Strong Electronic Correlations First-Principles ... · 2.2 Models of strongly correlated systems . . . . . . . . . . . . . . 10 ... of the electronic structure making the development

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Doctoral Thesis

First-Principles Simulations of Multi-Orbital Systems withStrong Electronic Correlations

Author(s): Surer, Brigitte

Publication Date: 2011

Permanent Link: https://doi.org/10.3929/ethz-a-6665042

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Diss. ETH No. 19902

First-principles simulations of

multi-orbital systems with strong

electronic correlations

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

Brigitte Surer

Dipl. Phys. ETH

born November 8, 1982

citizen of

Arisdorf BL

accepted on the recommendation of

Prof. Dr. M. Troyer, examiner

Prof. Dr. S. Biermann, co-examiner

Prof. Dr. P. Werner, co-examiner

2011

In memory of my mother

“Jenes ganze Spiel von Anfang bis zu Ende studiere ich jetzt,

das heisst, ich arbeite mich durch jeden seiner Satze durch,

ubersetze ihn aus der Spielsprache in seine Ursprache zuruck.”

Hermann Hesse, Das Glasperlenspiel

Abstract

One of the greatest challenges in theoretical solid state physics is the theory of

the electronic properties of materials in the strong correlation regime, which is

characterized by the interplay between itinerant and atomic behavior. These

atomic-like properties arise as a result of complex multi-orbital interactions as

it often concerns materials with partially filled shells and (almost) degenerate

orbitals.

Recent efforts to combine model descriptions of strongly correlated electronic

systems with Density Functional Theory open the possibility to study their

electronic structure from first principles. In this thesis we present such ab-

initio calculations for transition metal impurities and transition metal oxides

taking into account the full intra-atomic Coulomb interactions.

In a first part we consider impurity problems. We identify multiplet struc-

tures in the spectral function of Iron impurities on Sodium and compare

these to photoemission spectroscopies. In a study of the multi-orbital Kondo

effect of Cobalt impurities in as well as on Cupper we compute the Kondo

temperature from first principles.

In a second part we address transition metal oxides, which we treat within

the Dynamical Mean Field Approximation. We study the phasediagram of

the five band Hubbard model on the Bethe lattice, which serves as a generic

model of transition metal oxides and identify magnetic exchange mechanisms.

Concerning the transition metal monoxides Nickeloxide and Cobaltoxide we

compare different formulations of the on-site interaction and compare our

result to photoemission spectroscopies.

v

vi

Zusammenfassung

Eine der grossten Herausforderungen in der theoretischen Festkorperphysik

ist die Theorie der elektronischen Eigenschaften von Materialen im Regime

starker Korrelationen, welches durch das Zusammenspiel von itineranten und

atomaren Verhalten gekennzeichnet ist. Diese atomahnlichen Eigenschaften

ergeben sich aus komplexen multiorbitalen Wechselwirkungen, da es sich oft-

mals um Materialien mit halb gefullten Schalen und (beinahe) entarteten

Orbitalen handelt.

Neuerliche Bestrebungen die Modellbeschreibungen stark korrelierter elek-

tronischer Systeme mit Dichtefunktionaltheorie zu kombinieren, eroffnen die

Moglichkeit, deren elektronische Struktur von Grund auf zu bestimmen. In

dieser Dissertation prasentieren wir solche ab-initio Rechnungen fur

Ubergangsmetallstorstellen und -oxide unter der vollstandigen Beruck-

sichtigung der intra-atomaren Coulomb Wechselwirkungen.

In einem ersten Teil betrachten wir Storstellenprobleme. Wir identifizieren

Multiplett-Strukturen in der Spektraldichte von Eisen-Fremdatomen auf Na-

trium und vergleichen diese zu Photoemissionsspektroskopien. In einer Un-

tersuchung des multiorbitalen Kondo Effekts von Kobalt-Fremdatomen in

sowohl auf Kupfer berechnen wir von Grund auf die Kondotemperatur dieser

Systeme.

In einem zweiten Teil wenden wir uns Ubergangsmetalloxiden zu, welche wir

innerhalb der dynamischen Molekularfeld Approximation behandeln. Wir un-

tersuchen das Phasendiagramm des Funf-band Hubbard Modells auf dem

Bethe-Gitter, welches als generisches Modell fur Ubergangsmetalloxide dient

und identifizieren magnetische Austauschmechansimen. Fur die Ubergangs-

metall-Monoxide Nickeloxid und Kobaltoxid vergleichen wir verschiedene For-

mulierungen der lokalen Wechselwirkungen und vergleichen unsere Resultate

zu Photoemissionsspektroskopien.

vii

viii

Contents

Abstract v

Zusammenfassung vii

1 Introduction 1

1.1 Motivation and Outline . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Description of a solid . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 3

2 Strongly correlated systems 7

2.1 Introduction to strongly correlated systems . . . . . . . . . . . 7

2.1.1 First encounter with strongly correlated systems . . . . 7

2.1.2 Interplay between atomic and itinerant physics . . . . . 8

2.2 Models of strongly correlated systems . . . . . . . . . . . . . . 10

2.2.1 The Hubbard model . . . . . . . . . . . . . . . . . . . 10

2.2.2 Quantum impurity models . . . . . . . . . . . . . . . . 12

2.3 Extension to multi-orbital models . . . . . . . . . . . . . . . . 14

2.3.1 Coulomb interaction matrix . . . . . . . . . . . . . . . 15

2.4 Extension to material-specific descriptions . . . . . . . . . . . 18

2.4.1 Realistic hybridization functions . . . . . . . . . . . . . 18

2.5 Electronic spectroscopy . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Linear Response Theory . . . . . . . . . . . . . . . . . 20

2.5.2 Photoemission spectroscopy . . . . . . . . . . . . . . . 21

2.5.3 Scanning Tunneling Microscopy . . . . . . . . . . . . . 23

3 Impurity solvers 27

3.1 Monte Carlo sampling . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 29

ix

CONTENTS

3.1.2 Markov process . . . . . . . . . . . . . . . . . . . . . . 30

3.1.3 Continuous-time Quantum Monte Carlo . . . . . . . . 31

3.1.4 The Sign Problem . . . . . . . . . . . . . . . . . . . . . 32

3.2 Expansions for quantum impurity models . . . . . . . . . . . . 34

3.2.1 Partition function expansion . . . . . . . . . . . . . . . 34

3.2.2 Weak coupling expansion . . . . . . . . . . . . . . . . . 36

3.2.3 Hybridization expansion . . . . . . . . . . . . . . . . . 36

3.3 Variants of the hybridization expansion solver . . . . . . . . . 41

3.3.1 Density-density approximation . . . . . . . . . . . . . . 41

3.3.2 General local interactions . . . . . . . . . . . . . . . . 43

3.3.3 Krylov-implementation . . . . . . . . . . . . . . . . . . 44

3.3.3.1 Computational Complexity . . . . . . . . . . 47

3.3.3.2 Choice of Trace cutoff . . . . . . . . . . . . . 49

3.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.1 Green’s function measurements . . . . . . . . . . . . . 51

3.4.2 Measurements at τ = β/2 . . . . . . . . . . . . . . . . 52

4 Analytical continuation 53

4.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . 53

4.2 Maximum Entropy method . . . . . . . . . . . . . . . . . . . . 56

4.3 Stochastic optimization method of spectral analysis . . . . . . 59

4.4 Analytic continuation with Pade approximants . . . . . . . . . 61

4.5 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Transition metal impurities on metal surfaces 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Multiplet features in Fe on Na . . . . . . . . . . . . . . . 66

5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2 Model and Computational method . . . . . . . . . . . 67

5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.3.1 Identification of the transitions . . . . . . . . 67

5.2.3.2 Effect of the bath . . . . . . . . . . . . . . . . 69

5.2.3.3 Comparison to PES . . . . . . . . . . . . . . 70

5.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Kondo physics in Co on Cu and Co in Cu . . . . . . . . 72

5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 72


x

CONTENTS

5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.3.1 Quasiparticle spectra . . . . . . . . . . . . . . 77

5.3.3.2 Low energy Fermi liquid . . . . . . . . . . . . 79

5.3.3.3 Estimation of TK from QMC . . . . . . . . . 81

5.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.4.1 Charge fluctuations . . . . . . . . . . . . . . . 85

5.3.4.2 Spin and orbital fluctuations . . . . . . . . . 86

5.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Dynamical Mean Field Theory 91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 Self-consistency equations . . . . . . . . . . . . . . . . . . . . 94

6.3 5-band Hubbard model on the Bethe lattice . . . . . . . 98

6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3.3.1 Determination of the phases . . . . . . . . . . 99

6.3.3.2 Exchange mechanism . . . . . . . . . . . . . . 102

6.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Extension to real materials: LDA+DMFT . . . . . . . . . . . 106

6.4.1 The double-counting problem . . . . . . . . . . . . . . 106

6.4.2 Self-consistency equations . . . . . . . . . . . . . . . . 107

6.4.3 High-frequency expansion . . . . . . . . . . . . . . . . 107

6.5 Transition metal monoxides: NiO and CoO . . . . . . . 110

6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 110


6.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.3.1 Occupation . . . . . . . . . . . . . . . . . . . 112

6.5.3.2 Sector and State statistics . . . . . . . . . . . 112

6.5.3.3 Density of states . . . . . . . . . . . . . . . . 113

6.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Conclusion 117

xi

CONTENTS

xii

Abbreviations

CTQMC Continuous-time Quantum Monte Carlo

QMC Quantum Monte Carlo

ED Exact Diagonalization

DFT Density Functional Theory

LDA Local Density Approximation

GGA General Gradient Approximation

nn Density-Density interactions

SK Slater-Kanamori interactions

full U full four-fermion Coulomb interactions

U Coulomb interaction parameter

J Hund’s coupling

DMFA Dynamical Mean Field Approximation

DMFT Dynamical Mean Field Theory

LDA + DMFT Combination of DFT and DMFT

(I)PES (Inverse) Photo-emission spectroscopy

STM Scanning Tunneling Microscopy

(L)DOS (Local) Density of States

MaxEnt Maximum Entropy Method

xiii

CONTENTS

xiv

Chapter 1

Introduction

1.1 Motivation and Outline

The primary goal of solid state physics is the understanding of materials

and their phenomena. As the physical behavior of electrons underlies mate-

rial properties such as electrical conductivity, thermal conduction, magnetic

properties and many others their description rests upon the understanding

of the electronic structure making the development of theoretical approaches

and computational methods for electronic structure calculation a key task in

this field. Even though electronic structure calculations based on effective-

single-particle formulations are very successful in describing many materials

they fail for a class of materials where strong correlation among electrons turn

the problem into a complicated many body problem. These so-called strongly

correlated materials exhibit many physically as well as technologically inter-

esting properties, most notably the phenomena of high-temperature super-

conductivity (Nobel prize 1987, Bednorz and Muller) and giant magneto-

resistance (Nobel prize 2007, Fert and Grunberg) and much effort has been

put into determining their electronic structure in the last decades. Charac-

teristic for these strongly correlated systems is that the electrons are neither

fully localized nor fully delocalized - both kinetic and potential energy terms

are important rendering perturbative approaches impractical and quantum

Monte Carlo (QMC) methods which could in principle deal with interacting

many body problems suffer from the fermionic sign problem. The theory of

electrons in solids therefore poses one of the greatest challenges in theoretical

physics.

The track record of Dynamical Mean Field Theory (DMFT) in combina-

1

1.1 Motivation and Outline

tion with Density Functional Theory (DFT) has promoted the self-consistent

quantum impurity approximation to a standard tool for the investigations of

strongly correlated solids. The framework owes much of its success to the re-

cent development of continuous-time quantum Monte Carlo (CTQMC) impu-

rity solvers, which allow for a numerical exact solution of quantum impurity

models. In particular, the hybridization impurity solver opened the way to

accurately treat intra-atomic interactions on the impurity thereby accounting

for the fact that several orbitals are physically relevant for the description of

elements of the d- and f -series. This is not only important from a quantita-

tive point of view, because the enlargement of the local degrees of freedom

as well the role of degeneracy brings in new physical aspects in these multi-

orbital systems. The capabilities of these numerical methods open up new

possibilities to start exploring the performance of first-principles simulations.

Predicting physical properties of a material compound from first principles

means that only the atomic charges and crystal structures are provided with-

out using any empirically adjustable parameters for the calculation. Studies

benchmarking first-principles simulations mark an important first step to-

wards answering the question how crystal structures and atomic properties

affect the electronic behavior of strongly correlated materials and potentially

also how these properties could be brought together to design new materials

with desired properties.

This thesis is outlined as follows. After an introduction to materials cal-

culations we start in Chapter 2 with an introduction to strongly correlated

electron systems and introduce the models and experimental probes to study

their electronic structure. In Chapter 3 and 4 we explain the computational

methods to solve these models and analytical continuation methods to ob-

tain the observables which can be compared to experiments. In the second

part we present applications to multi-orbital systems with strong electronic

correlations. In Chapter 5 we will focus on genuine quantum impurity sys-

tems and study transition metal impurities in as well on non-magnetic host

materials. First, we study multiplet features in the spectral function of Fe on

Na and compare our quantitative predictions to photoemission experiments.

Second, we investigate the multi-orbital Kondo physics of Co impurities cou-

pled to a Cu host and determine the Kondo temperature from first-principles.

In Chapter 6 we present simulations of strongly correlated electron systems

within the Dynamical Mean Field Approximation (DMFA). We investigate

the 5-band Hubbard model on the Bethe lattice, which serves as a generic

2

Introduction

model for transition metal compounds, and show how magnetic exchange

mechanisms, which also occur in transition metal oxides, can be identified.

In a study of the transition metal monoxides NiO and CoO we compare

different formulations of the on-site interaction and compare our results to

photo-emission experiments. In the last Chapter we present a summary

and concluding statements of this thesis on “First-principles simulations on

multi-orbital systems with strong electronic correlations”.

1.2 Description of a solid

The microscopic theory of any material is in principle known and determined

by the Schrodinger equation of a many body quantum system. To model

a solid we assume that electrons are moving in a positively-charged back-

ground of lattice ions forming a potential Vlattice. Given the many-particle

wavefunction Ψ(x1, ..., xN , t), the properties of the system can be determined

by solving Hψ = i~∂Ψ∂t

[1] with

H = − ~2

2m

N∑

j=1

∆j +∑

i<j

Ve-e(xi − xj) +∑

j

Vlattice(xj) (1.1)

and Ve-e(xi − xj) =e2

4πǫ0|xi−xj |the Coulomb repulsion between the electrons.

For indistinguishable particles, such as electrons, the Hilbert space size grows

exponentially as dN with d the dimension of the single-particle Hilbert space.

Solving above equation for a macroscopic system of N = 1023 particles is in-

tractable and therefore requires an approximation describing the low-energy

physics of these systems. The most notable method which is applied not only

in many scientific fields but also in engineering is the DFT, for which Kohn

has been awarded the Nobel prize in 1998.

1.3 Density Functional Theory

Hohenberg and Kohn showed [2] that there exists a functional Φ of the elec-

tron density n(r) consisting of a universal and a material-specific part:

Φ [n(r)] = Φuniv [n(r)] +∫

drVlattice(r)n(r) (1.2)

which is minimized by the physical density belonging to the ground state of

the described electronic system. If the universal functional Φuniv was known

3


it would suffice to minimize the functional in order to obtain the density and

energy of the ground state. As the functional is not known, we approach the

problem by making an ansatz for Φuniv. We decompose it into a kinetic term

Φkin, a Hartree-term Φhartree and a term Φxc taking into account exchange

and correlation effects. With the Hartree-term:

Φhartree [n(r)] =e2

2

∫

drdr′n(r)n(r′)

|r− r′| (1.3)

the variational problem is then stated as:

δΦ [n(r)] =∫

drδn(r)

(

Vlattice + e2∫

dr′n(r′)

|r− r′| +δΦkin

δn(r)+δΦxc

δn(r)

)

= 0

(1.4)

with the additional constraint of particle number conservation:

∫

drδn(r) = 0. (1.5)

Given a basis φµ the variational equation has the same structure as for a

single-particle problem:

(

− 1

2m∇2 + Veff(r)

)

φµ(r) = ǫµφµ (1.6)

if we identify

Veff(r) = Vlattice + e2∫

dr′n(r′)

|r− r′| + vxc(r). (1.7)

The exchange-correlation potential vxc(r) =δExc

δn(r), which depends on the en-

tire density distribution

n(r) = 2

N/2∑

µ=1

|φµ(r)|2 , (1.8)

remains unknown and we have to choose an ansatz. One ansatz is the so-

called Local Density Approximation (LDA). We take the exchange-correlation

potential to have the same form as for the free electron gas. Under the as-

sumption that the electron density does not vary much we have

vxc(r) ∝ n(r)1/3 (1.9)

4

Introduction

with n(r) the electron density of the interacting system.

Eqs. (1.6), (1.7) and (1.8) are called the Kohn-Sham equations [3]. The

eigenenergies ǫµ can be used to approximate the physical band structure.

The Kohn-Sham equations allow to overcome the problem of minimizing

an unknown functional by reformulating the problem as an effective single-

particle problem with a self-consistency condition, which can be solved iter-

atively. We will encounter this idea of mapping a variational problem onto

an auxiliary problem, embedded in an iterative self-consistency cycle again

in Chapter 6, where we discuss the Dynamical Mean Field Approximation.

We have seen that DFT greatly reduces the computational complexity be-

cause we can treat the high-dimensional many-body problem by solving a

one-dimensional problem.

For many materials and applications DFT is amazingly accurate in predicting

physical properties. For strongly correlated systems however it does not

adequately capture the correlations among electrons which are often decisive

for the emergent phases in these systems.

5


6

Chapter 2

Strongly correlated systems

2.1 Introduction to strongly correlated systems

2.1.1 First encounter with strongly correlated systems

Electronic structure calculations were first realized with the development of

band theory by Bethe, Bloch and Sommerfeld [4–6] and allowed for a first

explanation for the existence of metals and insulators. The periodic ionic

potential gives rise to a discrete energy band structure. Electrons occupy

these bands up to the Fermi-level EF . Depending on whether the highest

occupied band is half-filled, electron transport is possible and the system is

metallic, otherwise the system is insulating. The classification within band

theory into metallic and insulating worked very well for systems with their

valence electrons in s- and p-orbitals. However, as realized first by de Boer

and Verwey [7] this classification fails for some transition metal compounds

with partially filled d-bands, which are either poor-conducting or even in-

sulating, It was argued in 1937 by Mott and Peierls [8] that for half-filled

d-bands strong Coulomb interactions on d-orbitals cause a suppression of

electron mobility. The suppression of mobility comes into existence because

an electron moving through the lattice will be exposed to the Coulomb in-

teraction when spending time on already-occupied orbitals. Such a large

Coulomb repulsion between electrons of opposite spin localized on the same

site can split a single metallic band. The resulting insulator is nowadays

known as a Mott-insulator.

7

2.1 Introduction to strongly correlated systems

2.1.2 Interplay between atomic and itinerant physics

Characteristic for strongly correlated systems is the interplay between atomic

and itinerant physics [9]. For a small wave-function radius compared to the

interatomic distance the system will be localized and atomic-like whereas for

an extended wavefunction radius orbitals start to overlap and we have elec-

tron transport in the formed bands. If we rearrange the periodic table by

r

R(r)

r

R(r)

Ce Pr NdPmSm Eu Gd Tb Dy Ho Er Tm

Th Pa U Np PuAmCmBk Cf Es FmMd

Sc Ti V Cr Mn Fe Co Ni

Y Zr Nb Mo Tc Ru Rh Pd

Lu Hf Ta W Re Os Ir Pt

partially filled shell

4f

5f

3d

4d

5d

Figure 2.1: Kmetko-Smith diagram: The periodic table is ordered according to

the degree of localization of the radial wavefunction, displayed on the left. The

table shows, where the behavior changes from magnetic and atomic-like (upper

right) to non-magnetic and itinerant (lower left). Strongly correlated electron

materials are usually those with elements at or close to the diagonal [10].

sorting according to the wavefunction radius we obtain the Kmetko-Smith

diagram [11] shown in Figure 2.1. In the lower-left corner we have non-

magnetic and itinerant behavior which is well-described by band theory. In

the upper right corner we have atomic like behavior. From atomic physics

we know that according to Hund’s rule spins will align such as to form a

local moment. The f-electrons in this regime do often not participate in

the bonding. Bonding and local moment formation are often not reconciled

as we know e.g. from the solution of the Heitler-London model of the H2-

molecule. Across the diagonal we have a cross-over from the metallic state

to the insulating magnetic state. In the cross-over regime on the diagonal

materials with these elements exhibit both itinerant and atomic-like features

at the same time [10]. Materials in this regime have interesting properties

such as high-temperature superconductivity, large magneto-resistance, struc-

8


tural instabilities, large effective masses in the case of Heavy-Fermions and

many other properties [10, 11]. Close to this line are also the well-known

ferromagnets Fe, Co and Ni, whose ferromagnetic behavior can be intuitively

understood from the Stoner criterion Uρ(EF ) > 1 as the localized nature of

the d-orbitals causes distinct peaks in the density of states (DOS) ρ(E) at

the Fermi-level [12].

The classification based on the wavefunction radius and interatomic distance

is very simplistic and the sketched line in Figure 2.1 should not be taken too

explicit. The position of this metal-insulator crossover will also depend on

the degeneracy of involved narrow bands, i.e., one has to take into account

whether the orbitals are degenerate or whether the degeneracy is lifted due

to crystal field effects or Jahn-Teller distortions as has been shown e.g. in a

study of 5d transition metal oxides [13]. As a matter of fact the important

role of degeneracy in these systems is one of the reasons why there is need of

computational methods which can accurately treat the multi-orbital aspects

of these systems.

In this intermediate regime the electronic behavior is very peculiar. What

happens is that when electrons move from atom to atom the motion is corre-

lated in a way to give the properties characteristic of the atomic theory. On

partially-filled orbitals electrons with opposite aligned spins will be repelled,

while others will be favored in order to satisfy the intra-atomic interactions.

These atomic properties persist beyond the time-scale τ = a0vF

of band mo-

tion determined by the Fermi-velocity vF and interatomic distance a0. It

therefore looks to us as if the properties such as spin are associated with

the atom and not the electron moving through the lattice. This correlated

manner of electron motion is the reason why we speak of strongly correlated

systems [14].

9

2.2 Models of strongly correlated systems


When studying strongly correlated systems we are interested in those sys-

tems, where the localized nature of the orbitals gives rise to an interplay be-

tween atomic and itinerant physics. Models describing these systems should

capture the localized nature of the bonding d- or f -electrons. In this Section

we will discuss the models which describe these systems.

2.2.1 The Hubbard model

The Hubbard model was proposed in 1963 independently by Hubbard, Kanamori

and Gutzwiller [14–16]. It was introduced as a model of strongly corre-

lated systems to describe metal-insulator transitions and ferromagnetism.

Besides ferromagnetism it also exhibits antiferromagnetism, ferrimagnetism

and paramagnetism and is used as a model in the field of high-temperature

superconductors [17,18]. We start from a one-band model and add electron-

U

↑↓

t

Figure 2.2: In the one-band Hubbard model electrons hop to neighboring sites

with a hopping amplitude t. If a site is doubly occupied the energy is increased

by the on-site Coulomb repulsion U .

electron interactions to the kinetic part:

H =∑

k,σ

ǫkc†k,σck,σ +

1

2

∑

k1,k2,σ,k1

′,k2′,σ

⟨

k1k2.

∣∣∣∣

1

r12

∣∣∣∣k1

′k2′

⟩

(2.1)

The operator c†k,σ (ck,σ) create (annihilate) a Bloch state with momentum

k, spin σ and eigenenergy ǫk and⟨

k1k2.∣∣∣

1r12

∣∣∣k1

′k2′⟩

denotes the matrix-

10


element of the Coulomb interaction of two each other repelling electrons at

a distance r12.

We want to make use of the localized nature of the orbitals and represent

the Hamiltonian in a Wannier-function basis:

H =∑

i,j,σ

tijc†i,σcj,σ +

1

2

∑

i,j,σ,i′,j′,σ′

⟨

ii′∣∣∣∣

1

r12

∣∣∣∣jj′

⟩

c†i,σc†i′,σ′cj′,σ′cj,σ (2.2)

where i, j denote different lattice sites at the position of the ions. Under

the assumption that the overlap⟨

ii′∣∣∣

1r12

∣∣∣ jj′

⟩

is small and negligible for

different sites compared to the on-site term Uiiii =⟨

ii∣∣∣

1r12

∣∣∣ ii

⟩

we arrive at

the Hamiltonian of the one band Hubbard model:

H =∑

i,j,σ

tijd†iσdjσ + U

∑

i

ni,↑ni,↓ (2.3)

where we switched the notation of the electron creation (d†iσ) and annihila-

tion (diσ) operator to emphasize that we are dealing here with electrons in

localized d- or f -orbitals. The number operator niσ = d†iσdiσ delivers the

occupation of the state with spin σ at site i. A sketch of this model is shown

in Figure 2.2. The hopping of electrons is described by the amplitude tij = t

and stands for a finite probability for an electron transfer from site i to j.

The hopping itself however need not to take place directly, e.g. in transition

metal oxides the electron transfer is mediated via an oxygen. If two elec-

trons reside on the same site there is an additional energy cost U due to the

Coulomb repulsion.

The Hubbard Hamiltonian conserves the particle number and preserves the

spin-rotation symmetry (SU(2)-symmetry), i.e., the occupation number op-

erator N and spin operators Sα, α = x, y, z commute with the Hubbard

Hamiltonian and therefore the operators N , S2, Sz (or other component α)

and the Hamiltonian H can be simultaneously diagonalized.

For large U and at half-filling the model can be simplified to the Heisenberg

model:

H = J∑

j

SjSi (2.4)

with spin operators S at sites i and j and J = 4t2

Ubeing an effective antifer-

romagnetic coupling due to the super-exchange mechanism in the Hubbard

model [19].

11


Even though the Hubbard model has a very simple form a perturbative ap-

proach in the regime between the (far) metallic and insulating regime fail,

because there is no weak coupling parameter that would allow for a well-

controlled diagrammatic expansion. The Hubbard model can however be

treated within the DMFA presented in Chapter 6.

2.2.2 Quantum impurity models

Quantum impurity models appear in two contexts in the field of strongly

correlated systems. As an auxiliary problem they are the working horse to

obtain solutions within the DMFA (see Chapter 6). As a physical system they

represent the most basic non-trivial system where the itineracy competes with

the atomic tendencies such as local moment formation. A quantum impurity

Ed + U

Ed

(a)

Ed + U

Ed

(b)

Ed

ρ(E)

E

(c)

Figure 2.3: We consider a transition metal impurity whose electron is in a bound

state with level energy Ed as can be seen from the potential in (a). If the impurity is

immersed in a metallic host (b) electrons will tunnel through the potential barrier

between the impurity and the conduction sea. In the spectral function (c) the level

broadens to a peak with width ∆.

describes an impurity of the d- or f -series which is immersed in a metal

host. In order to illustrate the physics in such a system we will consider a

transition metal impurity with level energy Ed (see Figure 2.3). When it is

placed in a metallic host, there will be a finite probability for the electron

on the impurity to tunnel through the potential barrier into the conduction

sea [20]. The electron has a finite life-time on the impurity, mirrored in

the fact that the level broadens to a resonance with width ∆, which sets a

new energy scale in the system. The electron transitions from and to the

conduction sea cause a mixing of the impurity states. The physics described

is captured by the Single Impurity Anderson model introduced by Anderson

12


in 1961 [21]:

H =∑

p

ǫpa†pap +

∑

p,j

(V d†σap + V ∗a†pdσ

)+ Un↑n↓ +

∑

σ

Edσd†σdσ (2.5)

with p = (k, σ) specifying momentum and spin of the conduction band elec-

trons, d†σ (dσ) creating (annihilating) impurity states and V the hybridization

describing the transition strength from the impurity to the bath and vice-

versa. A sketch is shown in Figure 2.4. The virtual exchange processes are

Impurity

Electron Reservoir

Vpe−

V ∗p

e−

Figure 2.4: In the Anderson model a single impurity with on-site Coulomb in-

teraction U is coupled to an electron reservoir with which it exchanges electrons.

The transition strength is determined by the hybridization Vp.

similar to the super-exchange process in the Hubbard model and also give

rise to an effective antiferromagnetic coupling. Kondo showed [22] that the

antiferromagnetic coupling causes the increase of resistivity towards very low

temperatures nowadays known as the Kondo effect. The temperature scale at

when this resistivity increase takes place is the so called Kondo temperature

TK associated with the quasi-particle width ∆.

Already in this seemingly simple impurity problem we encounter several en-

ergy scales set by the interaction U , the hybridization V and the dynamically

generated resonance width ∆ at high, intermediate and low energy scales re-

spectively.

A sketch of the spectral function of the Single Impurity Anderson model is

shown in Figure 2.5. At the Fermi-level at E = 0 we have the previously

discussed quasi-particle resonance. Further away we have lower and upper

Hubbard bands split by an energy U . They mark the transitions d1 →d0 + e− and d1 + e− → d2. If charge fluctuations become negligible at lower

temperatures we can translate the Single Impurity Anderson Model into an

effective spin model. Via a Schrieffer-Wolff procedure [23] the high-energy

states are projected out leading to a reduced energy window, where the charge

13

2.3 Extension to multi-orbital models

fluctuations occur as virtual processes only and spin degrees of freedom are

the only remaining degrees of freedom. The impurity system is in this case

well described by the Kondo model:

H =1

2N

∑

k,q

Jkq

[

Sz(c†qσckσ − c†qσckσ + S+c

†qσckσ + S−c

†qσckσ

]

(2.6)

where conduction electrons with momenta k and spin σ get scattered by the

localized spin via the exchange coupling Jkq. The idea behind the mapping

is sketched in Figure 2.5.

EF ω

Ad(ω)

Anderson

Kondo

d1 → d0 + e− e− + d1 → d2

Figure 2.5: Spectral function of the single impurity Anderson model. Lower

and upper Hubbard bands describing electron removal and addition processes are

separated by the energy U . At the Fermi-level EF there is a quasi-particle peak.

By projecting out high-energy states in the Anderson model only spin degrees of

freedom remain which are captured by the Kondo model.


Impurities with partially filled d- or f -orbitals often have (almost) degenerate

orbitals which results in more possible exchange processes, not only between

the impurity and the bath but also between the orbitals of the impurity. In

order to include all of the physically relevant orbitals we have to go beyond

the one-orbital case. A multi-orbital treatment requires to deal with intra-

atomic interactions having a more complex structure and we replace the

14


Coulomb interaction U by a more general four-fermion Coulomb interaction

matrix Uσσ′

αβγδ:

HCoul =∑

iαβγδσσ′

Uσσ′

αβγδd†iασd

†iβσ′diδσ′diγσ (2.7)

with Greek indices ασ denoting orbital and spin of the atom at site i.

With a generalized hopping amplitude tαβij the Hubbard model in its multi-

orbital description reads

H =∑

ijαβ

tαβij d†iασdjβσ +

∑

iαβγδσσ′

Uσσ′

αβγδd†iασd

†iβσ′diδσ′diγσ −

∑

iασ

µd†iασdiασ (2.8)

For multi-orbital extension of the Single-orbital Anderson impurity model

we will also replace the Coulomb interaction U by the four-fermion Coulomb

matrix Uσσ′

αβγδ and will describe the possible transitions from the impurity to

the bath by a hybridization matrix V αp . In the following we will often refer

to the multi-orbital Anderson impurity model in terms of the decomposition:

H = Hbath +Hhyb +Hloc (2.9)

with

Hbath =∑

p

ǫpa†pap (2.10)

Hhyb =∑

pασ

V αp d

†ασap + V α∗

p a†pdασ (2.11)

and

Hloc =∑

ασ

ǫαd†ασdασ +HCoul (2.12)

where we use one of the formulations Eq. (2.7), (2.17) or (2.18) for HCoul.

2.3.1 Coulomb interaction matrix

The orbital index can be specified by its magnetic quantum number m taking

the valuesm = −l,−l+1, ...,+l with l = 2 for d-orbitals. Given the Coulomb

repulsion Vee the Coulomb matrix element Uαβγδ = 〈m,m′′|Vee|m′′m′′′〉 canbe expressed through the Slater integrals F k [24]:

〈m,m′′|Vee|m′m′′′′〉 =∑

k

ak(m,m′, m′′, m′′′)F k. (2.13)

15


If the orbital-basis consists of spherical harmonics the coefficients ak are given

by

ak(m,m′, m′′, m′′′) =

k∑

q=−k

(2l + 1)2(−1)m+q+m′

×(l k l

0 0 0

)

×(

l k l

−m q m′

)

×(

l k l

−m′′ −q −m′′′

)

with the Wigner 3j-symbols

(j1 j2 j3m1 m2 m3

)

(2.14)

which are computed using the Racah formula [25].

For d-orbitals only the Slater integrals F 0, F 2 and F 4 are needed [24].

The F 0-term stabilizes a state with a particular number of d-electrons and

can be identified with the screened Coulomb interaction U . This U -value is

defined as the energy cost to move an electron between two atoms having

initially n electrons:

U = E(dn)−E(dn−1) + E(dn)− E(dn+1) (2.15)

In a solid this energy difference is reduced compared to the atomic case due

to screening.

The F 2- and F 4-term stabilize a particular (L, S) configuration of a dn-

configuration with spin S and angular momentum L satisfying the Hund’s

rule [26]. For 3d elements the ratio F 4/F 2 has been experimentally found

to be constant F 4/F 2 ∼ 0.625 [27]. The two values are associated with the

Hund’s coupling J via [28]

J = (F 2 + F 4)/14. (2.16)

In order to account for the screening effects the U and J parameters are

derived from a constrained LDA calculation using a supercell approximation

[29].

A special case of the four-fermion Coulomb interactions is a simplified scheme

16


in the style of the scheme proposed by J. Kanamori:

HCoul = U∑

α,σ

ndiασndiασ

+ (U − 2J)∑

α6=β,σ

ndiασndiβσ

+ (U − 3J)∑

α6=β,σ

ndiασndiβσ

+ J∑

α6=β,σ

(

d†iασd†iασdiβσdiβσ − d†iασd†iβσdiασdiβσ

)

,

(2.17)

with the spin index σ = −σ and ndiασ the occupation number operator of the

corresponding d-operator.

The terms only depend on whether the electrons belong to the same orbital

(first line) or to different orbitals (second and third line for parallel and an-

tiparallel spins) assuming that the repulsion between different orbitals α and

β is the same for all pairs of orbitals. However, since the Coulomb repulsion

between the dxz, dyz, dx2−y2 , dxy and the dz2 orbitals are not the same the

inter-orbital interaction U ′ = U − 2J is approximated by the average repul-

sion between all different orbitals. The Slater-Kanamori ansatz preserves the

multiplet-character of the electron-electron interaction but will lead to more

degenerate states compared to the general four-fermion Coulomb interaction

matrix.

In the density-density approximation the spin-exchange and pair-hopping

terms in the last line of HCoul are neglected and the terms with only the

occupation number operators are kept:

HCoul = U∑

α,σ

ndiασndiασ

+ (U − 2J)∑

α6=β,σ

ndiασndiβσ

+ (U − 3J)∑

α6=β,σ

ndiασndiβσ

(2.18)

In the density-density approximation the spin-rotation symmetry is broken.

However, this approximate form has been commonly used in QMC studies,

mainly due to technical limitations of the previously available algorithms.

17

2.4 Extension to material-specific descriptions

2.4 Extension to material-specific descriptions

Even though DFT fails to describe strongly correlated systems we can use

it as a starting point for the simulations of strongly correlated systems by

enhancing it with a local Hubbard-like interaction for the d-or f -orbitals.

The Kohn-Sham Hamiltonian thereby serves as the non-interacting reference

frame onto which we add local intra-atomic interactions. Since formulations

such as the Hubbard model or Anderson impurity model are model Hamil-

tonians, a combination with DFT allows to make use of its first-principles

capabilities to include material specific effects. It can therefore also be viewed

as a way to upgrade the model Hamiltonians in order to bring in realistic

microscopic details.

Instead of a dispersion defined by the Fourier transform of tαβij we can use

a realistic band structure obtained as the Kohn-Sham eigenvalues ǫKS(k)

of a DFT calculation. If also other bands than the narrow bands are rel-

evant to describe the system as e.g. in charge-transfer systems the Kohn-

Sham-Hamiltonian not only includes the correlated subspace (d) but also the

uncorrelated part (p):

ǫKS(k) =

[hddk,αβ hdpk,αγhpdk,γα hppk,γδ

]

(2.19)

and the p− d-Hubbard-Hamiltonian reads:

H =∑

k,σ

(hddk,αβd

†kασdkβσ + hppk,γδp

†kγσpkδσ+

hdpk,αγd†kασpkγσ + hpdk,γαp

†kγσdkασ

)+

∑

i,σ,σ′

Uσσ′

αβγδd†iασd

†iβσ′diδσ′diγσ.

(2.20)

The d(†)kασ and p

(†)kγσ operators are the Fourier transforms of the annihilation

and creation operator d(†)iασ and p

(†)iγσ.

2.4.1 Realistic hybridization functions

Similar to the material specific extension of Eq. (2.20) we can use DFT

to provide realistic hybridization functions. In order to do so we need a

projection scheme to express the hybridization within a basis of localized

orbitals. DFT formulations are often based on delocalized plane-wave basis

18


functions. This is convenient because the basis is simple and universal and

the convergence is controlled by the energy cutoff. An example of such a

DFT formulation is the projector augmented wave method (PAW). Since

the model Hamiltonians operate on localized orbitals we need to project the

Bloch states onto localized orbitals. A suitable choice of basis to represent

these are Wannier functions.

DFT enters the calculation by providing the non-interacting Greens functions

or hybridization functions in terms of local orbitals.

We define a set of localized orbitals |L〉 spanning the correlated subspace,

i.e. the Hilbert-space where many-body correlations are important to treat

and corrections to DFT necessary. From this basis set we construct a pro-

jector Pcorr =∑

L |L〉〈L| with which we can express the Kohn-Sham Green’s

function in the |L〉-basis:

GcorrKS (ω) = PcorrGfull

KS(ω)Pcorr (2.21)

with the Kohn-Sham Green’s function constructed from the Kohn-Sham

Hamiltonian eigenstates |K〉 in the following way

GKS(ω) =∑

K

|K〉〈K|ω + i0+ − ǫK

. (2.22)

In order to evaluate the Green’s function in terms of the correlated localized

orbitals projections of the type 〈L|K〉 need to be evaluated.

The computation of these projections depends on the construction of the un-

derlying Wannier functions. For the Projected Local Orbital Method, which

has been applied to obtain the hybridization functions used in the studies of

Chapter 5, these constructions and evaluations of 〈L|K〉 are detailed in the

paper by Karolak et al. [30].

The hybridization function ∆α(ω) which we provide as an input quantity

for our simulations is obtained from the Green’s function Gα(ω) in terms of

correlated orbitals α as

∆α(ω) = ω + iδ − ǫα −G−1α (ω) (2.23)

where the static crystal field ǫα is extracted from the limit

limω→∞ ω −G−1α (ω)→ ǫα.

19

2.5 Electronic spectroscopy


By electronic spectroscopy the electronic structure of a material can be mea-

sured. Here, we discuss the microscopic theory of these experimental methods

and establish the connection between simulations and experiments. The ex-

perimental tools to which we compare to are Scanning Tunneling Microscopy

(STM) and (Inverse) Photo-emission spectroscopy ((I)PES). We start with

the basic definitions in linear response theory and apply these to describe

(I)PES and STM. In particular, we show how the DOS can be extracted and

motivate the observables we are going to measure in order to estimate the

predictive power of the presented first-principles simulations. The presented

approaches in this Section are adapted from Nolting’s textbook on many

body physics [31].

2.5.1 Linear Response Theory

When performing experiments we want to learn how a physical system reacts

when a field or force is applied and how it can be manipulated in the context

of technological applications. Linear response theory describes in a first order

approximation how a physical system reacts upon a perturbation. By ”0”

we will denote the unperturbed system which is described by a Hamiltonian

H0. Given a field F (t) coupling to an observable B(t) via a perturbation

V (t) = BF we want to express the induced change 〈A〉 ≈ 〈A0〉 + ∆A of an

observable A(t). In linear response theory the response is given by

〈A(t)〉 = 〈A〉0 −i

~

∫ t

−∞

dt′F (t′)Tr

e−βH0

[

A(t), B(t)]

(2.24)

By introducing the two-sided retarded Green’s function

GretAB(t, t

′) = 〈〈A(t), B(t′)〉〉 = −iΘ(t− t′) 〈[A(t), B(t′)]〉0 (2.25)

where Θ(t− t′) is the heaviside function Eq. (2.24) can be rewritten as

∆A(t) =1

~

∫ ∞

−∞

dt′F (t′)′GretAB(t, t

′). (2.26)

The Green’s function can be expressed in the spectral representation, where

it is being related to the spectral density SAB. In the energy-domain this

relation reads

GretAB(E) =

∫ ∞

−∞

dE ′ SAB(E′)

E −E ′ + iδ. (2.27)

20


When we discuss (Inverse) Photo-emission spectroscopy and Scanning Tun-

neling Microscopy we will establish the connection between Green’s functions

and the spectra of the elementary excitations underlying the electronic spec-

troscopy methods.

2.5.2 Photoemission spectroscopy

With the help of one-particle spectroscopies we study excitation processes,

which add or remove a particle to or from the system. In Photo-emission

spectroscopy an incident photon is absorbed by an electron within an occu-

pied energy band. For a large enough energy transfer E = ~ω the electron

will be knocked out of the solid and collected by a detector of the experimen-

tal setup. By analyzing the electron’s kinetic energy it is possible to infer

the structure of the occupied bands. The transition from the system having

N electrons to having N −1 electrons is described by the transition operator

T−1 = dα, where dα annihilates an electron in the state |α〉. The elementary

process of PES is sketched in Figure 2.6.

EF

e−

~ω

E

EF

e−E

Figure 2.6: Elementary process of Photoemission Spectroscopy: An incident

photon with energy ~ω removes an electron from the system.

The reverse process takes place for Inverse Photo-emission Spectroscopy.

Whereas PES allows to study the structure of occupied bands, its comple-

mentary spectroscopy IPES address the structure of (partially) unoccupied

bands. Here, an incident electron will occupy an unoccupied state |β〉 of anenergy band. The released energy is emitted as a photon, whose energy ~ω

will be analyzed in the detector. With T+1 = d†β the electron addition is

described and sketched in Figure 2.7.

Since the particle number of the system changes for the transition con-

sidered we will study the system using a grand-canonical ensemble Ω =

Tr [exp(−βHµ)], with Hµ = H − µN whose eigenenergies and eigenstates

21


EF

e−

E

EF

~ω

E

Figure 2.7: Elementary process of Inverse Photoemission Spectroscopy: An

incident electron is added to the system. The excess energy is released as a photon

with energy ~ω.

will be denoted as

Hµ|En(N)〉 = (En(N)− µN)|En(N)〉 ≡ En|En〉. (2.28)

With probability p ∝ |〈Em|Ts|En〉|2, s±1 there will be a transition from state

|En > to state |Em >. Summing over all transitions with an excitation energy

within [E,E + dE] we will obtain the intensity Is(E) of these elementary

processes:

Is(E) =1

Ω

∑

m,n

e−βEn |〈Em|Ts|En〉|2 δ(E − (Em − En)). (2.29)

Since Ts = T †−s the intensities of PES and IPES are connected by the relation:

Is(E) = eβEI−s(−E). (2.30)

This symmetry relation motivates to combine both quantities into the so-

called spectral density:

1

~Aǫs(E) = I−s(E)− ǫIs(−E) =

(eβE − ǫ

)Is(−E), ǫ = ±1 (2.31)

As mentioned previously it is closely related to the retarded Green’s function

introduced in Eq. (2.25), as can be seen when Fourier-transforming Aǫs(E)

to the time-domain:

Aǫ(t, t′) =1

2π〈[Ts(t), Ts(t′)]−ǫ〉0. (2.32)

The intensities of the (Inverse) Photo-emission spectroscopies directly mea-

sure the spectral density A(E). If we compute the corresponding Green’s

function in our simulations we gain the information about the spectral den-

sity and compare in this way our results to experiments.

22


2.5.3 Scanning Tunneling Microscopy

The scanning tunneling microscopy has been invented by Rohrer and Binnig

[32] for which they were awarded the Nobel prize in 1986. With a scanning

tunneling microscope the surface of a sample is probed with a scanning tip. A

voltage V is applied and thereby a tunneling current I flows between the tip

and the sample as shown in Figure 2.8(a). The current-voltage measurement

not only provides information on the topology of the sample but also probes

its electronic structure. In Figure 2.8(b) we sketch the underlying microscopic

process, where an electron is transferred from the tip to the sample. The tip

stm

tip

I

sample

V

(a)

sample e−

tip

Vvacuum

µ1 µ2

distance

energy

(b)

Figure 2.8: (a) STM setup: A tip is brought close to the sample. When a

voltage is applied a tunneling current flows between the tip and the sample thereby

probing the surface of the sample. (b) Elementary process of STM: An electron is

transferred from the tip to the sample. The differences in Fermi-energies µ2 − µ1

is equal to the applied voltage V .

and the sample shall be described by a Hamiltonian H0 = Htip + Hsample

and the interaction describing the tunneling from the tip to the sample is

modeled as:

Htunnel = Γe−ieV t/~(d†a+ da†

)=: Htunnel + H†

tunnel (2.33)

where the operators d(†) create or annihilate an electron in the sample and

a(†) correspondingly in the tip. The coupling parameter Γ shall be weak.

We are interested in the tunneling current I(t) = 〈I〉, with the tunneling

current operator:

I(t) = −ieΓ(d†ae−ieV t/~ − da†e−ieV t/~

)(2.34)

23


We compute the expectation value 〈I〉 using linear response theory:

〈I〉 = 〈I0〉︸︷︷︸

=0

− i~

∫ t

−∞

dt′Tre−βH0 [I(t), Htunnel(t

′)]

(2.35)

The only terms from the commutator [I(t), Htunnel(t′)] which contribute to

the trace are[

Htunnel, H†tunnel

]

and we find

〈I〉 = 1

~

∫ t

−∞

dt′e−ieV t/~Tr

e−βH0

[

Htunnel, H†tunnel

]

=1

~

∫ ∞

−∞

dt′e−ieV t/~GretI (t, t′)

with the retarded Green’s function GretI (t, t′) introduced in Eq. (2.25).

Applying the Fourier transform GretI (t, t′) = 1

2π

∫dωe−iω(t−t

′)GretI (ω) we find

the current to be constant in time and equal:

I(t) =eΓ2

hGret

I (ω). (2.36)

Next, we want to show how I(t) relates to the local density of states (LDOS).

From the spectral representation of GretI (ω) we find that we can express

GretI (ω) as:

GretI (ω) =

(eβ − 1

)∫ ∞

−∞

dteiωt〈HtunnelH†tunnel〉0 (2.37)

and by making use of the fact that d(†) and a(†) operate on different spaces we

realize that I(t) depends on the sample and tip one-electron Green’s function:

I(t) =eΓ2

h

(eβ − 1

)∫ ∞

−∞

dteiωt〈d†(t)d(0)〉sample〈a†(0)a(t)〉tip. (2.38)

Using again the spectral representation we can express I(t) in terms of the

LDOS A(ω) of the sample and the tip:

I(t) =eΓ2

h

(eβ − 1

)∫ ∞

0

dω′Asample(ω)

1 + eβ~ω′

Atip(ω′ − eV

~)

1 + eβ~(ω′− eV

~). (2.39)

We proceed by considering the case at small temperatures. Approximating

above equations for β large and using(1 + eβ~(ω

′−eV/~))−1 ≈ Θ(ω′ − eV/~)

we obtain

I(t) =eΓ2

h

∫ eV/~

0

dω′Asample(ω)Atip(ω′ − eV

~). (2.40)

24


If we assume that Atip can be well approximated by its value at the Fermi-

level Atip ≈ Atip(EtipF ) = const. the expression simplifies to:

I(t) =eΓ2

hAtip

∫ eV/~

0

dω′Asample(ω′) (2.41)

and the differential conductance dI/dV is:

dI

dV(ω) =

2πe2Γ2

~2AtipAsample(ω) (2.42)

As can be seen from above expression the differential conductance is pro-

portional to the DOS and therefore STM provides an alternative method to

PES to measure the DOS.

25


26

Chapter 3

Impurity solvers

IMPURITY SOLVER

ED DMRG NRG QMC

DTQMC CTQMC

CT-INT CT-HYB

Segment Matrix Krylov

CT-AUX CT-J

Figure 3.1: Overview and classification of impurity solvers. The Krylov im-

plementation of the hybridization expansion impurity solver is classified as a

continuous-time quantum Monte Carlo solver (CTQMC) formulated in the hy-

bridization expansion (CT-HYB).

For the solution of Quantum impurity models several numerical methods

have been developed. An overview of the methods is illustrated in Fig-

ure 3.1. The numerical renormalization group (NRG) and Density matrix

renormalization group (DMRG) methods extract the low-energy description

within a smaller Hilbert space than in the original problem using an iterative

procedure [33, 34] – an idea, which has been motivated by Wilson’s renor-

malization procedure [35]. They can operate directly on real frequencies of

27

however a limited range. Especially NRG has been proven to be very suc-

cessful in the description of the low-energy behavior of the single impurity

Anderson model [33]. Similar to NRG and DMRG, Exact Diagonalization

(ED) works on the real axis directly. ED has been widely applied to Quan-

tum impurity models also in the context of DMFT-simulations [36]. The

bath is represented by a few bath sites only and therefore the system can be

represented by a cluster. The cluster comprises the impurity sites of energy

levels ǫd coupled to bath sites with energies ǫp via the hybridization strength

Vpd. In ED the hybridization function is approximated by the cluster hy-

bridization function ∆(iω) =∑

p|Vpd|

2

iω−ǫp. Due to the exponential scaling of

the Hilbert space dimension only a few bath sites are feasible to include. In

the 5-orbital studies presented in this thesis we compared to ED-calculations

where at most two bath sites had been coupled for each d-orbital.

The solver applied in this thesis is a quantum Monte Carlo (QMC) solver.

These methods can deal with a wide range of energy scales but come with the

drawback that they are formulated in imaginary time and require analytic

continuation to deliver the observables in real frequencies. The state-of-the-

art solver prior to continuous-time quantum Monte Carlo (CTQMC) methods

had been the Hirsch-Fye solver [37] which is based on a imaginary-time dis-

cretization (DTQMC) [38,39]. Depending on the quantity in which the per-

turbation series is expanded there exist different variants - CT-INT [40, 41],

CT-HYB [42, 43], CT-AUX [44], CT-J [45] - of CTQMC-sovlers, which are

reviewed in [46]. In this thesis the Krylov implementation [47] of the hy-

bridization impurity (CT-HYB) solver has been used. In this chapter we will

follow the thick blue line of Figure 3.1 to describe in detail how we arrive

at the Krylov implementation of the hybridization impurity solver. We will

briefly recapitulate Monte Carlo sampling and outline CTQMC methods in

Section 3.1.3. The description of the hybridization solver with its variant

follows in Sections 3.2.3 and 3.3

28

Impurity solvers

3.1 Monte Carlo sampling

3.1.1 Definitions

Given a configuration C with configurations x the expectation value of an

observable A with respect to a specific statistical ensemble p(x) is

〈A〉p =1

Z

∫

C

dxA(x)p(x). (3.1)

In many physics problems we are confronted with a high-dimensional phase

space and direct integration methods, which scale badly with the dimension,

are not the methods of choice. The idea behind Monte Carlo sampling is

to estimate Eq. (3.1) by an average of M measurements of A at randomly

chosen configurations xi:

〈A〉 ≈ A =1

M

M∑

i=1

A(xi). (3.2)

The error is given by the standard error of the mean:

∆ =

√

var(p)

M. (3.3)

with var(p) being the variance of the distribution p(x). For strongly peaked

probability density functions p(x) the variance and hence the error in Eq.

(3.3) will be large. We can reduce the error if we provide our insight how p(x)

is peaked in phase space. For this so-called Importance Sampling we choose

to generate the configurations x according to a distribution ρ(x) similar to

p(x) and reweigh the expectation value:

〈A〉 = 1

Z

∫

C

dxA(x)ρ(x) =

∫

CdxA(x)p(x)

ρ(x)ρ(x)

∫

Cdxp(x)

ρ(x)ρ(x)

=〈Ap

ρ〉ρ

〈pρ〉 (3.4)

The error will be reduced to

∆ =

√

var(p/ρ)

M. (3.5)

We will use the shift of probabilities not only in the case of Importance

Sampling but also to ensure the positivity of the probability distribution

p(x) see the Section 3.1.4 on the sign problem.

29


With Monte Carlo sampling we have a method at hand with which we can

measure observables with an accuracy ∆ ∝M−1/2 irrespective of the dimen-

sion of the phase space. We now have to address the question how we can

generate the configurations. There exist direct sampling techniques [48] but

the solvers treated in this thesis rely on Markov process techniques.

3.1.2 Markov process

The strategy is to use a Markov process to reach the equilibrium state spec-

ified by the statistical ensemble p(x). To do so, we have to formulate the

dynamics for probability distribution functions ps. The Master equation de-

fines the equation of motion [49]

∆ps =∑

s′

(Wss′ps′ −Ws′sps) (3.6)

with Wss′ denoting transition probabilities from state s to s′. The Master

equation describes the gain and loss of ps in terms of rates given by the

addend and subtrahend in Eq. (3.6). Since the diagonal elements in the

transition probabilities Ws′s cancel in Eq. (3.6) we can set them to zero.

Under certain conditions we can converge towards an equilibrium state by

iterating the Master equation t-times lim t → ∞. A stationary state exists

if∑

s′ Ws′s < 1 and if the sequence Wss1 ...Wssn remains finite for all pos-

sible combinations of transitions, implying that there are no disconnected

subspaces within the phase space and all states can be reached (Ergodicity

condition).

From the Master equation we see that the stationarity condition ∆p = 0 can

be fulfilled if:Wss′

Ws′s=psps′

(3.7)

which is known as the Detailed Balance condition.

Developing a Monte-Carlo algorithm amounts to specifying the transition

probabilities Wss′ which lead to the desired equilibrium state. Above consid-

erations show that the algorithm has to meet the requirements of Ergodicity

and Detailed Balance. The latter is automatically fulfilled for the Metropolis

and heat-bath algorithm.

Within our algorithm we have to discern between the probability W prop to

propose a configuration update and the acceptance probabilityW acc to accept

30

Impurity solvers

the proposed update. Given an update from configuration x to configuration

y the transition probability Wyx is decomposed as

Wyx = W propyx W acc

yx (3.8)

Using the Metropolis algorithm [50] a configuration update will be accepted

with probability:

W accyx = min [1, Ryx] (3.9)

with

Ryx =p(y)W prop

yx

p(x)W propxy

. (3.10)

3.1.3 Continuous-time Quantum Monte Carlo

The Monte Carlo solvers presented in this thesis perform the sampling in

continuous time. We will therefore specify the structure of the underly-

ing configurations and derive the acceptance probabilities for these CTQMC

methods. We consider partition functions of the form:

Z =

∞∑

k=0

∫ ∫ ∫ β

0

dτ1...dτkp(τ1, ..., τk) (3.11)

and interpret the collection c = (k, τ1, ..., τk) of k τ -points and additional

discrete variables α representing quantum numbers as configurations c ∈ Ccontributing a weight wc to the partition function. A graphical representa-

tion of these configurations is shown in Figure 3.2. Vertices are placed on

the τ -axis of length β at the points τ1, ..., τk, with k being the expansion

order of the term in the series. Such a diagram can be generated for any

0 β

0 β

0 β

0 β

τ1 τ2 τ3

τ1 τ2

τ1

Figure 3.2: Graphical representation of the configurations c = (0, ), c =

(1, τ1), c = (2, τ1, τ2), c = (3, τ1, τ2, τ3) for the expansion orders k = 0, 1, 2, 3

31


0 β 0 β

τ1 τ2 τk+1 τ1 τk

remove

insert

wc = dτ1dτ2...

dτk+1w(k+1, τ1, τ2, ..., τk+1)

wc = dτ1dτ2...

dτkw(k, τ1, τ2, ..., τk)

Figure 3.3: Configurations are updated by insertion or removal of vertices. The

corresponding weight of the configurations is written below.

arbitrary configuration by removing or inserting vertices. Hence an algo-

rithm based on these updates will be ergodic. In Figure 3.3 we depict the

two elementary updates to transient between two configurations of weight

wc = dτ1dτ2...dτkw(k, τ1, τ2, ..., τk) and

wc = dτ1dτ2...dτkdτk+1w(k, τ1, τ2, ..., τk, τk+1).

The proposal probability to insert a vertex somewhere on the τ -axis of length

β is

pins =dτ

β. (3.12)

To remove one of the vertices of a configuration with k + 1 vertices is equal

the probability:

prem =1

k + 1. (3.13)

According to Eq. (3.9) the accpetance probability is computed as

W acck→k+1 = min

[

1,dτ1dτ2...dτkdτk+1w(k + 1, τ1, τ2, ..., τk+1)

dτ1dτ2...dτkw(k, τ1, τ2, ..., τk)

1k+1dτβ

]

. (3.14)

In the resulting ratio the k+1 infinitesimals cancel and the acceptance proba-

bility is solely determined by the terms w(k, τ1, τ2, ..., τk), w(k+1, τ1, τ2, ..., τk+1),

k and β:

W acck→k+1 = min

[

1,w(k + 1, τ1, τ2, ..., τk+1)

w(k, τ1, τ2, ..., τk)

β

k + 1

]

. (3.15)

3.1.4 The Sign Problem

In the derivation in Section 3.1.2 we required that the weights wc can be inter-

preted as probabilities, as can be seen from the Metropolis acceptance ratio in

32

Impurity solvers

Eqs. (3.9) and (3.15), which only makes sense for well defined probabilities.

However, in fermionic problems the interchange of fermionic operators can

cause a negative sign in the weight wc potentially leading to a non-sensical

negative acceptance probability. As a work-around we use the shift of prob-

abilities of Eq. (3.4) and sample an observable G according to the absolute

value |wc|:

G =

∑

c∈C |wc|signcGc∑

c∈C |wc|signc=〈sign G〉〈sign〉 (3.16)

with 〈sign〉 the expectation value of the sign of wc. The drawback is that

we sample according to a system of hard-core bosons described by a free

energy F|wc| instead of the actual fermionic system, meaning that mostly

irrelevant configurations are sampled. The expectation value of the sign can

be expressed in terms of the difference in the free energy density ∆f =

(F − F|wc|)/V as

〈sign〉 = Z

Z|wc|= e−βV∆f . (3.17)

The error according to Eq. (3.5)

∆sign =

√

(〈sign2〉 − 〈sign〉2)/M〈sign〉 ≈ eβV∆f

√M

(3.18)

blows up with increasing inverse temperature and volume and expectation

values sampled according to Eq. (3.16) suffer from exponentially growing

errors. This so-called sign problem hampers the Monte Carlo simulations of

large fermionic systems at low temperatures and has been proven to be a

NP-hard problem [51].

33

3.2 Expansions for quantum impurity models


Partition functions of the type Eq. (3.11) are typical for partition function

expansions in the interaction picture. Depending on the quantity in which

we perform the expansion we will choose a specific representation for the

vertices. We will put the representation in concrete forms and discuss how

to efficiently compute the weights of configurations.

3.2.1 Partition function expansion

Often we cannot directly compute the partition function

Z = Tr[e−βH

](3.19)

for a system described by a Hamiltonian H . Instead, we decompose the

Hamiltonian into two parts H = H0+V and choose the interaction represen-

tation in which the time evolution of operators is set by O(τ) = eτH0Oe−τH0.

Using the time ordering operator Tτ we can rewrite Z as:

Z = Tr[

e−βH0Tτe−∫ β

0dτV (τ)

]

. (3.20)

For discrete time methods above equation is approximated using a Trotter

decomposition:

Z ≈ Tr∏

l

e−∆τlH0e−∆τlV (3.21)

where the imaginary time axis is discretized into time slices ∆τl. One ex-

ample of such a discrete-time quantum Monte Carlo solver is the Hirsch-Fye

algorithm [37]. Compared to continuous-time methods it has several defi-

ciencies. Due to the discretization systematic errors are introduced which

require a careful analysis of the limit ∆τ → 0. Since the algorithm scales

with the number of time slices it is much more inefficient at low temperatures

than CTQMC methods [52].

Turning back to Eq. (3.20) we can apply the time-ordering operator to the

exponential and expand it in powers of V :

Z =∞∑

n=0

∫ β

0

dτ1 . . .

∫ β

τn−1

dτnTr[

e−(β−τn)H0(−V ) . . .

. . . e−(τ2−τ1)H0(−V )e−τ1H0

]

. (3.22)

34

Impurity solvers

0 β

τ1

e−τ1H0

τ2

e−(τ2−τ1)H0

τn

e−(β−τn)H0

Figure 3.4: Graphical representation of a configuration at expan-

sion order n. Vertices represent the interaction V (τ), lines the

propagator exp(−∆τH0). The weight of the configuration is wc =

Tr[e−(β−τn)H0(−V ) . . . e−(τ2−τ1)H0(−V )e−τ1H0 ]dτn.

In the previous Section we have seen that we can regard such an expansion

as a series of diagrams and discussed how we can compute Z using Monte

Carlo sampling techniques. The vertices in the diagram shown in Figure 3.4

are placed whenever an operator V (τ) occurs in the expansion term. The

weight of a configuration of order n can be read off as

wc = Tr[e−(β−τn)H0(−V ) . . . e−(τ2−τ1)H0(−V )e−τ1H0 ]dτn. (3.23)

In connection with perturbation theory one usually thinks of V as a small ap-

plied perturbation to a non-interacting exactly solvable system described by

H0 and one would compute corrections up to some order. For the continuous-

time methods considered here we do not rely on a partitioning of the Hamilto-

nian into an interacting and non-interacting part. Instead, we choose decom-

positions H = Ha +Hb for which the weight wc can be efficiently computed.

One particular issue in the case of fermionic problems is to take care of the

sign originating from the interchange of fermionic operators. State-of-the art

fermionic QMC algorithms operate with contractions of operators which can

be combined into determinants [46].

The quantum impurity model discussed in Section 2.2 consists of a local

contribution describing the interaction on the impurity and contributions

coming from the bath and hybridization:

H = Hbath +Hhyb +Hloc. (3.24)

There is the possibility to expand in Hloc or Hhyb which are known as the

weak coupling and hybridization expansion.

35


3.2.2 Weak coupling expansion

The first continuous-time algorithm has been developed by Rubtsov and

Lichtenstein [40,41]. It is based on an expansion in the Coulomb interaction

U :

Hb = Hloc, Ha = Hbath +Hhyb (3.25)

The vertices are therefore operators of the form: Ud†σ(τi)d†σ(τi)dσ(τi)dσ(τi)

and the non-quartic terms are integrated out giving rise to bath Green’s

function lines. A sketch of the configurations in the weak coupling algorithm

are shown in Figure 3.5.

0 β

0 β

d↑(τ1) d†↑(τ1)

d†↓(τ1) d↓(τ1)

G0↑

G0↓

G0↑

G0↓

G0↑

G0↓

Figure 3.5: In the weak coupling algorithm vertices Ud†σ(τi)d†σ(τi)dσ(τi)dσ(τi)

are placed on the imaginary time axis. Integrating out the non-quartic terms give

rise to Green’s function lines G0(τ, τ ′) connecting the vertices at τ and τ ′.

3.2.3 Hybridization expansion

Complementary to the weak-coupling expansion is the hybridization expan-

sion, which has been developed by Werner et al. [42,43]. Here we choose the

following decomposition:

Hb = Hhyb, Ha = Hbath +Hloc. (3.26)

We further split Hhyb into two parts:

36

Impurity solvers

a↑

d†↑

a†↑d↑

Figure 3.6: Vertices of the hybridization expansion. They consist of an impurity

and bath operator and represent the process of electron hopping onto the impurity

(left) and back to the bath (right).

Hhyb =∑

p,j

V jp d

†jap + V j∗

p a†pdj = H†hyb + Hhyb (3.27)

The first term represents the electron hopping from the bath to the impurity,

the second term the reverse hopping process from the impurity back to the

bath. In Figure 3.6 we depict the vertices symbolizing these two types of

transitions.

Only diagrams with an equal number of H†hyb and Hhyb have a non-vanishing

trace and we write Z as:

Z =∞∑

k=0

∫ β

0

dτ1...

∫ β

τk−1

dτk

∫ β

0

dτ ′1...

∫ β

τ ′k−1

dτ ′k

× Tr[

e−β(Ha)TτH†hyb(τk)Hhyb(τ

′k)...H

†hyb(τ1)Hhyb(τ

′1)]

(3.28)

and using the definition of H†hyb and Hhyb we find:

Z =∞∑

n=0

∫ β

0

dτ1...

∫ β

τn−1

dτσn

∫ β

0

dτ ′1...

∫ β

τ ′n−1

dτ ′n∑

j1,...,jnj′1,...,j

′n

∑

p1,...,pnp′1,...,p

′n

V j1p1Vj′1∗

p′1...V jn

pn Vj′n∗p′n

× Tr[

e−βH1Tτdjn(τn)a†pn(τn)apn(τ

′n)d

†jn(τ ′n)...dj1(τ1)a

†p1(τ1)ap1(τ

′1)d

†j1(τ ′1)

]

(3.29)

There will be also an equal number of creation and annihilation operators

for both spin up and spin down because the spin σ is left invariant under

the action of Ha = Hloc +Hbath. Furthermore, since the impurity and bath

operators act on different spaces, Ha does not mix impurity and bath states,

37


0 β

0 β

0 βτ ′1 τ ′1 τ ′2 τ ′2 τ ′3 τ ′3

Trd[. . . ]

Tra[. . . ]

Figure 3.7: Hloc and Hbath do not mix impurity and bath states. The impurity

contribution Trd [...] (upper right) and bath contribution Tra [...] (lower right) can

be separated in Eq. (3.30).

which allows to separate the trace over bath states and impurity states:

Z =∞∑

n=0

∫ β

0

dτ1...

∫ β

τn−1

dτσn

∫ β

0

dτ ′1...

∫ β

τ ′n−1

dτ ′n∑

j1,...,jnj′1,...,j

′n

∑

p1,...,pnp′1,...,p

′n

V j1p1Vj′1∗

p′1...V jn

pn Vj′n∗p′n

× Trd

[

e−βHlocTτdjn(τn)d†jn(τ

′n)...dj1(τ1)d

†j1(τ ′1)

]

× Tra[e−βHbathTτa

†pn(τn)apn(τ

′n)...a

†p1(τ1)ap1(τ

′1)].

(3.30)

In Figure 3.7 we have sketched this separation.

We will first address the contribution of the bath operators. As Hbath is

non-interacting we can apply Wick’s theorem [53] and integrate out the bath

degrees of freedom.

We will consider first the lowest non-trivial perturbation order shown in

Figure 3.8 for which we compute the sum:

Z(1) =1

Zbath

∑

p1,p′1

V j′1∗p V j1

p Tra[e−βHbathTτa

†p1(τ1)ap1(τ

′1)]

(3.31)

where we introduced the bath partition function in the prefactor:

Zbath = Tre−βHbath = ΠσΠp

(1 + e−βǫp

)(3.32)

Evaluating Eq. (3.31) we find

Z(1) =∑

p

Vj′1∗p V j1

p

e−βǫp + 1×

−e−ǫp(τ1−τ ′1−β) τ1 − τ ′1 > 0

e−ǫp(τ1−τ′1) τ1 − τ ′1 < 0

(3.33)

38

Impurity solvers

0 β 0 β

Figure 3.8: Left: First order contribution from the bath. The bath contribution

is expressed through the hybridization function ∆(τ) defined in Eq. (3.34). Right:

The bath contribution is indicated by a line connecting two impurity operators.

This expression motivates the definition of the β-antiperiodic hybridization

function

∆lm(τ) =∑

p

V l∗p V

mp

e−βǫp + 1×

−e−ǫp(τ−β) τ > 0

e−ǫpτ τ < 0(3.34)

with which the first-order contribution of Z is rewritten as:

Z = Zbath

∫ β

0

dτ1

∫ β

τ ′1

dτ ′1∑

j1,j′1

Trd

[

e−βHlocTτdj1(τ1)d†j′1(τ ′1)

]

∆j1j′1(τ1 − τ ′1)

(3.35)

In order to represent the factor ∆lm we have added a hybridization line

connecting impurity creation and annihilation operator in the graphical rep-

resentation in Figure 3.8. For higher expansion orders there exists several

0 β

0 β

0 β

0 β

0 β

0 β

Figure 3.9: All possible contractions of bath operators for a given configuration.

Their contributions give rise to a determinant weight defined in Eq. (3.36).

possibilities to pair up bath creation and annihilation operator as shown in

39


Figure 3.9. The contributions of all these possible hybridization line configu-

rations have the structure of a determinant. For the example given they are

combined into

det(∆) =

∣∣∣∣∣∣

∆11′ ∆12′ ∆13′

∆21′ ∆22′ ∆23′

∆31′ ∆32′ ∆33′

∣∣∣∣∣∣

(3.36)

with the entries ∆lm(τl, τ′m) defined by the hybridization function. The weight

coming from the trace over bath states can therefore be absorbed into such

a determinant and the expansion can be formulated as:

Z = Zbath

∞∑

k=0

∫ β

0

dτ1...

∫ β

τ ′k−1

dτ ′k

∑

j1,...,jkj′1,...,j

′k

Trd

[

e−βHlocTτdjk(τk)d†j′k(τ ′k)...dj1(τ1)d

†j′1(τ ′1)

]

det(∆)(3.37)

In the trace factor the information is encoded how the impurity fluctuates

between different quantum states as electrons hop in and out whereas the

determinants sum up the bath evolutions compatible with these electron

hopping processes.

Having treated the contribution of the bath operators we will now address

how to compute the trace over impurity operators. We will distinguish three

approaches: the Segment, Matrix, and Krylov implementation.

40

Impurity solvers

3.3 Variants of the hybridization expansion solver

3.3.1 Density-density approximation

0 β

Figure 3.10: Configurations in the density-density approximation can be repre-

sented as configurations of segments.

If we restrict Hloc to density-density interactions there exists a very efficient

formulation of the hybridization algorithm, the so-called Segment implemen-

tation. Due to the restriction to density-density interactions Hloc is diagonal

in the occupation number basis. For a one orbital problem e.g. the eigenen-

ergies are E0 = 0, E1 = −µ and E2 = U−2µ and the time evolution operator

is simply given by diag(e−τE0, e−τE1, e−τE1 , e−τE2). The time evolution does

not flip spins and hence we can treat spin-up and spin-down species sepa-

rately. Because each creation operator need to be followed by an annihilation

operator and vice versa the resulting configurations consists of collections of

segments depicted in Figure 3.10. The sum of all segment lengths is a mea-

sure of the average occupation on the impurity, which enters the local energy

via the chemical potential term. Whenever two segments of opposite spin

and/or orbital overlap it costs an additional energy Oij. These energy costs

0 β

0 βO

L

Figure 3.11: Contributions to Trd [...] in the Segment representation. The length

L of a segment is equal to the time interval ∆τ an electron resides on the site,

the overlap O when the site is doubly occupied. From the sum of all lengths and

overlaps the contribution to Trd [...] is computed according to Eq. (3.38.)

41


are illustrated in Figure 3.11 and enter the trace factor as follows:

Wloc = Trd [...] = eµ∑

j Lje−∑

i<j UijOij (3.38)

Because the configurations are constrained to segments we have to adopt the

configuration updates accordingly. Instead of vertices, we remove or insert

segments, which changes the expansion order k to k ± 2 as shown in Figure

3.12.

When we propose to insert a segment we choose a τ -point with probabilitydτβ

and accept it as a valid point if it lies outside a segment. We denote the

remaining place until the next segment starts with lmax and randomly choose

the segment end-point with probability dτlmax

. The proposal probability to

insert a segment is therefore:

W insprop =

dτ 2

βlmax

(3.39)

and the proposal probability to remove a segment:

W remprop =

1

k/2 + 1(3.40)

is equal to choose one of the k/2 + 1 segments.

0 β 0 β

remove

insert

Figure 3.12: In the Segment picture the updates are based on segment insertion

and removal.

Again we establish the acceptance probability according to Eq. (3.15):

Wacc = min(1,1

k/2 + 1

lmax

β

W ′loc det∆

′

Wloc det∆). (3.41)

The segment code or density-density interactions case has the advantage that

the complexity only increases linearly with the number of orbitals. Adding an

additional orbital simply corresponds to including two additional time lines

with segment configurations. However, if we want to go beyond the density-

density approximation the local part Hloc is not diagonal anymore and the

computation of the trace becomes more involved. In the next section we will

outline how more general interactions can be treated.

42

Impurity solvers

3.3.2 General local interactions

0 β

Figure 3.13: Graphical representation of configurations of a multi-orbital impu-

rity with general interactions. Different symbols represent operators of different

spin-orbitals.

Simulations beyond the density-density approximation are technically chal-

lenging and computationally very demanding. Before the introduction of the

weak coupling and hybridization expansion solver, simulations were restricted

to the density-density case. The density-density approximation however is

problematic, because it neglects spin-exchange and pair-hopping terms pa-

rameterized in the considered models by J which is not necessarily a small

quantity compared to the Coulomb interaction U . Furthermore, the models

presented in Section 2.2 conserve the spin, which is artificially broken in the

density-density approximation. Therefore even if the neglection of J does

not change the internal energy U much, the breaking of the symmetry will

change the degeneracy of the states and thereby also the entropy S. Since

phase transitions are not determined by the internal energy U but the free

energy F = U − TS calculations aiming at giving quantitative correct pre-

dictions will have to correctly capture the degeneracy of the local energy

spectrum. The introduction of the weak and hybridization expansion solver,

which are able to treat general interactions, therefore mark an important

advancement towards realizing simulations of realistic multi-orbital systems

with strong electronic correlations.

In this Section we will explain for the hybridization expansion solver how to

treat general interactions for impurities consisting of several orbitals.

The indices of the impurity operators will not only denote spins but also

orbitals. In our graphical representation we emphasize this difference by in-

troducing different symbols for the operators as shown in Figure 3.13. Apart

from this extension of what the indices specify the derivations of the hy-

bridization expansion Eqs. (3.28) - (3.37) remain the same. The bath can

again be traced out giving rise to a determinant weight as before. Each or-

bital is coupled to its own reservoir, which ensures that the hybridization

43


Figure 3.14: Left: The Matrix Code operates in the eigenbasis of Hloc in which

the propagator (middle matrix) is diagonal. For the computation of Trd [...] dense

matrices of size d are multiplied (left and right matrices). Right: By block-

diagonalizing the matrices the O(d3)-scaling can be reduced.

is diagonal. Off-diagonal hybridization terms would lead to a sign-problem

otherwise [46]. The trace weight Trd [...] has to be explicitly computed and

we need to choose a basis representation for the operators and the propa-

gator exp(−βHloc). One approach is to use the eigenbasis of Hloc, in which

the propagator is a diagonal matrix and to express the annihilation and

creation operator in this basis. The resulting dense matrices are then multi-

plied and the trace is computed from the final matrix. At expansion order

n this amounts to n matrix-matrix multiplications of the operators times

the trivial time evolution of the propagator see Figure 3.14. Since the lo-

cal Hilbert space dimension d grows as d ∼ 4Norb for Norb orbitals and a

matrix-matrix multiplication requires O(d3) operations the trace calculationbecomes soon expensive for a few orbitals already and low temperatures,

where the expansion order is high. To remedy this costly scaling the oper-

ators are block-diagonalized, e.g. into blocks with given particle and spin

quantum number as depicted in Figure 3.14. Nevertheless, the calculation of

the trace presents the computational bottleneck of multi-orbital simulations

with general interactions.

3.3.3 Krylov-implementation

A different approach has been recently introduced by Werner and Lauchli [47]

which exploits the sparsity of the impurity operators andHloc. In the Krylov-

implementation we operate in the occupation number basis instead of the

eigenbasis of Hloc. The number of interactions in Hloc increases at most with

N2orb, whereas the dimension of the matrix grows as 4Norb . This implies that

not only creation and annihilation operator but also Hloc are sparse matrices

in the occupation number basis. In the Krylov approach the exponential of

Hloc is never computed directly. Instead, we apply the matrix exponential to

44

Impurity solvers

a vector, i.e., we compute exp(−τHloc)|ψ〉. For these evaluations there exist

efficient sparse-matrix algorithms based on Krylov-space methods, where the

local Hamiltonian is restricted to the Krylov space

K =|ψ〉, Hloc|ψ〉, H2

loc|ψ〉, ..., HNiter

loc |ψ〉

(3.42)

and constructed using a Lanczos recursion. It has been shown that the

representation of Hloc according to Eq. (3.42) efficiently approximates the

full matrix exponential exp(−τHloc)|ψ〉 for a small number Niter ≪ Ndim [54].

0

|ψ1〉 = e−τλ0 |ψ0〉

0

|ψ2〉 = O1(τ1)|ψ1〉

0

|ψ3〉 = e−(τ2−τ1)Hloc |ψ2〉

(a) (b) (c)

Figure 3.15: First three propagation step in the Krylov implementation. (a)

Trivial time evolution. (b) Application of the operator O1(τ1). (c) Propagation

using a Lanczos recursion.

We will now specify in detail the required steps to compute the trace. In the

beginning of the simulation we completely diagonalize Hloc and identify the

low-lying energy states for a given energy window. In order to evaluate the

trace

Trd [...] =∑

|ψ〉

〈ψ|e−(β−τk)HlocOk(τk)e−(τk−τk−1)Hloc...O1(τ1)e

−τ1Hloc |ψ〉 (3.43)

we start with an initial state |ψ00〉 with eigenenergy λ0 within the chosen

energy window. A sketch of the first three propagation steps is shown in

Figure 3.15. We propagate the state |ψ00〉 up to the first hybridization event

at τ1. This first time evolution is trivial and the state |ψ01〉 after this time

evolution is given by:

|ψ01〉 = e−τ1λ0 |ψ0

0〉. (3.44)

At τ1 we apply the operator O1(τ1) to the state |ψ01〉:

|ψ02〉 = O1(τ1)|ψ0

1〉. (3.45)

45


The next step requires again a propagation of the state:

|ψ03〉 = e−(τ2−τ1)Hloc|ψ0

2〉 (3.46)

At this point |ψ02〉 is not necessarily an eigenstate ofHloc anymore and we have

to construct the Krylov space in order to represent the matrix exponential.

The Lanczos recursion is specified by the following iterations:

βn+1vn+1 = Hlocvn − αnvn − βn (3.47)

with αn = v∗nHlocvn and βn = |v∗

nHlocvn−1|. In the case considered we start

with v0 = 0, v1 = |ψ02〉 and β0 = 0.

Hloc is then represented by the tridiagonal matrix:

Hloc =

α1 β2 0 · · · 0

β2 α2. . .

. . ....

0. . .

. . .. . . 0

.... . .

. . .. . . βNiter

0 · · · 0 βNiterαNiter

(3.48)

with eigenvalues ǫ1, ..., ǫNiter, where the˜denotes that the matrix is repre-

sented in the Krylov or so called chain basis. When the matrix exponential

D = diag(exp(−∆τǫ1), ..., exp(−∆τǫNiter)) is back-transformed to the chain-

basis we can read off the desired state |ψ03〉 = e−(τ2−τ1)Hloc|ψ0

2〉 from the first

column of D. The steps of applying the impurity operators and performing

the time evolution are repeated for all operators and ∆τ -intervals on the

time line. In our implementation we propagate the states forward as well

backward until τ = β/2. The contribution of the chosen initial state |ψ00〉

is added to the trace and we repeat the procedure with a different initial

state |ψ10〉, which has again an eigenenergy within the chosen energy window.

As the main contributions to the trace come from low-energy states at low

temperatures the trace is well approximated by

Trd [...] ≈Ntr∑

i=0

〈ψi0|e−(β−τk)HlocOk(τk)e−(τk−τk−1)Hloc...O1(τ1)e

−τ1Hloc|ψi0〉 (3.49)

for a small number of Ntr. Note that Ntr has to include at least the multiplet

of the ground state.

46

Impurity solvers

The Krylov method is advantageous for simulations at low temperatures for

several reasons. Even though we truncate the outer trace according to Eq.

(3.49) the high expansion order at low temperatures ensures that also higher

excited states will be visited. A crucial difference to the scheme used in [55] is

that it does not rely on a truncation of states of the local Hamiltonian. The

Krylov implementation allows to access all states during the time evolution.

In addition ∆τ will be small on average at low temperatures and therefore the

representation of the propagator only needs a small number of Krylov basis

states. However, arbitrarily low temperatures are not accessible because the

computational complexity still rises due to the increasing expansion order.

We will address the computational complexity in the next Section.

3.3.3.1 Computational Complexity

Given Nhyb hybridization events - this corresponds to an expansion order

k = Nhyb/2 - and Ntr states in the considered energy window, then operators

of size Ndim are applied NtrNhyb times to a state leading to a computational

complexity of O(Ndim ×Ntr ×Nhyb). Similarly, for the time evolution which

additionaly requires Niter Lanczos steps we have a computational complexity

of O(Ndim×Niter×Ntr×Nhyb) and hence the overall complexity is given by:

O(Ndim ×Ntr × [Nhyb +Niter ×Nhyb]). (3.50)

The case Ntr = Ndim and Niter = Ndim corresponds to the full computation of

the trace and we recover the N3dim-scaling of dense and not block-diagonalized

matrix-matrix multiplications. The main point is that in practice we can

operate with Ntr ≪ Ndim and Niter ≪ Ndim, which makes the method very

attractive for multi-orbital simulations. Werner and Lauchli benchmarked

the Matrix and Krylov Code and showed that the Krylov method becomes

favorable to the Matrix code for impurities consisting of 5 to 7 orbitals [47].

In order to get an estimate for typical numbers of Ntr, Niter and Nhyb we will

present as an example the case of Co in Cu at a chemical potential µ = 28,

which will be presented in Chapter 5. As discussed previously the number

Ntr depends on the chosen energy window.

In Figure 3.16 the eigenvalue spectrum of the local Hamiltonian of Co in Cu

µ = 28 is shown for up to 20 lowest eigenenergies of each sector. Taking

into account states within an energy window of Ec = 0.1 eV amounts to run

the outer trace over 9 initial states |ψ〉. Increasing the energy window to

47


0

10

20

30

40

50

60

70

80

90

100

110

120

130

0 5 10 15 20 25 30 35

Eig

envalu

es (

eV)

Sector

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35

0

0.05

0.1

0.15

0.2

0 10 20 30 40 50

Pro

bab

ilit

y

Order

β=20β=40β=80

Figure 3.16: Left: Eigenvalue spectrum of the local Hamiltonian Eq. (2.7) of Co

in Cu, µ = 28. The inset shows the eigenenergies within an energy window of 2 eV

above the ground state. There are 9 states within an energy window ∆E = 0.1 eV,

18 states within ∆E = 0.2 eV and 21 states within ∆E = 0.3 eV. Right: Increasing

expansion order of Co in Cu, µ = 28 eV with decreasing temperatures.

Ec = 0.2 eV (Ec = 0.3 eV) raises the number of states considered up to 18

states (21 states). All higher states remain accessible during the calculation

of the trace, but will not be considered as starting state in the Krylov forward

and backward propagation. By varying Ntr we decided to use Ntr = 18 for

β = 20 and Ntr = 9 for the lower temperatures. The underlying procedure

will be discussed later in this Section.

In Figure 3.16 the expansion order histogram is shown for the eg- and t2g-

orbitals at inverse temperatures β = 20, 40, 80. From the peak position of

t2g-orbitals we estimate the values of Nhyb = k/2 and find k ≈ 20 (β = 20),

k ≈ 30 (β = 40) and k > 50 (β = 80).

During the simulation we measure the number of Lanczos iterations Niter.

The average numbers for the examples considered are Niter = 23 (β = 20,

Ntr = 18 ), Niter = 36 (β = 40, Ntr = 9) and Niter = 51 (β = 80,

Ntr = 9). If we combine these numbers in the expression C = Ndim ×Ntr × [Nhyb +Niter ×Nhyb] appearing in the computational complexity in

Eq. (3.50) we can estimate the order of magnitude of operations required

for a trace weight calculation and find C ≈ 34 × 106, C ≈ 40 × 106 and

C > 11 × 107 operations on average. In our simulations we tried temper-

atures down to β = 80, however only for temperatures down to β = 40

we obtained good enough statistics. We therefore estimate that complexi-

ties C ∼ 11 × 107 and higher are not feasible to simulate with the current

48

Impurity solvers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35

Eig

envalu

es (

eV)

Sector

−1

−0.8

−0.6

−0.4

−0.2

0

0 5 10 15 20

G(τ

)

τ

9 states18 states21 states

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

δ

τ

Figure 3.17: Left: Eigenvalue spectrum of the local Hamiltonian Eq. (2.7) of

Co in Cu, µ = 28 for energies within an energy window ∆E = 2 eV above the

ground state. Right: Green’s function of the dxy-orbital at T = 0.05 eV. Green’s

function of different values Ntr agree. The inset shows the relative difference

δ =∣∣∣G−Gtr21

Gtr21

∣∣∣ < 0.1.

implementation and computing resources.

3.3.3.2 Choice of Trace cutoff

In the Krylov implementation we have to determine first to how many states

we can restrict the outer trace for a given set of parameters. We perform

test-runs of the system of different energy windows. As already mentioned in

the previous Section we inspect the eigenvalue spectrum and choose energy

windows such that we include degenerate states at each enlargement of the

energy window. Depending on the degeneracy of the states this easily leads to

a large number of states Ntr to consider. For the example of Co in Cu, µ = 28

considered previously we have performed test-runs using ∆E = 0.1 eV (9

states), ∆E = 0.2 eV (18 states) and ∆E = 0.3 eV (21 states). A comparison

of the resulting Green’s function is shown in Figure 3.17. The Green’s

function agree for different trace cutoffs and it is reasonable to restrict the

trace to Ntr = 9. There are other systems where a difference can be clearly

seen and we will present the case of Fe on Na, µ = 23 whose eigenvalue

spectrum is shown in Figure 3.18. We studied the eigenvalue spectrum and

chose the energy windows ∆E = 0.01 eV (4 states), ∆E = 0.02 eV (12 states),

∆E = 0.04 eV (16 states), ∆E = 0.06 eV (28 states) and ∆E = 0.5 eV (53

states). The Green’s function at β = 20 eV−1 shown in Figure 3.18 clearly

49


0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 35

Eig

envalu

es (

eV)

Sector

−1

−0.8

−0.6

−0.4

−0.2

0

0 5 10 15 20

G(τ

)

τ

4 states12 states16 states28 states53 states

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

δ

τ

Figure 3.18: Left: Eigenvalue spectrum of the local Hamiltonian Eq. (2.7) of

Fe on Na, µ = 23 for energies within an energy window ∆E = 0.5 eV above the

ground state. Right: Green’s function of the dz2-orbital at T = 0.05 eV using

different values Ntr. From Ntr = 16 on upwards the Green’s function remains the

same. The inset shows the relative difference δ =∣∣∣G−Gtr53

Gtr53

∣∣∣.

evolves with increasing Ntr. Including 16 states and more yields the same

Green’s function, which is the reason why we chose ∆E = 0.04 eV for the

production runs.

These examples illustrate that one has to carefully choose the trace cutoff Ntr

depending on the simulation temperature T (∆E ∼ T ) and the eigenvalue

spectrum of the system.

50

Impurity solvers

3.4 Measurements

3.4.1 Green’s function measurements

For the measurements of the Green’s function we want to make use of our

partition function configurations. The idea is to remove hybridization lines

from the configuration to arrive at Green’s function diagrams. In Figure

3.19 we show two diagrams with the same operator sequence. On the left

hand side a typical diagram in the expansion of G(0, τ) = −〈Tτ [d(τ)d†(0)]〉 ofweight w

(0,τ)c is shown, whereas on the right hand side we are dealing with a

diagram in the expansion of Z with creation operator d† at 0 and annihilation

operator d at τ having a weight wd†(0)d(τ)c .

0 β 0 β

Figure 3.19: Left: A typical diagram in the expansion of G(0, τ) Right: A similar

diagram in the expansion of Z, with d† at 0 and d at τ .

Writing the Green’s function with operators at 0 and τ we have:

G(0, τ) =1

Z

∑

c

wd†(0)d(τ)c =

1

Z

∑

c

w(0,τ)c

wd†(0)d(τ)c

w(0,τ)c

(3.51)

The weight consists of two contributions: the trace over all operators in the

sequence and the hybridization matrix determinant. Since the trace factors

are identical the ratio of these weights can be expressed in terms of the

hybridization matrix determinants det(M) with M = ∆−1 of Eq. (3.34).

Let i,j be the row and column in(

M(τ,0)c

)−1

corresponding to the operators

d†(0), d(τ). We arrive at the matrix (Mc)−1 by removing this row and column

from(

M(τ,0)c

)−1

. Therefore, det[(Mc)

−1] is a minorM of det

[(

M(τ,0)c

)−1]

and the ratio of the weights in Eq. (3.51) is:

wd†(0)d(τ)c

w(0,τ)c

=detM−1

c

det(M(0,τ)c )−1

= (M(0,τ)c )ji. (3.52)

Using all partition function configurations and not only those with d† at 0

and d at τ , we make use of the translational invariance and find each creation

51

3.4 Measurements

and annihilation operator pair to contribute with 1β∆(τ, τi − τj)(Mc)ji with

∆(τ, τ ′) =

δ(τ − τ ′) if τ − τ ′ > 0;

−δ(τ − τ ′ − β) if τ − τ ′ < 0.(3.53)

Finally, we have for the measurement of the Green’s function the formula:

G(τ) =

⟨∑

ij

1

β∆(τ, τi − τj)(Mc)ji

⟩

(3.54)

3.4.2 Measurements at τ = β/2

0 β

forward backward

β/2

0 β

Figure 3.20: Measurements in the Krylov implementation. For the trace com-

putation a forward propagation from τ = 0 to τ = β/2 and backward propagation

from τ = β to τ = β/2 is performed. Top: Operators whose expectation value is

wanted are evaluated at τ = β/2. Bottom: Measurements of correlation functions

are problematic due to the bias coming from the outer trace truncation.

For other observables than the Green’s function we explicitly insert the op-

erator on the imaginary time line. Close to τ = 0 and τ = β only states

within the chosen energy window occur in the Krylov implementation. We

therefore do not place operators close to these boundaries, whose expectation

value would be affected by a truncation error. Since we perform the time

evolution forward and backward in imaginary time we place the operators

instead at τ = β/2, where also higher excited states have been meanwhile

accessed during the time evolution. The measurement is depicted in Figure

3.20. The truncation error close to the boundaries is also the reason why

we cannot measure correlation functions in an unbiased way in the Krylov

implementation. The only exception is the Green’s function, since we can

compute it using determinant weights only according to Eq. (3.54).

52

Chapter 4

Analytical continuation

4.1 Statement of the problem

We have seen in Section 2.5 that the spectral function A is related to the

Green’s function G. After having performed a simulation yielding a Green’s

function we therefore want to solve

G(τ) =

∫ ∞

−∞

dωK(τ, ω)A(ω) (4.1)

with the kernel

K(τ, ω) =e−τω

1 + e−βω. (4.2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

K(ω

,τ)

ω

β=10

τ = 0.05βτ = 0.1βτ = 0.2βτ = 0.5β

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

K(ω

,τ)

ω

β=100

τ = 0.05βτ = 0.1βτ = 0.2βτ = 0.5β

Figure 4.1: Rapid decay of the kernel as a function of frequencies for different

values of τ at β = 10 eV (left) and β = 100 eV (right).

53

4.1 Statement of the problem

Since the Green’s function is only specified on a finite number of discrete

values there is an abundance of consistent solutions of A(ω) defined via Eq.

(4.1). Furthermore the values of G contain statistical errors and it is unclear

how one should deal with these when inverting Eq. (4.1).

A huge problem of this inversion is that it is ill-conditioned. As shown in

Figure 4.1 the kernel rapidly decays to zero and inverting it becomes very

close to singular as well as soon close to the limits of numerical precision.

From the rapid decay we can also see that two spectral functions which

may have a large difference in some frequency range ∆ω may correspond to

Green’s function which only differ very little from each other if the frequency

range is at large frequencies.

In order to illustrate this in more detail we assume to have two spectral

functions which are identical up to a double peak structure as sketched in

Figure 4.2.

ω0 ω0

∆ω

1e−20

1e−18

1e−16

1e−14

1e−12

1e−10

1e−08

1e−06

0.0001

0.01

1

0 1 2 3 4 5 6 7 8 9 10

∆G

τ

ω0 = 0.5 eVω0 = 1.5 eVω0 = 2.5 eVω0 = 3.5 eVω0 = 4.5 eV

Figure 4.2: Left: One peak and double peak structures at the frequency ω0.

The peaks in the double peak structure are separated by a gap of width ∆ω.

Depending on ω0 and ∆ω the differences ∆G in the Green’s function may be

too small to resolve the double peak structure. Right: Difference in the Green’s

function between a single peak and double peak-structure with a gap of ∆ω = 1 eV

for β = 20 eV−1. The peaks are located at ω0. Given statistical errors of the order

10−4 (line) there are only a few points carrying the information about the difference

between the double peak and single peak in the spectral function.

For simplicity the difference ∆A(ω) shall be a rectangle of width ∆ω at posi-

tion ω0 and we do not care about the normalization of A(ω). We can now ask

how the corresponding Green’s function will differ. Since the transformation

54


is linear the difference ∆G is:

∆G(τ) =

∫ ∞

−∞

dωK(τ, ω)∆A(ω). (4.3)

For a rectangle of width ∆ω = 1 eV we compute ∆G(τ) at an inverse temper-

ature β = 20 eV−1 and present ∆G for ω0 = 0.5, 1.5, 2.5, 3.5, 4.5 eV in Figure

4.2.

It can be seen that ∆G becomes small for values away from τ = 0, the

larger ω0 the steeper the decay of ∆G. Since there is a certain statistical

error associated with the values of G there will be only a few values which

can have the information about the double peak structure encoded. In our

simulations this statistical error is typically of the order 10−5 − 10−4 which

we have indicated in Figure 4.2 by a threshold line. If one is particularly

interested in reproducing a particular feature in a PES-spectra - especially

those which had not been reproduced by any previous QMC-simulation - it

is advisable to estimate the required data accuracy for a desired number of

information-relevant Green’s function values in the proposed way. In combi-

nation with test-runs one can extrapolate the required computing resources

using the fact that a ten times smaller error bar requires 100 times more

measurements according to Eq.(3.3). Having made these preliminary consid-

erations we will now outline three methods how the spectral function can be

computed from the imaginary-time Green’s function. We have used in this

thesis three approaches - the Maximum Entropy Method (MaxEnt) [56, 57]

the Stochastic optimization method [58] and the analytic continuation with

Pade approximants [59].

55

4.2 Maximum Entropy method


The Monte Carlo simulation yields an estimate G(τ) of the exact Green’s

function G(τ). Given a spectral function A(ω) we can ask about the prob-

ability P[A, G

]that A generates the Green’s function G(τ) for which we

provide the Monte-Carlo estimate G(τ). Bayes theorem [60] connects this

posterior probability to the prior probability P [A] and likelihood function

P[G|A

]

P[A, G

]∝ P

[G|A

]P [A] (4.4)

For the likelihood function P[G|A

]∼ e−L we use a least-square fitting mea-

sure

L =1

2χ2 =

1

2

∑

k

Gk − Gk

σ2k

(4.5)

in order to quantify how well G(τ) with statistical errors σk is reproduced by

G(τ) which is the Green’s function obtained by back-transforming the given

spectral function A(ω). Maximizing the likelihood function is equivalent to

minimizing the deviation measure L. In principle we could just stop here,

assume P[A, G

]∝ P

[G|A

]and simply select the spectral function A(ω)

which maximizes the likelihood function. We will do so for the Stochastic

optimization method of spectral analysis discussed in Section 4.3. Choosing

the spectral function, which produces a Green’s function G(τ) fitting G(τ)

best will still be different from the exact value G(τ). As we have seen from

our considerations in Section 4.1 tiny changes in G(τ) can result in large

differences in A(ω). Therefore, we will have most likely artificial features in

A(ω) which are not present in the ”true” spectral function. In the Maximum

Entropy method (MaxEnt) we seek to avoid the overfitting of the data via

regularization. In MaxEnt the prior probability is chosen as P [A] ∼ eαS

with the entropy S

S = −∫

dωA(ω) ln

[A(ω)

m(ω)

]

(4.6)

and the default model m(ω) serving as a reference system. By choosing

the entropy as a regularization function we ensure that correlations are not

introduced via the regularizer. Correlations are only produced if the data

supports them.

With the above choices for the likelihood function and prior probability we

can now explicitly maximize the posterior probability P[A, G

]by maximiz-

56


ing:

Q = αS − L. (4.7)

The parameter α plays the role of a regularization parameter and controls the

competition between the entropic solution and the data constraints imposed

by the fitting.

We will now outline the MaxEnt algorithm following [57]. As an input we

provide the Monte-Carlo Green’s function Gj = G(τj), j = 1, ..., Nd with

statistical errors σj . We decide on the frequency width ∆ωi and number of

frequency points N for which we want to obtain the spectral density Ai =

A(ωi)∆ωi. The Kernel in discretized form is given by the matrix:

Tij =e−τjωi

1 + e−βωi(4.8)

and yields the ”mock” Green’s function

Gj =∑

i

TijAi (4.9)

How well G matches the measured data is specified by the least-square fitting

Eq. (4.5). As a default model we use mi = m(ωi)∆ωi = const. and use the

entropy:

S =∑

i

Ai −mi − Ai log(Aimi

)

. (4.10)

We decompose the Kernel T using a singular value decomposition

T = V ΣUT (4.11)

Σ is a diagonal matrix diag(γ1, ...., γs, 0, ..., 0) with s non-zero singular values

γi. We can restrict V , Σ and U to this s-dimensional subspace:

T = Vs︸︷︷︸

Nd×s

Σs︸︷︷︸

s×s

UTs

︸︷︷︸

s×N

(4.12)

which makes the calculation of maximizing Q much more efficient.

In order to maximize Q we compute the derivative with respect to A:

α∇S −∇L = 0 (4.13)

Bryan showed that this equation can be solved using a Newton method pro-

ceeding in steps δu defined via [57]:

(αI +MK) δu = −αu− g (4.14)

57


with

M = ΣsVTs diag

(1

σ1, ...,

1

σNd

)

VsΣs

K = UTs diag (A1, ..., AN)Us

u = UTv, vi = lnAimi

g = ΣV Tw, wi =∆Gi

σ2i

(4.15)

At step p we update up+1 = up + δup and stop if δuTKδu = 0. We must

take care that the step length does not become too large. If δuTKδu > 1 we

solve instead of Eq. (4.14).

((α + µ)I +MK) δu = −αu− g (4.16)

choosing µ large enough to ensure δuTKδu . 1.

The final u yields the spectral function

A = meUTs u (4.17)

for a given value of α.

The optimal choice of α can again be found using Bayes Theorem. The

conditional probability P[α|G,m

]is proportional to

P[α|G,m

]∝ Πi

(α

α + λi

)1/2

eQP [α] (4.18)

with λi the eigenvalues of MK. The optimal value for α is found for:

−2αoptS =∑

i

λiλi + αopt

. (4.19)

Above equation can be viewed as a count Ng of good measurements, since

each λi ≫ α increases the left hand side by one.

Instead of using only this single αopt parameter we average over a range of

reasonably probable α according to their probability Pα = Πi

(α

α+λi

)1/2

eQ:

〈A〉 =∑

α PαAα∑

α Pα. (4.20)

For this we choose a step length ∆α and consider those αt for which lnPαt<

lnPαopt+∆ for a chosen threshold ∆.

58


4.3 Stochastic optimization method of spectral analysis

The stochastic optimization method of spectral analysis has been developed

by Mishchenko et al. [58]. and is based on χ2-minimization only, i.e., it does

not introduce any regularization. Its advantages is that unlike MaxEnt it

can restore sharp features in the spectral function and does not rely on a

discretized frequency space.

We define an optimization procedure in which we update the proposed spec-

tral function stochastically. We generate several valid spectral functions

which fulfill the condition that the back-transformed Green’s function de-

viates from the measured Green’s function at most by a threshold Dmin

We consider the deviation function

D(A) =

∣∣∣G(τ)− G(τ)

∣∣∣

G(τ)(4.21)

where we ensure by dividing by G(τ) that the Green’s function is also closely

approximated where it has very small values, e.g. for an insulating system

at intermediate τ .

We represent the proposed spectral function A(ω) as composed of rectangular

contributions:

˜A(ω) =K∑

t=1

χRt(ω) (4.22)

with rectangles Rt = ht, wt, ct defined as

χRt(ω) =

ht, if ω ∈ [ct − wt/2, ct + wt/2]

0, otherwise(4.23)

with positive values for the height ht and width wt and centers ct within the

considered frequency range.

The normalization is fulfilled by imposing:

K∑

t=1

htwt = 1 (4.24)

for a given configuration C = R1, ..., RK of K rectangles.

For the spectral function A(ω) of a given configuration C the corresponding

Green’s function is computed according to Eq. (4.9). For each configura-

tion we are therefore able to measure the deviation of the back-transformed

Green’s function to the measured Green’s function using Eq. (4.21).

59

4.3 Stochastic optimization method of spectral analysis

We will now discuss how we update the configuration of rectangles and ac-

cording to which criterion we accept new configurations within our optimiza-

tion procedure. There are updates changing only the shape of the rectangles

but keeping the total number fixed and updates for changing the number of

rectangles of a configuration. The moves consist of

1. Shifting a rectangle.

2. Reshaping a rectangle by keeping its area fixed, i.e. changing the width

and height simultaneously.

3. Reshaping two rectangles, such that the sum of both areas stay fixed.

4. Adding a new rectangle and simultaneously reducing the weight of an-

other randomly chosen rectangle accordingly.

5. Removing a rectangle by simultaneously enlarging another randomly

chosen rectangle accordingly.

6. Splitting a rectangle and shifting the two parts.

7. Glueing two rectangles together and reducing the width to again fulfill

the normalization condition.

We update a new configuration r′ with probability:

Pr→r′ =

1 if D [C(r′)] < D [C(r)]f (D [C(r)] /D [C(r′)]) , otherwise

(4.25)

with f(x) = x1+d using d ∈ [1, 2]. We therefore also allow for updates which

increase the deviation D, because we want to avoid getting stuck in local

minima and want to ensure that we can in principle generate any positive

and bounded spectral function A(ω). The stochastic update procedure is

very similar to the updates in Monte-Carlo algorithms. However, since we

are dealing with an optimization procedure, there is no need to fulfill the de-

tailed balance condition, which would be required if we measured observables

according to a given statistical ensemble.

60


4.4 Analytic continuation with Pade approximants

Instead of an optimization procedure such as MaxEnt or the Stochastic op-

timization method we can also interpolate the data with a polynomial and

analytically continue the polynomial. We present here the analytic continua-

tion with Pade approximants which are based on an interpolation of rational

polynomials [59]. By interpolating our data we do not take into account

any statistical errors which may lead to artificial features in the spectral

function. Nevertheless, the procedure turns out to be quite reliable for fre-

quencies close to the Matsubara axis, where the quality of the QMC data is

sufficiently good. We consider a rational polynomial r(x) of the form

r(x) = f0 +a1(x− x0)

1 +a2(x− x1)

1 +a3(x− x2)

1 +a4(x− x3)1 + ...

(4.26)

which interpolates given values of a function f(x). We choose N points of

the function of f(x) to be interpolated at the points S = x1, x2, ..., xN. Ina first step we define the initial reduced function g(0)(x) given the set S:

g(0)(xi) = f(xi)− f(x0), xi ∈ S. (4.27)

We will recursively construct the coefficients a1, a2, ..., aN specifying the

rational polynomial. At step j we compute

aj =g(j−1)(xj)

(xj − xj−1)(4.28)

and set

g(j)(xi) = ajxi − xj−1

g(j−1)(xi)− 1, i 6= j − 1 (4.29)

.

We repeat this procedure until for all xi ∈ S we find that either g(j−1)(xi) =

0 or =∞. From the coefficients a1, a2, ..., aN we construct recursively the

polynomial r(x) for any desired values x.

In order to evaluate the polynomial for a specific value of x we compute the

continued fraction in the following iterative way: We start with q1(x)/q0(x) =

1, i.e., q−1(x) = 0, q0(x) = 1 and compute

qi+1(x) = qi(x) + ai+1(x− xi)qi−1(x), i = 1, ..., t = N − 1 (4.30)

61

4.5 Benchmarks

or equivalently:

qi+1(x)/qi(x) =1 + ai+1(x− xi)qi(x)/qi−1(x)

(4.31)

From the final continued fraction qt+1(x)/qt(x) we compute the interpolating

rational polynomial r(x) as r(x) = qt+1(x)/qt(x)− 1 + f0.

We apply this interpolation scheme to the Green’s function G(iω). By con-

struction of the interpolating polynomial we will obtain the measured values

G(iωn) on the Matsubara frequency points

iπβ, i3π

β, ..., i (2N+1)π

β

. We are

now able to analytically continue G(iωn) by evaluating r(x) at x = iω + iδ

which yields an estimation of G(ω). We can in principle provide any de-

sired frequency range and resolution. However, the further away we are from

the interpolation points iωn the less trustworthy the generated values will

be. From the analytically continued Green’s function G(ω) we compute the

spectral function

A(ω) = −1

πImG(ω). (4.32)

Since we do not have a measure of the error for the values away from the

interpolation points, we have to examine whether the result is reasonable.

As a first check we study how including a different number N of Matsubara

frequency points will affect the result. Features which exist only for one

particular choice of N can in this way be detected as artificial. As a second

check we compare the result to MaxEnt and the stochastic optimization

method result.

4.5 Benchmarks

We found Pade to work well to resolve quasi-particle peaks for Kondo impu-

rity problems. It agrees nicely with MaxEnt and the stochastic optimization

method as shown in Figure 4.3 in the energy range [−1, 1] eV. If we compare

the stochastic optimization method to MaxEnt the spectral function looks

very noisy and it is not clear to decide which features are physical and which

artificial.

The three methods are very different in how they favor regularization ver-

sus fulfilling the data constraints, i.e. in how explicitly the data is taken.

We can arrange the methods along a line between these two extremes as

depicted in Figure 4.4. Pade means directly inverting the Green’s function,

the Stochastic optimization method is based on likelihood-minimization only

62


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−10 −8 −6 −4 −2 0 2 4 6 8 10

DO

S

Energy (eV)

PadeStochastic

MaxEnt

0

1

−1 0 1

Figure 4.3: Spectral function of Co on Cu, µ = 29 eV obtained via different

analytic continuation methods.

and MaxEnt balances the minimizing of the likelihood and maximizing of the

entropy with the parameter α. If a feature in the spectral function persists

over a wide range on this axis, it is very likely that the feature is really sup-

ported by the data. In other words it will reflect that the data statistics is

good enough such that the information of this feature is encoded in enough

data points as outlined in our considerations in Section 4.1. In conclusion we

suggest to examine first what the accuracy requirements for a studied energy

range are and to make out via a comparison of different methods whether

the data accuracy is good enough.

MaxEnt

large α small α

stochastic optimization

method

Pade

regularization data constraints

Figure 4.4: Characterization of the different analytic continuation methods in

terms of the two extremes of regularization and fulfillment of the data constraints.

63

4.5 Benchmarks

64

Chapter 5

Transition metal impurities on

metal surfaces

5.1 Introduction

Quantum impurity models are the most basic models capturing the compe-

tition between localization and delocalization and are thus very instructive

to learn about the interplay of atomic and itinerant physics in strongly cor-

related systems. With the quantum impurity solver used in this thesis, local

degrees of freedom can be incorporated, which allows to study the phenomena

originating from orbital degrees of freedom. Most importantly, the quantum

impurity solver used provides a numerically exact solution to quantum im-

purity models. It is therefore possible to directly test the accuracy of the

first-principles calculations providing the material specific input parameters.

The approach to incorporate material specific effects to Model-Hamiltonians

describing strongly correlated systems is also used in lattice calculations,

e.g. realized in “LDA+DMFT” first-principles simulations. As will be ex-

plained in Section 6 these simulations involve the DMFA and for a given

system it is often not investigated what the effect of this approximation is

and whether deviations from the experimental data should be accounted to

this approximation or other effects. By taking one step back to investigations

of quantum impurity systems we can benchmark the predictive power of the

“DFT+Model-Hamiltonian” approach.

65

5.2 Multiplet features in Fe on Na


5.2.1 Introduction

To the presented study the following persons have contributed: B. Surer,

T. Wehling, A. Belozerov, A. Poteryaev, A. Lichtenstein, M. Troyer, A.

Lauchli and P. Werner. Our aim in this study is to use the ”DFT+Model-

Hamiltonian” approach in combination with the Krylov implementation of

the Hybridization solver to reproduce the PES-spectra of a system, which

exhibits very rich features in the spectra. Carbone et al. [61] studied Fe

atoms on different alkali metals. The hybridization varies with the metallic

host - the heavier the alkali ions the lower the electron density and hence

also the weaker the hybridization. In Figure 5.1 a sketch is shown how the

spectral density varies with the strength of hybridization. At the bottom the

E

DOS

Figure 5.1: Sketch of the densities of states of a Fe impurity on different alkali

hosts. As the hybridization varies along the alkali series the DOS changes from

atomic-like (bottom) to a system with quasi-particle peak and Hubbard bands

(top).

hybridization is very weak (host: Cs) and the atomic-like features are hardly

changed. In the intermediate hybridization regime (host: Na) a quasi-particle

peak develops and some of the peaks get broadened to form an incoherent

feature, but still there are features reminiscent of the atomic system. At

strong hybridization all the atomic features are dissolved in a broad Hub-

bard band and the intense scattering processes between the impurity and

the conduction electrons is reflected in the distinct quasi-particle peak (host:

Li).

66

Transition metal impurities on metal surfaces

5.2.2 Model and Computational method

We choose to study the intermediate case represented by Fe on Na having a

feature rich spectra, which exhibits multiplet features as well effects from the

coupling to the bath. Since schemes in the density-density approximation fail

to correctly reproduce multiplet features we study the multi-orbital quantum

impurity system using the full Coulomb interaction matrix as defined in Eq.

(2.7). We study the system for the Coulomb interaction parameters U = 4 eV

and J = 0.9 eV and apply the Krylov hybridization solver which is able to

treat the local Hamiltonian on the five orbitals exactly. We compare our

results to ED-calculations of three cases. First, we address the local problem

only and identify which transitions the peaks in the spectra stand for. Second,

we couple the 5-orbitals to one bath sites each (’5+5’) as well two bath

sites each (’5+10’). This allows us to investigate the effect of coupling the

impurity with more bath sites, in which case the CTQMC-result represents

the coupling to an infinite number of bath sites. For the total occupation of

the Fe atom on a Na host DFT yields n . 7 and we therefore consider total

densities n ∼ 6− 7.

5.2.3 Results

5.2.3.1 Identification of the transitions

The spectral function has resonances at the excitation energies of electron

removal and addition processes as shown in Section 2.5. The energy range

below the Fermi-level (EF = 0) corresponds to the transitions dn → dn−1,

the energy range above EF to transitions dn → dn+1. The initial state is

the ground-state whose spin S and angular momentum L are in line with

Hund’s rule. In a multi-orbital system there are different possibilities of how

the electrons are arranged in the final dn±1-configuration depending on from

which (to which) orbital we remove (add) the electron. The final states can be

classified according to their angular momentum L′ and spin S ′. For the given

symmetry of the local Hamiltonian Hloc the set of states of a given (L, S)

value will be degenerate and therefore the peak in the spectra will represent

transitions to not only a single state but to a multiplet. In Figure 5.2 we show

the spectra of the isolated Fe atom with an occupation of n = 7. Since it is

not coupled to a bath, the system is solely described by Hloc. In the atomic

problem we can associate quantum numbers n, l,m to the eigenenergies Enlm.

67


0

0.5

1

1.5

2

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

DO

S

E (eV)

(3,3/2) −> (3,1)

(3,3/2) −> (4,1)

(3,3/2) −> (1,1)

(3,3/2) −> (5,1)

(3,3/2) −> (2,1)

(3,3/2) −> (2,2)(3,3/2) −> (3,1)

(3,3/2) −> (1,1)

(L,S) −> (L’,S’)

Figure 5.2: Spectra of the isolated Fe atom: The peaks mark the transition from

the (L,S) = (3, 3/2) ground state to the final states (L′, S′). The specific (L′, S′)

multiplet is indicated at the corresponding peak.

We subtract the ground-state energy EGS from the eigenenergies to obtain

all energy differences ∆E = En′l′m′ − EGS. We determine the transitions

by identifying the peak position Ep with the excitation energies ∆E which

allows us to tag the angular momentum L and spin S to the final state. In

the ground-state the seven electrons arrange to a spin S = 3/2 and angular

momentum L = 3. The peak closest to EF represents the transition to

S ′ = 2 and L′ = 2 state, which is the ground-state of a d6-configuration.

Other final d6 configurations have a higher excitation energy and therefore

lie at higher excitation energies further away from EF . We have 5 different

final multiplets with S = 1 and L = 1, 2, 3, 4, 5. The transitions to the d8

configurations have large excitation energies due to the Coulomb repulsion

and because the charge cannot be screened. In the presence of itinerant

electrons we expect that the environment will react to an additional charge.

As a next step we will therefore study what happens if we place the system

in a metallic host.

68


5.2.3.2 Effect of the bath

The coupling between the Fe atom and the Na host is encoded in the hy-

bridization function ∆. To obtain ∆ we performed DFT calculations using

a generalized gradient approximation (GGA) [62] as implemented in the Vi-

enna Ab-Initio Simulation Package (VASP) [63] with projector augmented

waves basis sets (PAW) [64,65]. For the simulation of the Fe impurity on the

(100) bcc Na surface we used a 2 × 2 supercell x 7 layer slab as a supercell

structure. The Fe adatom and the three topmost Na layers were relaxed until

the forces acting on each atom were below 0.02 eVA−1. Using the projection

method as outlined in Section 2.4.1 we extracted the hybridization functions

from our DFT calculations. In Figure 5.3 we show Im∆(ω) for Fe on Na.

The hybridization is especially strong at large positive frequencies. In the

ED calculation the parameters Vp and bath site energies ǫp are chosen such

to approximately cover the hybridization function. In Figure 5.3 we show the

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

−5 −4 −3 −2 −1 0 1 2

Im ∆

(eV

)

E (eV)

dxydx

2−y

2

dxz,dyzdz

2

0

0.5

1

1.5

2

2.5

3

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

DO

S

E (eV)

isolated Fe atom5 bath sites10 bath sitescontinuum

Figure 5.3: Left: Hybridization function Im∆(ω) for Fe on Na. Right: Evolution

of the spectral function of Fe on Na with increasing number of bath sites.

spectra obtained by ’5+5’ and ’5+10’ ED-calculations , as well the CTQMC

result (’5+∞’). At negative frequencies the peaks of the ED-calculation get

shifted by a constant shift of around 1 eV compared to the atomic case. This

implies that in the Green’s function G(iωn) = (iωn + ǫd −∆− Σ)−1 the term

−∆ − Σ is well represented by a static value and the level ǫd gets shifted to

the position ǫd −∆0 − Σ0. Apart from the shift the peak structure remains

qualitatively the same in the ED-calculation. At high frequencies the spec-

trum changes substantially. The strong hybridization allows the conduction

electrons to efficiently screen the additional charge. Including more bath sites

69


(’5+10’) does not change the spectra much and also the analytically contin-

ued CTQMC data (’5 +∞’) is in very good agreement with the ED-data.

The transition to the L′ = 2, S ′ = 2 state is clearly visible in the CTQMC

spectral function, whereas the different S = 1 multiplets get combined into

one broad feature. It cannot be made out whether this is due to the fact

that the bath strongly mixes the impurity states leading to a broadening or

whether this is due to the smoothing inherent to MaxEnt.

5.2.3.3 Comparison to PES

0

0.5

1

1.5

2

2.5

3

3.5

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

DO

S

E (eV)

CTQMCED 5+5

ED 5+10

0

0.5

1

1.5

2

2.5

3

3.5

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

DO

S

E (eV)

CTQMCED 5+5

ED 5+10

Figure 5.4: Spectral function of Fe on Na for the chemical potential µ = 22 eV

(left) and µ = 24 eV (right). The data for µ = 23 eV is shown in Figure 5.3.

In our simulations we have to adapt the chemical potential for a given total

density for which we use the DFT estimate n . 7. We performed calculations

for µ = 22, 23, 24 eV which yields total densities n = 6, 6.9, 7.0. For all these

choices we find very good agreement between ED and CTQMC see Figure 5.4.

We find best agreement with the the experimental Photo-emission spectrum

of Fe on Na for the density n = 6.9, i.e. µ = 23 eV. The comparison is shown

in Figure 5.5. In both experimental and computed spectra we find a quasi-

particle peak at EF , a multiplet peak at −0.5 eV - which is the transition to

the L′ = 2, S ′ = 2 multiplet - and a broad Hubbard band originating from

many multiplets at −2 to −4 eV. The spectra obtained from ED can even

resolve the double peak structure in the Hubbard band.

70


0

0.5

1

1.5

2

2.5

3

3.5

4

−4 −3 −2 −1 0

DO

S

E (eV)

H M QP

CTQMCED 5+10

PES

Figure 5.5: Comparison of the computed spectral function with the experimental

PES data [61]. Quasi-particle peak (QP), Multiplet (M) at −0.5 eV and Hubbard

band (H) between −2 and −4 eV can be reproduced.

5.2.4 Conclusion

We conclude that applying the material specific “DFT+Model Hamiltonian”

approach in the full Coulomb formulation applied to the Fe impurity on Na

system allows to reasonably reproduce the PES-spectra. Since the spectra

is sensitive to the choice of µ, which we cannot determine ab-initio with the

desired precision the simulations were not fully ab-initio and required us to

compare a certain density range to the experimental data. Improvements to

better predict the chemical potential are needed. We have studied the effect

of the reservoir of conduction electrons in the Na host and shown that it acts

to screen additional charges on the Fe atom. Concerning the methods we find

the combination of ED and CTQMC very powerful, because it enables us to

consider a continuous bath and to still identify the underlying transitions in

the spectra, as well to judge the quality of the analytical continuation of the

CTQMC data.

71

5.3 Kondo physics in Co on Cu and Co in Cu


B. Surer, T. Wehling, A. Wilhelm,

A. Lichtenstein, M. Troyer, A. Lauchli and P. Werner

submitted to Phys. Rev. B

5.3.1 Introduction

We investigate the electronic structure of cobalt atoms on a copper surface

and in a copper host by combining DFT calculations with a numerically

exact CTQMC treatment of the five-orbital impurity problem. In both cases

we find low energy resonances in the DOS of all five Co d-orbitals. The

corresponding self-energies indicate the formation of a Fermi liquid state at

low temperatures. Our calculations yield the characteristic energy scale – the

Kondo temperature – for both systems in good agreement with experiments.

We quantify the charge fluctuations in both geometries and suggest that

Co in Cu must be described by an Anderson impurity model rather than

by a model assuming frozen impurity valency at low energies. We show

that fluctuations of the orbital degrees of freedom are crucial for explaining

the Kondo temperatures obtained in our calculations and and measured in

experiments.

Figure 5.6: Sketch of Co impurities in a Cu host. We consider the two cases: (i)

the impurity buried in the bulk (Co in Cu), and (ii) on top of the Cu layer (Co on

Cu).

The Kondo effect, arising when localized spins interact with a metallic envi-

ronment, is a classic many body problem [66]. At present, idealized models

dealing with, for instance, a single spin degree of freedom screened by a sea

of conduction electrons are well understood. Spin S = 1/2 Kondo models or

single orbital Anderson impurity models have been widely considered to de-

scribe magnetic impurities with open d-shells in metallic environments and

72


proved helpful in qualitative discussions [67–74]. As realized, however, by

Nozieres and Blandin in 1980 [75], such idealized models may ignore impor-

tant aspects of the nature of transition metal impurities as they disregard

orbital degrees of freedom. This makes comparisons between theory and ex-

periment often very difficult. More realistic models accounting for the orbital

structure, Hund’s rule coupling, non-spherical crystal fields and an energy-

and orbital-dependent hybridization of the impurity electrons with the sur-

rounding metal are theoretically very demanding due to the multiple degrees

of freedom and multiple energy scales involved.

In recent years different attempts have been made to address this problem.

For the classic example of Fe in Au, which has been experimentally studied

since the 1930s, a model describing the low energy physics has been derived

[76] by comparing numerical renormalization group (NRG) calculations to

electron transport experiments. The Kondo temperature TK , below which

the impurity spin becomes screened and a Fermi liquid develops, served as

a fitting parameter in this study. A scaling analysis of multiple Hund’s

coupled spins in a metallic environment showed that Hund’s rule coupling

can strongly quench the formation of Kondo singlet states [77]. For highly

symmetric systems like Co adatoms on graphene [78] or Co-benzene sandwich

molecules in contact to metallic leads [79], the orbital degree of freedom has

been suggested to control Kondo physics down to the lowest energy scale.

However, a general strategy to assess which degrees of freedom are involved

in the formation of low energy Fermi liquids around magnetic impurities in

metals is still lacking.

Co atoms coupled to Cu hosts present another experimentally extensively

studied system which has been interpreted in terms of Kondo physics [68,

69, 71–74, 80]. Theoretical descriptions of this system have often been based

on single orbital Anderson impurity models [67–70] or Kondo models [71]

and the role of orbital fluctuations in these systems has remained rather un-

clear. Recently developed CTQMC [46] approaches allow to describe the full

orbital structure of magnetic impurities in metallic hosts, while accounting

for all electron correlations in a numerically exact way. So far, however,

such CTQMC studies have been limited to rather high temperatures [81],

well above typical Kondo scales on the order of 10K to 500K. The realistic

description of transition metal Kondo systems thus remains a long standing

open problem in solid state physics.

Here, we employ the recently developed Krylov CTQMC method [47] in com-

73


bination with DFT based first-principles calculations to achieve an ab-initio

description of two archetypical Kondo systems: Co adatoms on a Cu (111)

surface, as well as Co impurities in bulk Cu (see Figure 5.6). We consider the

energy dependent hybridization of the impurities with the surrounding host

material as well as the full local Coulomb interaction and find low energy

resonances developing in the spectral function as the temperature is lowered.

Such resonances are found in all impurity 3d-orbitals and our calculations in-

dicate that spin- and orbital-fluctuations are crucial for the formation of low

energy Fermi liquids involving all impurity 3d-orbitals. We also demonstrate

the intermediate-valence character of the Co impurity in bulk, which implies

that the physics cannot be correctly described by a low-energy Kondo model

which neglects charge fluctuations.


A realistic description of the Co atoms on Cu (111) and in bulk Cu including

all five Co 3d orbitals can be formulated in terms of a multi-orbital Ander-

son impurity model Eq.(2.9) with the full four-fermion Coulomb interaction

Eq.(2.7).

To specify the parameters of impurity models describing Co on Cu (111) as

well as Co in bulk Cu we performed first-principles calculations.

DFT calculations were performed to obtain relaxed geometries and the hy-

bridization functions for single Co atoms in Cu and on a Cu (111) surface.

The DFT calculations have been carried out using a generalized gradient

approximation (GGA) [62] as implemented in the Vienna Ab-Initio Sim-

ulation Package (VASP) [63] with projector augmented waves basis sets

(PAW) [64, 65]. For the simulation of a cobalt impurity in bulk Cu we em-

ployed a CoCu63 supercell structure. Co on Cu (111) was modeled using a

3× 4 supercell of a Cu (111) surface with a thickness of 5 atomic layers and

a Co adatom on the surface, see Figure 5.6. All structures were relaxed until

the forces acting on each atom were below 0.02 eVA−1. For Co in bulk Co the

entire supercell and for Co on Cu (111) the adatom and the three topmost

Cu layers were relaxed. The PAW basis sets provide intrinsically projections

onto localized atomic orbitals, which we used to extract the hybridization

functions:

∆α1α2(ω) =

∑

k

V ∗kα1Vkα2

iω − ǫk. (5.1)

74


from our DFT calculations as outlined in Section 2.4.1.

The impurity model (2.9) can be solved without approximations using the

CTQMC technique. Both the weak-coupling [41] and hybridization expan-

sion [42, 43] algorithms can treat multi-orbital systems with general four-

fermion interaction terms. However, a strong-coupling approach is advan-

tageous in the case of a strongly interacting five orbital model, because the

expansion in the hybridization leads to much lower perturbation orders than

an expansion in the various interaction terms. For the interaction parame-

ters of the Co impurities used in this study the order in the hybridization

expansion was found to be a factor 20 lower compared to a weak coupling

expansion [81]. Furthermore, if the hybridization function is diagonal in the

orbital indices, as it is to a good approximation the case in the Co/Cu sys-

tems studied here, no sign problem appears, in contrast to the weak-coupling

approach, where correlated hopping terms were found to lead to severe sign

cancellations [81].

5.3.3 Results

The DFT calculations yield the orbital dependent hybridization functions

shown in Figure 5.7. In bulk Cu, the environment of the Co impurities is

cubically symmetric and the hybridization function decomposes into three-

fold degenerate t2g and twofold degenerate eg blocks. In the bulk symmetry

forbids off-diagonal elements in the hybridization function. On the surface,

the symmetry is reduced to C3v. For Co on Cu the hybridization function

decomposes into two twofold degenerate blocks transforming according to the

E-irreducible representation of C3v (E1 (dxz, dyz) and E2 (dx2−y2 , dxy)) and

the dz2-orbital transforming according to the A1 representation. For Co on

Cu the hybridization functions contain small off-diagonal matrix elements.

These off-diagonal elements, which are smaller than the diagonal ones, will

be neglected in our simulations. As a general trend, one can already see that

the hybridization of the Co d-electrons is about twice larger in the bulk than

on the surface.

DFT calculations are also used to calculate the occupancy of the Co 3d

impurity orbitals. To this end, we performed spin-polarized DFT calculations

using GGA as well as GGA+U of Co in and on Cu with the full interaction

vertex defined via the average screened Coulomb interaction U = 4 eV and

the exchange parameter J = 0.9 eV. We obtained the occupancies of the Co

75


a) b)

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

e d+

Re

∆ (e

V)

E (eV)

Co in Cu

egt2g

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

e d+

Re

∆ (e

V)

E (eV)

Co on Cu

E2E1A1

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

Im ∆

(eV

)

E (eV)

Co in Cu

egt2g

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

−8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5

Im ∆

(eV

)

E (eV)

Co on Cu

E2E1A1

Figure 5.7: Hybridization functions and crystal fields for Co in bulk Cu (a)

and as ad-atom on Cu (111) (b). In the upper panel the dynamical crystal field

ǫd + Re∆ is shown. Lower panel: Im∆.

3d orbitals derived from the PAW projectors n = n↑ + n↓ and the impurity

spin Sz =12(n↑ − n↓) and present them in Tab. 5.1.

In all cases the average Co 3d occupancy suggested by our DFT calculations

is between n = 7 and n = 8. For Co on Cu, the impurity spin is Sz ≈ 1

which is well in line with a d8 configuration of the Co. In the bulk, the Co

spin is Sz ≈ 1 in GGA+U and Sz ≈ 1/2 in GGA.

In the following we study Co in and on Cu in the five-orbital Anderson impu-

rity model formulation Eq. (2.9). In this framework, the chemical potential

has to be chosen to fix the occupancy of the Co d-orbitals. Due to the well

know double-counting problem in LDA+DMFT type approaches [82], the

precise chemical potential µ and the Co d-occupancy are not known. There-

76


GGA GGA+U

n Sz n Sz n Sz n SzCo on Cu 7.3 0.96 7.6 1.00 7.4 0.96 7.7 1.00

Co in Cu 7.3 0.51 7.6 0.53 7.3 0.90 7.5 0.93

Table 5.1: Occupancies n and impurity spins S as obtained from our GGA and

GGA+U calculations. Values obtained directly from the PAW projectors (n, S)

and normalized by the integrated total Co d-electron DOS, N =∫ν(E)dE, are

shown (n = n/N , Sz = Sz/N ).

fore, we computed results in a range of chemical potential values which yield

a total d occupancy consistent with the estimates of the DFT calculations.

For both systems the results of the DFT calculations predict a total den-

sity n . 8 and suggest a spin S ≈ 1 or slightly below in the case of Co in

Cu. For µ = 26, 27, 28 eV (Co in Cu) and µ = 27, 28, 29 eV (Co on Cu) we

obtain total densities and spins close to these DFT estimates. The values

of both observables for the lowest simulation temperature T = 0.025 eV are

presented in Table 5.2.

Table 5.2: Total density and spin.

System µ (eV) 〈n〉〈S〉Co in Cu 26 7.51± 0.07 1.02± 0.02

Co in Cu 27 7.78± 0.05 0.92± 0.02

Co in Cu 28 8.06± 0.03 0.817± 0.007

Co on Cu 27 7.76± 0.05 1.07± 0.01

Co on Cu 28 7.93± 0.05 0.99± 0.01

Co on Cu 29 8.21± 0.03 0.860± 0.007

5.3.3.1 Quasiparticle spectra

We now analyze the excitation spectra of the Co impurities in order to under-

stand the dominant physics at different energy scales. For a first, qualitative

insight into the strength of many body renormalizations, we compare in Fig-

ure 5.8 the Co 3d-electron DOS obtained from our DFT calculations to the

Co 3d spectral functions obtained from analytical continuation of our QMC

results (Figure 5.8).

The non-spinpolarized GGA calculations used to determine the hybridization

functions yield — by definition — the LDOS corresponding to the Anderson

77


0

0.2

0.4

0.6

0.8

1

1.2

−6 −5 −4 −3 −2 −1 0 1 2 3 4

DO

S

E (eV)

Co in Cu

GGAGGA+Uµ=26eVµ=27eV

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−6 −5 −4 −3 −2 −1 0 1 2 3 4

DO

S

E (eV)

Co on CuGGA

GGA+Uµ=28eVµ=29eV

Figure 5.8: DOS of the Co impurities in bulk Cu (left) and on Cu (111) (right)

obtained from DFT (GGA and GGA+U) as well as QMC simulations at temper-

ature T = 0.025 eV. QMC results obtained at different chemical potentials µ are

shown.

model without two-particle interactions (U = J = 0 eV). For both, Co in and

on Cu, the GGA DOS exhibits a peak near the Fermi level (EF = 0). The

QMC DOS qualitatively reproduces the GGA DOS for the case of Co in Cu.

Here, the main difference between the two approaches is that QMC yields a

peak near the Fermi level which is approximately twice narrower and shifted

towards EF . GGA+U accounts for the local Coulomb interactions at the Co

atoms on a Hartree Fock level which leads to the destruction of the quasi-

particle peak near EF with all the spectral weight shifted to broad Hubbard

bands. The comparison to the QMC results shows that this destruction of

the quasi-particle peak is unphysical.

For Co on Cu the hybridization is weaker and the DOS from the QMC

simulations exhibits both quasi-particle peaks near EF as well as Hubbard

type bands at higher energies. The reduction of spectral weight of the quasi-

particle peak as compared to GGA is stronger here.

The orbitally resolved DOS of Co in and on Co is shown in Figure 5.9. For

Co in Cu the DOS of the eg and the t2g orbitals is very similar particularly

regarding the quasi-particle peak — despite the (energy dependent) crystal

field splitting on the order of some 0.1 eV.

The DOS of Co on Cu exhibits stronger differences between the E1, E2, and

A1 orbitals. The E2 orbitals, which spread out perpendicular to the z-axis,

show the weakest hybridization effects, but even here, a quasi-particle peak

78


0

0.2

0.4

0.6

0.8

1

1.2

−6 −5 −4 −3 −2 −1 0 1 2 3 4

DO

S

E (eV)

Co in Cu

egt2g

0

0.2

0.4

0.6

−6 −5 −4 −3 −2 −1 0 1 2 3 4

DO

S

E (eV)

Co on Cu

E2E1A1

Figure 5.9: Orbitally resolved DOS of the Co impurities in bulk Cu (left) and on

Co (111) (right) obtained from our QMC simulations at temperature T = 0.025 eV

and chemical potential µ = 27 eV and µ = 28 eV, respectively.

appears in all orbitals. The appearance of low energy quasi-particle peaks in

all orbitals is different from the behavior expected for a spin-1 two-channel

Kondo model, where a low energy quasi-particle resonance would be observed

in two orbitals (four spin-orbitals) only.

The DOS as obtained from our QMC calculations suggests a low-temperature

Fermi liquid state involving all orbitals for both, Co in and on Cu. We

investigate the nature of this state in the following sections by analyzing the

self-energies obtained from QMC and the statistics of relevant atomic states.

5.3.3.2 Low energy Fermi liquid

If a Fermi liquid develops, the self-energy takes the form

Σ(T, ω) = Σ(T, 0) + Σ′(T, 0)ω +O(ω2) (5.2)

with Σ(T, 0) and the first energy derivative Σ′(T, 0) being real for T → 0. In

this regime, the spectral weight Z associated with the quasi-particle peak is

determined by

Z = (1− ReΣ′(0))−1. (5.3)

Our QMC calculations yield the self-energy on the Matsubara axis. Analytic

continuation ω → iωn shows that Fermi liquid behavior manifests itself on

the Matsubara axis by

ImΣ(T, iωn) ≈ ImΣ(T, 0)− ImΣ′(T, 0)ωn (5.4)

79


at low frequencies with ImΣ(T, 0) ∼ T 2.

We now compare these relations to the frequency and the temperature depen-

dence of ImΣ(T, iωn) obtained from our QMC calculations. At the lowest ac-

cessible temperature, T = 0.025 eV, we obtained the Matsubara self-energies

depicted for Co in and on Cu in Figure 5.10. In both systems |ImΣ| clearlydecreases as ωn → 0, for all orbitals except for the E2 orbitals of Co on Cu at

µ = 27 eV. This is clearly different from the diverging Σ(iωn) ∼ 1iωn

, expected

for the localized moment of an isolated atom. For Co in bulk Cu, the eg and

t2g orbitals exhibit very similar self-energies, whose low-energy behavior is

consistent with the form expected for a Fermi-liquid (Eq. 5.4). For Co on Cu,

the self-energies differ considerably between the different orbitals with the E1

orbitals being least correlated and the E2 orbitals exhibiting the largest self-

energies at low frequencies. Our results indicate that a Fermi liquid develops

in all Co orbitals, also here, although the Kondo temperature appears to be

orbital dependent.

5.3.3.3 Estimation of TK from QMC

To define a Kondo temperature scale even in cases without well defined local

moment at intermediate temperatures, we define TK through the width of

the quasi-particle resonance in the single particle spectral function near EF ,

which is measured in STM experiments.

In our QMC simulations we determine TK from the quasi-particle weight

Z. The simulations yield the self-energy at the Matsubara frequencies ωn =(2n+1)π

β. Analytical continuation of Eq. (5.3) yields

Z ≈(

1− ∂ImΣ(iωn)

∂iωn

∣∣∣∣ωn=0

− Re∆′(0)

)−1

(5.5)

and we use Eq. (5.4) to evaluate the derivative. In Figure 5.11 we show the

quasi-particle weight of Co in Cu and Co on Cu for the different types of

orbitals as a function of the chemical potential µ. The values of degenerate

orbitals agree within an accuracy of 10−2 to which precision we are listing

them in Tab. 5.3 for T = 0.025 eV. The systems whose spin is closest to

S = 1 are found to have the lowest values of Z. Co in Cu clearly has higher

quasi-particle weights compared to Co on Cu. As we will see, this results

in a higher Kondo temperature TK , which is also confirmed experimentally.

In experiments using STM measurements the Kondo temperature has been

80


−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6

Im Σ

(eV

)

ωn (eV)

Co in Cueg

t2g

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6Im

Σ (

eV)

ωn (eV)

Co on CuE2E1A1

−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6

Im Σ

(eV

)

ωn (eV)

Co in Cueg

t2g

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6

Im Σ

(eV

)

ωn (eV)

Co on CuE2E1A1

−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6

Im Σ

(eV

)

ωn (eV)

Co in Cueg

t2g

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 1 2 3 4 5 6

Im Σ

(eV

)

ωn (eV)

Co on CuE2E1A1

Figure 5.10: Orbitally resolved self-energies for Co in Cu (left panel) and Co

on Cu (right panel). From top to bottom , ImΣ(iω) is shown for the chemical

potentials µ = 26, 27, 28 eV (Co in Cu) and µ = 27, 28, 29 eV (Co on Cu) .

81


0

0.2

0.4

0.6

0.8

1

25 26 27 28 29 30

Z

µ

egt2gE2E1A1

Figure 5.11: Orbitally resolved quasi-particle weight of Co in Cu and Co on Cu

at temperature T = 0.025 eV.

found to be TK = 655K ± 155K = 0.056 ± 0.013 eV for Co in Cu [74] and

TK ≈ 54± 5K = 0.0046± 0.0005 eV for Co on Cu [68, 71, 74, 80].

Following Hewson’s derivation of a renormalized perturbation theory of the

Anderson model [83] we use as a definition for the Kondo temperature:

TK = −π4ZIm∆(0). (5.6)

The values computed according to Eq. (5.6) are listed in Tab. 5.3 for all

impurity energy levels Eα+µ at temperature T = 0.025 eV. As for the quasi-

particle weights we find the lowest values of TK for µ = 26 eV (Co in Cu) and

µ = 28 eV (Co on Cu). Averaging over orbitals we obtain TK = 0.118 eV (Co

in Cu, µ = 26 eV, T = 0.025 eV) and TK = 0.016 eV (Co on Cu, µ = 28 eV,

T = 0.025 eV) with a ratio of T INK /TON

K = 7.4. This large difference between

the two Kondo temperatures is in fair agreement with experiments, where a

ratio of T INK /TON

K = 12.1 has been found [74]. As discussed in Section 5.3.4 the

physical quantities which determine the Kondo temperature scale enter in the

argument of an exponential function, suggesting that a comparison of the log-

arithms log(Texp)/ log(TK) is more appropriate when judging the predictive

82


power of our first-principles simulations. We find log(Texp)/ log(TK) = 1.4

(Co in Cu, µ = 26 eV, T = 0.025 eV) and log(Texp)/ log(TK) = 1.3 (Co on

Cu, µ = 28 eV, T = 0.025 eV).

Table 5.3: Kondo temperatures TK computed from the quasi-particle weight Z

and the hybridization function ∆(ω) according to Eq. (5.6) at the lowest sim-

ulation temperature T = 0.025 eV. The experimental Kondo temperatures are

TK = 0.056±0.013 eV (Co in Cu) and TK = 0.0046±0.0002 eV (Co on Cu) [68,74]

.

System µ (eV) Ei -Im(∆(0)) (eV) Z TK (eV)

Co in Cu 26 -0.288 0.43± 0.01 0.38 0.13

Co in Cu 26 -0.44 0.340± 0.009 0.39 0.10

Co in Cu 27 -0.288 0.43± 0.01 0.42 0.14

Co in Cu 27 -0.44 0.340± 0.009 0.47 0.12

Co in Cu 28 -0.288 0.43± 0.01 0.48 0.16

Co in Cu 28 -0.44 0.340± 0.009 0.56 0.15

Co on Cu 27 -0.12 0.124± 0.002 0.06 0.006

Co on Cu 27 -0.39 0.226± 0.002 0.19 0.03

Co on Cu 27 -0.34 0.197± 0.001 0.08 0.01

Co on Cu 28 -0.12 0.124± 0.002 0.05 0.005

Co on Cu 28 -0.39 0.226± 0.002 0.15 0.03

Co on Cu 28 -0.34 0.197± 0.001 0.10 0.01

Co on Cu 29 -0.12 0.124± 0.002 0.14 0.01

Co on Cu 29 -0.39 0.226± 0.002 0.26 0.05

Co on Cu 29 -0.34 0.197± 0.001 0.27 0.04

For both systems our computed Kondo temperatures are higher than the

experimentally determined values. This can be either due to the neglect

of spin-orbit coupling effects which lifts degeneracies and narrows the low

energy resonances or due to the Coulomb interactions being larger than the

U = 4 eV assumed here.

The temperature dependence of the self-energy allows for an alternative test

whether and at which energy scale a Fermi liquid emerges. This is illustrated

for Co in and on Cu in Figure 5.12. We find an almost temperature indepen-

dent behavior of ImΣ(T, iωn) for Co in Cu if T < 0.05 eV, which provides an

estimate of TK ≈ 0.05 eV. In the case of Co on Cu ImΣ(T, iωn) still evolves

as one lowers the temperature from T = 0.05 eV to T = 0.025 eV, which

suggests a lower Kondo temperature.

According to Eq. (5.4), the linear extrapolation of ImΣ(T, iω) to iωn → 0

83


−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Im Σ

(eV

)

ωn (eV)

Co on Cu

T=0.05eVT=0.025eV

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Im Σ

(eV

)ωn (eV)

Co in Cu

T=0.2eVT=0.1eV

T=0.05eVT=0.025eV

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Im Σ

(eV

)

ωn (eV)

Co on Cu

T=0.05eVT=0.025eV

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Im Σ

(eV

)

ωn (eV)

Co in Cu

T=0.2eVT=0.1eV

T=0.05eVT=0.025eV

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Im Σ

(eV

)

ωn (eV)

Co on Cu

T=0.05eVT=0.025eV

Figure 5.12: Left panel: Im(Σ) of Co on Cu for µ = 29 eV at orbital energies

Ei = −0.12 eV (dxy, dx2−y2), Ei = −0.39 eV (dyz , dxz) and Ei = −0.34 eV (dz2).

Right panel:Im(Σ) of Co in Cu for µ = 28 eV at orbital energies Ei = −0.288 eV(dxy, dyz , dxz) and Ei = −0.44 (dz2 , dx2−y2).

84


−1

−0.8

−0.6

−0.4

−0.2

0

0 0.05 0.1 0.15 0.2

Inte

rcep

t

T (eV)

ImΣ → aωn+bfit of Im Σ(0,T), t2g

−1

−0.8

−0.6

−0.4

−0.2

0

0 0.05 0.1 0.15 0.2

Inte

rcep

t

T (eV)

ImΣ → aωn+bfit of Im Σ(0,T), eg

Figure 5.13: Temperature dependent quasi-particle lifetimes for Co in Cu at

µ = 26 eV for the t2g (left) and eg orbitals (right).

yields the inverse lifetime ~τ−1 = ImΣ(T, 0) of quasi-particles at the Fermi

level if a Fermi liquid is formed. In Figure 5.13 we plot this extrapolation

for Co in Cu, µ = 26 eV and show ImΣ(T, 0) as a function of temperature.

We find a ImΣ(T, 0) ∼ T 2-behavior for both sets of orbitals, which corrobo-

rates the formation of a Fermi-liquid in the eg and t2g orbitals at the lowest

accessible temperatures.

5.3.4 Discussion

Our QMC calculations showed that, for both Co in and on Cu Fermi liquids

involving all orbitals of the Co impurity form at low temperatures. We now

want to understand these results on the basis of scaling arguments starting

with (higher energy) charge fluctuations going to (lower energy) spin and

orbital fluctuations.

5.3.4.1 Charge fluctuations

With our values of the Coulomb interaction strength in the local Hamiltonian

(U = 4 eV; J = 0.9 eV) the energies of removing (E−) or adding (E+) an elec-

tron to the impurity are E− ≈ E+ ≈ 2 eV. A first order expansion in the hy-

bridization gives a qualitative estimate of the role of charge fluctuations [75]:

The norm of the admixtures of d7 and d9 configurations to a predominantly

d8 ground state of the impurity is approximately Nn 6=8 = − 1πIm∆(0) 10

U/2. For

Co in Cu, −Im∆(0) ≈ 0.4 eV leads to Nn 6=8 ≈ 0.6. The hybridization of Co

85


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Pro

bab

ilit

y

Occupation

Co in Cu

µ=26eVµ=27eVµ=28eV

GGA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Pro

bab

ilit

y

Occupation

Co on Cu

µ=27eVµ=28eVµ=29eV

GGA

Figure 5.14: Left: Occupation statistics of Co in Cu for T = 0.025 eV . Right:

Occupation statistics of Co on Cu for T = 0.025 eV.

on Cu is about twice smaller, −Im∆(0) ≈ 0.2 eV, yielding a correspondingly

smaller weight of non-d8 configurations Nn 6=8 ≈ 0.3.

Our QMC calculations allow to quantitatively measure the charge fluctua-

tions. In Figure 5.14 we plot the “valence histogram” [84] for Co in/on Cu

for different choices of the chemical potential and compare it to the “valence

histogram” of the Slater determinant built from the lowest GGA eigenstates.

The histogram shows the weights which the eigenstates in the different charge

sectors n = 0, 1, . . . 10 contribute to the partition function. In all cases the

local Coulomb interaction in the QMC simulation leads to a narrower dis-

tribution of the occupancies as compared to the GGA valence histograms.

This effect is most pronounced in the case of Co on Cu (111), where the d8

configuration clearly dominates over the d7 and d9 configurations. For Co in

Cu, there are still noticeable correlations and the narrowing of the valance

histogram as compared to the GGA case but the d8 configuration contributes

only about 50% in the QMC simulations, with significant weight coming from

the d7, d9 and even the d6 and d10 configurations. The measured values for

Nn 6=8 (≈ 0.5 for Co in Cu and ≈ 0.3 for Co on Cu) agree surprisingly well

with the simple perturbative estimate.

A Schrieffer-Wolff decoupling of ionized impurity states to discuss the low

energy physics can be justified if the weight of non-d8 configurations is

Nn 6=8 ≪ 1. Hence, an intermediate energy scale of well formed (unscreened)

fluctuating spin or orbital moments at frozen impurity valency might be de-

fined in the case of Co on Cu (111) but clearly not in the case of Co in

86


Cu.

5.3.4.2 Spin and orbital fluctuations

While for Co in Cu charge, spin and orbital fluctuations will be present

down to lowest energies, for Co on Cu only fluctuations of the orbital and

the spin degree of freedom are expected to dominate the low energy physics.

To further investigate to which extent orbital and spin fluctuations might

determine the low energy behavior of the impurity we estimate Kondo tem-

peratures within simplified models and compare these estimates to our QMC

calculations as well as experiments.

Assuming well-defined magnetic moments (Nn 6=8 ≪ 1) a scaling analysis

(cf. Ref. [75]) allows to estimate the Kondo scale analytically in simplified

situations: If neither Hund’s rule coupling nor crystal field splitting or any

other symmetry breaking terms were present the spin and the orbital degrees

of freedom of the Co impurities could fluctuate freely and independently.

This would lead to a Kondo temperature [75] TK ∼ D0 exp [−1/2Nn 6=8],

where D0 ∼ min(E±,Λ) is related to the impurity charging energies (E±)

and the electronic bandwidth Λ. D0 is on the order of several eV. With

D0 = U/2 = 2 eV we estimate TK ≈ 0.4D0 = 0.9 eV in the case of Co in Cu

and TK ≈ 0.2D0 = 0.4 eV in the case of Co on Cu. This is in both cases (at

least) an order of magnitude larger than the Kondo temperatures obtained

from QMC and the experimentally measured Kondo temperatures.

The opposite limit is given by the case with strong Hund’s rule coupling

and supressed orbital fluctuations. Without orbital fluctuations, but still

disregarding the Hund’s coupling J , the Kondo temperature reads [66, 75]

TK ∼ D0 exp[

πU

8Im∆(0)

]

= D0 exp[

− 52Nn6=8

]

, which would lead to TK =

0.02D0 ≈ 0.04 eV for Co in Cu. The Hund’s rule coupling reduces the

Kondo temperature [77] further to T ∗K = TK(TK/JHS)

2S−1. With S = 1

and JH = 0.9 eV this would lead to T ∗K ≈ 0.002 eV. For Co on Cu, the as-

sumption of an orbital singlet yields TK = 0.0004D0 ≈ 0.001 eV and Hund’s

rule coupling further reduces the Kondo temperature to T ∗K ≈ 1µ eV. This

limit thus yields Kondo temperatures which are orders of magnitude smaller

than those obtained in our QMC calculations as well as the Kondo temper-

atures measured experimentally for Co in and on Cu.

It is thus the successive locking of the impurity electrons to a large spin by the

Hund’s rule coupling and the partial freezing out of orbital fluctuations that

87


0

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5 1

DO

S

E (eV)

J=0.9eVJ=0.0eV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

28.4 28.6 28.8 29

Z

µ+Eα (eV)

E2

A1E1

J=0.9eVJ=0.0eV

Figure 5.15: Comparison of the DOS (left) and quasi-particle weight Z (right)

of Co on Cu at U = 4 eV for the two values J = 0.0 eV and J = 0.9 eV.

determines the on-set of Fermi liquid behavior and the Kondo temperature

in realistic systems like Co in or on Cu.

With this in mind it is instructive to analyze the influence of a static crystal

field on the energy spectrum of otherwise isolated Co atoms. Without crystal

fields, in a d8 configuration our local Coulomb interaction (U = 4 eV; J =

0.9 eV) yields a 21 fold degenerate L = 3, S = 1 ground state which is

separated from the L = 2, S = 0 multiplet by an energy of EL=2,S=0 = 1.3 eV.

This is clearly larger than the crystal field acting on the Co impurities (Figure

5.7): The cubic crystal field (evaluated at the Fermi level) of Co in Cu leads to

the eg states being 0.18 eV higher in energy than the t2g states. In this crystal

field, the resulting d8 ground state is an orbital singlet. Excitations to higher

crystal field split states require energies on the order of 0.2 eV. This is larger

but comparable in order of magnitude to the Kondo temperatures obtained

in experiments and simulation. However, fluctuations to these higher crystal

field split states must be taken into account to explain the characteristic

temperature of the low energy Fermi liquid formed at Co impurities in Cu.

In our model of Co on Cu, the static crystal fields also lift the degeneracy

of the ground state multiplet but a double degeneracy in the orbital space

remains. Excitations to higher crystal field split states require 0.03−0.08 eV.In this model, even the ground state multiplet allows for fluctuations of the

orbital degree of freedom.

In order to examine the effect of constraining orbital fluctuations we consider

the case of Co on Cu which exhibits a strong reduction of the quasi-particle

88


peak compared to the GGA spectral function representing the U = 0, J = 0

case. Turning off the Hund’s coupling J allows the orbital and spin degrees of

freedom to fluctuate more freely and given the scaling considerations should

result in a higher Kondo temperature as well as a broader quasi-particle peak.

We study the effect of J = 0 for Co on Cu, T = 0.025 eV, µ = 29 eV and

present the comparison of the quasi-particle weight and peak in Figure 5.15.

In line with our statement that the Kondo temperature is determined by the

locking of the impurity electrons to a larger spin and possible restrictions of

the orbital fluctuations, we find a broadening of the quasi-particle peak and

increase of the quasi-particle weight Z as J → 0.

5.3.5 Conclusions

For Co in and on Cu we found that a Fermi liquid is formed at low T in-

volving all impurity d-orbitals. The example of Co on Cu shows that the

characteristic temperature, TK , associated with the onset of Fermi liquid be-

havior can differ between the impurity orbitals. The comparison of our QMC

calculations and scaling arguments further demonstrates that fluctuations in

the orbital degree of freedom and Hund’s rule coupling are crucial in deter-

mining TK in realistic systems. This is well beyond the physics of simple

“spin-only” models. The understanding of magnetic nanostructures based

on 3d adatoms on surfaces as well as magnetic impurities in bulk metals re-

quire us to account for the orbital degrees of freedom. DMFT provides a link

between quantum impurity problems and extended lattices of atoms which

are subject to strong electron correlations.

It remains thus a future challenge to understand how the orbital degree

of freedom controls the quenching of magnetic moments and eventually the

formation of low energy Fermi liquids in realistic extended correlated electron

systems as well as magnetic nanostructures.

89


90

Chapter 6

Dynamical Mean Field Theory

U

↑↓

t

Time

Electron Reservoir

Vpe−

V ∗p

e−

Figure 6.1: Sketch of the DMFT-mapping: A site within the lattice is singled

out and described as a quantum impurity system. The dynamics on the impurity

is adjusted to mimic the lattice system.

6.1 Introduction

The Hubbard model has been proposed as a relatively simple model to de-

scribe strongly correlated systems. Despite its simplicity the Hubbard model

is very difficult to solve analytically as well numerically in two and three

dimensions and requires approximate methods. It has been first realized by

Muller-Hartmann [85] and Metzner and Vollhardt [86] that its diagrammat-

ics simplifies in the limit of infinite dimension. Later, Georges, Kotliar and

coworkers showed that the solution of this quantum many body problem

can be formulated as the solution of a quantum impurity system subject to a

91

6.1 Introduction

self-consistency condition which connects the impurity problem to the lattice

problem [87–89].

The key concept of the DMFA is sketched in Figure 6.1. In DMFA the lattice

model is mapped onto a single-site problem and a self-consistency condition.

The idea can be well explained by considering the classical analogy of the

Ising model. In the mean-field theory of the Ising model we map the lattice

model:

H = −J∑

〈ij〉

SiSj (6.1)

with coordination number z onto a single-site effective model describing a

single spin S0 in an effective field heff = Jzm

Heff = −heffS0. (6.2)

The spin degrees of freedom of the neighboring lattice sites have therefore

been replaced by an effective field heff. Within this model we can easily

evaluate the effective magnetization meff:

meff = tanh(βheff) (6.3)

and we connect back to the lattice model by identifying the magnetization

m per site with meff:

m!= meff = tanh(βJzm) (6.4)

Above identification imposes a condition for the magnetization with the aim

to obtain a magnetization which is consistent with the lattice model.

Analogously, we single out a lattice site in the DMFA and couple it to a bath

of non-interacting fermions. The Hubbard model

H =∑

ijαβ

tαβij d†iασdjβσ +

∑

iαβγδσσ′

Uσσ′

αβγδd†iασd

†iβσ′diδσ′diγσ −

∑

iασ

µd†iασdiασ (6.5)

is thereby mapped onto a quantum impurity model:

H =∑

p

ǫpa†pap +

∑

pασ

V αp d

†ασap + V α∗

p a†pdασ

+∑

ασ

ǫαd†ασdασ +

∑

iαβγδσσ′

Uσσ′

αβγδd†iασd

†iβσ′diδσ′diγσ

(6.6)

and we enforce the electron transitions from and to the impurity to behave as

the electron motion on the lattice by identifying the lattice Green’s function

92


with the impurity Green’s function:∑

k

G(k, ω)!= Gimp(ω) (6.7)

The bath will thus mimic the electron motion on the lattice.

Also within DFT we encountered a similar mapping strategy. In DFT we map

a many-body problem onto a single-effective particle problem and adjust the

Hartree and exchange-correlation potential consistently with the density. It

is therefore not astonishing that we can phrase the self-consistency conditions

to solve the Hubbard model within a Functional Formalism as well.

In analogy to the Hohenberg-Kohn functional the Luttinger Ward-Functional

[88]

Φ [G] = Φuniv [G] + Trln [G]− Tr[G−1

0 G]

(6.8)

is a functional of the Green’s function, where G0 is the Green’s function of

the non-interacting reference system. In this non-interacting reference system

(U = 0) the Greens function is given by

G0(k, ω) =1

ω + µ− ǫk. (6.9)

The effects of the interaction are encoded in the self-energy Σ(k, ω) and the

Greens function of the interacting model is given by

G(k, ω) =1

ω + µ− ǫk − Σ(k, ω)(6.10)

The Dyson equation relates G and G0 to the self-energy:

Σ(k, ω) = G0(k, ω)−1 −G(k, ω)−1. (6.11)

Similar to the Hohenberg Kohn functional the Luttinger Ward functional

Φ [G] is extremal at the physical correct Green’s function. As in DFT the

functional has a universal part Φuniv and a material specific part. Since Φuniv

is the sum of all vacuum to vacuum diagrams the functional derivative:

δΦuniv [G]δG

= Σ (6.12)

delivers the self-energy, which is the conjugate observable of the Green’s

function. Using a Legendre transform the functional can also be expressed

as a functional of the self-energy:

Φ [Σ] = Φuniv [Σ]− Trln[G−1

0 − Σ]

(6.13)

93

6.2 Self-consistency equations

Motivated by the observation that the self-energy becomes momentum-independent

in the limit of infinite coordination number we replace the self-energy Σ(k, ω)

by a momentum independent self-energy Σ(ω) and arrive at an approximate

Luttinger-Ward functional:

Φapprox [Σapprox] = Φapproxuniv [Σapprox]− Trln

[G−1

0 − Σapprox]. (6.14)

Even though this approximation is a priori uncontrolled, it is possible to

control this approximation by reintroducing the momentum dependence with

a small parameter [90, 91].

Since there is no spatial information encoded in Σ(ω) the functional deriva-

tive:δΦuniv [Σapprox]

δΣapprox= Gimp (6.15)

yields a Green’s function describing a quantum impurity.

The functional is stationary at the Dyson equation:

Σ−G−10 + G−1 = 0 (6.16)

where in DMFA we now provide the Green’s function by solving a quantum

impurity model.


For the first iteration step we provide an initial guess for the hybridization

function ∆

∆(iωn) =∑

p

V 2p

iωp − ǫp(6.17)

or bath Green’s function G0

G0 = (iωn + µ−∆(iωn))−1 (6.18)

and simulate the Quantum impurity models defined in Section 2.2 using one

of the Impurity solvers presented in Chapter 3.

In the case of the CTQMC impurity solver we will compute the impurity

Green’s function in imaginary time:

Gimp(τ) = −〈Tτd(τ)d†(0)〉 (6.19)

94


Σ(iωn) = iωn + µ−∆(iωn)−G−1(iωn)

G(iωn) =∑

k

(iωn + µ− ǫ(k)− Σ(iωn)

)−1

∆(iωn) = iωn + µ− Σ(iωn)− G−1(iωn)

∆(iωn)

∆(τ)

G(τ)

G(iωn)

momentum-average

Dyson

Fourier transform

Impurity Solver

Fourier transform

Dyson

momentum-average

DMFT

Self-Consistency Loop

Figure 6.2: Iterative procedure of a DMFT-simulation. The impurity solver on

the left hand side provides the impurity Green’s function G(τ). Applying the self-

consistency equations on the right hand side yields a new hybridization function

∆(τ). The cycle is repeated until convergence is reached.

By Fourier transforming Gimp to the Matsubara frequency domain we can

extract the self-energy Σimp using the Dyson equation:

Σimp(iωn) = G−10 (iωn)−G−1

imp(iωn) (6.20)

In the DMFA we approximate the lattice self-energy Σ(k, iωn) by the mo-

mentum independent self-energy Σ(iωn) using Eq. (6.16). With Σ(k, iωn)←Σ(iωn) the momentum averaged Green’s function of Eq. (6.7) reads:

G(iωn) =∑

k

1

iωn + µ− ǫ(k)− Σ(iωn)(6.21)

or written as an integral over the lattice DOS D(ǫ):

G(iωn) =

∫ ∞

−∞

D(ǫ)

iωn + µ− ǫ− Σ(iωn)(6.22)

Applying again the Dyson equation with G = G

G−10 (iωn) = Σimp(iωn) + G−1 (6.23)

95


we obtain a new bath Green’s function or using Eq. (6.18) a new hybridiza-

tion function which enters again the impurity solver.

Figure 6.3: Sketch of the Bethe lattice.

For a Bethe lattice [92] shown in Figure 6.3 with lattice DOS

D(ǫ) =

√

4− (ǫ/t)2

2πt(6.24)

Eq. (6.22) simplifies to

G0(iωn) = iωn + µ− t2G(iωn). (6.25)

The iterative procedure of a DMFT self-consistency routine is summarized in

the diagram in Figure 6.2. Computationally most expensive is the solution

of the impurity problem.

The converged solution contains the information about the lattice in the sense

that the exchange of electrons between the impurity and the bath will emulate

the topology of the lattice. Within the DMFA the lattice behavior is however

only captured to some extent, e.g. spatial fluctuations which are known to

play an important role in low-dimensional systems are by construction - the

quantum impurity model we are dealing with is a zero-dimensional system -

not incorporated [90].

A converged solution is usually achieved within 10-20 iterations depending

on whether the system is close to a phase transition or deep within a phase.

The converged solution can be processed further. From the Green’s function

G(τ) the spectral function can be computed via the analytic continuation

methods presented in Chapter 4 and compared to experimental data such

as (I)PES spectra or the differential conductance of STM measurements as

explained in Section 2.5.

Application wise we will first present a DMFT study of a model-Hamiltonian

system, namely the 5-band Hubbard model on the Bethe lattice which is

96


presented in the next Section. Second, we discuss the extension to describe

realistic materials and present a study of the two transition metal monoxides

NiO and CoO.

97

6.3 5-band Hubbard model on the Bethe lattice

6.3 5-band Hubbard model on the Bethe lat-

tice

6.3.1 Introduction

The 5-band Hubbard model on the Bethe lattice provides a non material-

specific model and gives insight into generic features of strongly correlated

multi-orbital systems. In particular, it serves as a generic model of transition

metal compounds whose electronic behavior is determined by the d-orbital

states close to the Fermi level.

We investigate the effect of the Hund’s coupling J on the magnetic and

metal-insulator behavior of the 5-band Hubbard model on the Bethe lattice.

Using single-site DMFT we explore the occurring phases as a function of the

chemical potential for two cases of the Hund’s coupling.

We will not only describe the variety of phases we find but also show how

starting from the valence statistics we can identify the exchange mechanisms

underlying the magnetic phases.

6.3.2 Model

We study the 5-orbital Hubbard model Eq. (2.8) in the Slater-Kanamori

formulation Eq. (2.17) within the single-site DMFA and use the Krylov

implementation of the hybridization algorithm to solve the impurity problem.

For a given U and J the 5-orbital system is at half-filling for the chemical

potential µhalf = 4.5U − 10J [47]. For the scan at different µ we restrict

ourselves to the range of densities n ∈ [0, 5] since the model is particle-hole

symmetric. We explore the phase diagram for two Hund’s coupling values

J = U/4 and J = U/6 at a Coulomb interaction U = 12t and temperature

T = 0.02t.

Even though the study of magnetic ordering induced by spontaneous sym-

metry breaking in principle requires to simulate a quantum impurity model

for each sublattice, Chan et al. [93] showed that it is possible to study the

magnetic and orbital ordering by analyzing the iterative behavior of the

DMFT-cycles. In this study we applied this inexpensive method to estimate

two-sublattice order.

We use the self-consistency equation Eq. (6.25) to impose the self-consistency

condition on the Bethe lattice. It is often customary to enforce symmetries

98


within a DMFT self-consistency step and to symmetrize the Green’s and

hybridization function over degenerate spin-orbitals. Here, however, we do

not perform such a symmetrization and as a result encounter oscillations in

the DMFT solution as a function of iterations. The crucial point here is that

these oscillations indicate the presence of two-sublattice order which we use

to determine the magnetic phases.

6.3.3 Results

6.3.3.1 Determination of the phases

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10 12 14 16 18

Orb

ital

Occ

upat

ion

Iteration

(a)

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14 16 18

Orb

ital

Occ

upat

ion

Iteration

(b)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18

Orb

ital

Occ

upati

on

Iteration

(c)

Figure 6.4: Orbital occupations as a function of DMFT-iterations. A paramag-

netic (a), ferromagnetic (b) and antiferromagnetic phase (c) can be distinguished.

99


In Figure 6.4 we show the orbital occupation of the 10 spin-orbitals as a

function of DMFT iterations of three cases representing different magnetic

ordering. At the very low density n = 0.81 corresponding to a chemical

potential µ = 1t all the orbital occupancies fluctuate around ni ≈ 0.08

(Figure 6.4(a)). No specific pattern can be observed from which we infer

that the system is paramagnetic. In Figure 6.4(b) we show the example of

ferromagnetic ordering. It can be clearly seen how the orbital occupancies

for spin-down (blue) and spin-up (orange) start to differ in the first iteration

steps and stabilize at the two values ni ≈ 0.42 and ni ≈ 0.01. Since orbitals of

only one spin-species are dominantly occupied the system is magnetic. Also

the system presented in Figure 6.4(c) exhibits magnetic ordering. Here we see

how the orbital occupancies of different spin species oscillate between n = 1

and n = 0. As shown by Chan et al. [93] oscillations signal a two-sublattice

ordering and we conclude that the system is antiferromagnetic.

As can be seen from these three representatives the paramagnetic (PM),

ferromagnetic (FM) and antiferromagnetic (AFM) phases can be clearly dis-

tinguished by this analysis. In this way we inspected the DMFT-iterations

for all J and µ studied and present the densities associated with these phases

in Figure 6.5.

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

n(µ

)

µ

PMFM

AFM

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

n(µ

)

µ

PMFM

AFM

Figure 6.5: Density as a function of chemical potential for J = U/4 (left) and

J = U/6 (right). The corresponding magnetic phase is indicated.

For both J = U/6 and J = U/4 we find paramagnetism for total densities

n < 1, which is in agreement with the three-orbital study by Chan et al. [93].

At half-filling the system is antiferromagnetic. According to Hund’s rule

the spins at a single site will arrange such that the total spin is maximized.

100


At half filling this implies that each orbital is occupied by one specific spin

orientation and due to the Pauli blocking electron exchange with the bath is

only possible for electrons with opposite spins. It is therefore favorable that

neighboring sites have opposite spin orientation and hence antiferromagnetic

order is favored. For J = U/6 we find antiferromagnetism for n = 4 also. At

other intermediate densities n ∈ (1, 5) we find the system to be ferromagnetic.

Compared to the three-orbital case ferromagnetism persists over a wider

range.

For the ferromagnetic order we additionally distinguish whether the system

is fully-polarized or not.

Close to the integer densities n=3,4 the ferromagnetic systems are fully-

polarized resulting in insulating orbitals for one of the spin-species, whereas

at lower densities in the ferromagnetic regime the system is not fully-polarized.

We find the fully-polarized regime to be sensitive to the data accuracy. In

a first attempt to explore the phase diagram with short simulation runs

we found the oscillatory behavior shown in the right panel of Figure 6.6

suggesting according to Ref. [93] a two-sublattice orbital ordering pattern.

However, this turned out to be an artifact, since the oscillatory behavior

is not reproduced when increasing the simulation time as shown in the left

panel of Figure 6.6.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18

Orb

ital

Occ

upati

on

Iteration

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18

Orb

ital

Occ

upati

on

Iteration

Figure 6.6: Orbital occupations as a function of DMFT iterations in the fully

polarized regime. Long simulation runs (left) yield a ferromagnetic phase. For

shorter simulation runs (right) and therefore reduced accuracy of measurements

the system becomes unstable mocking a two-sublattice orbital ordering pattern.

In addition to the magnetic phases we determined the metallic and insulating

101


behavior by investigating the slope of the Green’s function. In the insulat-

ing case the Green’s function has a steep slope and decays exponentially

to G(τ) → 0 at intermediate τ -values, whereas in the metallic case G(β/2)

remains finite and no steep slope is observed.

A Mott-insulating state exists at half-filling as well at n = 4 for J = U/6, so

essentially the Mott insulating state is accompanied by an antiferromagnetic

phase, whereas the metallic phase is either paramagnetic or ferromagnetic.

Since orbital degeneracy stabilizes the metallic phase [94] it becomes plausible

that the ferromagnetic and metallic phase is enlarged compared to the three-

orbital case.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

2

3

µ

J

n = 1 n = 2 n = 3 n = 4 n = 5

PM FP FP MI,AFM

Figure 6.7: Phase diagram of the 5-band Hubbard model on the Bethe lattice.

We find a paramagnetic (PM) (green), a ferromagnetic (blue) with fully polarized

parts (FP) (intermediate and dark blue) and an antiferromagnetic (AFM) phase

(orange), where the system is also insulating (MI).

In Figure 6.7 we have summarized the observed phases and sketched the

phase diagram of the 5-orbital Hubbard model on the Bethe lattice. Shaded

green is the paramagnetic phase, orange the antiferromagnetic and Mott-

insulating phase, the remaining part is ferromagnetic. In blue we highlighted

the fully-polarized phase within the ferromagnetic phase and have used a

lighter blue, where the fully-polarized phase starts to develop.

6.3.3.2 Exchange mechanism

When comparing the two different Hund’s coupling strengths J = U/6 and

J = U/4 we see a striking difference at n = 4. Whereas for the larger Hund’s

102


coupling J = U/4 the system is ferromagnetic we observe antiferromagnetism

for J = U/6. In order to understand the origin of the different magnetic

phases we looked at the valence histogram.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Pro

bab

ilit

y

Occupation

J=U/6

d4 p2 d4

Figure 6.8: Top: Valence histogram for J = U/6 at n = 4. With probability

Pd8 ∼ 0.91 the system is found in a d4 configuration. Bottom: For systems with

fixed charge occupancy antiferromagnetism due to super-exchange is favorable ac-

cording to the Goodenough-Kanamori-Anderson rules [95]. The antiferromagnetic

ordering is also confirmed by our simulation.

For J = U/6 at density n = 4 the system is most dominantly in a d4 con-

figuration and there are only little contributions from the n = 3 and n = 5

sector as shown in Figure 6.8. Thinking in terms of the lattice system and

considering two sites which are in reach via hopping processes and connected

with each other via some mediator site there will be two 5-orbital impuri-

ties on these sites, which have most likely (with probability P 2d8 ∼ 0.8) the

same valence, namely ntot = 4. The spins on the impurity will arrange such

that the total spin is maximal and at the same time such that the two im-

purities are able to exchange electrons. Concerning the kind of the effective

exchange coupling Goodenough, Kanamori and Anderson [96–98] established

rules (GKA rules) to determine these from the electronic configuration. At

n = 4 the exchange between different sites will be mostly exchange between

103


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Pro

bab

ilit

y

Occupation

J=U/4

d4 p2 d3 d3 p2 d4

Figure 6.9: Top: Valence histogram for J = U/4 at n = 4. The system ex-

hibits mixed-valence character. Bottom: For atoms of different valence the double

exchange mechanism leads to an effective ferromagnetic coupling [95]. The ferro-

magnetic ordering is also confirmed by our simulation.

half-filled and half-filled orbitals, which according to the GKA rules leads to

a strong and antiferromagnetic exchange coupling. In Figure 6.8 we depicted

this super-exchange mechanism for a transition metal oxide. Only two or-

bitals of the five orbitals are drawn but the spin orientation of the left-out

three orbitals is shown. The transition metal atoms exchange electrons via

an intermediate oxygen 2p orbital. The favorable antiferromagnetic ordering

can be easily read off from the sketch.

For J = U/4 the valence histogram for n = 4 is qualitatively very different

as shown in Figure 6.9. Still most contributions come from d4, but it is also

quite likely to find the system in a d3 or d5 configuration. Relating again back

to the lattice system this implies that we will have atoms of different valence.

In Figure 6.9 we sketch this exchange mechanism for a doped transition metal

oxide. The underlying exchange mechanism is double exchange and leads to

an effective ferromagnetic exchange interaction as can be seen from Figure

6.9.

104


These two examples demonstrate that based on the measured valence his-

togram we can reason the exchange processes obeying Pauli’s exclusion prin-

ciple as well fulfilling Hund’s rule and thereby we can explain the resulting

nature of the magnetic exchange coupling.

6.3.4 Conclusion

In this study we have investigated the phase diagram of the 5-band Hubbard

model on the Bethe lattice for two Hund’s coupling values J = U/4 and

J = U/6 and determined the magnetic as well as metal-insulator behavior

for the range of densities n ∈ [0, 5]. In contrast to a study of the 3-band

version of this model we do not find an orbitally ordered phase, but find a

fully-polarized ferromagnetic phase mocking an orbitally ordered phase when

the accuracy of measurements is reduced.

Depending on the strength of the Hund’s coupling J we have found different

magnetic phases at the density n = 4. We could amount this difference

to the different exchange mechanisms between the atoms and have shown

how starting from the valence statistics we can reason the nature of the

magnetic exchange coupling. As the 5-band Hubbard model on the Bethe

lattice serves as a generic model of transition metal oxides we have thereby

obtained a qualitative picture of various phases which can potentially occur

in these materials.

105

6.4 Extension to real materials: LDA+DMFT


We have argued in Chapter 2 that in order to describe strongly correlated

systems we have to treat atomic-like and band-like behavior on an equal

footing, since these systems are at the brink of localization and delocaliza-

tion. This requirement is met by the Hubbard model which can be solved

approximately using the DMFA.

In order to do first-principles simulations we have to bring in material-specific

properties to this model Hamiltonian. This is achieved by combining the

framework with DFT in the Local Density Approximation and goes under

the name ”LDA+DMFT”. The idea of this approach is to use the band

structure given through the Kohn-Sham eigenvalues for the dispersion ǫ(k)

in Eq. (6.21). What we essentially do is to put the local Coulomb interactions

on top of the non-interacting DFT-solution for the localized orbitals.

6.4.1 The double-counting problem

There is however a problem with this modular building block framework.

Since there are already some correlations considered in LDA via the exchange-

correlation potential some of the interaction contribution are counted twice

and we therefore need to subtract a double counting correction term Edc from

the Hamiltonian. Within LDA the electron density is not given by the density

of the localized orbitals but the total density, which is the reason why this

double counting correction term is not known exactly. We use the Hartree

ansatz for the double counting correction but there are different ansatzes to

compute EDC [99, 100].

For a given compound we usually have a rough estimate of the average oc-

cupation nc in the d-orbitals from its electronic configuration.

The Coulomb energy ECoulomb of these orbitals is given by:

ECoulomb =U

2nc(nc − 1) (6.26)

where we use the screened Coulomb repulsion U . We subtract this expression

from the impurity energy level given by LDA

ǫc =d

dnc(ELDA − ECoulomb) (6.27)

106


which gives rise to a double counting shift

EDC = U(nc −1

2). (6.28)

In addition, one also has to subtract the energy contribution coming from

the Hund’s coupling J . As the precise amount is not known, we will scan

possible values close to the estimate of Eq. (6.28) as presented in Section

6.5.

6.4.2 Self-consistency equations

With the newly introduced expressions ǫKSk and EDC for the ”LDA+DMFT”-

framework the k-summation of Eq. (6.21) is adapted to

G(iωn) =∑

k

(iωn + µ− ǫKSk − Σimp(iωn)− EDC

)−1(6.29)

The Kohn-Sham equations form a matrix in orbital space including more

than the correlated (d- or f -) bands, whereas the self-energy and double

counting corrections have non-zero entries only in the blocks corresponding

to the correlated orbitals.

6.4.3 High-frequency expansion

For realistic band structures ǫKSk or lattice DOS other than the semi-circular

DOS, for which the simple Eq. (6.25) holds we have to carry out a Fourier

transform of G(τ) first. In order to compute the Fourier transform we want

to make use of the high-frequency behavior of the Green’s function and hy-

bridization function.

Given a function F (iω) =∫F (τ)eiωnτdω we find by integration by part:

F (iω) = −F (β) + F (0)

iω+F ′(β) + F ′(0)

(iω)2− F ′′(β) + F ′′(0)

(iω)3−

∫F ′′′(τ)

(iω)4eiωτ

(6.30)

At high frequencies the first three terms will be dominant and we write the

high-frequency asymptotic expansion as:

F asym(iω) + F rest(iω) =c1iω

+c2

(iω)2+

c3(iω)3

+O( 1

(iω)4) (6.31)

with the coefficients ci identified with the numerators of the terms in Eq.

(6.30).

107


If we Fourier transform F (iω) to the imaginary time we obtain:

F (τ) = −c12+c24(−β − 2τ) +

c34(βτ − τ 2) + Frest(τ) (6.32)

The Fourier transform of the Green’s function G(τ) and the hybridization

function ∆(τ) enter the self-consistency equations. What we want to do is

subtract the asymptotic part F asym(τ) from the input function in imaginary

time F (τ) and numerically compute the Fourier transform of Frest(τ). To

F rest(iω) we again add the high-frequency behavior F asym(iω) in order to

obtain F (iω). The advantage is that the numerical Fourier transform is

much more well-behaved for this remaining term.

One possibility is to use the numerical derivatives to compute the coefficients,

which may however be afflicted with a large error and we will use the analytic

expressions for c1 and c2 of the Green’s function.

Given the definition of the Green’s function

G(τ) = −〈Tτ [d(τ)d†(0)]〉 (6.33)

with d(†) given in the interaction representation d(τ) = eHτdeHτ we can

formally expand the exponential in a Taylor series:(

1 +Hτ +H2τ 2

2!+ ...

)

d

(

1−Hτ + H2τ 2

2!+ ...

)

=

d+ τ [H, d] +τ 2

2![H, [H, d]] + ...

(6.34)

If we insert this expression in the definition of the Green’s function in Eq.

(6.33) we obtain for the coefficients:

c1 = −G(β)−G(0) = −〈d, d†〉c2 = −G′(β)−G′(0) = −〈[H, d], d†〉c3 = −G′′(β)−G′′(0) = −〈[H, [H, d]], d†〉

(6.35)

From the fermionic anti-commutation rule, we immediately find c1 = 1.

For the higher moments we have to be explicit on the Hamiltonian we choose.

We consider a Hamiltonian containing the four-fermion Coulomb interaction

matrix.

H =∑

σ

ǫdσd†σdσ+

∑

k,σ

ǫk,σc†kσckσ+

∑

k,σ

(Vkσckσdσ + h.c)+∑

i,j,k,l

Uijkld†iσd

†jσ′dlσ′dkσ

(6.36)

108


which we investigated in a study of transition metal monoxides described in

Section 6.5. We will first consider those terms in the Hamiltonian with two

fermionic operators. The bath and hybridization terms do not contribute

since [c†kσckσ, dσ′] = 0 and [ckσdσ, dσ′ ], d†σ′ = 0. Only the part describing

the impurity delivers

ǫdσ[d†σdσ, dσ′

]= ǫdσdσ. (6.37)

For the evaluation 〈[HU , d], d†〉 of the four-fermion part

HU =∑

i,j,k,lUijkld†iσd

†jσ′dlσ′dkσ it is useful to expand the expression in an-

ticommutators of the form

d(†)α , d

(†)β

which we can immediately evaluate

using the fermionic anticommutation rules. The following identities are use-

ful to determine the contribution of HU to c2: For the commutator [H, d] we

use

[ABCD,E] = AB [CD,E] + [AB,E]CD

[AB,C] = A B,C+ A,CB(6.38)

and

ABC,D = AB C,D − A B,DC + A,DBC (6.39)

for the anti-commutator 〈[H, d], d†〉. Carrying out the commutator expan-

sion and evaluation and combining the result with Eq. (6.37) we obtain:

c2 = −ǫd −∑

j

(Uijij − δσσ′Uijji) 〈ninj〉 (6.40)

with

〈ninj〉 = 〈d†iσd†jσ′djσ′diσ〉 (6.41)

109

6.5 Transition metal monoxides: NiO and CoO


6.5.1 Introduction

To the presented study the following persons have contributed: B. Surer, A.

Poteryaev, Jan Kunes, M. Troyer, A. Lauchli and P. Werner.

Transition metal oxides are a class of materials which has been investigated

extensively, both experimentally and theoretically. These compounds exhibit

a multitude of phases with remarkable properties, including high tempera-

ture superconductivity, colossal magnetoresistance and Mott metal-insulator

transitions [101]. Central to the physics of TMOs is the question of electronic

correlations and their interplay with chemical bonding. The late transition

metal monoxides are prototypical systems for the investigation of this is-

sue. Classified as charge-transfer insulators in the Zaanen-Sawatzky-Allen

scheme [102], they are characterized by the strong influence of the hybridiza-

tion between the transition metal 3d and oxygen 2p orbitals on their physical

properties. In this study we will consider two prominent charge-transfer in-

sulators, NiO and CoO.

The basic understanding of the electronic spectra of these materials dates

back to the cluster ED studies of Fujimori et al. [103] and Sawatzky and

Allen [104]. The first material specific calculations based on ab-initio DFT

were those of Anisimov, Zaanen and Andersen [105] using the so called

LDA+U method. Compared to pure DFT calculations, LDA+U signifi-

cantly improved the description of physical properties related to the elec-

tronic ground state and their adiabatic changes such as equilibrium lattice

constants, bulk moduli or phonon spectra, but this description fails to cap-

ture some qualitative features in the electron excitation spectra. Recently,

the application of many-body techniques in the framework of DMFT lead

to substantial improvement of the theoretical description of charge-transfer

systems.

The DMFT results obtained so far still involve several approximations, whose

effect is not well understood. In this study we address one of them, namely,

the form of the on-site interaction. This is a particularly important ques-

tion since it involves symmetry. We use the Krylov implementation of the

Hybridization expansion solver with full Coulomb interaction (having the

SU(2) symmetry) to perform LDA+DMFT calculations on the paramag-

netic phases of NiO and CoO. The results are compared with calculations in

110


the density-density approximation, calculations performed with ED as well

to Photo-emission spectra of the two systems.


In order to account for the charge-transfer character of the studied transition

metal monoxides we investigate an eight-band p-d Hamiltonian defined in

Eq. (2.20). For the construction of the effective Hamiltonian, for which LDA

provides the band structure, the projection onto Wannier functions has been

applied [106].

We investigate the systems CoO and NiO for three different formulations of

the Coulomb interaction. These three cases include the density-density case

of Eq. (2.18), the Slater-Kanamori case of Eq. (2.17) and the full Coulomb

interaction case of Eq. (2.7).

We solve Eq. (2.20) within the single-site DMFA using the Krylov implemen-

tation of the hybridization expansion to solve the quantum impurity model.

As shown in previous Sections this method is capable of treating a five orbital

model with general four-fermion interaction terms in the strong correlation

regime. We examined the effect of truncating the outer trace by comparing

the results of a single DMFT iteration for different values of Ntr as discussed

in Section 3.3.3.2. We considered states up to an energy +2 eV above the lo-

cal ground state energy and confirmed by comparing the observables that it is

sufficient to perform the trace over the states of the degenerate ground-state.

The simulations were performed at a temperature β−1 = 0.1 eV and interac-

tion values parametrized by U = 8 eV and J = 1 eV for NiO and U = 8.8

eV and J = 0.92 eV for CoO. The Green’s function was accumulated on 200

time slices on the imaginary time axis, which is appropriate at this temper-

ature. Since the electronic configuration of NiO is 2s22p63d8 and 2s22p63d7

for CoO, we expect a total number of 14 (NiO) and 13 (CoO) electrons in

the O 2p and Ni 3d orbitals. In the DMFT calculation, the chemical poten-

tial is adjusted such that the self-consistent total density corresponds to this

expected total number of electrons.

For the double counting value we have scanned possible values around the

estimate of Eq. (6.28) yielding EDC ∼ 60 eV (NiO) and EDC ∼ 57 eV (CoO)

and chose the values EDC = 59 eV (NiO) and EDC = 57 eV (CoO). Higher

double counting values yield a metallic system.

NiO and CoO have a rock-salt crystal structure. Due to the presence of the

111


oxygens the degeneracy of the 3d orbitals is lifted and the levels are split into

two eg and three t2g orbitals. LDA yields for the crystal field splittings the

values ∆cf = 0.566 eV (NiO) and ∆cf = 0.273 eV (CoO).

6.5.3 Results

6.5.3.1 Occupation

With the chosen double counting we find the hybridized Ni 3d - O 2p (Co 3d

- O2p) orbitals to have an occupation n = 8.21(1) (n = 7.14(1)) where the t2gorbitals have an average occupation of nt2g = 0.99982(7) (nt2g = 0.8180(8))

and neg = 0.553(2) (neg = 0.558(2)) for the eg orbitals. In Figure 6.10

we have depicted the electron configuration of NiO and CoO. According to

Hund’s rule the d8-configuration (NiO) is expected to have spin S = 1 and

the d7-configuration (CoO) spin S = 3/2.

eg

t2g∆cf

eg

t2g∆cf

Figure 6.10: Diagram of the electron configuration of NiO (left) and CoO (right).

6.5.3.2 Sector and State statistics

The degeneracy of the spin triplet state in the spin-rotation symmetry pre-

serving simulation can be seen in the left panel of Figure 6.11 presenting the

sector statistics. The sector statistics provides the statistics about the proba-

bility of the impurity to be in different quantum states. For the analysis, it is

most convenient to group the different eigenstates according to the conserved

quantum numbers (nup, ndown), n = 0, . . . , 5, resulting in a histogram repre-

senting the weight in 36 quantum number sectors. The spin triplet ground

state has 2 holes in the eg orbital for spin up (↓↓〉), 2 holes for spin down

(↑↑〉) and one hole in the eg orbital for both spin directions (↑↓ + ↓↑)/√2.

In the density-density case the spin rotation symmetry is broken and the

ground state is only doubly degenerate, because the (↑↓ + ↓↑)/√2 state lies

higher in energy. The preserved degeneracy of the spin triplet ground state

is reflected in the fact that the three sectors (3, 5), (4, 4), (5, 3) belonging to

the d8 configuration dominate the histogram with three equal contributions.

112


0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25 30 35

Pro

babilit

y

Sector

sector statistics

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25 30 35

Pro

babilit

y

Sector

sector statistics

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3 4 5

Pro

bab

ilit

y

State

Sector (2,5)Sector (3,4)Sector (4,3)Sector (5,2)

Figure 6.11: Sector statistics of NiO (left) and CoO (middle) in the full Coulomb

interaction formulation. Right: Atomic weights of the states within the 7-particle

sectors of CoO. The weights of the states above the 12-fold degenerate ground-state

are negligible. Only the 5 lowest states are shown.

The spin rotation symmetry is also preserved in the full Coulomb matrix

simulation of CoO and seen in the degeneracy of the atomic weights, displayed

in the middle panel of Figure 6.11.The degeneracy of the dominant sectors

(2, 5), (3, 4), (4, 3), (5, 2) of the d7 configuration are well in line with a spin

quartet S = 3/2 state. Applying Hund’s rules to the electron configuration

of CoO we expect a hole in one of the three t2g levels of the Co 3d orbitals

giving rise to a three-fold degeneracy for each of the 4 sectors, see sketch in

Figure 6.10. Indeed, as can be seen in the right panel of Figure 6.11, which

plots the weights of individual states in the 7-particle sectors, the resulting

12-fold degeneracy of the ground-state is confirmed by the degeneracy of the

atomic weights.

6.5.3.3 Density of states

To obtain the single-particle spectral functions analytic continuation to real

frequencies is performed using MaxEnt as explained in Chapter 4. The spec-

tral functions of NiO within the density-density, Slater-Kanamori and full

Coulomb interaction formulation are shown in Figure 6.12. There is hardly

any difference in the spectral function computed within the density-density

approximation and the two SU(2) invariant formulations of the Slater-Kanamori

and the full Coulomb interaction. In order to exclude the possibility that

the conformity of spectral functions is due to any smearing out of spectral

features inherent to MaxEnt we have made a comparison of these three for-

mulations using ED as an impurity solver. The advantage is that ED works

directly with real frequencies and does not require analytic continuation of

the data. Unlike CTQMC it can incorporate only a few bath sites, hence

113


0

0.1

0.2

0.3

0.4

0.5

−20 −15 −10 −5 0 5 10 15 20

Spec

tral

den

sity

Energy (eV)

nn

egt2g

total

0

0.1

0.2

0.3

0.4

0.5

−20 −15 −10 −5 0 5 10 15 20

Spec

tral

den

sity

Energy (eV)

SK

egt2g

total

0

0.1

0.2

0.3

0.4

0.5

−20 −15 −10 −5 0 5 10 15 20

Spec

tral

den

sity

Energy (eV)

full U

egt2g

total

0

0.1

0.2

0.3

0.4

0.5

−20 −15 −10 −5 0 5 10 15 20

Spec

tral

den

sity

Energy (eV)

full UnnSK

Figure 6.12: Spectral function of NiO in the density-density (nn) approximation,

with the Coulomb interaction in the Slater-Kanamori (SK) form and with the full

four-fermion (full U) Coulomb interaction.

effects of the hybridization will not be adequately captured. However, con-

cerning differences in the local interaction we expect ED to detect the dif-

ferences very clearly. As a matter of fact we expect it to rather overestimate

these differences because the on-site part is fully incorporated whereas the

bath and hybridization is only partially considered.

The comparison of different Coulomb interaction formulation computed within

ED is shown in Figure 6.13. Again, we encounter qualitatively very similar

spectral functions for the three formulations. Detecting the tiny differences

of the presented spectral functions also within CTQMC is very unlikely due

to the effect of the bath as well due to analytic continuation. In hindsight it

is therefore not surprising that all three formulations yield qualitatively the

same result in the CTQMC-calculations.

114


0

0.2

0.4

0.6

0.8

1

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6

DO

S

E (eV)

Full USlater KanamoriDensity−Density

0

0.2

0.4

0.6

0.8

1

1.2

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6

DO

S

E (eV)

CTQMCED

PES

Figure 6.13: Left: Spectral function of NiO obtained from ED for all three

formulations of the interaction. Right: ED and CTQMC Spectral function of NiO

compared to PES [104].

We compare the computed spectral function of NiO to the PES data of

Sawatzky et al. [104]. Eastman and Freeouf studied the 2p contribution of

the oxygen in the spectra of NiO and showed that it decreases with increasing

photon energy [107]. At 120 eV photon frequency the O 2p contribution is

negligible at which frequency the PES-data by Sawatzky et al. has been ob-

tained. The comparison between ED, CTQMC and PES is shown in Figure

6.13. First of all the agreement between the ED and the analytically contin-

ued CTQMC data is good except for the double peak structure at around

−2 eV and −4 eV. As we have already noted for the spectral function of Fe

on Na presented in Section 5.2 the analytically continued CTQMC data fails

to resolve a similar double peak structure at these frequencies. When com-

paring the computed spectral function to the PES data we see that both ED

and CTQMC underestimate the experimentally found gap of NiO of the or-

der of ∆ ∼ 4 eV [104,108] which suggests that the estimate for the Coulomb

interaction U is below the appropriate value. At energies below the Fermi-

level (EF = 0) the theoretical curves agree quite well with the experimental

data. The spectral features at −2 eV and the broad part at around −8 eVto −10 eV are reproduced by the QMC and ED data, where ED in addition

captures the feature at −4 eV.In Figure 6.14 the spectral functions of CoO within the full Coulomb inter-

action formulation is shown. The spectral function has a similar shape as the

NiO data with the difference that the incoherent part extends to frequencies

115


0

0.1

0.2

0.3

0.4

0.5

−20 −15 −10 −5 0 5 10 15 20

Spec

tral

den

sity

Energy (eV)

egt2g

total

0

0.2

0.4

0.6

0.8

1

−16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6

DO

S

E (eV)

CTQMCLDA O p

PES

Figure 6.14: Left: Spectral function of CoO with the full Coulomb interaction.

Right: Spectral function of CoO compared to PES [108].

down to −15 eV. We compare our data to the experimental data of Parmi-

giani [108] shown in Figure 6.14. Also the incoherent part of the PES-data

has weight at these large frequencies. The computed spectral function how-

ever looks qualitatively very different from the PES-spectra presented in the

right panel of Figure 6.14. Ref. [108] did not provide information on how

much spectral weight in the spectra comes from the oxygen. From the LDA

calculation we see that the energy range where the O 2p contributions would

enter is exactly where the CTQMC data lacks the spectral weight compared

to the PES-data. We find the gap to be of the order ∆ ∼ 2.8 eV which is in

good agreement with the experimental value of ∆ ∼ 2.6 eV [108].

6.5.4 Conclusion

We showed that the ground-state degeneracy of both NiO and CoO is cor-

rectly captured by our spin-rotation-symmtery preserving CTQMC-simulation.

For the spectral function of NiO within the different formulations of the

Coulomb interaction we find neither in CTQMC nor in ED any appreciable

difference. Pronounced features in the PES data are well reproduced by the

computed spectral functions, where mismatches can be probably related to

the additional weight coming from the oxygens. We find that simulations of

charge-transfer systems constitute an ambitious goal, because it is difficult

to capture physical effects coming from both the Ni 3d (Co 3d) and the O 2p

orbitals in a correct balance as well to disentangle these effects in the PES

116


data, which we rely on for our comparison. These systems therefore seem

not to be an optimal starting point for first-principles simulations as there

are too many adjustable parameters such as the double counting value in

addition to the material specific input parameters.

117


118

Chapter 7

Conclusion

In this thesis we have studied transition metal compounds from first-principles

taking into account the full intra-atomic Coulomb interactions of these 5-

orbital systems. As the physics of these transition metal compounds is de-

termined by strong electron correlations we have treated these systems in a

many-body formulation. We have explored the first-principles capabilities of

“DFT+Model Hamiltonian” descriptions of quantum impurity systems and

applied the recently developed Krylov implementation of the hybridization

expansion algorithm to solve the quantum impurity models numerically ex-

act. We have addressed both lattice systems, i.e., solids which we treated

within the DMFA as well as genuine quantum impurity systems.

Concerning genuine quantum impurity systems we have studied Iron impu-

rities on Sodium (Fe on Na) as well as Cobalt impurities in a Cupper host

(Co in and on Cu ).

For the Fe on Na system we have identified the transitions in the spectral

function and have shown how the reservoir of conduction electrons of the

Na host acts to screen additional charges on the Fe impurity. If we finetune

the occupation on the Fe impurity, we found our CTQMC calculations to

reasonably reproduce the PES-spectra and conclude that improvements to

better predict the chemical potential are needed. In this study we have

found ED and CTQMC a synergetic combination to study spectral functions,

because it enabled us to consider a continuous bath and to still identify the

underlying transitions in the spectra, as well as to judge the quality of the

analytical continuation of the CTQMC data.

In a study of Co in and on Cu we investigated the multi-orbital Kondo

effect and computed the Kondo temperature from first principles. We treated

119

these systems as multi-orbital Anderson impurity models with the full four-

fermion Coulomb interaction and observed that their physical behavior is

beyond the physics of simple “spin-only” models. In our study we have

shown that the Kondo temperature, TK , associated with the onset of Fermi

liquid behavior can differ between the impurity orbitals. The comparison

of our QMC calculations and scaling arguments further demonstrated that

fluctuations in the orbital degree of freedom and Hund’s rule coupling are

crucial in determining TK in realistic systems. We believe that the role of

orbital degrees of freedom is also of importance for the understanding of

other magnetic nanostructures based on 3d adatoms on surfaces as well as

magnetic impurities in bulk metals.

Concerning strongly correlated materials we have focussed on transition metal

oxides and studied a generic model of transition metal oxides (five band Hub-

bard model on the Bethe lattice) as well as the transition metal monoxides

Nickeloxide (NiO) and Cobaltoxide (CoO).

In the study of the 5-band Hubbard model on the Bethe lattice we have

determined the magnetic as well as the metal-insulator behavior for different

densities and compared two cases for the Hund’s coupling J . We could

amount differences in the magnetic phases to different exchange mechanisms,

which also occur in transition metal oxides. We have shown how starting from

the computed valence statistics we can reason the nature of the magnetic

exchange coupling and conclude that the 5-band Hubbard model on the Bethe

lattice is potentially very interesting to reproduce and provide insights to

aspects typical of transition metal oxides.

Concerning our study of NiO and CoO we have demonstrated that the

ground-state degeneracy of both NiO and CoO is correctly captured by our

spin-rotation-symmtery preserving CTQMC-simulation. We have found that

pronounced features in the PES data are well reproduced by the computed

spectral functions and motivated why mismatches to the experimental data

can probably related to the additional weight coming from the oxygens.

Concerning the aim to establish first-principles simulations we suggest to

further explore quantum impurity systems whose studies are by far not yet

exhausted. They provide an optimal starting point for first-principles simu-

lations as they have only few parameters to be determined ab-initio. We

believe that detailed benchmarks and the resulting improvements to the

“DFT+Model Hamiltonian” approach represent an indispensable step to-

wards achieving first-principles simulations of strongly correlated materials.

120

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List of Publications

6 B. Surer, T. O. Wehling, A. Wilhelm, A. I. Lichtenstein, M. Troyer,

A. M. Lauchli, P. Werner,

“Multi-orbital Kondo physics of Co in Cu hosts”,

accepted for publication in Phys. Rev. B

5 D. Koop, E. Santos, P. Mates, H. T. Vo, P. Bonnet, B. Bauer,

B. Surer, M. Troyer, D. N. Williams, J. E. Tohline, J. Freire, C. T.

Silva,

“A Provenance-Based Infrastructure to Support the Life Cycle of Exe-

cutable Papers”,

Procedia Computer Science 4, 648-657 (2011)

4 B. Bauer, et al. (ALPS collaboration),

“The ALPS project release 2.0: open source software for strongly cor-

related systems”,

Journal of Statistical Mechanics: Theory and Experiment 2011, P05001

(2011)

3 E. Gull, P. Werner, S. Fuchs, B. Surer, T. Pruschke, M. Troyer,

“Continuous-time quantum Monte Carlo impurity solvers”,

Computer Physics Communications 182, 1078-1082 (2011)

2 B. Surer, H. G. Katzgraber, G. T. Zimanyi, B. A. Allgood, G. Blatter,

“A Reply to the Comment by A. Mobius and M. Richter”,

Phys. Rev. Lett. 105, 039702 (2010)

1 B. Surer, H. G. Katzgraber, G. T. Zimanyi, B. A. Allgood, G. Blatter,

“Density of States and Critical Behavior of the Coulomb Glass”,

Phys. Rev. Lett. 102, 067205 (2009)

133

Curriculum Vitae

Personal Data

Name: Brigitte Surer

Date of Birth: November 8, 1982

Nationality: Swiss

Education

2011 PhD Thesis on ”First-principles simulations of multi-

orbital systems with strong electronic correlations” under

the supervision of Prof. M. Troyer

2007 - 2011 Doctoral studies at ETH Zurich at the Institute for The-

oretical Physics

2007 Diploma thesis on ”Electron Glasses” under the supervi-

sion of Prof. H. G. Katzgraber

2002 - 2007 University studies in physics at ETH Zurich

2001 Matura with distinction (Basler Maturandenpreis)

1998 - 2001 High school education at Gymnasium Muttenz (BL), fo-

cus on Latin (Typus B)

135

Acknowledgements

Completing this thesis could not have happened without the support of many

people, to whom I would like to express my sincere thanks and gratefulness.

First of all, I would like to express my gratitude to my advisor Matthias

Troyer for giving me the opportunity to carry out this PhD thesis and to work

on a research topic to which I can attach much importance and meaningful-

ness. I very much enjoyed working in a field very close to both theoretical

and experimental physics, where Computational Physics serves as a bridge

between these two worlds. Being in the fortunate situation to have a lead-

ing expert in Computational Condensed Matter Physics, High-Performance

Computing and Software design as my advisor, I very much appreciate that

he shared his expertise, scientific ideas and visions with all of our Computa-

tional Physics group. Letting me be part of this group and common projects,

where I felt much of the enthusiastic spirit he conveyed, was a great source

of motivation for me.

I am deeply indebted to Philipp Werner who has guided and mentored me

in many ways. Introducing me to my collaborators and initiating most of

the research projects he profoundly influenced my research activities. I am

very grateful for all the time he spent answering my countless questions. Our

discussions on quantum impurity solvers and the physics of strongly corre-

lated electron systems immensely helped to clarify my thoughts. With his

valuable advice, numerous helpful suggestions and comments he contributed

at various stages to this thesis.

I would like to thank Silke Biermann for taking the time to read my thesis

manuscript and accepting to be a co-referee at my examination as well as for

very helpful discussions and insights to LDA+DMFT simulations.

Very influential already during my studies and in the beginning of my PhD

was Emanuel Gull who introduced me to Computational Physics and in par-

ticular to simulations of strongly correlated systems. His support tremen-

137

dously facilitated getting started with my research and I thank him for intro-

ducing me to the field and for always being a valuable partner for discussions.

The presented projects would not have come into existence without the help,

input and work of my collaborators and I am very grateful for everything

that I had learned by working with them.

In the scope of the transition metal monoxide project I owe my gratitude to

Jan Kunes for helping me in establishing the LDA+DMFT framework and

for his hospitality during my visit in Prague.

The projects on the quantum impurity systems were done in collaboration

with the group of Alexander Lichtenstein and Alexander Poteryaev. I had

the great pleasure to meet and talk to all of them during my visit in Hamburg

and thank them very much for their hospitality and insightful discussions.

Thanks to Alexander Poteryaev’s ED calculations of NiO and Fe impurities

on Na I regained confidence in my CTQMC-calculations. My special thanks

go to Tim Wehling. I benefitted very much from his profound knowledge and

experience of genuine quantum impurity systems and Kondo physics and I

thank him for explaining me many aspects of this field. It is a pleasure to

work with him and I appreciated very much his organizational skills, as the

sustained coordination of our efforts in both Zurich and Hamburg were very

decisive for the realization of our project.

With my group colleagues I had a very good time discussing physics, com-

puter science as well as talking about anything and everything and I would

like to thank them for the happy times we spent together at the Institute as

well as at hikes and dinners, which were very precious moments for me.

I would also like to thank everyone of the Condensed Matter and Compu-

tational Condensed Matter Theory groups for stimulating discussions in our

offices, at informal seminars or poster sessions. Knowing I could ask for

support and advice at anytime was helpful and a comfort.

My warmest thanks go to my beloved family and Urs. With his incisive

questions Urs always pushed me to make the physics world intelligible beyond

mathematical formulas and to refine my conception of the world. My family

and Urs always encouraged and supported me in all my endeavors and I

would like them to know that they mean very much to me. As I cannot

hug my mother anymore to thank her for the fantastic physics books she

found and bought, for keeping her finger’s crossed on important occasions

and for all her emotional support especially in difficult times during my PhD

I decided to thank her by dedicating my thesis to her.

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Strong Electronic Correlations First-Principles ... · 2.2 Models of strongly correlated systems . . . . . . . . . . . . . . 10 ... of the electronic structure making the development

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Strong Electronic Correlations First-Principles ... · 2.2 Models of strongly correlated systems . . . . . . . . . . . . . . 10 ... of the electronic structure making the development