Fermions en interaction: Introduction à la théorie

de Champ Moyen Dynamique(DMFT)

Cours 3 – Différentes perspectives sur l’établissement des équations de la théorie de champ moyen dynamique et la limite de

grande dimensionCycle 2018-2019

21 mai 2019

Chaire de Physique de la Matière Condensée

Antoine Georges

Lecture mostly on BOARD – see recording

Interacting Fermions: Introduction to

Dynamical Mean-Field Theory (DMFT)

Lecture 3 – Different perspectives on the DMFT equations, their derivation and the

limit of large dimensions.

2018-2019 LecturesMay 21, 2019

Chaire de Physique de la Matière Condensée

Antoine Georges

Slides will be in EnglishPlease don’t hesitate to ask questions in French or English

Today’s seminar (11:30)

Jan KunešInstitute of Solid-State Physics, TU Wien

Institute of Physics, Czech Academy of Sciences, Praha

Excitonic Condensation of Strongly Correlated Electrons

Exceptional seminar Wednesday May 22 at 11:00

Salle 5

Andrew J. MillisColumbia University

and CCQ-Flatiron Institute, Simons Foundation, New York

Correlated Electrons and the Lattice

`Impurity’ (DMFT) SolversAlgorithms

Quantum Monte Carlo

NRG, DMRGED,…

ApproximateSolvers, NCA etc.

Cuprates

GW+DMFTOther

EmbeddingsSEET, DMET

Quantum Chemistry Cluster

EmbeddingsDCA,C-DMFT,

Nested… Out of Equilibriumt-DMFT

`Extended’ DMFT

(EDMFT)

DMFTConceptual

Core

Applications To Models

DFT+DMFTApplications to

Materials

Oxides

Rare-Earths

Actinides

OrganicMaterials

Etc…

Mott/Hubbard

Hund

Kondo

Etc…

MoleculesMaterials

Vertex-basedExtensions

DΓA, TRILEX, Dual methods

Extensions

Pump-probe

NESS

Disorder +Interactions

A theoretical description of the solid-state based on ATOMS

rather than on an electron-gas picture: « Dynamical Mean-Field Theory »

Dynamical Mean-Field Theory:A.G. & G.Kotliar, PRB 45, 6479 (1992)Correlated electrons in large dimensions:W.Metzner & D.Vollhardt, PRL 62, 324 (1989)

Early review: Georges et al. Rev Mod Phys 68, 13 (1996)

Important intermediate steps by: Müller-Hartmann, Schweitzer and Czycholl, Brandt and Mielsch, V.Janis

Dynamical Mean-Field Theory:viewing a material as an (ensemble of) atoms coupled to a

self-consistent medium

EffectiveMediumAtom

Solid: crystal lattice of atoms

Correlated electrons in large dimensions: W.Metzner & D.Vollhardt, 1989Dynamical Mean-Field Theory: A.G. & G.Kotliar, 1992

Example: DMFT for the Hubbard model (a model of coupled atoms)

Focus on a given lattice site:“Atom” can be in 4 possible configurations:Describe ``history’’ of fluctuations between those configurations

Imaginary-time effective action describing these histories:

Effective `bare propagator’ self-consistently determined, hence depends on U, and other parameters.

The amplitude Δ(τ) for hopping in and out of the selected site is self-consistently determined: it is the quantum-mechanical Generalization of the Weiss effective field.

Shyb =

Z �

0d⌧

Z �

0d⌧ 0

X

�

d+� (⌧)�(⌧ � ⌧ 0)d�(⌧0)

<latexit sha1_base64="kOEYH/buVnWp8YGOsnMMUbJmnpI=">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</latexit>

S = Sat + Shyb<latexit sha1_base64="GJnZnbCTf7e/IUx2BEe3jHKm9yQ=">AAAB/nicbVDLSgNBEOz1GeMrKp68LAZBMIRNFPQiBr14jMQ8ILsss5NJMmT2wUyvEJaAv+LFgyJe/Q5v/oWf4GwiookFTRdV3UxPeZHgCi3rw5ibX1hcWs6sZFfX1jc2c1vbDRXGkrI6DUUoWx5RTPCA1ZGjYK1IMuJ7gjW9wVXqN++YVDwMbnEYMccnvYB3OSWoJTe3W7ML53ah5iYER0e69YfeKOvm8qWiNYb5Q6xpJX/xCWNU3dy73Qlp7LMAqSBKtUtWhE5CJHIq2Chrx4pFhA5Ij7U1DYjPlJOMzx+ZB1rpmN1Q6grQHKu/NxLiKzX0PT3pE+yraS8V//PaMXbPnIQHUYwsoJOHurEwMTTTLMwOl4yiGGpCqOT6VpP2iSQUdWJpCNbM32dCaJSLpeNi+eYkX7mcpAEZ2IN9OIQSnEIFrqEKdaCQwAM8wbNxbzwaL8brZHTO+N7ZgT8w3r4AbveV8A==</latexit>

G�10 (i!n) = i!n + µ��(i!n)

<latexit sha1_base64="CNIe8DqQj2ct2rzqI/nLssqUaJM=">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</latexit>

Sat =

Z �

0d⌧

X

�

d+� (⌧)

✓@

@⌧� µ

◆d�(⌧) + U

Z �

0d⌧ n"(⌧)n#(⌧)

<latexit sha1_base64="pvmSAi3Ww6z7W/aK1fjCCXwSbxU=">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</latexit>

The self-consistency equation and the DMFT loopApproximating the self-energy by that of the local problem : à fully determines both the local G and Δ:

EFFECTIVE QUANTUM IMPURITY PROBLEM

THEDMFTLOOP

Local G.FBath

SELF-CONSISTENCY CONDITION

GRR is related to the exact self-energy of the lattice (solid) by:

In which is the tight-binding band (FT of the hopping tRR’)

Let us now make the APPROXIMATION that the lattice self-energy is k-independent and coincides with that of the effective atom (impurity problem):

This leads to the following self-consistency condition:

High-frequency à

Δ(ω): generalizing the Weiss field to the quantum world

Einstein, Paul Ehrenfest, Paul Langevin, Heike Kammerlingh-Onnes, and Pierre Weiss at Ehrenfest's home, Leyden, the Netherlands. From Einstein, His Life and Times,

by Philipp Frank (New York: A.A. Knopf, 1947). Photo courtesy AIP Emilio Segrè Visual Archives.

AlbertEinstein

Paul Langevin

Heike Kammerlingh-Onnes

Pierre Weiss

PaulEhrenfest

Pierre Weiss1865-1940« Théorie du Champ Moléculaire »(1907)

Weiss mean-field theoryDensity-functional theoryDynamical mean-field theory

rely on similar conceptual basis

- Exact representation of local observable:- Generalized ``Weiss field’’ - Self-consistency condition, later approximated

- Exact energy functional of local observable

see e.g:A.GarXiv cond-mat0403123

The DMFT construction is EXACT:

• For the non-interacting system (U =0 à Σ = 0 !)

• For the isolated atom (strong-coupling limit t=0 à Δ = 0)

àHence provides an interpolation from weak to strong coupling

• In the formal limit of infinite dimensionality (infinite lattice coordination) [introduced by Metzner and Vollhardt, PRL 62 (1989) 324]

Proofs: LW functional, Cavity construction (more on board)

Physical Limits in which DMFT is accurate: small correlation lengths

0 10 20 30 40β6t

0

0.5

1

1.5

s

DMFT10th order8th order6th orderPadéHeisenberg model (QMC)ln(2)

U/6t=2.5, µ=U/2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

T/t0 0.5 1 1.5s

-0.8

-0.6

-0.4

-0.2

0

T/t0 0.5 1 1.5

E/t

lattice QMCDMFTextrapolated DCAHTSEln 2

Hubbard d=3 ½ filledU/t=15De Leo et al.PRA 83 (2011) 023606

Hubbard d=3 ½ filledU/t=8Fuchs et al.PRL 106 (2011) 030401

Doped: smaller correlation length

cf. A.G. Ann. Phys (Berlin) 523 (2011) 672

The Luttinger-Ward Functional

Does it really exist ? J

Classical StatMech Model in which an infinite series of terms must be summed in d=infty: fully frustrated Ising

Density of states of a tight-binding band on a d-dimensional cubic lattice, as d increases.

MERCI POUR VOTRE ATTENTION !

PROCHAINE SEANCE:MARDI 28 Mai 9:30

Two Lectures:- Atom in a bath: introduction to the

Anderson impurity model with a DMFT perspective

- The Mott transition

No seminar