eptibility for the

mixture of heavy and light atoms whi h an be des ribed by the Fali ov-Kimball

Hamiltonian. First, we start with the harge sus eptibility for itinerant ele trons.

The CDW phase transition arises from the distortion of harge, so the harge sus-

eptibility an be de�ned as the derivative of the harge density with respe t to h as

one adds a �eld −∑i hini, whi h ause the symmetry breaking,

χccij =

1

2N + 1

∫ β

0

dτ [Tr〈e−βHnc

i(τ)ncj(0)〉

Z − Tr〈e−βHnc

i〉Z Tr

〈e−βHncj〉

Z ], (5.16)

where Z is the partition fun tion and nci =

∑2N+1σ=1 c†iσciσ is the harge density whi h

an be presented in the Heisenberg representation as

ni(τ) = exp[τ(H− µN )]nci exp[−τ(H− µN )]. (5.17)

Sin e nci = T

∑

nGi(iωn), we an rede�ne the sus eptibility as

χccij =

T

2N + 1

∑

nσ

dGjjσ(iωn)

dhi= − T

2N + 1

∑

nσ

∑

lm

Gjlσ(iωn)dG−1

lmσ(iωn)

dhiGmjσ(iωn),

(5.18)

where G−1lmσ(iωn) = [iωn + µ + hl − Σllσ(iωn)]δlm − tlm. Using the hain rule for the

derivative, we get

χccij =

T

2N + 1

∑

nσ

[−Gijσ(iωn)Gjiσ(iωn) +∑

lmσ′

Gjlσ(iωn)GljσdΣllσ(iωn)

dGllσ′

dGllσ′(iωn′ )

dhi].(5.19)

Now, we an perform a Fourier transform into momentum spa e and we de�ne Γcclm

Γcclm =

1

T

1

2N + 1

∑

σσ′

dΣσ(iωl)

dGσ′(iωm), (5.20)

and we have de�ned,

χccl (k) =

1

L

1

2N + 1

∑

j−q,σ

dGjjσ(iωn)

dhqeik.(j−q), (5.21)

111

and

χcc0l = − 1

L

1

2N + 1

∑

j−q,σ

Gjqσ(iωn)Gqjσ(iωn)eik.(j−q), (5.22)

where L is the number of latti e sites. Then, the harge sus eptibility be omes,

χccl (k) = χ

cc0l (k)− T

∑

m

χccl0(q)Γ

cclmχ

ccm(k), (5.23)

whi h is the Dyson equation for sus eptibility and known as the Bethe-Salpeter equa-

tion. Now, we need to ompute the vertex Γccl0 term. For the Fali ov-Kimball model,

we have obtained the self-energy in hapter 2 as

Σ(iωn) = − 1

2G(iωn)+U

2± 1

2G(iωn)

√

1− 2(1− 2w1)UG(iωn) + U2G2(iωn)

(5.24)

whi h expli itly has G(iωn) dependen e and impli it w1 dependen e, so we have

Γccl0 =

1

(2N + 1)T

∑

σσ′

{

(∂Σlσ

∂Glσ

)w1δσσ′δlm + (∂Σlσ

∂w1

)Glσ(∂w1

∂Gmσ′

)}

. (5.25)

With introdu ing new variable γ(k) de�ned as

γ(k) =∑

l

χccl (k)

∑

σ

(∂w1

∂Glσ), (5.26)

we get,

χccl (k) = χ

ccl0(k)

1− (∂Σlσ1/∂w1)Glσ1γ(k)

1 + χccl0(k)(∂Σlσ1/∂Glσ1)w1

(5.27)

where σ1 is a parti ular spin state that derivative has been al ulated. Now, if we mul-

tiply both sides by Σσ(∂w1/∂Glσ) and perform the summation over l, we should get

the equation for γ(k). Using the hain rule ∂w1/∂Glσ =∑

m[∂w1/∂Zm][∂Zm/∂Glσ],

where Zlσ = iωn + µ− λlσ = G−1lσ + Σ, we have

∂w1/∂Glσ = −∂w1

∂Zl[1−G2

l (∂Σl

∂Gl)w1]

G2l [1−

∑

m∂w1

∂Zm)(∂Σm

∂w1)Gl

]. (5.28)

112

Substituting in γ(k) equation, we get

γ(k) =

∑

lσ ∂w1/∂Zlσ[1−G2l (∂Σl/∂Gl)w1]/[1 +Glηl(k)−G2

l (∂Σl/∂Gl)w1]

1−∑lσ ∂w1/∂ZlσGlηl(k)(∂Σl/∂w1)Gl/[1 +Glηl(k−G2

l (∂Σl/∂Gl)w1],(5.29)

where we have introdu ed new de�nition,

ηl(k) = Gl[−1

G2l

− 1

χccl0(k)

]. (5.30)

Now, the harge sus eptibility be omes

χcc(k) = −T∑

l

[1− γ(k)(∂Σl/∂w1)Gl]G2

l

1 +Glηl(k)−G2l (∂Σ1/∂Gl)w1

. (5.31)

Now, using the identity that we have obtained for w1 In the Fali ov-Kimbalmodel, one

an ompute the derivative terms. Similarly, one an al ulate the harge sus eptibility

for lo alized parti les and other orrelations, for more details see Ref. [18℄.

5.3 Enhan ing quantum order with fermions by in reasing spe ies

degenera y

While the o�-diagonal long-range order in old bosoni atomi gases has been

observed many years ago, quantum magnetism in fermioni gases is still a hallenge

for experimentalists. Despite the possibility to ontrol the intera tion between spin

states of atoms in an opti al latti e [101℄, the temperatures required to obtain mag-

neti ordering remain lower than those a hievable with urrent te hniques. Therefore,

it is easier to demonstrate the presen e of magneti orrelations, before the true

long-range magneti order is established. Using the spin-sensitive Bragg s attering

of light, antiferromagneti orrelations in a two-spin- omponent Fermi gas, magneti

This se tion is reprinted from Khadijeh Naja�, M. M. Ma±ka, Kahlil Dixon, P. S. Juli-

enne, J. K. Freeri ks, Copyright Physi al Review A 96 (5), 053621.

113

orrelations have been observed at a temperature 40% higher than the putative tem-

perature for the transition to the antiferromagneti state in three dimensions [20℄. In

this experiment, the two lowest hyper�ne ground states of fermioni

6Li atoms in a

simple ubi opti al latti e were labeled as spin-up and spin-down states. The repul-

sive intera tion between atoms in these states was ontrolled by a magneti Feshba h

resonan e. Sin e the magneti superex hange intera tion is given by J = 4t2/U , the

experiment ontrolled the value of J and in a parti ular regime, it measured anti-

ferromagneti orrelations as extra ted from the spin stru ture fa tor. Very re ently

it was demonstrated that spin (and harge) orrelations an be dete ted also with

the help of site-resolved imaging. In Refs. [96, 102, 103℄ quantum gas mi ros opy

was used to determine spatial orrelations for fermioni atoms in a two-dimensional

opti al latti e. While there is no phase transition in 2D, the measurements have

shown an in rease of the orrelation length as the temperature was lowered. Similar

antiferromagneti orrelations extending up to three latti e sites have also been

observed in a 1D system [104℄.

The simplest many-body model that has a nonzero phase transition in two dimensions

is the Ising model [105℄. Its fermioni analog, the Fali ov-Kimball model [2℄, also

displays a nonzero transition temperature in two dimensions, whi h behaves Ising-like

when the intera tion strength be omes large. This system an be easily simulated

with mixtures of old atoms on opti al latti es, be ause it involves mobile fermions

intera ting with lo alized fermions [106℄. One simply needs to have the hopping of

the two atomi spe ies to be drasti ally di�erent. The simplest ase of one trapped

atomi state for ea h of the fermioni spe ies maps onto the spinless version of the

Fali ov-Kimball model. This model has been solved exa tly in in�nite dimensions via

114

dynami al mean-�eld theory (DMFT) [107, 108℄ and numeri ally in two dimensions

with Monte Carlo (MC) [109℄.

Our motivation for this work stems from the DMFT solution to the problem.

There, one an derive a ondition for the transition to an ordered phase with a

he kerboard pattern [107, 108℄, whi h takes the form 1 =∑

n γ(n), with the sum

running over all integers [whi h label fermioni Matsubara frequen ies iωn = πi(2n+

1)T , with T the temperature℄. The fun tion γ(n) is a ompli ated fun tion that

is onstru ted from the mobile fermion Green's fun tion, its self-energy, the on-site

interation between the lo alized and mobile fermions U and the density of the lo alized

fermions w1. The important point to note, is that if we in rease the degenera y of the

mobile fermions (while enfor ing that they do not intera t with themselves), then the

Tc equation is modi�ed by γ(n) → Nγ(n), where N is the number of degenerate states

for the mobile fermions [108℄. Sin e one an immediately show that

∑

n γ(n) → C/T

for T → 0 and

∑

n γ(n) → C ′/T 4for T → ∞ [107℄, we expe t that the transition

temperature for the degenerate system will initially grow linearly in N and then turn

over to a slower in rease, proportional to N1/4for larger N . It is the rapid growth with

degenera y for small N , whi h makes these e�e ts so spe ta ular. (These ideas are

further supported by the observation that in reasing spe ies degenera y lowers the

�nal temperature after the opti al latti e is ramped up in alkaline-earth systems [110℄)

The argument that Tc grows linearly with the degenera y at low temperature an

be made more general. We start with the Hamiltonian for the Fali ov-Kimball model

on a latti e Λ that has |Λ| latti e sites. The Hamiltonian for a given on�guration of

115

the heavy atoms {w} is

H({w}) = −t∑

〈ij〉

N∑

σ=1

c†iσciσ + U∑

i

N∑

σ=1

niσwi

=

N∑

σ=1

Hσ({w}), (5.32)

where σ denotes the N di�erent ��avors� of the mobile fermions and wi = 1 or 0,

denotes whether site i has a lo alized fermion on it, or not, respe tively (the lo alized

fermions ontinue to be spinless). The hopping matrix is hosen to be nonzero only

for nearest neighbors, and we set t = 1 as our energy unit (we also set kB = 1). We

de�ne Ei ≡ εi − µ, with µ the hemi al potential and {εi} the set of (degenerate)

eigenvalues of Hσ({w}), whi h is independent of the spe i� value of σ be ause the

mobile fermions are nonintera ting amongst themselves, and they share the same

intera tion with the lo alized fermions. Here, the index i runs over i = 1, . . . , |Λ| (we

will be working on a square latti e of edge L whi h then has |Λ| = L× L).

The orresponding grand partition fun tion is given by

Z =∑

{w}

|Λ|∏

i=1

[1 + e−βEi({w})]N , (5.33)

with β = 1/T the inverse temperature. Introdu ing the free energy F , Eq. (5.33) an

be rewritten as

Z =∑

{w}e−βF({w}), (5.34)

where

F({w}) = −Nβ

∑

i

ln[1 + e−βEi({w})]

= N∑

i

Eiθ [−Ei({w})]

− N

β

∑

i

ln[1 + e−β|Ei({w})|] , (5.35)

116

and θ(. . .) is the Heaviside unit step fun tion. In the low-temperature limit the se ond

term on the RHS vanishes. Inserting the limiting form of F into Eq. (5.34) yields

Z =∑

{w}e−βN

∑

i Eiθ[−Ei({w})]. (5.36)

Note, that this result an be re ognized to be the ondition for the �lling of mobile

fermions into the Fermi sea determined by the bandstru ture orresponding to the

parti ular on�guration of the lo alized fermions, as given by the on�guration {w}.

Sin e, in the low-temperature limit F does not depend on temperature, the partition

fun tion depends on temperature only through the term βN . This means that the

thermodynami s of the system depends only on the ratio T/N , with initial orre tions

expe ted to be small as T rises (be ause they will be proportional to T/TF with some

suitably large Fermi temperature TF ). As a result the riti al temperature Tc in the

low-temperature limit will ne essarily in rease linearly with in reasing degenera y N .

This is an exa t result, independent of the details of the latti e or the dimensionality�

it only requires there to be a phase transition.

There are two assumptions that went into this anaylsis, whi h turn out not to

hold when we a tually al ulate the maximal Tc as a fun tion of N . First, the lowest

Tc values are not so low, so the linear regime fairly rapidly rosses over to a slower

in reasing behavior and se ond, the intera tion value Umax(N), where the maximal

Tc,max(N) o urs, a tually hanges with N (see the inset in Fig. 5.3), so the arguments

about the pre ise fun tional dependen e of the Tc,max(N) on N turns out not to hold

in the a tual data; our arguments assumed we ompared systems with the same U .

The �rst e�e t is to redu e how Tc in reases with N , while the se ond enhan es how

Tc in reases with N .

Corre tions to the linear dependen e of Tc on N ome mostly from states lose

to the Fermi level [Ei ≈ 0, see the se ond term in the RHS of Eq. (5.35)℄. Therefore,

117

Figure 5.2: (Color on-line) Comparison of the 2D DMFT (solid red line) and MC

(blue dots, onne ted with a solid line as a guide to the eye) riti al temperatures to

the he kerboard density wave at half �lling for both spe ies on a square latti e with

N = 1. The lines marked as "Ising mean-�eld" (green) and "Ising exa t" (bla k) show

the riti al temperatures for the orresponding Ising model, whi h be ome exa t for

the respe tive theories when U → ∞. Left panel shows the riti al temperature for

N = 1 whi h is the below of the urrent a hievable temperature at experiment(1.4TN).Middle panel shows the enhan ement of the riti al temperature for N = 3 whi h is

lose to urrent a hivable temperature and the right panle for N = 4, manifest the

enhan ement whi h is well above the urrent a hivable temperature

we an expe t that the linear se tion of the Tc(N) urve an be longer for bipartite

latti es for whi h the density of states is redu ed lose to the Fermi energy, e.g., for a

hexagonal latti e. Also in 3D, where there is no Van Hove singularity (the singularity

for the square latti e is redu ed by the intera tion with the heavy atoms) the linear

part an persist to even higher temperatures.

In Fig. 5.2, we plot the transition temperature to the he kerboard density wave

on a square latti e with N = 1. The top urve is for the DMFT approximation, while

the bottom urve is for the exa t MC results. Note that the intera tion strength for

the peak of the urve lies in the range of U ≈ 4−5 with the maximal U value slightly

118

Figure 5.3: (Color on-line.) The maximal riti al temperature Tc plotted as a fun -

tion of mobile fermion degenera y N (as al ulated with MC). The dashed line shows

the orresponding DMFT Tc al ulated at U = Umax(DMFT). The solid lines show

MC Tc's for di�erent values of U . The bla k dotted line shows Tc(DMFT)×0.75, whi hagrees well with nearly all the MC results. In the inset, Umax(DMFT) is plotted as a

fun tion of degenera y N , indi ating it hanges signi� antly with N .

higher for DMFT versus MC. The DMFT results are semiquantitative, and learly

overestimate the Tc, but the overall error is not that large.

As N in reases, we �nd that the maximum Tc in reases as does the value of

the intera tion strength where the Tc(U) urve is maximized. The full urve out to

N = 100 is plotted in Fig. 5.3. The DMFT results are al ulated for ea h N by

�rst �nding the intera tion strength at the maximum of the Tc urve. For the MC

results, we work with �xed U , varying N and then onstru ting the �maximal hull�

of the data. It turns out that these MC results are nearly perfe tly �t to the DMFT

results when the latter are renormalized by a fa tor of 0.75. The DMFT urve initially

grows linearly with N , but then settles into an in rease that grows proportional to

√N − 1.7, whi h is in between our linear and 0.25 power results, as we expe ted, due

to the fa t that Umax in reases with N .

119

We �nd the enhan ement of the maximal Tc for higher N versus N = 1, given

by Tc,max(N)/Tc,max(1) satis�es: 1.98 (MC, N = 2), 1.899 (DMFT, N=2); 2.84 (MC,

N = 3), 2.651 (DMFT, N = 3); and 3.60 (MC, N = 4), 3.287 (DMFT, N = 4).

Sin e the maximal Tc(DMFT) for the Fali ov-Kimball model is about one half the

maximal Tc(DMFT) for the orresponding Hubbard model, we need to be able to have

a degenera y of N ≥ 3 before this e�e t will have a high enough Tc that it an rea h

urrent experimentally a essible values for the 3D ase. We fo us the remainder of

this letter on dis ussing possible experimental realizations for su h higher degenera y

mixtures.

The Fali ov-Kimball model has zero intera tion between the mobile fermions. One

an argue, on rather general grounds, that the modi� ation of Tc due to a nonzero

intraspe ies intera tion u will have orre tions to Tc of order u2. Hen e, if u is small,

the e�e t we dis uss here should ontinue to hold, with only slight redu tions. This

allows us to formulate our sear h riterion for physi al systems that will show this

degenerate spe ies e�e t.

In sear hing for appropriate mixtures, we want to �nd systems that (i) an have

a degenera y of three or more for the light fermioni spe ies, (ii) have a similar inter-

spe ies intera tion U between the mobile and lo alized fermions, whi h will be tuned

either via an interspe ies Feshba h resonan e, or via the depth of the trapping poten-

tial for the light spe ies; and (iii) have a small intraspe ies intera tion u between the

mobile fermions. We also note, that as long as the lo alized parti le is nondegenerate,

then it an a tually be either Bose or Fermi, sin e its statisti s does not enter the

analysis be ause it does not move. (However, if the heavy parti le is a boson, we do

need its intraspe ies intera tion to be large and positive, so it generi ally forms a

Mott insulator with at most one parti le per site and it does not Bose ondense on

the latti e.)

120

We start with examining some prototypi al systems whi h have already been

demonstrated to be trapped on opti al latti es. The �rst hoi e to examine is mixtures

of

40K (mobile fermion) and

87Rb (lo alized boson) [111℄. If we ould trap the mF =

−5/2,−7/2, and −9/2 states of K, we would have an N = 3 mixture. This system

is ni e, in the sense that it has a tunable interspe ies intera tion via a Feshba h

resonan e, and the intraspe ies intera tions for K have a s attering length on the

order of 100 a0 (in some ases one of the pairs an be tuned to zero s attering length).

The hallenge is that the Rb-Rb intera tion is too small (on the order of 100 a0), and

is not tunable, whi h would make it di� ult to satisfy the required onditions for

this e�e t. If we instead try

133Cs (lo alized boson) [112℄, we �nd that the Cs-Cs

intera tion is large, with a s attering length near 2000 a0 at B ≈ 260 G, but the

interspe ies intera tion is small (≈ −40 a0) and not simultaneously tunable for all

three K spe ies.

Moving on to other possibilities, if we use mixtures of

171Yb or

173Yb (mobile

fermion) [113, 114℄ and

133Cs (lo alized boson) [115, 116℄, we only have a degenera y

of N = 2 for 171Yb, even though its intraspe ies s attering is small, while for

173Yb the

intraspe ies s attering length is ≈ 200 a0, whi h is still viable, given the potentially

large Cs-Cs s attering length, but it would require a tunable Cs-Yb s attering length

that is large, and although this has not yet been measured, we do not anti ipate

that there is any reason why it should be parti ularly large. If we tried Rb as the

lo alized boson [117℄, it su�ers from the same issues as with K-Rb�namely, the Rb-

Rb s attering is too small.

Using

6Li as the mobile fermion appears attra tive [118, 119℄. However, the inter-

spe ies s attering length is only small for low �elds, and when a mixture is formed

from the N = 3 trappable state, at least one intraspe ies intera tion will be large

(although the other two an be lose to zero). So, this ase is suboptimal.

121

Next, we onsider mixtures of

87Sr (light fermion) whi h has up to N = 10 and a

Sr-Sr s attering length on the order of 100 a0 [120, 121℄. If we use Cs as the (lo alized

boson), then if the Cs-Cs s attering length an be set to the order of a few 1000 a0,

and the Sr-Cs s attering length is on the order of 500 a0, then this system might work

to illustrate this degenerate spe ies e�e t, and it has the potential to be spe ta ularly

large.

The remaining hoi es that might be workable seem to be longshots, but annot

yet be ruled out be ause we do not have enough information about their interspe ies

intera tions. We dis uss some of these possibilities next.

43Ca is a fermion with a nu lear spin of 7/2 [122, 123℄,

25Mg is a fermion with

a nu lear spin of 5/2 [124℄, Ba has two spin 3/2 fermioni spe ies [125℄, and

201Hg

is also spin 3/2 [126℄. It is unknown what the intraspe ies intera tions are amongst

these di�erent spin states, how many an be trapped, and what their interspe ies

intera tions are with potential heavy parti les. So they all are possible, but at this

stage quite di� ult systems to work with. Finally, there are all of the magneti -dipole

systems, like Er [127, 128℄, Dy [129, 130℄, and Cr [131, 132, 133℄. These systems

often have haoti intraspe ies intera tions due to a huge number of resonan es, but

they might show some small intera tions at low �elds, and hen e may also be viable

andidates for the light fermions.

In summary, we have illustrated the idea that by enhan ing spe ies degenera y,

one an enhan e Tc for fermioni neutral atoms trapped on opti al latti es su h that

their Tc to an ordered state an be raised high enough that they would be a essible

to explore with urrent experimental te hnology in ooling. This idea omes at this

problem from a di�erent angle than the many di�erent ooling strategies that have

been proposed, and ould provide the ability to truly study spatially ordered quantum

phases. The hallenge is to �nd the right mixture of atoms where this e�e t an be fully

122

exploited. We have suggested some possible systems, with Yb-Cs and Sr-Cs mixtures

as the most promising, but it is lear the experiments will be hallenging to arry out.

We want to end by ommenting that similar work has examined SU(N) symmetri

Hubbard models. The repulsive ase a tually sees a de rease in the antiferromagneti

Tc with in reasing N [134℄, while the attra tive ase sees an enhan ement similar to

what we see for the density-wave instability [135℄, we do not know of any large N > 3

systems with attra tive intera tions. Furthermore, there are hallenges with �nding

atomi systems with a small enough U value (for large N), sin e a maximal hopping

is required to have an a urate single-band des ription.

123

6

Nonequilbrium dynami s of XY hain

�A Physi al law must posses mathemati al beauty.�

� Paul Dira

Understanding the nonequilibrium dynami s of a many body quantum systems

has be ome one of the most intriguing and fas inating problems in ondensed matter

physi s and has attra ted the attention of physi ists from di�erent perspe tives . On

the one hand, understanding the time evolution of an intera ting quantum system and

the me hanism in whi h the systems thermalize, is one of the fundamental questions

in quantum me hani s. Sin e 1929, when von Neumann proposed the equilibration

of a quantum system, the problem remained unsolved [136℄ until re ently, where the

advent of ultra old atoms has opened a new horizon on realizing isolated quantum

systems with highly ontrollable parameters and paved the path for investigating the

dynami s of a quantum systems [137℄. On the other hand, �nding a systemati way to

hara terize the nonequilibrium dynami s of quantum system plays a ru ial role in

new te hnologies su h as a quantum simulator whi h is a type of quantum omputer

[26℄. In the following, we will elaborate on the importan e of this problem from both

fundamental point of view and it's appli ation in te hnology.

From the theoreti al side, one of the main goals of quantum me hani s is to

understand the time evolution of a generi quantum system starting from its initial

nonequilibrium state. A simple paradigm of su h problem is known as the quen h

124

problem; where one studies the dynami s of the system often by hanging one of

the system's parameters. In this hapter, we will onsider the XY spin hain in one

dimension whi h an be onsidered as a simple laboratory to study di�erent orre-

lation fun tions after a quantum quen h. The XY spin hain is one of the basi and

most studied systems that an shed light on fundamental on epts su h as dynami s

and propagation of information in spin systems [27, 195℄. The XY spin hain be omes

even more important as the spin hain is one the primary andidates for onstru ting

and building quantum simulators [26, 28, 29℄.

Furthermore, the XY hain manifests a quantum phase transition [22℄. The

quantum phase transition happens at zero temperature where the thermal �u tua-

tions are absent and the ground state energy of a system varies abruptly as one of the

ouplings su h as the external magneti �eld hanges [23℄. Similar to thermal phase

transition, at riti al points, some of the properties hange so drasti ally that they

annot be des ribed analyti ally [138℄. Furthermore, in the vi inity of the riti al

point, the orrelation length diverges and the system be omes s ale invariant in that

it an be des ribed by a simple �eld theory in two-dimensional Eu lidean spa e

(x, τ), where τ = it is the imaginary time. Due to this s ale invarian e, the system

manifests universal behavior and onsequently, various quantities su h as di�erent

sus eptibilities and orrelation fun tions an be des ribed by a power law behavior.

For example, lose to riti al point, the orrelation length diverges as ξ = |λ− λc|−ν

with ν orresponding to a universal riti al exponent. Another important quantity is

known as the hara teristi energy s ale ∆ that vanishes at a riti al point and s ales

as ∆ ≈ |λ − λc|zν, where z is known as the dynami al exponent. Su h a universal

behavior is independent of the mi ros opi properties of the system and only depends

on underlying symmetries, dimensionality and the range of intera tion in systems.

125

In addition, it turns out that systems that are invariant under s ale transforma-

tion, translation, and rotation are also invariant under a onformal transformation.

The onformal invarian e be omes more ru ial in two dimensions as the onformal

�eld theory be omes invariant under an in�nite-dimensional symmetry group whi h

makes the onformal �eld theory (CFT), a powerful tool to des ribe orrelation fun -

tion lose to riti al points. The details of how CFT an be used to des ribe systems

at riti al points is beyond the s ope of this thesis and we refer the interested reader

to Ref. [139℄.

The rest of this hapter is organized as follows. First, in se tion 6.1 , we will

introdu e the XY model as the simplest model to des ribe a spin intera tion whi h

manifests a quantum phase transition. We will further explain the mapping onto

the free fermioni system and des ribe the diagonalization pro edure. In se tion 6.2,

we will explain some of the orrelation fun tions that we are interested to study

and mention some of the relevant experiments. Finally, we will present some of our

published results on the in�uen e of the initial state on the revival probability and

the light- one velo ity, and the formation probabilities and Shannon information in

last three se tions, respe tively [21, 31, 32℄.

6.1 XY spin hain

The problem of an intera ting spin hain is one of the oldest problems in quantum

me hani s whi h has been proposed by Heisenberg and Dira in order to explain the

magneti properties of materials [141, 142℄. The Heisenberg Hamiltonian is de�ned

as

H = J∑

i

Si.Si+1, (6.1)

126

whi h des ribes the e�e tive intera tion between nearest neighbor spins with oupling

J . In 1931, Hans Bethe solved the Heisenberg spin hain by onstru ting its many

body fun tion [143℄. Although his solution opened a new path into a �eld of exa tly

solvable models, the solution is still remains quite involved. Here, instead, we will use

the XY model as a simple paradigm of an exa tly solvable model where it an be

simply realized by turning o� the intera tion in the z dire tion and onsequently the

Hamiltonian be omes

HXY = −L∑

j=1

[

(1 + a

2)σx

j σxj+1 + (

1− a

2)σy

jσyj+1 + hσz

j

]

, (6.2)

where σx,y,zj are the Pauli matri es. We have hosen the oupling onstant equal to

+1 for a ferromagneti intera tion. Moreover, a indi ates the anisotropy intera tion

between spins, and h denotes the transverse magneti �eld. We will use the Jordan-

Wigner transformation and introdu e fermioni operators as cj =∏

m<jσzm

σxj −iσy

j

2and

N =∏L

m=1σzm = ±1 whi h maps the Hilbert spa e of a quantum hain of a spin 1/2

into the Fo k spa e of spinless fermions [22, 144℄. Then, the Hamiltonian be omes

H =

L−1∑

j=1

[

(c†jcj+1 + ac†jc†j+1 + h.c.)− h(2c†jcj − 1)

]

+N (c†Lc1 + ac†Lc†1 + h.c.). (6.3)

where c†L+1 = 0 and c†L+1 = ±c†1 for open and periodi boundary onditions respe -

tively. The phase diagram of the XY- hain is shown in �gure (6.4) where di�erent

phases arise from di�erent values of a and h.

In the following we will show a generi way to diagonalize a free fermion system

whi h, we will losely follow the method in Ref. [22, 144℄. First, we onsider a generi

(real) free fermion Hamiltonian:

H =∑

ij

[c†iAijcj +1

2c†iBijc

†j +

1

2ciBjicj ]−

1

2TrA (6.4)

where c†i and ci are fermioni reation and annihilation operators and in our ase A

and B are real matri es. By hoosing the proper matri es A and B, one an re over

127

a

h

1

−1

1

criticalX

X

critical XY

critical XY

Ising

critical Ising

Figure 6.1: (Color online)Di�erent riti al regions in the quantum XY hain. In the

riti al XX h = 0 while for riti al XY, we have a = ±1.

the XY Hamiltonian from the free fermioni Hamiltonian. We will present the A

and B below. To diagonalize the Hamiltonian, we use the following unitary anoni al

transformation

†

= U

†

η

η†

. (6.5)

with

U =

g h

h g

, (6.6)

Then, we an write the diagonalized from of the Hamiltonian as follows

HD =∑

k

|λk|η†kηk + const, (6.7)

128

This requires that [ηi, H ] = |λi|ηi, or in other words

(

λg λh

)

=(

g h

)

A B

−B −A

, (6.8)

we emphasize that λ is the diagonal matrix made of positive eigenvalues |λk|. Then,

by onsidering the real A and B matri es, we have g and h an be derived from the

following equations:

g =1

2(φ+ψ), (6.9)

h =1

2(φ−ψ). (6.10)

where we have

(A+B)φk = |λk|ψk, (6.11)

(A−B)ψk = |λk|φk. (6.12)

or

(A−B)(A+B)φk = |λk|2φk, (6.13)

(A+B)(A−B)ψk = |λk|2ψk. (6.14)

When λk 6= 0, φk and λk an be al ulated by solving the eigenvalue equation (6.13),

then ψk an be determined using equation (6.11). When λk = 0, φk and ψk an be

dedu ed dire tly from equations (6.11) and (6.12). To evaluate the onstant term, we

need to take the tra e of the Hamiltonian in the c and η representations

H =∑

k

|λk|η†kηk −1

2Trλ, (6.15)

It's worth to omment about a few important issues whi h one needs to take into

a ount properly. In this method, all λk's are onsidered positive. This is due to the

fa t that in this way one an �nd the ground state |G〉 by a ting with an η ve tor,

ηk|G〉 = 0 (6.16)

129

Subsequently, one an build the higher ex ited states by a ting with η†k.

6.2 Dynami s of observables in XY spin hain

Most of interesting quantities in physi s an be expressed in the form of orrelation

fun tions. While a one point orrelation fun tion su h as magnetization, provides

information about the lo al properties of system, higher orrelation fun tions an

provide information about the non-lo al properties of many body systems. In this

hapter, we will study the dynami s of di�erent quantities and orrelators. We start

with two point orrelation fun tion whi h is de�ned as

Cln ≡ 〈c†l cn〉,

C†ln ≡ 〈c†ncl〉,

Fln ≡ 〈clcn〉,

F †ln ≡ 〈c†l c†n〉, (6.17)

Noti e that for onserved number of parti les, we have:Fln = F †ln = 0. By introdu ing

new operators ai = c†i + ci and bi = c†i − ci, one may re�ne the following fun tions,

Gbaij ≡ 〈biaj〉,

Gabij ≡ 〈aibj〉,

Gaaij ≡ 〈aiaj〉,

Gbbij ≡ 〈bibj〉, (6.18)

whi h are known as G matri es. We will introdu e the G matrix orresponding dif-

ferent boundary onditions and geometry in the next se tion. In addition, we will

show how one an use the G matrix to obtain higher orrelation fun tions and other

interesting quantities su h as formation probabilities, Shannon information, omplete

130

revival probabilities, and the light one velo ity. We have performed all of our al-

ulation dire tly in the on�guration basis, whi h are the eigenstate of σz and it has

a few advantages. First, by dire tly working in the on�guration basis, we are able

to map the omputational basis of a free fermioni system into a fermioni oherent

state whi h provides a powerful tool to arry out the al ulations. Se ond, most of

the experiments are performed in the omputational basis. Consequently, all of the

di�erent quantities that we introdu e in this hapter an be dire tly be ompared

with orresponding experimental results.

One of the earliest non-lo al orrelation fun tions that has been studied vastly is

an emptiness formation probability whi h is the probability of �nding a onse utive

sequen e of up spins with size l in a spin hain. The analyti al result for this quantity

has been derived for up and down spin on�gurations in Refs. [145, 146, 147℄. We have

generalized the study of the formation probability to various number of on�gurations

whi h we all them � rystal on�gurations�. The results of this study will be presented

in the next se tion [21℄. In fa t, our method, allows us to dire tly measure the prob-

ability of all on�gurations and onsequently, we were able to study the Shannon

information and the it's time evolution after a quantum quen h. Furthermore, as we

mentioned above our al ulations an be dire tly ompared with experimental data.

For example, in a re ent study Zhang and his ollaborators in Monroe's group at JQI,

were able to dete t the many body dynami al phase transition (DPT) in a quantum

simulator with 53 qubits [25℄.

In this experiment, they dete ted the DPT after a quantum quen h in the trans-

verse Ising model with long range intera tions whi h is intra table with onventional

omputational methods, see �gure 6.2. To dete t the DPT, they have studied the

average magnetization and espe ially the average two spin orrelators. Furthermore,

they were able to study higher order orrelation fun tions as the probability of a

131

Figure 6.2: S hemati set up of quen h of trapped ion experiment. a) A one dimen-

sional spin hain is prepared in a produ t state, then the quen h pro edure is per-

formed by swit hing the spin-spin intera tion and transverse magneti �eld and �nally,

the magnetization an be measured. Di�erent olors of spin indi ates di�erent spin

states. For example, the blue spins indi ate the initial spin dire tion, and red olor

indi ate the down spin for whi h the proje tive measurement is performed. Other

olors indi ate the superposition of spins. b) The time evolution of the average of

the spin magnetization is shown in the Blo h sphere. The ompetition between Ising

intera tion along the x axis and external magneti �eld along the z axis, ause the

os illation and the relaxation. The average magnetization for small and large trans-

verse �eld is shown in blue and green olor respe tively. Figure taken from Ref. [25℄.

132

domain of up spins whi h we have proposed in our study [21℄, see �gure 6.3. We have

Figure 6.3: Distribution of domain size of a one dimensional spin hain with 53 spins.

The top �gure indi ates the bright state or all up spin state (| ↑〉x). The other threespin on�gurations in the top and bottom, presents di�erent on�gurations omposed

of up and down spins. In the enter, the statisti of domains sizes is shown for di�erent

values of transverse magneti �eld. The boxes in the image indi ate two large domain

sizes orresponding to di�erent values of the transverse �eld. Noti e that, how for the

bigger value of transverse �eld, the probability of �nding a large size domain is very

low. The dashed lines indi ates a exponential �t. Figure taken from Ref. [25℄

also studied some other quantities su h as post measurement entanglement entropy

in a quantum spin hain and full ounting statisti s and the distribution of subsystem

energy in a free fermioni system whi h are beyond the s ope of this thesis and the

interested reader is referred to Refs. [33, 34℄.

6.3 On the possibility of omplete revivals after quantum quen hes

to a riti al point

This hapter is reprinted from This hapter is reprinted from K Naja�, MA Rajabpour,

On the possibility of omplete revivals after quantum quen hes to a riti al point, Phys.

Rev. B 96:014305, Copyright(2017)

133

The quantum me hani al version of Poin aré re urren e theorem guarantees that

any system with dis rete energy eigenstates, after a su� iently long but �nite time,

will return to a state whi h is very lose to it's initial state. Although this seems a

natural expe tation for many body systems, it usually takes astronomi al times to see

an (almost) omplete revival. However, in some systems, partial revivals are possible

whi h makes the problem a very interesting subje t, see for example[149, 150, 151,

152, 153, 154, 155, 156, 157, 158, 159, 160, 161℄. The problem of revivals or related

phenomena also appears in many other on epts su h as dynami al transition and

quantum speed limit [162, 163, 164, 165℄. Quite naturally, one usually is interested

in the problem of revivals when the number of parti les is limited or in other words

when there is a �nite size e�e t. The presen e of the �nite size e�e t usually makes

the exa t al ulations di� ult, however, it has the bene�t of being a essible by the

numeri al means.

In a re ent letter [148℄, the author has made an interesting analyti al al ulation

by using onformal �eld theory for a �nite system size and observed that the omplete

revivals are possible for Los hmidt amplitude in riti al systems in a period whi h

is a fra tion of the system size. This is quite unexpe ted be ause any full revival

requires a nearly perfe t �ne-tuning of phase onditions. To the best of our knowledge,

this is the �rst example of the predi tion of full revivals in many-body quantum

systems in an a essible time. In this brief arti le, we make a loser look to this

phenomena in mi ros opi systems. In parti ular, we show that the full revivals of

[148℄ are impossible in mi ros opi realizations of those onformal �eld theories that

were studied in [148℄. In the next se tion, �rst, we brie�y review the arguments in favor

of and against the presen e of omplete revivals in riti al systems. Then in se tion

three, we study the Los hmidt e ho (�delity) in a quantum XY hain and �nd an

exa t determinant formula for a parti ular initial state. In se tion four, we al ulate

134

the �delity at the riti al point of the periodi (open) transverse �eld Ising hain

analyti ally (numeri ally), and show that the omplete revivals dis ussed in [148℄

are absent. In se tion �ve, we explore the other parts of the phase diagram of the

XY hain. In parti ular, we study the quasi-parti le pi ture for di�erent post-quen h

Hamiltonians and show that the pi ture an determine the periods of the revivals

just in some part of the phase diagram. Finally, in the last se tion, we on lude our

paper.

6.3.1 Revivals in onformal field theories

Consider a one-dimensional periodi quantum hain of length L and the Hamiltonian

H . Assume an initial state whi h is very lose to a onformally invariant state |B〉.

Sin e the onformal states are non-normalizable, one needs to introdu e a parameter

β whi h is alled extrapolation length and then one an write the initial state as

|ψ0〉 ≈ e−β4H |B〉. The extrapolation length is usually of the order of a few latti e sites,

in other words, we have L ≫ β. The parameter β an be estimated by al ulating

the expe tation value of the Hamiltonian as 〈ψ0|H|ψ0〉 = πcL6β2 , see [148℄. To show the

omplete revival, the referen e [148℄ al ulates the �delity de�ned as

F (t) = |〈ψ0|e−iHt|ψ0〉| (6.19)

using the well-established re ipe, see for example [161℄. Based on his argument, for

minimal models at large

Lβ, there must always be omplete revivals at multiples of

t = M L2, where M ∼ 24

1−c. Not surprisingly in the regime L ≫ β, there is no e�e t

of the extrapolation length in the revival times. Although there is nothing wrong in

the CFT al ulations in [148℄, this e�e t an not be seen in a mi ros opi quantum

hain. There are two good reasons: The �rst reason, whi h is already noti ed in [161℄,

is related to the presen e of the ex ited states in any global quantum quen h whi h in

135

prin iple an not be des ribed by CFT. The se ond reason is that [148℄ assumes that

there is a one to one orresponden e between an initial state in a mi ros opi system

and onformal boundary states whi h in general is not true. There are many dis rete

initial states, probably exponentially growing with the system size, that �ow to the

same onformal boundary states either with the same β or di�erent extrapolation

lengths. This means that although in a CFT setup the system omes ba k to itself

with probability one, in the dis rete model it an be in a state whi h is ompletely

di�erent but still with the same ontinuum des ription. Although in prin iple, this

problem an be resolved by onsidering all the possible irrelevant perturbations of the

CFT, see for example [148, 166℄ it will eventually a�e t the omplete revivals anyway.

The referen e [148℄ dis usses, in parti ular, a quen h from the disordered phase in the

transverse �eld Ising hain and shows that there should be omplete revivals at times

t = nL2for even n, while for odd n the omplete revivals are suppressed. In the next

se tion, we will show that the omplete revivals are absent in the riti al transverse

�eld Ising hain.

6.3.2 Los hmidt e ho in quantum XY hain

In this se tion, we study the revivals in the quantum spin hain when the initial state

is the ase with all spins σzare up or down. The Hamiltonian of XY- hain is as

follows:

HXY = (6.20)

−JL∑

j=1

[

(1 + a

2)σx

j σxj+1 + (

1− a

2)σy

j σyj+1

]

− h

L∑

j=1

σzj .

Di�erent phases of the model for J = 1 are shown in the Figure 6.4. The line

a = 1 is the transverse �eld Ising hain. The h = 1 line is riti al for all the values of

136

a and we all it XY riti al line. On the ir le a2 + h2 = 1, the wave fun tion of the

ground state is fa torized into a produ t of single spin states [167℄.

a

h

1

1

CriticalX

XCritical XY

Ising

Critical Ising

Figure 6.4: Di�erent regions in the phase diagram of the quantum XY hain. The

riti al XX hain has entral harge c = 1 and riti al XY line has c = 12. The region

a2 + h2 < 1 is depi ted with the yellow olor.

After using the Jordan-Wigner transformation c†j =∏

l<j σzl σ

+j , whi h maps the

Hilbert spa e of a quantum hain of a spin 1/2 into the Fo k spa e of spinless fermions,

the new Hamiltonian be omes

H = J

L−1∑

j=1

(c†jcj+1 + ac†jc†j+1 + h.c.)−

L∑

j=1

h(2c†jcj − 1) (6.21)

+NJ(c†Lc1 + ac†Lc†1 + h.c.),

where c†L+1 = 0 and c†L+1 = N c†1 for open and periodi boundary onditions respe -

tively with N =∏L

j=1 σzj = ±1. The above Hamiltonian an be written as:

H =

†.A. +1

2

†.B. † +1

2 .BT . − 1

2TrA, (6.22)

137

with appropriate A and B matri es as:

A =

−2h J 0 . . . NJ

J −2h J 0 0

0 J −2h J 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

NJ 0 . . . J −2h

,

B =

0 aJ 0 . . . −aJN

−aJ 0 aJ 0

0 −aJ 0 aJ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

aJN 0 . . . −aJ 0

. (6.23)

To al ulate the Los hmidt e ho, �rst we de ompose e−iHtusing the Balian-Brezin

formula [168℄ as:

e−iHt = e12

†X

†

e †Y e−

12TrYe

12 Z , (6.24)

where X, Y, Z an be al ulated from the blo ks of matrix T de�ned as

T = e

−it

A B

-B -A

=

T11 T12

T21 T22

, (6.25)

Then we have

X = T12(T−122 ), Z = (T−1

22 )T21, e-Y = T

T22. (6.26)

Note that we always have Z = −X. Finally, the �delity for the desired initial state

(all the spins up) will be:

F (t) = |〈0|e−iHt|0〉| = | det(T22)|12 . (6.27)

138

The same formula is also valid for the ase when the initial state is all the spins down,

in other words when all the sites are �lled with fermions |1〉. In the next subse tions,

we will use the above formula to study the revivals in di�erent phases of the quantum

spin hain. Note that although we will keep the J oupling expli itly in some of the

formulas for numeri al al ulations we always take J = 1.

6.3.3 Revivals in riti al transverse field Ising hain

In this se tion, �rst we al ulate the �delity for the periodi riti al Ising point exa tly,

then we will study the open hain numeri ally.

Periodi riti al transverse field Ising hain

Consider a periodi riti al Ising model Hamiltonian in the Ramond se tor after

Jordan-Wigner transformation, equation (6.36) with N = J = a = h = 1. Sin e in

this ase the matri es A and B ommute, one an al ulate the �delity exa tly. Note

that in this ase the matrix

A B

-B -A

is a ir ulant matrix whi h guarantees the

exa t al ulation of the eigenvalues of the matrix T22 with lassi al methods. After

expanding (6.25) we have

T22 = T

∗11 = cosh(2t

√A) +

i√A

2sinh(2t

√A), (6.28)

T12 = −T21 = −itB[sinh(2t

√A)

2t√A

]. (6.29)

Although it is not needed for our future dis ussion, we also report the exa t form of

the matri es X:

X =−iB

2√A coth[2t

√A] + iA

, (6.30)

139

Sin e the eigenvalues of the matrix A are λj = −2 + 2 cos 2πjL, where j = 1, 2, ..., L ;

logarithmi �delity for the riti al periodi Ising hain an be written expli itly as

ln[F (t)] =1

4

L−1∑

j=0

ln[1− cos2[πj

L] sin2[4t sin[

πj

L]])]. (6.31)

In Figure 6.5, one an see that although there are partial revivals at multiples of t = L4,

whi h an be understood with the quasi-parti le pi ture, the omplete revivals do not

happen. Of ourse if one waits enough time, there will be always almost omplete

revivals but they are usually expe ted to happen in mu h larger times that are usually

ina essible. Note that for the onsidered initial state, we have 〈ψ0|H|ψ0〉 = L whi h

means that β =√

π12

or in other words we are in a regime that

Lβis very large whi h

is well inside the regime onsidered in [148℄.

0 25 50 75 100 125 150 175 200 225 250t

0

0.002

0.004

0.006

0.008

0.01

F(t)

L = 100

Figure 6.5: (Logarithmi �delity for the periodi riti al Ising hain starting from

the all aligned spins σzinitial state.

Open riti al transverse field Ising hain

In this subse tion, we repeat the analyses of the previous se tion for the open hains

to see the e�e t of the boundary ondition on the revivals. Unfortunately, we were

not able to provide an exa t result in this ase so the al ulations are based on a

numeri al evaluation of the determinant in the equation (6.27). The main reason for

140

our failure at the riti al Ising point an be tra ed ba k to this fa t that in this ase

the two matri es A and B do not ommute and so the expansion method gets too

ompli ated after few steps. Also note that in this ase the matrix

A B

-B -A

is not a

ir ulant matrix and so the ommon methods of diagonalization an not be applied.

The numeri al results depi ted in Figure 6.6 on�rms the absen e of the omplete

revivals introdu ed in [148℄ and the usefulness of the quasi-parti le pi ture. We will

ome ba k to a more detailed study of the quasi-parti le pi ture in the next se tion.

0 25 50 75 100 125 150 175 200 225 250t

0.0

5.0×10-6

1.0×10-5

1.5×10-5

2.0×10-5

F(t)

L=100

Figure 6.6: Logarithmi �delity for the open riti al Ising hain starting from the

all aligned spins σzinitial state.

6.3.4 Revivals and quasi-parti le pi ture

In this se tion, we extend the analyses of the previous se tion to the other parts of

the phase diagram of the XY hain. In addition, we also examine the appli ability of

the quasi-parti le pi ture in determining the periods of the revivals in the Los hmidt

e ho. First, we make a brief omment on the quasi-parti le pi ture, see [211℄. Based

on this semi- lassi al pi ture the pre-quen h state has more energy than the post-

quen h Hamiltonian ground state and so onsequently, the initial state plays the role

of a sour e of quasi-parti les. The quasi-parti les with the maximum group velo ity

usually are the ones that an be onne ted to the saturation of the entanglement

141

entropy [212℄ or the revivals in the Los hmidt e ho [161℄. The dispersion relation and

the group velo ity of the Hamiltonian (6.36) are

ǫk = J√

(cosφk − h)2 + a2 sin2 φk, (6.32)

vg = J sin φk2a2 cos φk − cosφk + h

√

(cosφk − h)2 + a2 sin2 φk

. (6.33)

where φk = 2πL(k + N−1

4) with k = 0, ..., L − 1. In Figure 6.7, we depi ted vg for

di�erent values of a and h. Note that for su� iently large L, there is no signi� ant

di�eren e between open and periodi ases. Using the above formula one an derive

0 π/2 π 3π/2 2πϕ

-1.5

0

1.5

ν g

a=0.2a=0.4a=0.6a=0.8a=1.0

-1.5

0

1.5

ν g

a=0.2a=0.4a=0.6a=0.8a=1.0

-2

0

2

ν g h=0.2h=0.4h=0.6h=0.8h=1.0

h=1

h=0.8

a=1

Figure 6.7: Group velo ity vφ with respe t to φ for di�erent values of a and h.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a

1.61.71.81.9

22.1

ν gmax

h=1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a

0

0.5

1

1.5

2

ν gmax

h=0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2h

0

0.5

1

1.5

2

ν gmax a=1

Figure 6.8: Maximum group velo ity for di�erent values of a and h.

the maximum group velo ity vmaxg for di�erent values of a and h, see Figure 6.7.

142

Having these group velo ities for quasi-parti les, one an guess the following periods

for the appearan e of revivals in the Periodi and Open quantum hains:

Tp =L

2vg, To =

L

vg, (6.34)

where vg is usually the velo ity of the fastest quasi-parti les vmaxg . However, this is

not a rule and sometimes other quasi-parti les an arry more information than the

fastest quasi-parti les as it was dis ussed in the ontext of entanglement entropy in

[171℄ and in the ontext of Los hmidt e ho in [161℄. In those ases vg in the equation

(6.34) will be di�erent from vmaxg . We are not aware of a riterion whi h one an

use a priori to de ide what is the most important group velo ity. In the next three

subse tions, we study the revivals and the quasi-parti le pi ture in di�erent regimes.

Ising line: a = 1

In Figure 6.9, we depi ted the Los hmidt e ho for di�erent values of h. Two omments

are in order: �rst of all, at non- riti al points similar to the riti al point we have

partial revivals. Apart from the period of the revivals, there is no signi� ant di�eren e

in the form of the Los hmidt e ho at and outside of the riti al point. Se ondly, the

period of the revivals an be understood by taking the maximum group velo ity

vmaxg = 2h as the relevant velo ity.

6.3.5 Criti al XY line: h = 1

This is a riti al line whi h it is in the same universality lass as the riti al Ising

hain. The results for the Los hmidt e ho on di�erent points are shown in Figure 6.10.

The interesting fa t is that in this ase, the relevant group velo ity is learly |vfg | = 2a

whi h is the Fermi velo ity. As it was already dis ussed in the ontext of the Los hmidt

e ho after lo al quen hes in [161℄, it is not the maximum group velo ity for a <√32.

143

0 20 40 60 80 100 120 140 160 180 200t

0

0.0006

0.0012

0.0018h=1

0.0004

0.0008 h=0.75

0.002

0.004h=0.5

0.003

0.006 h=0.25

a=1F(

t)

0 20 40 60 80 100 120 140 160 180 200t

0

0.00012

0.00024h=1

6e-05

0.00012 h=0.75

0.00015

0.0003 h=0.5

0.0002

0.0004 h=0.25

a=1

F(t)

Figure 6.9: Los hmidt e ho on the Ising line for periodi and open hains in left and

right panel, respe tively(L = 80). The revivals are ompatible with quasi-parti le

pi ture with maximum group velo ity vmaxg = 2h.

This is an interesting example of a ase whi h the relevant group velo ity is di�erent

from the maximum group velo ity.

0 20 40 60 80 100 120 140 160 180 200t

0

0.0006

0.0012

0.0018a=1

0.005

0.01 a=0.75

0

0.004

0.008 a=0.5

0.05

0.1a=0.25

h=1

F(t)

0 20 40 60 80 100 120 140 160 180 200t

0

0.0001

0.0002 a=1

8e-06

1.6e-05 a=0.75

0.0016

0.0032 a=0.5

0.03

0.06a=0.25

h=1

F(t)

Figure 6.10: Los hmidt e ho on the XY riti al line for periodi and open hains

in left and right panel, respe tively(L = 80). The revivals are ompatible with quasi-

parti le pi ture with the Fermi group velo ity vfg = 2a.

144

Non- riti al regime: 0 < h, a < 1

Based on the results of the above two subse tions one might be tempted to guess that

sin e this region is non- riti al, the quasi-parti le pi ture with the maximum group

velo ity should be appropriate to guess the form of the partial revivals. However,

strikingly as one an see in Figure 6.11, there are two ompletely di�erent regimes

with very di�erent behaviors. In the region a2 + h2 > 1 the quasi-parti le pi ture

with the maximum group velo ity works perfe tly, however, in the region a2+h2 ≤ 1

there seems to be no lean way to attribute the revivals to quasi-parti les with �xed

velo ities. One might understand it as a regime that there are more than one type of

important quasi-parti les whi h their velo ity di�eren e kill lean periodi revivals. It

is absolutely un lear why the line a2 + h2 = 1 should separate these two regimes. To

keep the �gure simple, we have only depi ted four di�erent points, however we have

he ked many other di�erent points and on�rmed numeri ally that indeed this line

is at the border between the two di�erent regimes.

0 20 40 60 80 100 120 140 160 180 200t

0

0.0012

0.0024 a=1

0

0.0004

0.0008 a=0.8

0.005

0.01a=0.5

0.04

0.08 a=0.25

h=0.8

F(t)

0 20 40 60 80 100 120 140 160 180 200t

0

3e-05

6e-05 a=1

0

4e-05

8e-05 a=0.8

0

0.0003

0.0006a=0.5

0

0.006

0.012a=0.25

h=0.8

F(t)

Figure 6.11: Los hmidt e ho on the line h = 0.8 for periodi and open boundary

ondition. The revivals are ompatible with quasi-parti le pi ture with maximum

group velo ity (see Figure 6.8 far as a2 + h2 > 1. For the region a2 + h2 < 1

145

6.3.6 Con lusions

In this paper, we studied revivals in the XY hain starting from an initial state with all

the spins σzin the dire tion of the transverse �eld. Our on lusions are the following:

�rst of all, we proved that omplete revivals in the times introdu ed by [148℄, annot

happen in the mi ros opi quantum riti al hains. Se ondly, for the onsidered initial

state we showed that there are three interesting regimes. For a2+h2 > 1, one an use

the quasi-parti le pi ture to predi t the period of the partial revivals. On the riti al

XY line h = 1, one must use the Fermi velo ity vfg = 2a to al ulate the revivals.

However, for the other points, the maximum group velo ity vmaxg is the important

group velo ity. For the region a2 + h2 ≤ 1, the revivals do not follow a lean periodi

stru ture whi h indi ates the presen e of more than one type of important quasi-

parti les. It will be interesting to study the stru ture of revivals in also other models,

espe ially intera ting models su h as the XXZ hain. Los hmidt e ho in the non-

riti al phase of the XXZ hain has been already studied in [172, 173℄, however, it

seems that the problem of revivals in the riti al regimes of intera ting models has

not been studied in full detail so far. In parti ular, it is very important to study the

e�e t of the initial state in these models.

6.4 Light- one velo ities after a global quen h in a non-intera ting

model

In a seminal ontribution Lieb and Robinson [174℄ proved that in a non-relativisti

quantum system, the operator norm of the ommutator between two lo al observ-

ables A(x, t) and B(y, 0) is exponentially small as long as |x − y| ≥ vLRt, for

This hapter is reprinted from K.Naja�, M.A. Rajabpor, J.Vitt, Arxiv 1803.03856,Copy-

right(2018)

146

some vLR > 0. We refer to [175, 176, 177℄ for a more pre ise mathemati al state-

ment or re ent viewpoints. The Lieb-Robinson theorem holds in any dimension and

for a translation invariant Hamiltonian with arbitrary but �nite ranged intera -

tions. The parameter vLR is alled Lieb-Robinson velo ity and depends only on the

Hamiltonian [174, 175, 176, 177℄ driving the time evolution; in parti ular it is state-

independent. The existen e of a �nite vLR implies that information annot propagate

arbitrarily fast [174℄.

The Lieb-Robinson result represents nowadays a powerful tool to prove rigorous

bounds for orrelation and entanglement growth [178℄. Moreover, formidable exper-

imental [179, 180, 181, 182, 183℄ and theoreti al (for an overview [184, 185, 186℄)

progress in many-body quantum dynami s, have further underlined its striking phys-

i al impli ations. Among them, the emergen e of light- one e�e ts in orrelation fun -

tions of time-evolved lo al observables for systems that are not manifestly Lorentz

invariant, like quantum hains [187, 188, 189, 190, 191, 192, 193, 194℄. For instan e,

the Lieb-Robinson theorem has been invoked to explain the ballisti spreading of or-

relation fun tions in paradigmati ondensed matter models su h as the XXZ spin

hain [195℄ or the Hubbard model [196℄ after a global quen h; see also [197℄ for a ped-

agogi al survey. Similar onsiderations apply to the linear growth of the entanglement

entropies [198, 199, 200, 201, 202, 203, 204, 205, 206℄. Quantitative predi tions on the

spreading of orrelations and the entanglement entropy are also of lear experimental

interest; see [207, 208, 209, 210℄ for experimental veri� ations of light- one e�e ts

in many-body systems. A omplementary physi al interpretation of these emergent

phenomena is based on the idea [211, 212℄ that in a quen h problem the initial state

a ts as a sour e of pairs of entangled quasi-parti les. The quasi-parti les have oppo-

site momenta and move ballisti ally; see [213, 214, 215℄ for appli ations to integrable

models. Re ently it has also been pointed out that light- one e�e ts an be re ov-

147

ered from �eld theoreti al arguments without relying on parti ular properties of the

post-quen h quasi-parti le dynami s [216℄.

It is important to stress that the Lieb-Robinson theorem proves the existen e

of a maximal velo ity for orrelations to develop. However the observed propagation

velo ity is a tually state-dependent and non-trivially predi table. In other words, it is

not dire tly related to vLR that rather furnishes an upper bound. For one-dimensional

spin hains, a state-dependent light- one velo ity was noti ed �rst in [195℄. In a on-

text of integrability, it was pointed out how the dispersions of the quasi-parti les

in the stationary state [217℄ (the so- alled Generalized Gibbs Ensemble [218, 219℄)

was initial-state dependent. A predi tion for their velo ities was then proposed and

numeri ally tested. This idea has lead to many subsequent ru ial developments in

the �eld [213, 214, 220, 221, 222℄.

In the simpler setting of a non-intera ting model, the dispersion of the quasi-

parti les after the quen h annot be state-dependent, however as we will dis uss the

symmetries of the initial state and in parti ular translation invarian e an introdu e

additional sele tion rules on their momenta. As a matter of fa t, the light- one velo ity

an be state-dependent also in absen e of intera tions. A ir umstan e that, to our

best knowledge, has been not noti ed so-far.

The paper is organized as follows. In Se . 6.4.1 we introdu e the XY spin hain and

dis uss a spe ial lass of initials states for whi h time-evolution of lo al observables

an be al ulated easily. In Se . 6.4.2 and 6.4.4 we analyze the propagation velo ity

both for physi al two-point orrelation fun tions and the entanglement entropy. As

a by-produ t of our analysis we dis uss in Se . 6.4.3 an approximation of fermioni

orrelation fun tions at the edge of the light- one valid for large spa e-time separa-

tion. Su h an approximation shows that typi ally orrelations s ale as integer powers

of t−1/3. A similar statement was made previously [192℄ in the ontext of prethermal-

148

ization and weak breaking of integrability. After the on lusions in Se . 6.4.5, two

appendi es omplete the paper.

6.4.1 XY hain: notations and set up

In this se tion, we brie�y remind the XY hain, its fermioni representation and we

then introdu e the initial states dis ussed in the rest of the paper. The Hamiltonian

of the XY- hain [223℄ is

HXY =− JL∑

j=1

[

(1 + γ

2)σx

j σxj+1 + (

1− γ

2)σy

j σyj+1

]

− JhL∑

j=1

σzj , (6.35)

where the σαj (α = x, y, z) are Pauli matri es and J, γ, h are real parameters; in

parti ular J > 0 and onventionally h is alled magneti �eld. The XY model redu es

to the Ising spin hain for γ = 1 and it is the XX hain when γ = 0. We will also hoose

periodi boundary onditions for the spins, i.e. σαj = σα

j+L. Introdu ing anoni al

spinless fermions through the Jordan-Wigner transformation, c†j =∏

l<j σzl σ

+j , (6.35)

be omes

HXY = JL∑

j=1

(c†jcj+1 + γc†jc†j+1 + h.c.)− Jh

L∑

j=1

(2c†jcj − 1) (6.36)

where c†L+1 = −N c†1. Here N = ±1 is the eigenvalue of operator

∏Lj=1 σ

zj that is

onserved fermion parity. The above Hamiltonian an be written as

HXY =

†A +

1

2

†B

† +1

2 B

T − 1

2TrA, (6.37)

with appropriate real matri es A and B that are symmetri and antisymmetri ,

respe tively. In the se tor with an even number of fermions (N = 1), the so- alled

Neveau-S hwartz se tor, the Hamiltonian an be diagonalized in Fourier spa e by a

unitary Bogoliubov transformation. In parti ular, there are no subtleties related to

149

the appearan e of a zero-mode and one has HXY =∑L

k=1 εk(b†kbk − 1

2). The anoni al

Bogoliubov fermions b's have the following dispersion relation and group velo ities

εk = 2J√

(cos φk − h)2 + γ2 sin2 φk, (6.38)

v(φk) = 2J sinφkγ2 cosφk − cosφk + h

√

(cos φk − h)2 + γ2 sin2 φk

, (6.39)

where φk = 2πL(k − 1

2) and k = 1, ..., L . We will assume L even from now on. The

diagonalization pro edure will be also brie�y revisited in Se . 6.4.2.

In this paper, we are interested to study the time evolution (global quantum

quen h [198℄) of su h a system from initial states that are eigenstates of the lo al

σzj operators. For example, the initial state |ψ0〉 an be a state with all spins pointing

up (no fermions) or down (one fermion per latti e site); other possibilities an be

the Néel state | ↓↑↓↑ ... ↓↑〉 and alike. Moreover, all the states that we study have

a periodi pattern in real spa e with a �xed number of spin up. We label our rys-

talline initial states |ψ0〉 as (r, s), where r is the spin up (fermion) density and s is

the number of spin up in the unit ell of the rystal. For example, the Néel state is

labelled by (12, 1) and the state | ↓↓↑↑↓↓ ... ↓↓↑↑〉 will be (1

2, 2). It is also onvenient to

de�ne p ≡ srthat for simpli ity we restri t to be a positive integer (i.e. s is a multiple

of r). Although the lass of initial state onsidered is not omprehensive, it turns out

to be enough for the up oming dis ussion.

6.4.2 Evolution of the orrelation fun tions

Light- one e�e ts an be studied, monitoring the onne ted orrelation fun tion of

the z- omponent of the spin Sz = σz/2,

∆ln(t) = 〈Szl (t)S

zn(t)〉 − 〈Sz

l (t)〉〈Szn(t)〉. (6.40)

For our initial states, (6.40) is zero at time t = 0; however, a ording to [174℄, after

a ertain time whi h depends on |l − n|, it starts to hange signi� antly.

150

For instan e [195℄, su h a time an be hosen the �rst in�e tion point τ . Varying

|l− n| in Eq. (6.40), one an numeri ally evaluate τ and provide a predi tion for the

speed vmax

at whi h information spreads in the system determining the ratio

|l−n|2τ

. We

will all vmax

the light- one velo ity. A ording to [189, 190, 191, 211, 212℄, after the

quen h and in absen e of intera tions pairs of entangled quasi-parti le with opposite

momenta move ballisti ally with a group velo ity �xed by their dispersion relation.

Within this framework, one should expe t τ to be state-independent and vmax = vg

where vg > 0 is the maximum over the k's of Eq. (6.39). We will a tually show that

vg is rather an upper bound for the observed vmax

whi h an indeed be dependent on

the symmetries of the initial state. Finally observe that we do not expe t, in absen e

of intera tions, the light- one velo ity to be dependent on �nite size e�e ts as long as

L≫ |l − n|.

To study the time-evolution of the orrelation fun tion (6.40), �rst, we need to

analyze the propagators

Fln(t) = 〈ψ0|c†l (t)c†n(t)|ψ0〉, (6.41)

Cln(t) = 〈ψ0|cl(t)c†n(t)|ψ0〉, (6.42)

where |ψ0〉 is a state of the type introdu ed in Se . 6.4.1. If there is no ambiguity,

we will drop it from the expe tation values. From Eqs. (6.41)-(6.42) and the Wi k

theorem, whi h applies to our states [224℄, it follows

∆ln(t) = |Fln(t)|2 − |Cln(t)|2. (6.43)

Note that as L×L matri es whose matrix elements are given in Eqs. (6.41)-(6.42),

F and C satisfy F

T = −F and C

† = C.

151

Generi Quadrati Hamiltonian

T11, T12, T21 and T22 are omplex matri es.

T21 = (T12)∗

T22 = (T11)∗

(T11)T = T11

(T22)T = T22

T22T12 +T21T22 = 0

Table 6.1: Properties of the four L × L blo ks of the matrix T for a quadrati

Hamiltonian (6.37). The notation is obvious and time dependen e is omitted here.

It is straightforward to show that for a fermioni model with a quadrati Hamil-

tonian as in Eq. (6.37) one has,

(t)

†(t)

= e

−it

A B

-B -A

︸︷︷︸

T (t)

(0)

†(0)

(6.44)

where = {c1, c2, ..., cL} and

† = {c†1, c†2, ..., c†L} are ve tors of length L; i.e. T is a

2L × 2L matrix. Exploiting the properties of the four L × L blo ks of the matrix T

olle ted in Tab. ??, Eq. (6.44) an be written as,

(t)

†(t)

=

T

∗22(t) T12(t)

T

∗12(t) T22(t)

(0)

†(0)

. (6.45)

Finally, after some easy algebra, expli it expression for the time-evolved matri es

F and C an be omputed and read respe tively (time dependen e is dropped from

the T 's)

F(t) =T∗12F

†(0)T†12 +T

∗12C(0)T22 +T22T

†12

−T22CT (0)T†

12 +T22F(0)T22, (6.46)

152

C(t) =T∗22F

†(0)T†12 +T

∗22C(0)T22 +T12T

†12

−T12CT (0)T†

12 +T12F(0)T22. (6.47)

Eqs. (6.46)-(6.47) are valid in prin iple for any free fermioni systems with Hamilto-

nian (6.37); however, they have mu h simpler forms in the XY hain for our parti ular

hoi e of initial states as we now dis uss.

Evolution of the orrelation fun tions in the XY hain

In the periodi XY hain, it turns out [A,B] = 0; these matri es are indeed trivially

diagonalized by the unitary transformation with elements Ulk = 1√Le−ilφk

and φk

given, for N = 1, below Eq. (6.39). Consequently, the four blo ks of T are mutually

ommuting and this leads to further simpli� ations. In parti ular, it is easy to verify

that

T11 = cos[t√

A

2 −B

2]− iA√

A

2 −B

2sin[t

√

A

2 −B

2], (6.48)

T12 =−iB

√

A

2 −B

2sin[t

√

A

2 −B

2]. (6.49)

The other two blo ks an be found observing that T22 = T

∗11 and T21 = −T12. The

eigenvalues of the matri es A and B are

λAk = 2J(−h + cosφk), (6.50)

λBk = −2iJγ sin φk. (6.51)

From Eqs. (6.50)-(6.51) and omparing with Eq. (6.38), it follows εk =√

(λAk )2 − (λBk )

2.

Re alling then Eqs.(6.48)-(6.49) we �nally obtain

λT11

k = [λT22

k ]∗ = cos(tεk)− iλAkǫk

sin(tεk) (6.52)

λT12k = −λT21

k = −iλBk

εksin(tεk). (6.53)

153

The time-evolved matri es F(t) and C(t) an now be al ulated for the lass of repre-

sentative initial states |ψ0〉 introdu ed in Se . 6.4.1. A trivial ase is |ψ0〉 = | ↓↓ . . . ↓↓〉,

i.e. a state without fermions; here F(0) = 0 and C(0) = 1. Then using unitarity of

the matrix T we obtain

[F(t)]ln =1

L

L∑

k=1

λT22k λT12

k e−iφk(l−n)(6.54)

[C(t)]ln = δln −1

L

L∑

k=1

(λT12

k )2e−iφk(l−n). (6.55)

A ording to Eqs. (6.52)-(6.55), the time evolution of ∆ln(t) in Eq. (6.40) is des ribed

by the ballisti spreading of quasi-parti les with dispersion relation εk as in (6.38).

As it an be also easily he ked numeri ally, orrelations spread at the maximum

group velo ity vg obtained from (6.39), in agreement with a standard quasi-parti le

interpretation.

For the initial states labelled by (1/p, 1), the matrix C(0) has elements

[C(0)]ln = δln

[

1− 1

p

p−1∑

j=0

e−2πi ljp

]

, (6.56)

f whereas F(0) = 0. From the expressions in Eqs. (6.46)-(6.47) and inserting the

unitary matrix U that diagonalize simultaneously all four blo ks of T we derive

[F(t)]ln =1

L

L∑

k=1

λT ∗12

k λT22k e−iφk(l−n)

− 1

Lp

p−1∑

j=0

e−2πinj

p

L∑

k=1

λT ∗12

k λT22

−Ljp+ke−iφk(l−n)

+1

Lp

p−1∑

j=0

e−2πinj

p

L∑

k=1

λT22k λT12

−Ljp+ke−iφk(l−n), (6.57)

154

and

[C(t)]ln =1

L

L∑

k=1

|λT22

k |2e−iφk(l−n)

− 1

Lp

p−1∑

j=0

e−2πinj

p

L∑

k=1

λT ∗22

k λT22

−Ljp+ke−iφk(l−n)

+1

Lp

p−1∑

j=0

e−2πinj

p

L∑

k=1

λT12k λT12

−Ljp+ke−iφk(l−n). (6.58)

In the next subse tions, we will pass to study Eqs. (6.57)-(6.58) in details. Similar

al ulations an be also arried out for the initial states labelled by (s/p, s) where

C(0) has matrix elements

[C(0)]ln = δln

[

1− 1

p

p−1∑

j=0

s−1∑

l′=0

e−2πi (l+l′)jp

]

; (6.59)

we will also brie�y investigate su h a possibility.

Evolution of the orrelation fun tions in the XX hain

We onsider preliminary the simple ase of the XX hain (γ = h = 0 and J = −1)

where the fermion number is onserved and B (and F) vanishes. Then the matrix C

for the initial states labelled by (1/p, 1) an be rewritten as

Cln = δln −1

Lp

p−1∑

j=0

e−2πinj

p

L∑

k=1

e−i

[

φk(l−n)+

(

εk−ε−

Ljp +k

)

t

]

, (6.60)

where εk = −2 cosφk and φk = 2πkL

(k = 1, . . . , L). Noti e also that vg = 2. For the

sake of determining the light- one velo ity, the last exponential in Eq. (6.60) implies

that

εe�k,j(p) ≡1

2

(

εk − ε−Ljp+k

)

, (6.61)

an be interpreted as an e�e tive dispersion relation. The e�e tive dispersion origi-

nates from the dis rete translational symmetry of p latti e sites of the initial state

155

that allows quasi-parti les to be produ ed with momenta shifted by multiples of 2π/p.

An e�e tive group velo ity an be de�ned from Eq. (6.61) as

ve�k,j(p) = sin φk − sin

(

φk −2πj

p

)

; (6.62)

whose maximum value (over j and k) is then the light- one velo ity

vmax

= 2 when p is even, (6.63)

2 cos

(π

2p

)

when p is odd. (6.64)

Note that the e�e tive maximum group velo ity o urs when j = p2and j = p−1

2for

even and odd p, respe tively. In parti ular, from Eq. (6.63) follows that the a tual

light- one velo ity is state-dependent and annot be faster than the maximum group

velo ity vg. The predi tion in Eq. (6.63) is of ourse in agreement with a numeri al

estimation of vmax, obtained from the in�e tion points of the orrelations ∆ln; see

Fig. 6.12 for examples with states labelled by (1/2, 1), (1/3, 1) and (1/2, 2). It is

also interesting to observe that the absolute minimum visible in the se ond and third

panel in Fig. 6.12 is a �nite size e�e t, indeed the envelope of |∆ln(t)| is monotoni ally

de reasing in the limit L→ ∞ after rea hing the �rst maximum.

A similar analysis for the initial states (s/p, s), shows that

Cln = δln −1

Lp

p−1∑

j=0

Ajs e− 2πinj

p

L∑

k=1

e−i[φk(l−n)+2tεe�k,j (p)](6.65)

where Ajs =∑s−1

q=0 e− 2πijq

p. It is lear that as long as Ajs 6= 0 for the values of j

orresponding to the e�e tive maximum group velo ity, Eq. (6.63) remains valid.

However, one an verify that Ajs is a tually zero in some ases. For example, for the

state labelled by (1/2, 2), we have A22 = 0 and therefore the light- one velo ity is

obtained from the maximum over of Eq. (6.62) at j = 1 and p = 4; namely vmax =√2.

156

14 16 18 20 22 24 26t

-0.012

0

-0.006

0

∆ ln(t

)

-0.011

0

40 44 48 52 56 60|l-n|

10

12

14

16

18

20

22

τ

(1/2 , 1)

(1/3 , 1)

(1/2 , 2)

Figure 6.12: ∆ln(t) for the XX hain with di�erent initial states. The verti al lines

are indi ating analyti al values of the time T . Here we have L = 144 and |l−n| = 60.

Evolution of the orrelation fun tions for arbitrary values of the

parameters

The results of the previous subse tion an be extended to arbitrary values of the

parameters in the XY hain. Consider the on�gurations (1/p, 1). Ea h term in the

sum over j in Eqs. (6.57)-(6.58) is asso iated to a time propagation with an e�e tive

dispersion

ǫe�k,j,±(p) =1

2

(

εk ± εLjp−k

)

, j = 0, . . . , p− 1, (6.66)

where it is understood that for j = 0 we get ba k Eq. (6.38). Moreover, also the

�rst line in Eqs. (6.57)-(6.58) ontains a state-independent ontribution whose time

evolution expands over the usual dispersion. Therefore as long as B 6= 0, one should

expe t a state-independent light- one velo ity vmax = vg for all the fun tions (6.41)-

(6.42). However, interestingly for p = 2, the state-independent terms are an elled

exa tly by the j = 0 ontributions of the sums in Eqs. (6.57)-(6.58). The latter

157

observation follows from

λT ∗22

k λT22

−k + (λT12

k )2 = 1, (6.67)

that is a onsequen e of the unitarity of the matrix T . For arbitrary values of the

parameters γ and h, the e�e tive maximum group velo ity (i.e. the light- one velo ity)

an be then obtained as

vmax

= maxj 6=0,±,k

dεe�k,j 6=0,±(p)

dk. (6.68)

A tually if p = 2, only j = 1 is allowed in Eq. (6.68), however as we will dis uss,

in some ases, the same result applies to p > 2. In the Ising hain (γ = 1) vmax is

obtained sele ting the negative sign in Eq. (6.66), and φk as lose as possible to

π2.

The expli it value is

vmax =2Jh√1 + h2

. (6.69)

Similarly, in the regions of the parameters h = 1 and γ < γ∗, one �nds

vmax =2J

√

1 + γ2; (6.70)

where γ∗ is the solution of γ∗ = 1√1+(γ∗)2

. However, there is not a losed formula

for vmax; see the �rst and se ond panel in Fig. 6.17 for the numeri al estimations. In

Fig. 6.13, we plot ∆ln(t) for di�erent values of γ and h for quen hes from the initial

on�guration (1/2, 1). The light- one velo ities estimated from the in�e tion points

are in agreement with Eqs. (6.69)-(6.70) and more generally with Eq. (6.68).

For p > 2, the determination of vmax is more involved. Let us fo us on the the

riti al Ising hain. Sin e the �rst terms in Eqs. (6.57)-(6.58) are now not an elled

expli itly we should expe t that the C and F matrix elements will propagate with

velo ity vg. This is indeed orre t, see for instan e the red and blue urves in the two

panels of Fig. 6.14 with initial states p = 3 and p = 4. However unexpe tedly, when

158

0 2 4 6 8 10 12t

-0.007

0

0.007

-0.012

0

0.012

∆ ln(t

)

-0.012

0

0.012

15 20 25 30|l-n|

4

6

8

10

12

τ

( γ = 1 , h = 0.75 )

( γ = 0.5 , h = 1 )

( γ = 0.5 , h = 0.5 )

Figure 6.13: ∆ln(t) for the XY hain with various γ and h parameters with the

initial state (12, 1). The verti al lines are indi ating analyti al values of the time τ .

Here we have L = 96 and |l − n| = 12.

ombined into ∆ln, the two signals almost exa tly an el around the �rst maximum,

leaving a light- one velo ity slower than vg. The latter an be still al ulated as in

ase p = 2 from Eq. (6.68). See the green urve in Fig. 6.14 for an illustration. This

unexpe ted result holds also for any h ≤ 1 (ferromagneti phase) as we support

analyti ally in the Appendix 6.5.7. For h > 1 (paramagneti phase), Appendix 6.5.7

shows that su h a an ellation does not happen, therefore vmax = vg. This asymmetry

in the light- one velo ity between the two phases is hard to spot numeri ally, sin e

the state-independent term is of order 1/h2 and the di�eren e between Eq. (6.68) and

vg drops to zero fast as h in reases.

We studied the orrelation fun tion ∆ln for several di�erent values of the param-

eters γ and h, and for the initial states (1p, 1) with p > 2 we found numeri ally vmax

always to be given by Eq. (6.68) or vg. However a lear pattern does not emerge from

the numeri al analysis. For the on�guration (1/2, 2) with p = 4 again the terms that

159

are independent from the initial states an el out expli itly. For instan e in the Ising

hain the light- one velo ity is

vmax

=

√2Jh

√

1−√2h + h2

; (6.71)

a result that an be veri�ed numeri ally from the in�e tion points. Similar arguments

are also valid for all the on�gurations with (s/p, s).

10 12 14 16 18 20 22 24 26 28 30t

-0.006

-0.003

0

0.003

0.006| F

ln |

2

| Cln

|2

∆ln

( 1/3 , 1)

10 12 14 16 18 20 22 24 26 28 30t

-0.004

0

0.004

0.008| F

ln |

2

| Cln

|2

∆ln

( 1/4 , 1)

Figure 6.14: |Cln|2, |Fln|2, and ∆ln(t) for the XY hain with γ = 1 and h = 1. Theverti al lines in Maroon and Magneta, are indi ating analyti al values of the time τ al ulated from the maximum group velo ity and maximum e�e tive group velo ity,

respe tively. Here we have L = 144 and |l − n| = 60.

6.4.3 Airy s aling of orrelation fun tions at the edges of the light-

one

To understand analyti ally the behaviour of the fermioni propagators is more on-

venient to onsider the in�nite volume limit L→ ∞. We will now re- ast the results

in Se . 6.4.2 dire tly in Fourier spa e and obtain an approximation of orrelation

fun tions near the boundary of the light- one involving an Airy fun tion.

We onsider then fermioni operators ci and c†i with {c†i , cj} = δij , {ci, cj} =

{c†i , c†j} = 0 de�ned for any i, j ∈ Z. The onvention for the Fourier transforms are

cj =

∫ π

−π

dk√2π

eikjc(k), c(k) =1√2π

∑

j∈Ze−ikjcj, (6.72)

160

from whi h follows that {c†(k), c(k′)} = δ(k−k′) and {c(k), c(k′)} = {c†(k), c†(k′)} =

0. To be de�nite we only onsider initial on�gurations labelled by the pair (1/p, 1).

In XY hain, the time evolved operators c(−k, t) and c†(k, t) are linearly related (see

Eqs. (6.90)-(6.92)) to the orrespondent operators at time zero. In parti ular one has

c†(k, t)

c(−k, t)

=

T11(k, t) T12(k, t)

T21(k, t) T22(k, t)

c†(k)

c(−k)

(6.73)

where the matrix elements are

T11(k, t) = cos(ε(k)t) + i cos(θ(k)) sin(ε(k)t) (6.74)

T12(k, t) = sin(ε(k)t) sin(θ(k)). (6.75)

and T11(k, t) = [T ∗22(k, t)], T21(k, t) = −T12(k, t). From unitarity it follows more-

over |T11|2 + T 212 = 1 and T ∗

11T12 + T ∗21T22 = 0. The Bogolubov angle is θ(k)/2, see

Appendix 6.5.7 for expli it expressions. Fermioni orrelation fun tions are double

integrals in Fourier spa e. For instan e let us denote by fαa fermioni operator with

f+ ≡ c and f− ≡ c†, from the de�nition of the Fourier transform (6.72) we have

〈fαl (t)f

βn (t)〉 =

∫ ∫dkdk′

2πeiαkl+iβk′n〈fα(k, t)fβ(k′, t)〉, (6.76)

with integrals in the domain k, k′ ∈ [−π, π]. The time evolution of the matrix ele-

ment in (6.76) is obtained from Eq. (6.73) as a linear ombination of four matrix

elements of the same type at time t = 0. Among those four the only non-trivial on

the lass of initial states we are onsidering is g(k, k′) = 〈c†(k)c(k′)〉. Noti e indeed

that 〈c†(k)c†(k′)〉 = 〈c†(k)c†(k′)〉 = 0 and 〈c(k)c†(k′)〉 an be obtained from the anti-

ommutation relations. For our initial state (1p, 1) the fun tion g is given by

g(k, k′) =1

2π

∑

n∈Zeinp(k−k′) =

1

p

p−1∑

j=0

δ

(

k − k′ − 2πj

p

)

. (6.77)

161

It is then straightforward to derive integral representations for the orrelators Fln(t)

and Cln(t) that we write as

Fln(t) =

p−1∑

j=0

F jln(t) and Cln(t) =

p−1∑

j=0

Cjln(t). (6.78)

Expli it expressions are in Eqs. (6.93)-(6.96). Ea h integral F jln(t) and C

jln(t) des ribes

a time propagation with a velo ity that an be derived from the e�e tive dispersion

relation

εe�j,±(k, p) =1

2

[

ε(k)± ε

(

−k + 2πj

p

)]

, j = 0, . . . , p− 1, (6.79)

that is Eq. (6.66) in the limit L → ∞. As in Se . 6.4.2, the predi ition of the light-

one velo ity follows from al ulating the maximum e�e tive group velo ities obtained

from Eq. (6.79). In parti ular, it an be easily veri�ed (see Eqs. (6.93) and (6.95) in

parti ular) that for p = 2, the maximum e�e tive group velo ity is always smaller

than vg obtained from Eq. (6.39).

As shown in Fig. 6.14, the numeri s indi ates that a an ellation of the fastest

j = 0 ontributions in Eq. (6.79) appears also for p 6= 2 in the riti al Ising hain.

This observation extends to the whole ferromagneti phase h ≤ 1. It an be under-

stood analyti ally omparing the behaviours near their in�e tion points of |F 0ln(t)| and

|C0ln(t)| in (6.78) and showing that they are the same. At the edge of the light- one

those fun tions an be approximated by an Airy fun tion with in reasing a ura y as

t→ ∞. The te hni al details are presented in Appendix 6.5.7: we refer to Eqs. (6.106)-

(6.107) for the ferromagneti phase h < 1; Eqs. (6.109)-(6.110) for the paramagneti

phase h > 1 and Eq. (6.113) for the quantum riti al point h = 1.

Here, we demonstrate the validity of the Airy-approximation of free-fermioni

orrelation fun tions, through the neatest example of a quen h from the Néel state

(p = 2) in the riti al Ising hain with J = 1/2. In this ase the integrals F 0ln(t) and

C0ln(t) vanish for any time. For t → ∞ and x/t �nite, being x ≡ l − n > 0, using the

162

te hniques des ribed in the Appendix 6.5.7 we obtain

(4t)2/3∆ln(t) ≃ −Ai2(−X), t≫ 1. (6.80)

In (6.80), X = 2vmax

t−x[−te′′′(kmax)]1/3

∈ R, kmax = π/2 and a ording to (6.79)

e(k) ≡ εe�1,−(k, 2) =1

2[ε(k)− ε(−k + π)] (6.81)

vmax

= maxk∈[−π,π]

de(k)

dk=

1√2. (6.82)

The value of vmax is of ourse the same as in Eq. (6.68) taking J = 1/2. A omparison

of the stationary phase approximation in Eq. (6.80) with a numeri al evaluation of

the orrelation fun tion ∆ln(t) is presented in Fig. ??. It is �nally worth to remark

that for X = 0 (i.e. t = |l−n|2v

max

), Eq. (6.80) is within the 10% from the exa t value

already for t = 50. Noti e also that Ai

′′(0) = 0 whi h explains why for large times

the in�e tion point of the signal is lose to

|l−n|2v

max

.

Figure 6.15: The ontinuous blue urve is the fun tion −Ai2(−X). The points rep-resent numeri al evaluationof (4t)2/3∆ln(t) for a quen h from the Neel state in the

riti al Ising hain. Here |l − n| = 2ve�maxt− [−te′′′(kmax)]1/3X and t = 500

163

6.4.4 Evolution of the entanglement entropy

In this �nal se tion before the on lusions, we present numeri al results for the time

evolution of the entanglement entropy of a subsystem of size l for the di�erent initial

states dis ussed in Se . 6.4.1. Based on a quasi-parti le pi ture [211, 212℄, we expe t

the entanglement entropy, denoted by S(t), to grow linearly in time up to τs whi h is

approximately

l2vg

. After τs, whi h we will all the saturation time, the entanglement

entropy onverges [213℄ to the von Neumann entropy of the stationary state [217℄. In

the numeri s τs is obtained as the earliest time where the se ond derivative of the

signal hanges.

One an al ulate the entanglement entropy from the orrelation fun tions as

follows [263℄

S = −Tr

[1 + Γ

2ln

(1 + Γ

2

)]

, (6.83)

where Γ is a 2l × 2l blo k matrix whi h an be written as

Γmn =

⟨(axmaym

)

(axn ayn)

⟩

− δmn12×2

=

〈axmaxn〉 − δmn 〈axmayn〉

〈aymaxn〉〈aymayn〉 − δmn

. (6.84)

Here axm = c†m + cm and aym = i(cm − c†m) and the indexes m,n belong to the one-

dimensional subsystem of size l. One an easily �nd all the di�erent elements of the

matrix Γ as,

Γ11 = F + F † +C −CT , (6.85)

Γ12 = i(1−C −CT − F + F †), (6.86)

Γ21 = −i(1−C −CT + F − F †), (6.87)

Γ22 = −F − F † +C −CT . (6.88)

164

At this stage we have all the ingredients to build the matrix Γ and onsequently to

al ulate the entanglement entropy from Eq. (6.83). In fa t, one an al ulate the

entanglement from

S = −l∑

j=1

[1 + νj

2ln

(1 + νj

2

)

+1− νj

2ln

(1− νj

2

)]

, (6.89)

where νj's are the positive eigenvalues of the matrix Γ. It is tempting to onje ture

that the saturation time ould be given by

l2vmax

, with vmax is the light- one velo ity

extra ted from the orrelation fun tions, dis ussed in Se . 6.4.2. It is indeed evident

from the numeri s that, as long as p > 1 and γ 6= 0, vg does not ne essarily play a

role into the time evolution of the entanglement entropy. See in parti ular Fig. 6.16

and ompare the blue urve, orresponding to S(t) for p = 1 with the green that

refers instead to p = 3; here γ = 2 and h = 1.5. It is lear that starting from an

initial state with p 6= 1, the saturation time an in rease. Noti e also that for p = 1,

vmax = vg whereas for p = 3, vmax is obtained from Eq. (6.68) that is already non-

trivial. However, one should be areful in drawing on lusions based on the numeri s.

Consider for instan e the ase p = 2 with γ = 2 and h = 1.5. A ording to the

analysis in Se . 6.4.2, the light- one velo ity vmax is given by Eq. (6.68), sin e the

fastest terms travelling with velo ity vg are zero. However the entanglement entropy

displays a saturation time larger than

l2vmax

as it an be learly seen in Fig. 6.16 (red

urve). At present we do not have an analyti al understanding of this e�e t.

6.4.5 Con lusions

In this paper we have analyzed the in�uen e of the initial state on the maximum

speed at whi h orrelations an propagate, a ording to the Lieb-Robinson bound.

We investigated the XY hain and global quen hes from a lass of initial states that

are fa torized in the lo al z- omponent of the spin and have a rystalline stru ture.

165

0 10 20 30 40 50 60t

0

10

20

30

40

50

60

70

S(t)

( 0 , 1 )( 1/2 , 1 )( 1/3 , 1 )

Figure 6.16: ( olor online) The evolution of the entanglement entropy in the XY

hain with J = 1 and system size L = 600 and l = 72. Entanglement entropy as a

fun tion of time at γ = 2 and h = 1.5. The verti al dashed lines are in orresponden eof t = l

2vmax, being vmax the light- one velo ity for the orrelation fun tions obtained

in Se . 6.4.2. Comparing the blue (p = 1) and green (p = 3) urves is evident thee�e t of the initial state on the saturation times. The red urve orresponds to p = 2and shows instead that in su h a ase the saturation time is learly larger than

l2vmax

.

We demonstrated expli itly that momentum onservation in the rystal leads to a

state-dependent light- one velo ity vmax that rules how fast orrelations spread. We

have given, and he ked numeri ally, analyti al predi tions for the light- one velo ities

for several values of the parameters γ, h and p. We also dis ussed an approximation

of fermioni orrelations fun tions in in�nite volume that shows, in agreement with

previous results in [192℄, that the behaviour at the light- one edge an be hara terized

by integer powers of t−1/3. In parti ular this is the ase when the light- one velo ity

is a maximum with vanishing se ond derivative (and non-zero third derivative) of the

e�e tive dispersion. The degree of universality of the t−1/3-s aling and in parti ular

its dependen e on the initial state, however, have been not lari�ed yet [228, 229, 230,

231, 232℄.

166

We have then studied numeri ally the evolution of the entanglement entropy and

showed that the hoi e of the initial state a�e ts also the saturation time. When om-

pleting this paper, a preprint[233℄ appeared that analyzes entanglement dynami s in

the XX hain (γ = 0) for the lass of initial states here labelled as (1/p, 1). In parti -

ular an interesting semi lassi al interpretation in terms of entangled p-plets of quasi-

parti les is proposed. Our al ulations in Se . 6.4.2 for the light- one velo ity in the

XX hain are in agreement with su h a quasi-parti le pi ture. It will be important to

investigate how this an be adapted to determine the light- one velo ity vmax and the

linear growth of the entanglement entropy also for γ 6= 0. Our analysis suggests that

these observables are not easily predi table on the whole parameter spa e, therefore

a generalized quasi-parti le pi ture will be likely initial state dependent. Finally, it

will be relevant to study the e�e t of the initial state on �nite size e�e ts.

6.4.6 Additional details

Bogolubov rotation. The Bogolubov rotation in the XY hain is

b†(k)

b(−k)

=

R(k)︷︸︸︷

cos(θ(k)/2) −i sin(θ(k)/2)

−i sin(θ(k)/2) cos(θ(k)/2)

c†(k)

c(−k)

(6.90)

with cos θ(k) = 2J(cos(k)−h/J)ε(k)

, sin θ(k) = 2Jγ sin(k)ε(k)

and ε(k) = 2J√

(cos(k)− h/J)2 + γ2 sin2(k).

The Bogolubov fermions b(k) and b†(−k) have simple time evolution

b†(k, t)

b(−k, t)

=

U(k,t)︷︸︸︷

eiε(k)t 0

0 e−iε(k)t

b†(k)

b(−k),

(6.91)

and therefore we obtain Eq. (6.73) of Se . 6.4.3 as

c†(k, t)

c(−k, t)

=

T (k,t)︷︸︸︷

R†(k)U(k, t)R(k)

c†(k)

c(−k)

. (6.92)

167

Fermioni orrelators. Expli it expressions for the fun tions F jln(t) and C

jln(t) are

F 0ln(t) =

∫dk

2πe−ik(l−n)

[

T12(k, t)T11(−k, t)+

1

p(T11(k, t)T12(−k, t)− T12(k, t)T11(−k, t))

]

, (6.93)

F j 6=0ln (t) =

e−2πinj

p

p

∫dk

2πe−ik(l−n)

[

T11(k, t)×

×T12(

−k + 2πj

p, t)

− T12(k, t)T11

(

−k + 2πj

p, t)]

; (6.94)

and analogously

C0ln(t) =

∫dk

2πeik(l−n)

[

T22(−k, t)T11(k, t)(

1− 1

p

)

+

1

pT12(k, t)

2

]

, (6.95)

Cj 6=0ln (t) =

e−2πinj

p

p

∫dk

2πeik(l−n)

[

T21(−k, t)×

×T12(

k +2πj

p, t)

−T22(−k, t)T11(

k +2πj

p, t)]

. (6.96)

Noti e that, onsistently with the dis ussion in Se . 6.4.2, the integrals in Eqs. (6.93)

and (6.95) are vanishing for p = 2.

Behaviour at the in lination point. Consider an integral of the form

I(x, t) =

∫ π

−π

dk

2πH(k)e2itε(k)−ikx

(6.97)

where we assume x > 0 and H(k) a fun tion with support k ∈ [−π, π]. As it is

dis ussed for instan e in [226, 227℄, for large x, t with their ratio �xed we an approx-

imate (6.97) by an Airy fun tion. At the boundary of the light- one, the solution kmax

of the stationary phase equation

ε′(kmax) =x

2t, (6.98)

168

satis�es by de�nition ε′′(k

max

) = 0. A remarkable ex eption in presen e of intera tions

is ontained in [231℄. At k = kmax, ε′(k) has a maximum if t > 0 and a minimum if

t < 0. Therefore in a Taylor expansion near k = kmax of the phase in (6.97) we need to

keep the third order term. If H(kmax) 6= 0 we then obtain the following approximation

of the integral I(x, t) near the boundary of the light- one

I(x, t) ≃ e2itε(kmax)−ikmaxxH(kmax)

[−tε′′′(kmax)]1/3Ai(−X), (6.99)

being X = 2ε′(kmax)t−x

[−tε′′′(kmax)]1/3. It should be noti ed that

1. Eq. (6.99) is determined in the limit x, t → ∞ but it gives a fairly good approx-

imation of the integral as long as

|2ε′(kmax)t− x| ≪ [−tε′′′(kmax)]1/3. (6.100)

2. The in lination point onsidered in the main text orresponds to X = 0. It

follows that I(x, t) an be approximated at the in lination point t = τ as

I(x(τ), τ) ≃ e2iτε(kmax)−ikmaxx(τ) H(kmax)

[−τε′′′(kmax)]1/3Ai(0), (6.101)

with x(τ) = 2τε′(kmax) as in (6.98).

Example: Ising hain (J = 1/2; γ = 1). The fun tions F 0ln(t) and C0

x(t) an be

written as

F 0ln(t) = H0 +

∑

σ=±

∫ π

−π

dk

2πHσ(k)e

2iσε(k)t−ik(l−n), (6.102)

C0ln(t) = K0 +

∑

σ=±

∫ π

−π

dk

2πKσ(k)e

2iσε(k)t−ik(l−n); (6.103)

where H± and K± are obtained expanding the integrands in (6.93) and (6.95) and

we also used C0ln(t) = C0

nl(t). The onstant terms H0 and K0 an be also determined

169

expli itly from the residue theorem and they are given by

H0 = K0 = −hx−2(h2 − 1)(p− 2)

8p, h ≤ 1; (6.104)

H0 = −K0 = −h−x+2(h2 − 1)(p− 2)

8p, h > 1, (6.105)

where we de�ned x ≡ (l − n) ≥ 2. They are then vanishing at h = 1 or are exponen-

tially small with the distan e x when h 6= 1. Sin e we will onsider only large values

of the in lination point, it turns out that x ≫ 1 also (see below Eq. (6.101)).

We fo us �rst on the ase h < 1. Here we have kmax = arccos(h), ε′(kmax) ≡ vg =

h, ε′′′(kmax) = −h; using (6.99) we obtain for x → ∞ the approximations at the

in lination point τ

F 0ln(τ) ≃

2H+(kmax) cos[x(τ)φ(h)]

[x(τ)/2]1/3Ai(0) (6.106)

C0ln(τ) ≃

2K+(kmax) cos[x(τ)φ(h)]

[x(τ)/2]1/3Ai(0). (6.107)

where φ(h) =√

1/h2 − 1− arccos(h) and x(τ) is de�ned below (6.101). It turns out

that H+(kmax) = H−(−kmax) = −iK+(kmax) = −iK−(−kmax) and

H+(kmax) = −ip− 2

4p, (6.108)

Therefore at the in lination point |F 0ln|2 and |C0

ln|2 have the same approximation in

terms of an Airy fun tion. Although the argument applies for large t, we believe

that it justi�es the di� ulty in the numeri s to spot the the fastest group velo ity

ontribution when h < 1.

We now pass to dis uss the ase h > 1. Here we have kmax = arccos(1/h),

ε′(kmax) ≡ vg = 1, ε′′′(kmax) = −1. A similar expansion gives the approximations

F 0ln(τ) ≃

eix(τ)φ(h−1)H+(kmax) + e−ix(τ)φ(h−1)H−(kmax)

[x(τ)/2]1/3Ai(0) (6.109)

C0ln(τ) ≃

2K+(kmax) cos[x(τ)φ(h−1)]

[x(τ)/2]1/3Ai(0), (6.110)

170

where now K+(kmax) = K−(kmax) and

H±(±kmax) = −i(h∓√h2 − 1)(p− 2)

4h2p(6.111)

K+(kmax) =p− 2

4h2p. (6.112)

There is no more a an ellation of the two terms as for h < 1. The absen e of a

peak at τ = l−n2

(l ≫ n) in the observable ∆ln(t) might be however understood as

a ombination of two e�e ts. For large h ≫ 1, the values at the in lination point of

|F 0ln|2 and |C0

ln|2 are suppressed by a fa tor 1/h2 and 1/h4 respe tively. On the other

hand, as h→ 1, an exa t an ellation between |F 0ln| and |C0

ln| must happen.

For h = 1, the two distin t stationary points ±kmax a tually merge at k = 0 and

the asymptoti expansion involve only one term. One has vg = 1 and ε′′′(0+) = −1/4,

leading to the �nal result

F 0ln(τ) ≃ −i p− 2

2p[x(τ)]1/3Ai(0), C0

ln(τ) = −iF 0ln(τ), (6.113)

that ones again shows the an ellation of the fastest parti le ontribution for arbitrary

values of p. Analogous approximations near the orrespondent in lination points ould

be al ulated for F jln(t) and Cj

ln(t) with the same te hnique. In the main text we

estimate, for instan e, ∆ln(t) for p = 2 and h = 1.

171

Figure 6.17: Up(left)ve�max(2) and ve�

max(1) for di�erent values of a and h. Up(right)d = ve�max(1)− ve�max(2) for di�erent values of a and h. Down(left)ve�max(3) and v

e�

max(1)for di�erent values of a and h. Down(right) d = veffmax(1)− veffmax(3) for di�erent valuesof a and h..

172

6.5 Formation probabilities and Shannon information and their time

evolution after quantum quen h in the transverse-field XY

hain

Studying orrelation fun tions in many-body systems has been onsidered one of the

main topi s in statisti al me hani s and ondensed matter physi s for many years.

Although, for a long-time the main quantities of interest were the orrelation fun tions

of lo al observables the re ent interest in al ulating non-lo al quantities, espe ially

the entanglement entropy, has made signi� ant hanges. One of the main reasons for

this interest (at least in 1 + 1 dimensional riti al systems) is that by al ulating

some of the non-lo al observables one an derive the entral harge of the system

without referring to the velo ity of sound, for the ase of entanglement entropy see

[234℄. Another non-lo al quantity whi h has been studied for many years with Bethe

ansatz te hniques and some other methods is emptiness formation probability [145,

146, 147, 235, 236, 237, 238℄. In the ase of spin hains, it is the probability of �nding

a sequen e of up spins in the system (note that almost all of the studies in this

regard on entrate on the σzbases). These studies show that this probability, with

respe t to the sequen e size, de reases like a Gaussian in the ase of systems with

U(1) symmetry [236℄ and exponentially in other ases [145, 146, 238℄. In the riti al

ases, the Gaussian and exponential are a ompanied with a power-law de ay with a

universal exponent. In a re ent development [147℄ it was shown that for those riti al

systems without U(1) symmetry this universal exponent is dependent on the entral

harge of the system. The argument is based on onne ting the on�guration of all

spins up to some sort of boundary onformal �eld theory. One should noti e that the

This hapter is reprinted from K Naja�, MA Rajabpour, Formation probabilities and

Shannon information and their time evolution after quantum quen h in the transverse-�eld

XY hain, Physi al Review B 93, 12:125139, Copyright(2013)

173

argument works just for those bases that an be onne ted to boundary onformal

�eld theory.

In apparently onne ted studies re ently many authors investigated the Shannon

information of quantum systems in di�erent systems [239, 240, 241, 242, 243, 244,

245, 246, 247, 248, 249℄. The Shannon information is de�ned as follows: Consider the

normalized ground state eigenfun tion of a quantum spin hain Hamiltonian |ψG〉 =∑

I aI |I〉, expressed in a parti ular lo al bases |I〉 = |i1, i2, · · ·〉, where i1, i2, · · · are

the eigenvalues of some lo al operators de�ned on the latti e sites. The Shannon

entropy is de�ned as

Sh = −∑

I

pI ln pI , (6.114)

where pI = |aI |2 is the probability of �nding the system in the parti ular on�guration

given by |I〉. As it is quite lear this quantity is bases dependent and to al ulate it

one needs in prin iple to know the probability of o urren e of all the on�gurations.

The number of all on�gurations in reases exponentially with the size of the system

and that makes analyti al and numeri al al ulations of this quantity quite di� ult.

Note that the emptiness formation probability is just one of the whole possible on-

�gurations. The Shannon information of the system hanges like a volume law with

respe t to the size of the system so in prin iple one does not expe t to extra t any

universal information by studying the leading term. The universal quantities should

ome from the subleading terms. To study subleading terms, it is useful to de�ne yet

another quantity alled Shannon mutual information. By onsidering lo al bases it is

always possible to de ompose the on�gurations as a ombination of the on�gura-

tions inside and outside of a subregion A as |I〉 = |IAIA〉. Then one an de�ne the

marginal probabilities as pIA =∑

IApIAIA

and pIA =∑

IApIAIA

for the subregion A

174

and its omplement A. Then the Shannon mutual information is

I(A, A) = Sh(A) + Sh(A)− Sh(A ∪ A), (6.115)

where Sh(A) and Sh(A) are the Shannon information of the subregions A and A.

From now on instead of using pIAIAwe will use just pI . Sin e in the above quantity the

volume part of the Shannon entropy disappears Shannon mutual information provides

a useful te hnique to study the subleading terms. Note that one an similarly de�ne

the above quantity also for two regions A and B, i.e. I(A,B) that are not ne essarily

omplement of ea h other. The Shannon mutual information has been studied in

many lassi al [250, 251, 252, 253℄ and quantum systems [254, 255, 256, 257, 258℄.

Numeri al studies on variety of di�erent periodi quantum riti al spin hains show

that for parti ular bases (so alled onformal bases) we have [255, 257, 258℄

I(A, A) = β ln[L

πsin(

πl

L)], (6.116)

where L and l are the total size and subsystem size respe tively and β is very lose

to

c4, with c the entral harge of the system. Note that if one takes an arbitrary base

(non- onformal bases) the oe� ient β is nothing to do with the entral harge. It is

worth mentioning that based on [257℄ the onformal bases are those bases that an

be onne ted to some sort of boundary onformal �eld theory in the sense of [147℄.

For example in the transverse �eld Ising model the σxand σz

bases are the onformal

bases. Note that if one onsiders the Shannon information of the subsystem we will

have Sh(A) = αl + β ln[Lπsin(πl

L)]. Sin e to extra t the Shannon information one

needs to use all the probabilities the only way to onsider all of them in the numeri al

al ulations is exa t diagonalization. This makes the numeri al al ulation for large

sizes very di� ult. The results of the arti les [255, 257, 258℄ are all for periodi systems

with L = 30 whenever there are spin one-half system and smaller sizes for systems

175

with bigger spins. In a re ent work [256℄ the author was able to onsider an in�nite

system and study the Shannon information of the subsystem up to the size l = 40.

It was on luded that for the XX hain β is

c4with c = 1 but for the Ising model

although it is very lose to

c4with c = 1

2the results are suggesting that probably β is

not exa tly onne ted to the entral harge. Noti e that all of the above al ulations

are done by onsidering periodi boundary onditions for the onne ted regions A

and A. It is not lear how the equation (6.116) might hange if one onsiders open

boundary onditions. Finally it is worth mentioning that some of the the above results

have re ently been extended to dis onne ted regions in [259℄.

Motivated by the studies of emptiness formation probability and Shannon infor-

mation of the subsystem we study here some related quantities. First of all, as it is

natural one might be interested in studying the s aling limit of some other on�gu-

rations with respe t to the size of the subsystem. For example, onsider an antiferro-

magneti on�guration in the Ising model or any other on�guration with a pattern.

It is very important to know that these on�gurations are also �owing to some sort

of boundary onformal �eld theory or not. This study will learly also help to under-

stand the nature of the Shannon mutual information. In addition, this kind of study

also is very useful in the al ulations of post-measurement entanglement entropy and

lo alizable entanglement entropy [260, 261℄. Having the above motivations in mind,

we study formation probabilities , Shannon information and their evolution after a

quantum quen h in the quantum XY hain in the σzbases.

The outline of the paper is as follows: In the next se tion we will �rst de�ne our

system of interest, i.e. XY hain and then we will provide a method to al ulate the

probability of any on�guration in the free fermioni systems. In se tion three we

will list all the known analyti al results regarding emptiness formation probability

for in�nite systems and also �nite systems with periodi and open boundary ondi-

176

tions. Expli it distinguishment is made between riti al Ising model and XX hain

with U(1) symmetry. In se tion IV, we will de�ne many di�erent on�gurations with

spe ially de�ned pattern and al ulate their orresponding probabilities numeri ally.

Here again, we dis uss Ising and XX universalities separately for in�nite systems and

for the systems with boundary. We also dis uss on�gurations that do not have any

pattern. In se tion V, we study the Shannon information in the transverse �eld riti al

Ising model and also riti al XX hain. We will lassify the on�gurations based on

their magnetization and show that in prin iple just a small part of the on�gurations

an have a �nite ontribution to Shannon information in the s aling limit. Se tion

VI is devoted to the evolution of formation probabilities and Shannon information

after a quantum quen h. We prepare the system in a parti ular state and then we

let it evolve with another Hamiltonian and study the time evolution of the formation

probabilities and espe ially Shannon information. Finally, the last se tion is about

our on lusions and possible future works.

6.5.1 Formation probabilities from redu ed density matrix

In this se tion, we �rst de�ne the system of interest and after that using the redu ed

density matrix of this system we will �nd a very e� ient method to al ulate formation

probabilities for systems that an be mapped to free fermions. The Hamiltonian of

the XY- hain is as follows:

H = −L∑

j=1

[

(1 + a

2)σx

j σxj+1 + (

1− a

2)σy

j σyj+1 + hσz

j

]

. (6.117)

After using Jordan-Wigner transformation, i.e. cj =∏

m<jσzm

σxj −iσy

j

2and N =

∏L

m=1σzm = ±1 with c†L+1 = 0 and c†L+1 = N c†1 for open and periodi boundary

177

onditions respe tively the Hamiltonian will have the following form:

H =L∑

j=1

[

(c†jcj+1 + ac†jc†j+1 + h.c.)− h(2c†jcj − 1)

]

+N (c†Lc1 + ac†Lc†1 + h.c.). (6.118)

The above Hamiltonian has a very ri h phase spa e with di�erent riti al regions [262℄.

In �gure 1 we show di�erent riti al regions of the system. To al ulate probability of

formations for di�erent patterns we �rst write the redu ed density matrix of a blo k

of spins D by using blo k Green matri es. Following [263, 264℄, we �rst de�ne the

operators

ai = c†i + ci, bi = c†i − ci. (6.119)

The blo k Green matrix is de�ned as

Gij = tr[ρDbiaj ]. (6.120)

The elements of the Green matrix an be al ulated following [265℄ and we will men-

tion their expli it form for open and periodi boundary onditions later.

To al ulate the redu ed density matrix after partial measurement we need to �rst

de�ne fermioni oherent states. They an be de�ned as follows

|ξ >= |ξ1, ξ2, ..., ξN >= e−∑N

i=1 ξic†i |0 >, (6.121)

where ξi's are Grassmann numbers following the properties: ξnξm + ξmξn = 0 and

ξ2n = ξ2m = 0. Then it is easy to show that

ci|ξ >= −ξi|ξ > . (6.122)

Using the oherent states (6.121) the redu ed density matrix has the following form

[264℄

ρD(ξ, ξ′) = < ξ|ρD|ξ′ >

= det1

2(1−G)e

12(ξ∗−ξ′)TF (ξ∗+ξ′), (6.123)

178

where F = (G+1)(1−G)−1. One an use the above formula to extra t the formation

probability for arbitrary on�guration in the σz bases as follows: �rst of all to extra t

the probability of parti ular on�guration we need to look to the diagonal elements

of ρD(ξ, ξ′). When the spin in the σz

dire tion is up in the fermioni representation

it an be understood as the la k of a fermion in that site whi h in the language of

oherent states means that the orresponding ξ is zero in the equation (6.123). After

putting some of the ξ's equal to zero one will have a new redu ed density matrix in the

oherent state basis with this onstraint that some of the spins are �xed to be up. In

other words in the equation (6.123) instead of F we will have F whi h is a sub-matrix

of the matrix F . The elements of the new redu ed density matrix will be ρD(η,η′),

where we put the ξ's orresponding to the sites �lled with fermions equal to η. To

extra t the probability of formation one just needs to integrate over all the η's that

orrespond to the down spins. In other words after using formulas of the Grassmann

Gaussian integrals the formation probability will have the following formula

P (Cn) = det[1

2(1−G)]MCn

F , (6.124)

whereMCnF is the minor of the matrix F orresponding to the on�guration Cn. Noti e

that we just need prin ipal minors of the matrix F . Sin e the sum of all the prin ipal

minors of the matrix F is equal to det(1 + F ) the normalization is ensured. We have

(lk

)number of rank-k minors for matrix F with size l. Summing over the number of all

prin ipal minors, one an obtain 2l whi h is the number of all possible on�gurations.

Con�gurations with the same minor rank have the same number of up spins and

by in reasing k we a tually in rease the number of down spins in the orresponding

on�guration. For example k = 0(l) is the ase with all spins up (down) and sometimes

it is alled emptiness formation. The above formula gives a very e� ient way to

al ulate the formation probabilities in numeri al approa h. Sin e, as we will show, all

179

of these probabilities are exponentially small with respe t to the size of the subsystem

it is mu h easier to work with the logarithm of them and de�ne logarithmi formation

probabilities

Π(Cn) = − lnP (Cn). (6.125)

All of the al ulations done in this paper are based on the formula (6.124) and are

performed using Mathemati a. In the next se tions, we will study the formation

probabilities for di�erent on�gurations with the rystal order (pattern formation

probabilities) with respe t to the size of the subsystem for some parti ular riti al

regions of the system.

6.5.2 Emptiness formation probability: known results

Before presenting our results, we �rst review here the well-known fa ts regarding

emptiness formation probability (k = 0 and l). For reasons that will be lear in

the next se tion, we will all these two on�gurations x = 0 and 1 respe tively. Using

Fisher-Hartwig theorem the emptiness formation probability was already exhaustively

studied in Ref. [145, 146℄. The results were generalized to arbitrary onformal riti al

systems in [147℄. Due to the U(1) symmetry there is an important di�eren e between

the XX riti al line and the Ising riti al point. For this reason, we report the results

regarding these two possibilities separately.

Criti al Ising point

We list here the results regarding the logarithmi emptiness formation probability at

the riti al Ising point (a = h = 1).

180

Con�guration Cx=0, i.e. (| ↑, ↑, ... ↑>): This on�guration orresponds to k = 0 and

has the highest probability and using the equation (6.124) one an easily show that

P (x = 0) = det[1

2(1−G)] (6.126)

The above formula is valid independent of the boundary onditions and the size of

the system. For the in�nite system at the riti al Ising point the G matrix has the

following form:

Gij = − 1

π(i− j + 1/2). (6.127)

Sin e the above matrix is a Toeplitz matrix using Fisher-Hartwig onje ture in [? ℄

it was shown that the logarithmi probability for this on�guration hanges with the

subsystem size as follows:

Π(x = 0) = αl + β ln l + γln l

l+O(1), (6.128)

where β = 116

= 0.0625, and α and γ are some non-universal numbers. At the riti al

point of the Ising model these numbers are known α = ln 2 − 2C/π = 0.11002, with

C the Catalan onstant and γ = − 132π

= −0.00994. The ln llterm is the result of the

paper [147℄. Using onformal �eld theory te hniques in [147℄ it was argued that for

generi riti al systems the oe� ient of the logarithm should be β = c8, where c is

the entral harge of the riti al system. For the Ising universality lass c = 12.

When the size of the total system is �nite L depending on the form of the boundary

onditions, periodi or open; G has the following two forms

GPij = − 1

L sin(π(i−j+1/2)L

), (6.129)

GOij = − 1

2L+ 1

( 1

sin(π(i−j+1/2)2L+1

)+

1

sin(π(i+j+1/2)2L+1

)

)

.

(6.130)

181

Noti e that for L→ ∞ the �rst equation redu es to (6.127) and the se ond equation

will give the result for a semi-in�nite hain. The results for emptiness formation

probabilities for the above two ases are [147℄

ΠP (x = 0) = αl + β ln[L

πsin

πl

L] + γπ cot(

πl

L)ln l

L+O(1), (6.131)

ΠO(x = 0) = αl + βo ln[4L

π

tan2 πl2L

sin πlL

] + γoπ2− cos(πl

l)

sin πll

ln l

L+O(1), (6.132)

where β = c8and βo = − c

16. It is also onje tured that γ = − c

8πaand γo = c

32πa. The

above two equations are derived using boundary onformal �eld theory te hniques

and in prin iple they are valid in the Ising ase be ause x = 0 on�guration is related

to the free onformal boundary ondition in the onformal Ising model.

Con�guration Cx=1, i.e. | ↓, ↓, ... ↓>: This ase, whi h is also studied in [145, 146℄

and [147℄, orresponds to k = l and has the lowest probability. One an easily show

that

P (x = 1) = det[1

2(1−G)] detF = det[

1

2(1 +G)]. (6.133)

For the in�nite system it follows similar formula as (6.128) with also an extra ν (−1)l√l

term, in other words,

Π(x = 1) = αl + β ln l + ν(−1)l√

l+ γ

ln l

l+O(1), (6.134)

where β = c8. At the riti al Ising point α = ln 2+2C/π = 1.27626 and ν = −0.21505

and γ is unknown. The os illating ν term is mathemati ally explained by using gen-

eralized Fisher-Hartwig onje ture in [145, 146℄. To the best of our knowledge its

presen e at the riti al point has not been understood by physi al arguments [266℄.

Our numeri al results in the next se tion will show that the term is present whenever

182

the parity of the number of down spins hanges with the subsystem size. The term is

very important to be onsidered in numeri al al ulations to get reliable results for β

whi h is the universal and the most interesting term.

When the size of the system is �nite depending on the type of the boundary

onditions boundary hanging operators an play an important role. The following

formulas are presented in [147℄:

ΠP (x = 1) = αl + β ln[L

πsin

πl

L] + ..., (6.135)

ΠO(x = 1) = αl + βo1 ln[

L

πsin

πl

L] + βo

2 ln[L

πtan

πl

2L] + ...,

(6.136)

where β = c8, βo

1 = c16

and βo2 = 4h− c

8with h = 1

16being the onformal weight of the

boundary hanging operator. The dots are the subleading terms.

XX riti al line

The riti al XX hain a = 0 has U(1) symmetry whi h as it is already dis ussed

extensively in the literature is the main reason for having Gaussian de aying emptiness

formation probability [147, 236℄. Sin e in this model < c†ic†j >=< cicj >= 0 the

equation (6.123) has simpler form

ρD(ξ, ξ′) = det(1− C)eξ

∗Fξ′(6.137)

where Cij =< c†icj > and F = C(1− C)−1. Finally we have

P (Cn) = det[1− C]MCnF (6.138)

The form of the C matri es in the periodi and open ases are [267℄:

CPij =

nf

πδij + (1− δij)

sin(nf(i− j))

L sin(π(i−j)L

), (6.139)

COij =

(1

2− (

L

2(L+ 1)−n′f

π))

δij + (1− δij)1

2(L+ 1)

(sin(n′f (i− j))

sin(π(i−j)2L+2

)−

sin(n′f (i+ j))

sin(π(i+j)2L+2

)

)

,

183

where nf = πL

(

2⌈ L2π

arccos(−h)⌉ − 1)

is the Fermi momentum and n′f = π

2(L+1)

(

1 +

2⌊ (L+1)π

arccos(−h))⌋)

with ⌈x⌉(⌊x⌋) as the losest integer larger (smaller) than x.

The all spins up and down on�gurations do not lead to onformal boundary

onditions and so none of the equations that we mentioned in the last subse tion are

valid. However, using Widom onje ture it is already known that, see for example

[238℄, the probabilities for both Cx=0 and Cx=1 show Gaussian behavior. For systems

with U(1) symmetry one expe ts the following behavior for logarithmi emptiness

formation probability [236℄:

Π(x = 0) = α2l2 + αl + β ln l +O(1), (6.140)

where β = 14for riti al XX hain.

6.5.3 Logarithmi pattern formation probabilities

In this se tion, we study the logarithmi pattern formation probability de�ned as

Π(C) = − lnP (C) with respe t to the size of the subsystem. The easiest on�gurations

to study are those that have some kind of rystal stru ture. Although everything is

already known and he ked numeri ally for the emptiness formation probabilities we

will also report the results on erning these ases as ben hmarks. Here we introdu e

the on�gurations that we studied numeri ally. None of these on�gurations have

been onsidered before in the literature.

We study here the on�gurations with k = l2, l3, l4, ..., l

10with rystal pattern and

we all them on�gurations x = 12, 13, 14, ..., 1

10and for some ranks we will study the

two most basi ases. For example we will study

Con�gurations with k = l2:

184

a (| ↓, ↑, ↓, ↑, ... >)

b (| ↓, ↓, ↑, ↑, ↓, ↓, ↑, ↑, ... >)

Con�gurations with k = l3:

a (| ↑, ↑, ↓, ↑, ↑, ↓, ↑, ↑, ... >)

b (| ↑, ↑, ↑, ↑, ↓, ↓, ↑, ↑, ↑, ↑, ↓, ↓, ... >)

All the on�gurations with the same k belong to the ases with an equal rank of the

minor in the equation (6.124). Note that in all of the up oming numeri al al ulations

in every step we in rease the size of the subsystem with a number whi h is devidable to

the length of the base of the orresponding on�guration. For some parti ular k's the

a and b on�gurations di�er by the parity e�e t. For example, in k = l2a depending

on l = 4i or l = 4i − 2 with i = 1, 2, ... the subsystem has even or odd number of

down spins. This means that the parity of the number of down spins hanges with

the subsystem size for this on�guration. However, for k = l2b this is not the ase

be ause in order to have "perfe t rystal" in the subsystem we need to onsider a

subsystem with l = 4i with i = 1, 2, ... whi h has always even number of down spins

inside. Be ause of this di�eren e in parity e�e t for k = l2a and k = l

2b we expe t

di�erent subleading behavior for these two ases. Finally noti e that one an simply

de�ne on�gurations like k = l2c and k = l

3c by simply taking bigger bases for the

rystals. For example, k = l2c an be understood as a on�guration with the base:

three down spins and then three up spins.

185

Transverse-field Ising hain

Using the equation (6.124) and (6.127) we �rst studied the rystal on�gurations

introdu ed in the previous subse tion for the ase of in�nite hain. To al ulate the

formation probability for every on�guration we �rst use the matrix G introdu ed

in (6.127) to �nd the matrix F . Then for every on�guration we use an appropriate

minor to al ulate the orresponding probability in (6.124). For example, in the ase

of k = l2a this an be done by just �nding a minor of F whi h an be derived by

al ulating the determinant of a submatrix F obtained from F by removing every

other row and olumn. The results for α and β (the most interesting quantities in

this study) are shown in the Table 6.2. Based on the numeri al results one an derive

the following on lusions regarding rystal on�gurations:

1. All the rystal on�gurations follow either the equation (6.128) or (6.134) with

β = 116.

2. Whenever the parity of the number of down spins in a on�guration hanges

with respe t to the size of the subsystem we have the os illating term

1√l. For

example, in the ase of k = l2a we have the subleading term

(−1)l2√

lbut

1√l

orre tion is absent in k = l2b. It appears again for k = l

3a in the form of

(−1)l3√

l.

Generalization to other on�gurations is strightforward. .

3. Although in some ases α for bigger x is smaller than α with smaller x in average

α in reases with x.

We then studied the same on�gurations for the periodi boundary ondition. In

the �gure 6.18, it is shown that all of the on�gurations follow the formula (6.131)

. The ase of the open boundary ondition is more tri ky and depending on the

186

Con�guration α βx = 0 0.110025 0.062498x = 1 1.276267 0.062465x = 1

2(a) 0.984708 0.062462

x = 12(b) 0.755726 0.062496

x = 13(a) 0.818715 0.062468

x = 13(b) 0.542109 0.062491

x = 14(a) 0.710620 0.062481

x = 14(b) 0.434286 0.062524

x = 15(a) 0.634016 0.062495

x = 16(a) 0.576551 0.062509

x = 17(a) 0.531651 0.062523

x = 18(a) 0.495482 0.062537

x = 19(a) 0.465643 0.062549

x = 110(a) 0.440555 0.062562

Table 6.2: Fitting parameters for the logarithmi formation probabilities of di�erent

rystal on�gurations of the riti al Ising hain dis ussed in the text. All the data were

extra ted by �tting the data in the range l ∈ (2000, 2500) to αl+β ln l+γ ln ll+ δ 1

l+η

for those ases that do not show parity e�e t and to αl+β ln l+γ ln ll+ν (−1)m√

l+δ 1

l+η

(with suitable m) for those ases that show parity e�e t [268℄.

0 500 1000 1500 2000l

-0.2

0

0.2

0.4

0.6

Π(l

,L)-

α l

CFTx=0x=Lx=1/2(a)x=1/2(b)x=1/3(a)x=1/3(b)

Figure 6.18: Π(l, L)−αl for periodi system with total length L = 2000 with respe tto l for di�erent on�gurations. The dashed lines are the results expe ted from CFT,

i.e.

116ln[L

πsin πl

L] + η.

187

0 500 1000 1500 2000l

0

0.5

1

1.5

2

2.5

3

Π(l

,L)

- α

l

CFTx=1x=1/2(a)x=1/3(a)x=1/4(a)

(a)

0 500 1000 1500 2000l

-1

-0.8

-0.6

-0.4

-0.2

0

Π(l

,L)

- α

l

CFTx=0x=1/2(b)x=1/3(b)x=1/4(b)

(b)

Figure 6.19: Π(l, L)−αl for open system with total length L = 2000 with respe t tol for di�erent on�gurations. a) on�gurations without boundary hanging operators

and b) on�gurations with boundary hanging operators. The dashed lines are the

results expe ted from CFT.

on�guration we have two possibilities: When the parity of the number of down spins

is independent of the size of the subsystem (for example in the on�gurations x = 12b

and

13b) we have the formula (6.132) but when we have the possibility of having odd

or even number of down spins in a on�gurations (for example x = 12a, 1

3a) we have

the formula (6.136). The results are shown in the Figure 6.19. This behavior ould

be anti ipated based on the di�eren e between on�gurations k = 0 and k = l that

we dis ussed before. It looks like that the boundary hanging operator plays a role

whenever there is the parity e�e t in the on�guration. Looking to the problem in

the language of Eu lidean two dimensional lassi al system one an argue that in the

ase of open boundary ondition we have a strip with a slit on it [147℄. However,

the boundary onditions on the boundary of strip an be di�erent from the boundary

ondition on the slit, onsequently, one needs to onsider boundary hanging operator

on the point where the boundary ondition hanges. However, in general it is not lear

188

whi h on�gurations lead to di�erent boundary onditions on the slit and on the

boundary of the strip. Our numeri al results indeed give a hint that depending on the

bahavior of the parity of the number of down spins in a on�guration the onformal

boundary ondition on the slit an be di�erent. In the next two subse tions, we

will �rst omment on the validity of the above results in other ases su h as non-

rystal on�gurations. Then we will also indi ate the possible universal behavior of

our results.

Logarithmi formation probability of non- rystal onfigurations

The number of rystal on�gurations is mu h smaller than the number of the whole

on�gurations. In fa t, the number of rystal on�gurations grows polynomially with

the subsystem size but the number of whole on�gurations grows exponentially. How-

ever, numeri ally it is very simple to he k the formula for many on�gurations that

have a small deviation from the rystal states. For example, one an onsider the ase

k = 1 with all spins up ex ept one and al ulate the logarithmi formation proba-

bility using the equation (6.124). It is lear that one does not expe t the result be any

di�erent from the equation (6.128) and indeed numeri al results on�rm this expe -

tation. The important on lusion of this numeri al exer ise is that there are many

on�gurations " lose" to rystal on�gurations that indeed follow either the equation

(6.128) or equation (6.134) with all having the same β's but di�erent α's.

The above results strongly suggest that all of the rystal and non- rystal on-

�gurations dis ussed in this se tion are �owing to some sort of onformal boundary

onditions in the s aling limit.

189

Universality

To he k that the above results are the properties of the Ising universality lass we

also studied the riti al XY- hain whi h has also entral harge c = 12. The Green

matrix, in this ase, is given by

Gij =

∫ π

0

dφ

π

(cosφ− 1) cos[(i− j)φ]− a sinφ sin[(i− j)φ]

√

(1− cosφ)2 + a2 sin2 φ.

Our numeri al results depi ted in the Figure 4 show that the oe� ient of the

logarithmi term is a universal quantity whi h means that it has a �xed value on

the riti al XY-line. The oe� ient of the linear term hanges by varying a whi h

indi ates its non-universal nature.

0.5 0.6 0.7 0.8 0.9 1a

0.058

0.06

0.062

0.064

0.066

0.068

β

x=1/2(a)x=1/3(a)0.0625

Figure 6.20: The oe� ient of the logarithmi term in (6.128) for two on�gurations

x = 12a and x = 1

3a for di�erent values of a. The dashed line is the CFT result. The

size of the largest subsystem was l = 500 and all the results were extra ted by �tting

the data to αl+β ln l+γ ln ll+ν (−1)m√

l+δ 1

l+η with suitablem in the range l ∈ (100, 500).

Estimated errors in the numeri s are in the order of the size of the markers.

XX hain

We repeated the al ulations of the last se tion for also riti al XX hain. The entral

harge of the system is c = 1. The results of logarithmi formation probabilities for

190

di�erent magneti �eld h are shown in the Table 6.3 and Table 6.4. Based on the

numeri al al ulations we on lude the followings:

1. The on�gurations with x =nf

πfollow the equation (6.128) with β = 1

8. This

means that in the s aling limit most probably all of these on�gurations �ow

to some sort of bounday onformal onditions. Note that as far as there is no

boundary hanging operator in the system the equation (6.128) is valid for any

CFT independent of its stru ture.

2. All the other on�gurations follow the equation (6.140) with β whi h is di�erent

for di�erent on�gurations.

As we mentioned earlier XX hain has a U(1) symmetry whi h means that the number

of parti les is onserved. The only on�gurations that respe t this symmetry in the

subsystem level are the on�gurations with x =nf

π. Any inje tion of the parti les

into the subsystem hanges drasti ally the formation probability. In the ase of x =

0 this phenomena is already explained in [147℄ based on ar ti phenomena in the

dimer model. It is quite natural to expe t that similar stru ture is valid for all the

on�gurations with x 6= nf

π. Note that based on our results the oe� ient of the ln is

nf -dependent and stri tly speaking is not a universal quantity.

nf Con�guration α βx = 1/2(a) 0.3465735 0.124998

π/2 x = 1/2(b) 0.5198604 0.124997x = 1/2(c) 0.7127780 0.124597

π/3 x = 1/3(a) 0.3662041 0.124987π/4 x = 1/4(a) 0.3432345 0.125024

Table 6.3: Fitting parameters for the logarithmi formation probability of antifer-

romagneti on�gurations with di�erent �lling fa tors. All the results were extra ted

by �tting the data in the range l ∈ (100, 300) to αl + β ln l + γ ln ll+ δ 1

l+ η.

191

Con�guration α2 α βx = 0, 1 0.346573 0.000000 0.250054x = 1/3(a) 0.035191 0.366228 0.524293x = 1/3(b) 0.035188 0.597663 1.578683x = 1/4(a) 0.080911 0.346599 0.829767x = 1/4(b) 0.080910 0.587114 2.744492x = 1/5(a) 0.118119 0.321924 1.144949x = 1/6(a) 0.147178 0.298674 1.465399x = 1/7(a) 0.170072 0.278052 1.788319x = 1/8(a) 0.188433 0.260021 2.112155x = 1/9(a) 0.203427 0.244263 2.436061x = 1/10(a) 0.230432 0.230432 2.759481

Table 6.4: Fitting parameters for di�erent on�gurations with x < 12in the XX hain

with nf = π2. All the data were extra ted by �tting the data in the range l ∈ (100, 300)

to α2l2 + αl + β ln l + η.

We also studied the �nite size e�e t in this model. The results of the numeri al

al ulations of periodi boundary ondition for x = 12are shown in the Figure 6.21.

It is shown that all of the on�gurations with x = 12follow the formula (6.131) with

c = 1.

We also repeated the same al ulations for open boundary onditions. We �rst

performed the al ulations for semi-in�nite open hain and �tted the results to (6.128)

and extra ted the β. The oe� ient of the logarithm not only depends on the rank

of the on�guration but also to the on�guration itself. It also hanges with nf . We

were not able to �nd any universal feature in this ase.

The above results suggest that most probably all of the rystal on�gurations with

x =nf

πin the periodi boundary ondition �ow to a boundary onformal �eld theory.

In the language of Luttinger liquid, the orresponding boundary ondition should be

192

0 50 100 150 200 250 300l

0.4

0.8

1.2

1.6

Π(l

,L)

- α

l

CFTx=1/2(a)x=1/2(b)

Figure 6.21: Π(l, L)−αl for periodi system with total length L = 300 with respe t

to l for di�erent on�gurations for riti al XX- hain with nf = π2. The dashed lines

are the results expe ted from CFT.

the Diri hlet boundary ondition [147℄. The ase of the open boundary ondition is

intriguing and we leave it as an open problem.

6.5.4 Shannon information of a subsystem

In this se tion, we study Shannon information of a subsystem in transverse-�eld

Ising model and XX hain. For both models, the Shannon information is already

al ulated in [256℄ up to the size l = 40 whi h it seems to be the urrent limit for

lassi al omputers. The reason that we are interested in revisiting this quantity is to

have a more detailed study of the ontribution of di�erent on�gurations. This will

give an interesting insight regarding the possible s aling limit for this quantity.

193

Criti al Ising

In the last se tion, we studied many di�erent on�gurations in the riti al Ising model

and we found that all of them follow PCn = eαnl

lc8

with c = 12. The natural expe tation

is that if we plug this formula in the de�nition of the Shannon information we get

Sh(l) = αl +c

8ln l + ... (6.141)

where the dots are the subleading terms. The above formula is onsistent with [255℄.

However, one should be areful that although there are a lot of rystal on�gurations

(polynomial number of them) and " lose" to rystal on�gurations that are onne ted

to the entral harge it is absolutely not lear what is going to happen in the s aling

limit. For example we repeated the al ulations of [256℄ and realized that extra tion

of the oe� ient of the ln in the above equation is indeed very di� ult, see appendix.

Here we show where one should look for the most important on�gurations. After a

bit of inspe tion and numeri al he k, one an see that the on�guration with the

highest probability is the x = 0. Although the proof of the above statment doesn't

look straightforward one an understand it qualitatively by starting from the ground

state of the Ising model with h → ∞ and approa hing to the riti al point h = 1.

The ground state of the Ising model with h → ∞ is made of a on�guration with

all spins up. When we de rease the transverse magneti �eld the other on�gurations

start to appear in the ground state. Although the amplitude of the on�guration with

all spins up de reases by de reasing h it still remains always bigger than the other

on�gurations. Another way to look at this phenomena is by looking at the variation

of the expe tation value of the Hamiltonian H = 〈C|H |C〉 for di�erent on�gurations

C. It is easy to see that H is minimum for the on�guration C with all spins up. This

simply means that most probably when one onstru t the ground state of the Ising

model using the variational te hniques this on�guration playes the most important

194

role. The least important on�guration is x = 1 with the lowest probability. This an

also be understood with the same heuristi argument as above. For every rank k of

the minors, as we dis ussed, we have

(lk

)number of on�gurations whi h means that

for every x for large l we have ef(x)l with f(x) = −x ln x− (1−x) ln(1−x) number of

on�gurations. It is obvious that the number of on�gurations in every rank should

be high enough to ompensate the exponential de rease of probabilities. We realized

that in every rank the on�gurations a has the lowest probabilities. One an again

understand this fa t using the variational argument. In this ase it is mu h better to

make �rst the anoni al transformation: σx → −σzand σz → σx

in the Hamiltonian

of the rti al Ising model. Then one an simply argue that H is big if there are a

lot of domain walls, i.e. 〈C|σzjσ

zj+1 |C〉 = −1 in the system whi h is the ase for the

on�gurations a. Other important on�gurations are the on�gurations whi h divide

the subsystem to two onne ted regions with in one part all the spins are up and in

the other part all the spins are down. These on�gurations are interesting be ause

they have the biggest probabilities among all the on�gurations orresponding to their

minor rank. Note that in this ase we have just one domain wall. It is not di� ult

to see that the probability of all of these on�gurations de ay exponentially with the

following oe� ient α:

αmin(x) =4C

πx+ ln 2− 2C

π, (6.142)

where C is the Catalan onstant. In the two extreme points, we re over the pre-

vious results. We also he ked the validity of the above formula numeri ally. Having

the biggest and smallest probabilities for every rank, we an now easily read the

most important ranks. In Figure 6.22, we depi ted the αmax and αmin for di�erent

on�gurations. We also depi ted the graph of the number of on�gurations in every

rank. The Figure learly show that the on�gurations with x > 12 an not have any

195

signi� ant ontribution in the s aling limit be ause the number of on�gurations is

not enough to ompensate the exponential de ay of the probabilities. A similar story

seems to be valid also for the values of x lose to zero. The reason is that the number

of on�gurations with small x is su h low that an not ompensate exponential de ay

of the probabilities in this region to have a signi� ant ontribution in the Shannon

information. Just the region between the points that the two lines ut ea h other

will most likely survive in the s aling limit. The numeri al results indeed prove our

0 0.2 0.4 0.6 0.8 1x

0

0.5

1

1.5

2

f(x)α

minα

max

Figure 6.22: Values of f(x), αmin and αmax with respe t to x. The red urve is the

fun tion f(x) = −x log x − (1 − x) ln(1 − x) and the blue line is the linear fun tion

(6.142). The separated points are the αmax regarding the on�gurations dis ussed in

the text.

expe tation. In Figure 6.23, we depi ted the ontribution of every rank Shk(l) in

Shannon information for two di�erent sizes. As it is quite lear the most important

ontributions ome from 0 < x < 12. The ontribution of the on�gurations with

x > 12is exponentially small. This means that ignoring a lot of on�gurations will

produ e a very small amount of error in the �nal result of Shannon information. To

quantify this argument we al ulated the amount of error in the evaluation of the

Shannon information if we just keep the on�gurations with ranks up to xm. Suppose

Sh(l, xm) is the ontribution of the on�gurations with all ranks equal or smaller

196

than xm. Then the error of trun ation an be al ulated by E(xm, l) = Sh(l,1)−Sh(l,xm)Sh(l,1)

.

Interestingly we found that the logarithm of the error fun tion is a linear fun tion of

l, see Figure 6.24. In other words

ln E(xm, l) = −λ(xm)l + δ(xm), (6.143)

where λ(xm) is equal to zero and in�nity for xm = 0 and xm = 1 respe tively. λ(xm)

for the other values are shown in the inset of the Figure 6.24. The above formula

shows that one an al ulate Shannon information with a ontrollable a ura y by

ignoring non-important on�gurations. Although the above trun ation method help

to al ulate the Shannon information with good a ura y (espe ially the oe� ient

of the linear term α) it is still not good enough to al ulate the oe� ient of the

logarithm with ontrollable pre ision.

0 5 10 15 20 25 30k

0

0.5

1

1.5

2

2.5

3

Shk(l

)

l=14l=26

Figure 6.23: (The ontributions of di�erent ranks k in the Shannon information for

two sizes l = 14 and 26.

XX hain

The Shannon information of the subsystem in the XX- hain is already dis ussed

in [256℄ and based on numeri al results it is on luded that the equation (6.142)

is valid with β = 18whi h is onsistent with the onje ture in [255℄. Here we just

197

0 5 10 15 20 25 30l

-6

-5

-4

-3

-2

-1

0

lnε(

x m,l)

xm

=1/2

xm

=1/3

xm

=1/4

0 0.1 0.2 0.3 0.4 0.5x

m

0

0.05

0.1

0.15

0.2

0.25

λ(x m

)

Figure 6.24: The error E(xm, l) in the evaluation of Shannon information oming

from the trun ation at the rank k = xml. Inset: −λ(xm) with respe t to xm.

omment on the ontribution of di�erent ranks whi h shows very di�erent behavior

from the transverse �eld Ising hain ase. First of all, as we dis ussed in the previous

se tion when the external �eld is zero the only on�gurations that de ay exponentially

are those that respe t the half �lling stru ture of the total system. The rest of the

on�gurations s ale like a Gaussian whi h simply indi ates that their ontribution is

very small in the Shannon information. This is simply be ause the number of these

on�gurations s ale just exponentially. Based on this simple fa t one an anti ipate

that the only on�gurations that an survive in the s aling limit are those with k = l2.

Numeri al results depi ted in the Figure 6.25 indeed support this idea. Although the

k = l2is only one among l possible minor ranks the number of on�gurations with

this rank is highest with respe t to the others whi h an be one of the reasons that

one an obtain a good estimate for the oe� ient of the logarithm in (6.142) with

relatively modest sizes.

198

0 6 12 18 24k

0

2

4

6

8

Shk(l

)

l=12l=24

Figure 6.25: The ontributions of di�erent ranks k in the Shannon information for

two sizes l = 12 and 24 in the XX- hain.

6.5.5 Evolution of Shannon and mutual information after global

quantum quen h

Inspired by experimental motivations, the �eld of quantum non-equilibrium systems

has enjoyed a huge boost in the re ent de ade [269℄. One of the interesting dire tions

in this �eld is the study of information propagation after quantum quen h, see for

example [270, 271, 272, 273, 274, 275℄. Based on semi lassi al arguments and also

using Lieb-Robinson bound it is shown [270, 272℄ that in one dimensional integrable

system one an understand the evolution of entanglement entropy of a subsystem

based on quasi-parti le pi ture [270℄. The argument is as follows: after the quen h,

there is an extensive ex ess in energy whi h appears as quasiparti les that propagate

in time. The quasi-parti les emitted from nearby points are entangled and they are

responsible for the linear in rease of the entanglement entropy of a subsystem with

respe t to the rest. In this se tion, we �rst study the time evolution of formation prob-

abilities and subsequently Shannon and mutual information after a quantum quen h.

199

One an onsider this se tion as a omplement to the other studies of information

propagation after quantum quen h. To keep the dis ussion as simple as possible, we

will on entrate on the most simple ase of XX- hain or free fermions. Following [276℄

onsider the Hamiltonian

H = −1

2

+∞∑

m=−∞tm(c

†mcm+1 + c†m+1cm). (6.144)

The time evolution of the orrelation fun tions in the half �lling are given as

0 5 10 15 20 25Time

0

90

180

270x=025.30

8.75

10x=1/2(a)10.25

8

12

16

20x=1/2(b)13.58

Figure 6.26: (The evolution of logarithmi formation probability of di�erent on�g-

urations with respe t to time t after quantum quen h. The size of the subsystem is

taken l = 20.

Cmn(t) = in−m∑

jl

ij−lJm−j(t)Jn−l(t)Cjl(0), (6.145)

where, J is the Bessel fun tion of the �rst kind. Here we onsider the dimerized initial

onditions with t2m = 1 and t2m+1 = 0. The dimerized nature of the initial state will

help later to onsider di�erent possibilities for the initial Shannon mutual information.

Then at time zero we hange the Hamiltonian to tm = 1 and let it evolve. The time

evolution of the orrelation matrix is given by [276℄

Cmn(t) =1

2

(

δm,n +1

2(δm+1,n + δm−1,n) + e−iπ

2(m+n) i(m− n)

2tJm−n(2t)

)

.(6.146)

200

To al ulate the time evolution of the probability of di�erent on�gurations one

0 2 4 6 8 10 12 14Time

6

9

12

15

13.2186.628l=20l=10

Figure 6.27: (The evolution of Shannon information of a subsystem with di�erent

sizes with respe t to time t after quantum quen h. The full lines are the equation

(6.147). The saturation points t∗ = l2are marked by verti al arrows.

just needs to use the above formula in (6.138). The results for few on�gurations

are shown in the Figure 6.26. Of ourse sin e the sum of all the probabilities should

be equal to one some of the probabilities in rease with time and some de rease. All

the probabilities hange rapidly up to time t∗ ≈ l2and after that saturate. One

an also simply al ulate the evolution of the Shannon information with the tools of

previous se tions. In Figure 6.27, we depi ted the evolution of Shannon information of

a subsystem with respe t to the time t. The numeri al results show an in rease in the

Shannon information up to time t∗ ≈ l2and then saturation. This is similar to what we

usually have in the study of the time evolution of von Neumann entanglement entropy

after quantum quen h [270℄. However, one should be areful that in ontrast to the

von Neumann entropy the Shannon information of the subsystem is not a measure

of orrelation between the two subsystems. In addition, the in rease in the Shannon

information of the subsystem is not linear as the evolution of the von Neumann

entanglement entropy. Our numeri al results indi ate that apart from a small regime

201

at the beginning the Shannon information in reases as

Sh(l) = altb − dt t < t∗ (6.147)

where b ≈ 0.15(2) and a and d are positive l independent quantities.

0 10 20 30 40 50 60 70Time

0

0.0012

0.0024d=60

0

0.0015

0.003 d=400

0.003

0.006 d=20

Figure 6.28: (The mutual information between a pair of dimers lo ated at distan e

d with respe t to time.

0 5 10 15 20Time

0

0.1

0.2

0.3

0.4

0.5

0.6

I(l,l

)

l=12(a)l=20(a)l=12(b)l=20(b)

Figure 6.29: The evolution of mutual information between two adja ent subsystems

in two di�erent ases: when at the boundary between the two subsystems there is

a dimer, ase a and when there is no dimer, ase b. In the �rst ase the mutual

information starts from a non-zero value but in the se ond ase it starts from zero.

To study the time evolution of orrelations, it is mu h better to study another

quantity, Shannon mutual information of two subsystems. To investigate this quantity

we �rst studied the time evolution of Shannon mutual information of a ouple of

202

dimers lo ated far from ea h other. The results depi ted in the Figure 6.28 show

that the Shannon mutual information of the dimers are zero up to time t∗ ≈ l2and

after that in reases rapidly and then again de ays slowly. This pi ture is onsistent

with the quasiparti le pi ture. The two regions are not orrelated up to time that

the quasiparti les emitted from the middle point rea h ea h dimer[277℄. However, the

similarity between the evolution of the von Neumann entropy and mutual Shannon

information ends here. To elaborate on that we onsider the mutual information of

two adja ent regions with sizes

l2. Be ause of the dimerized nature of the initial

state there are two possibilities for hoosing the subsystems: at time zero at the

boundary between the two subsystems there an be a dimer or not. In the �rst ase

at time zero the Shannon mutual information between the two subsystems is not

zero but in the se ond ase it is zero. In the se ond ase naturally one expe ts an

overall in rease in the mutual information but in the �rst ase a priory it is not

lear that the mutual information should in rease or de rease. In the Figure 6.29,

we have depi ted the results of the numeri s for the two adja ent subsystems for

di�erent sizes. The numeri al results show that for the un orrelated initial onditions

the Shannon mutual information �rst in reases rapidly and then it de ays and �nally

saturates at time t∗ = l2. In the orrelated ase, we have overall de ay in the mutual

information and �nally the saturation again at time t∗ = l2. This behavior is very

di�erent from the quantum mutual information of the same regions whi h for the

onsidered initial states �rst in reases linearly and then saturates at time t∗ = l2. The

interesting phenomena is that after a short initial regime that the Shannon mutual

information is initial state dependent the system enters to a regime that this quantity

is ompletely independent of initial state and it de reases "almost" linearly and then

saturates. This an be also easily seen from the equation (6.147), where we an simply

203

drive

I(l, l) = −dt t < t∗. (6.148)

The saturation regime is independent of the size of the subsystem, this is simply

be ause in the equlibrium regime the Shannon mutual information follows the area-

law [250℄ and so it is independent of the volume of the subsystems.

6.5.6 Con lusions

In this paper, we employed Grassmann numbers to write the probability of o ur-

ren e of di�erent on�gurations in free fermion systems with respe t to the minors of

a parti ular matrix. The formula gives a very e� ient method to study the s aling

properties of logarithmi formation probabilities in the riti al XY- hain. In parti -

ular, we showed that the logarithmi formation probabilities of rystal on�gurations

are given by the CFT formulas for the riti al transverse �eld Ising model. This is

he ked by studying the probabilities in the in�nite and �nite (periodi and open

boundary onditions) hain. In the ase of riti al XX- hain whi h has a U(1) sym-

metry just the on�gurations with x =nf

πfollow the CFT formulas. The rest of the

on�gurations de ay like a Gaussian and do not show mu h universal behavior. We

also studied the Shannon information of a subsystem in the transverse �eld Ising

model and XX- hain. In parti ular, for the Ising model, we showed that in the s aling

limit just the on�gurations with a high number of up spins ontribute to the s aling

of the Shannon information. In prin iple, if one onsiders all the on�gurations, with

our method one an not al ulate the Shannon information with lassi al omputers

for sizes bigger than l = 40 in a reasonable time. However, if one admits a ontrollable

error in the al ulation of Shannon information it is possible to hire the results of

se tion V to go to higher sizes. It would be very ni e to extend this aspe t of our

204

al ulations further to al ulate the universal quantities in the Shannon information

with higher a ura y. For example, one interesting dire tion an be �nding an expli it

formula for the sum of di�erent powers of prin ipal minors of a matrix. This kind of

formulas an be very useful to al ulate analyti ally or numeri ally the Rényi entropy

of the subsystem.

Finally, we also studied the evolution of formation probabilities after quantum

quen h in free fermion system. In this ase, we prepared the system in the dimer

on�guration and then we let it evolve with homogeneous Hamiltonian. The evolution

of Shannon information of a susbsytem shows a very similar behavior as the evolution

of entanglement entropy after a quantum quen h. Espe ially our al ulations show

that the saturation of the Shannon information of the subsystem o urs at the same

time as the entanglement entropy. This is probably not surprising be ause the t = t∗

is also the time that the redu ed density operator saturates.

It will be very ni e to extend our al ulations in few other dire tions. One dire tion

an be investigating the evolution of mutual information after lo al quantum quen hes

as it is done extensively in the studies of the entanglement entropy [278, 279, 280℄.

The other interesting dire tion an be al ulating the same quantities in other bases,

espe ially those bases that do not have any dire t onne tion to CFT.

6.5.7 Additional details: Shannon information for riti al transverse-

field Ising model

In this subse tion, we will provide more details regarding the Shannon information of

transverse riti al Ising hain and XX hain. The data regarding Shannon information

for a subsystem with length l up to l = 39 is listed in the Table A1. Having the

data, we he ked many di�erent fun tions with di�erent parameters to study the

oe� ient of the logarithm. Needless to say in reasing the possible parameters an

205

make a di�eren e in the �nal result. In [256℄ the results �tted to

Sh(l) = αl + β ln l +5∑

n=1

bnln

+ δ (6.149)

show that the best value is β = 0.060. This an be also he ked using the data provided

in the TableA1. It is worth mentioning that one an also get reasonable results using

the data up to l = 40 for some formation probabilities (not all) if we onsider extra

terms

∑5n=1

bnlnin the �tting pro edure. Although we found that the equation (6.149)

is the most stable �t with the least standard deviation based on our results in the

main text we found it is hard to ex lude the term

ln llbe ause it is present in all the

on�gurations studied there. If one in ludes this term and does not add the terms

∑5n=1

bnlnthe β oe� ient will be 0.0617. If one keeps all the terms

∑5n=1

bnlnthe result

will be β = 0.060. The �nal on lusion is that as far as one justi�es the presen e of

the terms

∑5n=1

bnln

in the Shannon information formula the best value for β with the

urrent available data is 0.060.

206

l Shannon l Shannon

1 0.473946633733778 21 9.094267377324401

2 0.925441055292197 22 9.520258511384927

3 1.367970612016317 23 9.946131954351737

4 1.805854593071358 24 10.37189747498959

5 2.240889870728481 25 10.79756367448224

6 2.674003797245196 26 11.22313816533366

7 3.105734740754158 27 11.64862771729621

8 3.536422963908594 28 12.07403837729498

9 3.966297046625437 29 12.49937556879184

10 4.395517906953372 30 12.92464417475445

11 4.824203084194648 31 13.34984860684562

12 5.252441034545332 32 13.77499286454422

13 5.473946633733777 33 14.21026702317442

14 6.107833679024358 34 14.62511508430369

15 6.535085171703405 35 15.05009939651494

16 6.962089515106671 36 15.47503630258492

17 7.388875612253789 37 15.89992835887542

18 7.815467577831834 38 16.32477792018708

19 8.241885740227190 39 16.74960160654153

20 8.668147394540807

Table A1: Shannon information al ulated for sizes l = 1, 2, ..., 39.

207

7

Con lusion

�Learn from yesterday, live for today, hope for tomorrow. The important

thing is not to stop questioning.�

� Albert Einstein

With the progressing development in experimental te hniques su h as pump-probe

spe tros opy, designing and manufa turing two-dimensional materials, the advent of

the highly tunable ultra old opti al latti es an array of ions and atoms, investigating

the dynami s of strongly orrelated systems have be ome one of the most a tive

and e�e tive parts of the ondensed matter physi s. Motivated by numbers of re ent

experiments [8, 16, 20, 25, 30℄, in this thesis, we have used various omputational

and exa t formalisms su h as DMFT, sum rules, nonequilibrium Green's fun tion and

quantum �eld theory to investigate a wide range of systems su h as high Tc super-

ondu tors, quantum ele troni devi es, mixtures of fermioni and bosoni atoms in

ultra old opti al latti es, quantum spin hains, and quantum simulators. Below, we

will provide a review of the main results of ea h proje t and brie�y mention some of

the remaining open questions for future dire tion.

In hapter 3, we �rst, derived a general formalism for the nth derivative of a time-

dependent operator in the Heisenberg representation and we employ this identity to

�nd the spe tral sum rules for retarded Green's fun tion up to the third moment

and onsequently, we used the results to obtain the �rst moment of the self-energy

208

in the normal state. These sum rules provide a powerful self- onsisten y he k for

the omputational or experimental results, and it may provide insight into the re ent

experiment of the e�e t of the pump on the weakening of the ele tron-phonon inter-

a tion. In fa t, our results in the atomi limit suggest that the hanges in spe tral

weight in di�erent time intervals ompensate ea h other and the integral over the

frequen y remains onstant as it is expe ted from the sum rules. Furthermore, our

results agree with a re ent omputational study in Ref. [68℄ whi h suggests that one

an dete t the weakening in tr-Arpes of the pump-probe experiment as a result of

the indu ed pump without any hange in the ele tron-phonon intera tion. In order

to investigate the pump e�e t in the super ondu ting state, the next step would be

generalizing the sum rules into the super ondu ting state. One possible approa h is

using the Nambu-Gorkov formalism in whi h the normal Green's fun tion does not

hange, but one needs to modify the sum rules for the anomalous Green's fun tion.

Furthermore, there has been a few studies re ently, indi ating an enhan ed transient

ele tron-phonon oupling in whi h the me hanism for su h enhan ement still remains

un lear [94℄. Re ently, the role of nonlinear ele tron-phonon oupling in the indu e-

ment of transient ele tron-phonon oupling has been suggested in Ref. [95℄. So, one

may examine the sum rules for Hubbard-Holstein model with an addition of extra

nonlinear ele tron-phonon oupling.

Then, we devoted hapter 4 to study the nonlinear response of a multilayer devi e

by obtaining the urrent-voltage pro�le. The devi e that we study onsists of a single

barrier with a number of metalli leads on both sides whi h are atta hed to the left

and right bulk. We use the urrent-voltage biasing previously proposed by Freeri ks

in Ref [17℄ to obtain the urrent and �lling a ross the devi e. Due to s attering in

barrier region, the urrent and �lling drop passing through the barrier. Then, in order

to sustain the urrent and �lling onserved, we apply a lo al ele tri �eld a ross the

209

barrier and furthermore, we have developed an optimization pro edure to obtain the

best values for the ele tri �eld. We report the results for single barrier plane after

applying an ele tri �eld. Our results show that in fa t, the optimization pro edure

improves the urrent and �lling onservation. Then, we report the urrent-voltage

pro�le for metalli U = 1 and insulating U = 4 phase. As a next step, one may add

more insulating planes in the barrier region and improve the results of the urrent

and �lling pro edure by taking into a ount more parameters.

In hapter 5, we address the problem of dete ting quantum ordering su h as anti-

ferromagneti ordering in the ultra old atoms as it requires a hieving very low tem-

peratures whi h are not a essible with urrent ooling te hniques. Based on the

DMFT solution of the Fali ov-Kimball model [18℄, we have proposed a new method

to enhan e the riti al temperature of quantum ordering by in reasing the degenera y

of the light parti le in the mixture of light-heavy mixtures. Our method suggests that

the enhan ement for N = 3 is 1.4Tc, whi h is lose to the urrent a hieved tempera-

ture, and for N = 4, the enhan ement will be ompletely a essible with the urrent

te hnique. We further, proposed new mixtures su h as Y b−Cs and Sr−Cs as some of

the promising mixtures with whi h one may dete t the enhan ement of the quantum

order.

Finally, in hapter 6, we study dynami al properties of the XY hain as one of the

most studied quantum spin hains. This model provides a strong platform both for

the theoreti al study of dynami s of quantum system out of equilibrium and pra ti al

point of view in whi h an be realized in trapped ion or other experimental realization

whi h play an important role in the simulation of quantum systems [25, 30℄. We

have studied di�erent dynami al properties su h as the probability of revivals, the

light one velo ity in whi h we analyti ally derive an expression that is dependent

on the initial state [32, 33℄. We have also studied the formation probability of the

210

so- alled rystal on�gurations and the Shannon information of the transverse Ising

hain [31℄. Sin e all of our al ulations have been performed in a omputational basis,

it has advantages that are very ompatible with urrent experimental setup su h as

trapped ions and a one-dimensional array of Rydberg atoms [25, 30℄. We have also

al ulated the post-measurement entanglement entropy and full ounting statisti s

whi h we refer the interested reader to Refs.[34, 35℄.

211

8

Publi ation List of Khadijeh Najafi

1. J. K. Freeri ks, K. Naja�, A. F. Kemper, and T. P. Devereaux, Nonequilibrium

sum rules for the Holstein model, in Femtose ond ele tron imaging and spe tros opy:

Pro eedings of the onferen e on femtose ond ele tron imaging and spe tros opy,

FEIS 2013, 2013 Key West, FL, USA, 2013.

2. K. Naja�, MM Ma±ka, K Dixon, PS Julienne, JK Freeri ks, Enhan ing quantum

order with fermions by in reasing spe ies degenera , Phys Rev A 96 (5), 053621.

3. K. Naja�, MA Rajabpour, Formation probabilities and Shannon information

and their time evolution after quantum quen h in the transverse-�eld XY hain,

Physi al Review B 93, 12:125139, 2016.

4. K. Naja�, MA Rajabpour, On the possibility of omplete revivals after quantum

quen hes to a riti al point, Physi al Review B 96, 1:014305, 2017.

5. K. Naja�, MA Rajabpour, J. Viti, Light- one velo ities after a global quen h

in a non-intera ting model,arXiv:1803.03856.

6. K. Naja�, MA Rajabpour, Entanglement entropy after sele tive measurements

in quantum hains, JHEP, 12:124, 2016.

7. K. Naja�, MA Rajabpour, Full ounting statisti s of the subsystem energy for

free fermions and quantum spin hains, Phys. Rev. B 96, 235109, 2017.

In preparation:

8. K. Naja�, J. A. Ja oby, and J. K. Freeri ks, Nonequilibrium sum rules for the

Holstein-Hubbard model.

212

9. K. Naja� and J. K. Freeri ks, Nonequilibrium urrent-voltage pro�le of a

strongly orrelated materials heterostru ture using non-equilibrium dynami al mean

�eld theory.

10. F. Yang, J. Cohn, K. Naja�, and J. K. Freeri ks, Keldysh-Eigenstate thermal-

ization quantum omputing.

11. K. Naja�, MA Rajabpour, J. Viti, Los hmidt e ho and revivals in the XY-

hain.

213

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