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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)
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G�10 (i!n) = i!n + µ��(i!n)
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Sat =
Z �
0d⌧
X
�
d+� (⌧)
✓@
@⌧� µ
◆d�(⌧) + U
Z �
0d⌧ n"(⌧)n#(⌧)
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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
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