Dynamical Mean Field Theory - manep-nccr · 2017. 1. 27. · Dynamical mean field theory:--approximation to electron self energy obtained from solution of auxiliary quantum impurity

Copyright A. J. Millis 2013 Columbia University

Dynamical Mean Field Theory

Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)

Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, Rev. Mod. Phys. 77, 1027 (2005)

K. Held, II. A. Nekrasov, G. Keller, V. Eyert, N. Bluemer, A. K. McMahan, R. T Scalettar, T.h. Pruschke, V. I. Anisimov, and D. Vollhardt, D.; Phys. Status Solidi 243, 2599-2631 (2006)

G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006)

Review articles

http://phys.columbia.edu/~millis/MillisTLH.pdf


Dynamical Mean Field TheoryMany-body formalism: analogous to Hohenberg-Kohn

H =�

�,�

E��†�� +

�

��

I��†��†

�� + ...

G0: Green function of noninteracting reference problem (contains atomic positions)

F[{�}] = Funiv[{�}]�Trln[G�10 ��]

=>Luttinger-Ward functional

Funiv: determined (formally) from sum ofdiagrams. Depends only on interactions I��


ExampleSecond order termHubbard model

12

�LW [{G}] defined as sum of allvacuum to vacuum diagrams(with symmetry factors)

��G

= �

Funiv = �LW[{G}]�Tr [�G]


More formalities

�Funiv

��= G

Diagrammatic definition of Funiv =>

Thus stationarity condition�F��

= 0 => G =�G�1

0 ��1

F[{�}] = Funiv[{�}]�Trln[G�10 ��]

Difficulties with this formulation:--dont know Funiv (exc. perturbatively)--cant do extremization


1992 BreakthroughKotliar and Georges found analogue of Kohn-Sham steps: useful approximation for F and way to carry out minimization via auxiliary problem

Key first step: infinite d limit Metzner and Vollhardt, PRL 62 324 (1987)


Modern interpretation of Kotliar and Georges idea

Parametrize self energy in terms of small number N of functions of frequency

��(�) =�

ab

f��ab �ab

DMFT(�)

� = 1...M; a = 1...N << M

parametrization function f determines ‘flavor’ of DMFT (DFT analogue: LDA, GGA, B3LYP, ....)

Also must truncate interactionI�� “appropriately”


Approximation to self energy + truncated interactions imply approximation to Funiv

Approximated functional Fapproxuniv is functional of finite (small) number of functions of frequency , thus is the universal functional of some 0 (space) +1 (time) dimensional quantum field theory. Derivative gives Green function of this model:

�ab(�)

�Fapproxuniv

��ba(�)= Gab

QI(�)


Specifying the quantum impurity model

Need

--Interactions. These are the ‘appropriate truncation’ of the interactions in the original model

--a noninteracting (‘bare’) Green function G0

Then can compute the Green function and self energy

GQI =�G�10 ��

��1


Analogy:

Density functional <=> ‘Luttinger Ward functional

Kohn-Sham equations <=> quantum impurity model

Particle density <=> electron Green function


Quantum impurity model is in principle nothing more than a machine for generating self energies (as Kohn-Sham eigenstates are artifice for generating electron density)

Useful to view auxiliary problem as ‘quantum impurity model’ (cluster of sites coupled to noninteracting bath)

As with Kohn Sham eigenstates, it is tempting (and maybe reasonable) to ascribe physical signficance to it


In Hamiltonian representation

Impurity Hamiltonian

Coupling to bath+

�

p,ab

�Vp

abd†acpb + H.c

�. + Hbath[{c†pacpa}]

Important part of bath: ‘hybridization function’

�ab(z) =�

p

Vpac

�1

z� �bathp

�Vp,†

cb

HQI =�

ab

d†aE

abQIdb + Interactions

G�10 = � �Eab

QI ��ab(�)


Thus

F� Fapproxuniv [{�ab}]�Trln[G�1

0 ��

ab

f��ab �ab]

and stationarity implies

�F��ba

= GabQI �Tr��f��

ab

�G�1

0 ��

cd

f��cd �cd

��1

= 0

�G�10

�ab = �ab +

�

�Tr��f��ab

�G�1

0 ��

cd

f��cd �cd

��1�

��1

ab

so G0 is fixed

or, using GQI =�G�10 ��ab

��1from ‘impurity solver”


In practice

Guess hybridization function

Solve QI model; find self energy

Use extremum condition to update hybridization function

Continue until convergence is reached.

This actually works


Technical noteFrom your ‘impurity solver’ you need G at ‘all’ interesting frequencies. Solution ~uniformly accurate over whole relevant frequency range.

This is challenging.

Severe sign problem if ��, E

��= 0

=>reasonably high symmetry desirable so hybridization function commutes with impurity energies

For models with rotationally invariant multiplet interactions, cost grows exponentially with number of orbitals. 5 do-able with great effort more with Ising approximation (typically bad)

For simple 1 band Hubbard model, can access up to 16 orbitals at interesting temperatures.


advantages of method

*‘Moving part’ Trp [�a(p)Glattice(�approx)]

some sort of spatial average over electron spectral function--but still a function of frequency

*Computational task: solve quantum impurity model: not necessarily easy, but do-able

=>releases many-body physics from twin tyrannies of --focus on coherent quasiparticles/expansion about

well understood broken symmetry state--emphasis on particle density and ground state properties


In formal terms:

--Approximation to full M x M self energy matrix in terms of N(N+1)/2 functions determined from solution of auxiliary problem specified by self-consistency condition.

--Auxiliary problem: find (at all frequencies) Green functions of N-orbital quantum impurity model.

Question (practical): for feasible N, can you get the physics information you want?

``Exponential wall’’ remains:


Questions (not addressed here)

•What choices of parametrization function f are admissible

•What kinds of impurity models

•What can be solved• Where is the exponential wall (how large can N be?)• what kinds of interactions can be included

•What else (besides electron self energy) can be calculated

•Quality of approximation


Impurity model is just the Anderson model, with conduction band density of states fixed by self-consistency condition.

Regimes of Anderson model correspond to different physical behavior

Single-site (N=1) DMFT of Hubbard modelimpurity model: One non-degenerate d orbital per unit

cell

A =�

d�d��d†

�(�)G�10 (� � �

�)d�(�

�) + U

�d�n�(�)n�(�)

�

DMFT impurity model

H = ��

ij

ti�jc†i�cj� + U

�

i

ni�ni�


Phase diagram and spectral function at n=1

X. Y. Zhang, M. J. Rozenberg, and G. KotliarPhys. Rev. Lett. 70, 1666 (1993


interaction driven MIT via pole splitting

-4 -2 0 2 4Energy

-15

-10

-5

0

5

Im Σ

U=0.85Uc2

-6 -4 -2 0 2 4 6Energy

0

0.5

1

Im G

U=0.85Uc2

P. Cornaglia dataU=0.85 Uc2

’mass’ renormalizationdiverges as approach insulator

µ renormalization0 if p-h symmetry

�(z) � �21

z � �1+

�22

z � �2

�1,2 � 0 as U � Uc2

�(z � 0) = ��21�2 + �2

2�1

�1�2� z

�21�

22 + �2

2�21

�21�2

2


doping driven transition:also coexistence regime;

2 pole structure

-6 -4 -2 0 2 4 6ω

0

5

10

15

20

25

30

35

40

ImΣ(ω)

δ=0.041152δ=0.056402δ=0.106226δ=0.13505

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

0

2

4

6

8

10


Many regimes of behavior

From A Georges College de France lecture spring 2010 16 June 2010

Key Point: theory captures incoherent dynamics at high frequency and crossover to coherent low frequency behavior


Regimes of behaviorconfigurations of impurity |0>, | >, | >, |2>

Strong correlationsgeneric occupancy

T

All configs; equal prob.

|0>, |2> freeze outlocal momentincoherent transport

T

All configs; equal prob

|0>, |2> freeze outlocal momentgap opens

Mott insulator

resonance forms

fermi liquid transport

Strong correlationsinteger occupancy


Orbital degeneracy: t2g orbitals

Chan, Werner, Millis, Phys. Rev. B 80, 235114 (2009)

J=U/6

Mott, magnetic and orbitally ordered phases.

Also ``spin freezing’’


Spin freezing line: associated with larger spin. =>power law self energy

Werner, Gull, Troyer, Millis, Phys. Rev. Lett. 101, 166405 (2008)


The power law self energy does not extend to lowest energies

Georges, Medici, Mravlje, Ann Rev Cond Mat Phys (2012)

Higher spin: extraordinarily low quasiparticle scale


Phys. Rev. Lett. 100 016404 (2008)

Third example: nickelate superlattices

Idea:

Bulk LaNiO3 Ni [d]7

(1 electron in two degenerate eg bands). In correctly chosen superlattice structure, split eg bands, get 1 electron in 1 band--”like” high-Tc

Chaloupka-Khalliulin


Pseudocubic LaNiO3

Relevant orbitals: eg symmetry Ni-O antibonding combinations

3z2-r2

x2-y2

Hybridizes strongly along zHybridizes weakly in x-y

Hybridizes strongly along x-yHybridizes very weakly in z

2 orbitals transform as doublet in cubic symmetry


One electron (band theory) physics

Pseudocubic LaNiO3

Hamada J Phys. Chem Sol 54 1157

Two-fold degenerate eg band complex

Hamada’s result:


Superlattice

MJ Han, X Wang, C. Marianetti and A. J. Millis, arXiv:1105.0016

Superlattice: La2AlNiO6

LaAlO3 layer: insulating barrier

Question (Khalliulin): how much orbital polarization can we get?

P =nx2�y2 � n3z2�r2

nx2�y2 + n3z2�r2


Band theory of La2NiAlO6 heterostructure:

Hansmann et al PRL 103 016410

degeneracy of eg bands lifted: 3z2-r2 band moves up

La2NiAlO6

LaNiO3


Band theory of La2NiAlO6 heterostructure:

Hansmann et al PRL 103 016410

La2NiAlO6 LaNiO3

2d fermi surface 3d fermi surface

No qualitative difference

Hamada J Phys. Chem Sol 54 1157


DFT+DMFT: Correlate frontier orbitals

P. Hansmann, X. Yang, A. Toschi, G. Khaliullin, O. K. Andersen,and K. Held, Phys. Rev. Lett. 103, 016401 2009 and Phys. Rev. B 82, 235123 2010

Map conduction bands to (multiple-orbital) Hubbard model. Solve with single-site DMFT

Interaction (phenomenological)

Disperson: fit band theory


Result: Proximity to Mott insulator=>enhanced polarization

Phys. Rev. Lett. 103, 016401 2009

Interactions enhance ‘orbital polarization’

U=0

Interactions drive ‘orbitally selective metal-insulator transition

U=6eV

Phys. Rev. B 82, 235123 2010


SummaryDynamical mean field theory:

--approximation to electron self energy obtained from solution of auxiliary quantum impurity problem

--can treat high temperature and incoherent phases; metal insulator transition and low scales

--limits of accuracy of approximation: subject of different set of lectures

--meshes localized and k-space physics

?how to go beyond phenomenologically defined models?


Green function:

H0 =�

i

P 2i

2me+ Vext(ri) Bare Hamiltonian:

kinetic energy and lattice potential

G(r, r�;�) =

�{�(r),�†(r

�)}

�

�

=��1� H0 ��(r, r

�;�)

��1

DFT: static approximation to self energy

Many body theory: supplement DFT self energy with additional terms expressing ``beyond DFT’’ physics

GDFT (r, r�;�) =

��1� H0 ��DFT (r, r

�)��1

General formalism


Express beyond DFT physics as ‘dynamical’ self energy

�(r, r�;�) = �DFT(r, r

�) + �dyn(r, r

�;�)

Cannot treat many body physics of all electronic degrees of freedom=>must select subset of states for which dynamical self energy is computed*different ways to do this


Frontier Orbital Approach

Express G in band basis. DFT part is diagonal. Many-body (``dynamical’’) self energy is in general a matrix

GDFT (k, �) =��

� � �KSn (k)

��nm � �dyn(k, �)

��1

�nmdyn(k, �) =

�drdr

��KS

nk (r)�dyn(r, r�;�)�KS

mk (r�)

*Keep only self-energy and interaction matrix elements within frontier orbital basis.(Interaction: phenomenological U,J or screened coulomb interaction projected onto these states)


Issues:

relevant (mainly d) bands entangled with p bands.

Practical question: how to separate bands.

Principle question: how important are the O states

Red: Ni-d

Blue : O2p�

Cubic LaNiO3Fat bands

J. Rondinelli prvt comm

``d’’ orbital corresponding to the p-d antibonding bands has large spatial extent. Is it reasonable to ascribe just an on-site interaction U

OK Andersen JPhys. Chem. Solids 56, 1573 ~1995


Atomic Orbital Approach

alternative: define ‘atomic like’ orbital centered on sites iproject self energy onto this basis

�iat(r)

G(r, r�,�) =�� HKS � �dyn(r, r�;�)

��1

�dyn(r, r�;�) �

�

ij

|�iat(r) > �ij

dyn(�) < �jat(r

�)|

Interaction: intra-d interactions already discussed


In real materials applications

retain only on-site (but orbitally dependent) terms in self energy and only on-site Slater-Kanamori terms in interaction

�dyn(r, r�;�)�

�

iab

|�iad (r) > �ab

dyn(�) < �ibd (r

�)|

�double�count(r, r�)�

�

iab

|�iad (r) > �ab

dc < �ibd (r

�)|

G(r, r�,�) =�� HKS ��dc(r, r

�)��dyn(r, r�;�)

��1


Additional step (not often taken in practice): Full charge self consistency

Recall Kohn-Sham potential VKS[{n(r)}] is a functional of density.

=>need to use density computed from interacting G

n(r) =� µ d�

�ImG(r, r,�)

to construct VKS.


Two ways to define atomic orbital

*A’priori: simply choose a wave function you think is appropriate (e.g. free atom hartree fock d-wave function)

*Wannier function

0.0 0.2 0.4 0.6 0.8 1.0

�0.4

�0.2

0.0

0.2

0.4

0.6

0.8

If isolated band (index n) with wave function �nk(r)

then real space wave function centered in unit cell i (pos Ri)

�i(r) =�

BZ(dk)eik·Ri�nk(r)


Maximally localized Wannier function

If many bands in a energy window, then many ways to construct Wannier functions--parameterized by family of unitary transformations depending on k

J. Rondinelli ANL prvt comm

LaNiO3

�ai (r) =

�

BZ(dk)eik·Ri

�

n

�Uan(k)�nk(r)

�

Marzari-Vanderbilt: showed how to choose unitary transformation to make wannier functions correspond (as closely as Fourier’s theorem allows) to atomic-like functions centered on particular atoms (e.g. O 2p and Ni 3d)


If you have a set of bands reasonably well separated from other bands (e.g. p-d complex in many transition metal oxides)

Write G HKS and self energy as matrices in Wannier basisagain retain only terms in dynamical self energy corresponding to orbitals where correlations are important

G(�) =��1� HKS

n � �dyn(�)��1

HKSab =

�drdr

��a(r)

��2

2m+ Vlatt + �DFT(r, r

�)�

�b(r�)

HKSab is formal definition of tight-binding model corresponding to ``ab-initio’’ wave functions


Note!

All of these procedures assume that the wave functions of band theory have some meaning in the actual correlated material


Note!

atomic orbital approach

�dyn(r, r�;�) �

�

ij

|�iat(r) > �ij

dyn(�) < �jat(r

�)|

will in general lead to hartree shift which moves atomic (here, d) orbitals relative to the p orbitals


What do people do in practice*DFT+U (Anisimov 1998)

Identify atomic-like d-orbital. Keep on-site d-d self energyarising from Hartree approx to Slater interactions

Va� = U < na� > +(U� 2J)�

b �=a

< nb� > +(U� 3J)�

b �=a

< nb� >

This calculation ``knows’’ about correlations in the d-multiplet only via ordering. In non-ordered state, shift is same for all orbitals

As with most Hartree approximations, tendency to order is overestimated

Crucial physical effect: hartree shift of entire d-multiplet by amount of order (U-2.5J)*n/spin/orbital


Hartree term: shifts d-multiplet relative to p-states.this is important to the physics!!

d-d transition: 2dN->dN-1dN+1

E

d-p transition: dNp2->dN+1p1

E

Energy �Energy U

� > U: Mott-Hubbard � < U: Charge Transfer

Lower energy controls the interesting physics

Shift of d relative to p changes the physics


thus the crucial question:In a beyond-DFT calculation: where is the renormalized d energy, relative to the ligands


thus the crucial question:In a beyond-DFT calculation: where is the renormalized d energy, relative to the ligands

This is sometimes called ``the double counting problem’’, based on the idea that DFT includes some of the intra-d interactions and one does not want to count them twice. But the real question is--how much does the d-level shift relative to band theory.

a large literature exists (see, e.g. Karolak et al, Journal of Electron Spectroscopy and related phenomena, 181 (2010) 11–15) but there is as yet no generally agreed upon prescription. Results are sensitive to choice of double counting correction.


p-d covalency

My point of view (not universally accepted): parametrize the renormalized d energy by d occupancy

key issue: renormalized d energy, relative to the ligands.

Question: what does ‘‘renormalized d energy’’ mean

0.0 0.2 0.4 0.6 0.8 1.0

�0.4

�0.2

0.0

0.2

0.4

0.6

0.8

Unhybridized bands: upper one is pure d. Increase hybridization, increase d-character of occupied states


d-occupancy:* Intuitive notion: e.g. La3+Ni3+O32- =>Ni d7 (Nd=6+1)

*Theoretically needed (if you want to put correlations on d-orbital you need to know what this orbital is and how much it is occupied)

*definition:

In terms of exact Green function G(r, r�;�)

and predefined d-wave function �d

Nd =�

a,�

�d�

�f(�)

�d3rd3r

�Im

�(�a

d(r))� G�(r, r�,�)�a

d(r�)�


Notes

*d occupancy depends on how orbital is defined (as does the entire edifice of DFT+....)

Expect: all reasonable definitions give consistent answers (more on this later)

*should focus only on occupancy of ‘relevant’ orbitals


Notes

*Density functional theory only gives you

n(r) =�

a,�

�d�

�f(�)

�d3r Im [ G�(r, r,�)]

Thus even exact density functional, solved exactly, is not guaranteed to give you the exact Nd

But: seems plausible that in most cases DFT is a reasonably good approximation (more on this later)


Notes

*Nd: theoretically precisely defined;experimental measures are indirect

--Intensities in resonant absorption spectra--Sizes of moments/knight shifts--peak positions (``many-body ed-ep)

(more on this later)


In practice:Available double counting corrections give Nd fairly close to Nd defined from DFT band theory calculations

Example: rare earth nickelates ReNiO3

LuNiO3: insulator. 2 inequivalent Ni sites.

Ni1 Ni2GGA 8.21 8.20GGA+U 8.24 8.22

Charge fixed by electrostatic effects.Very robust


*DFT+DMFT (Georges04,Kotliar06,Held06)

Identify atomic-like d-orbital. Keep on-site d-d self energyarising from single-site DMFT approx using Slater-kanamori interactions

GabQI =< �ia

d (r)|G(r, r�,�)|�ibd (r

�) >

DMFT self consistency eq:

Same double counting issue as in DFT+U


Another example: High Tc cuprates

Green dots: fully charge self-consistent DFT+DMFT calculations for U=7,10eV. Vertical lines: DFT band theory (Wein2K and GGA)


Nickelate superlattices

Phys. Rev. Lett. 100 016404 (2008)

Phys. Rev. Lett. 103, 016401 2009

U=0

U=6eV

Frontier orbital approach


Atomic-like d orbitalsM. J. Han, X. Wang, C. A. Marianetti, A. J. M, Phys. Rev. Lett. 107 206804 (2011).

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

*LaNiO3/LaXO3 (mainly X=Al)*VASP/Wannier DFT+DMFT (Non-self-consistent) *Computed ‘orbital polarization’*Several combinations of U and ed-ep

M

I


Results 1: Define P by integrating spectral functionfor superlattice

Spectral functions

P=0.05 P=0.09

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -8 -6 -4 -2 0 2 4

DO

S (

sta

tes/e

V)

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -8 -6 -4 -2 0 2 4

(b)Op

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -8 -6 -4 -2 0 2 4

DO

S (

sta

tes/e

V)

Energy (eV)

(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -8 -6 -4 -2 0 2 4

Energy (eV)

(d) x2-y

2

3z2-r

2

P=0.2 P=0.07

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

Phase Diagram

M

I

Interactions decrease P


Results 2: How many sheets to the Fermi surface

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

Phase Diagram

M

I

P=0.05 P=0.09

P=0.2 P=0.07

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

(a) (b)

(c) (d)

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π


Results 2: Relation to Nd

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

Phase Diagram

M

I

Nd=2

Nd=1.98

Nd=1.5

Nd=1.43

P=0.05 P=0.09

P=0.2 P=0.07

-8 -4 0 4 ! (eV)

0

4

8

12

U(e

V)

(a) (b)

(c) (d)

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π

!1.0 !0.5 0.0 0.5 1.0!1.0

!0.5

0.0

0.5

1.0

kx!Π

k y!Π


Key point: Ni-O covalency

La2InNiO6

Occupancy per orbitalBand theory: electron moves from O to Ni,

configuration is d8L

d8 is high spin:both orbitals occupied

According to band theory, LaNiO3 and derivatives are close to being negative charge transfer energy materials


d8 (high spin)=> no distortionTwo electrons in high spin state=>no energy gain from distortion

Negligible response to lattice distortion

Over wide range of many-body phase space, Nd is close enough to d8 that low-spin J-T state is disfavored

2-band Hubbard model with 1 el/Ni unitcannot represent d8L physics


Covalency in perovskitesGenerally important

IM

Band theory or standard double counting indicates that La2CuO4 is not a Mott insulator


Example: NiO

Fit full p-d band complex:Add ‘Slater-Kanamori’U-J interactions on d-only. Solve.

Karolak et al Journal of Electron Spectroscopy and Related Phenomena 181 (2010) 11–15

DFT

many-bodycalc

choice of double counting (covalency) affects results (perhaps less strongly)


Conclusion

•Wide variety of behaviors•Important role played by instability of d valence•Energetics and state of the art of realistic theory of correlated electron materials•Key (and still ill-understood) role of double counting correction•Important open question: is the ``standard model’’ (atomic d, local correlation) the most correct and useful description?

Dynamical Mean Field Theory - manep-nccr · 2017. 1. 27. · Dynamical mean field theory:--approximation to electron self energy obtained from solution of auxiliary quantum impurity

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