Dynamical mean-field theory
|
|
- Lucy Dean
- 5 years ago
- Views:
Transcription
1 Dynamical mean-field theory Marcus Kollar Theoretical Physics III, University of Augsburg, Germany Autumn School: Hands-On DMFT DFG-Forschergruppe 1346 Forschungszentrum Jülich August 4-7, 2011
2 Outline I. Introduction Green functions Useful concepts II. Fermions in infinite dimensions Free fermions Many-body theory III. Dynamical mean-field theory Mapping onto impurity models A solvable example Impurity solvers
3 Electrons in solids condensed matter: electrons in a ionic potential individual atoms φ α (r) s, p, d, f,... condensed matter Bloch:ψ nk (r) Wannier:φ n (r R) unbound electrons 1 Jellium, V e ikr Coulomb interaction:v ee (r r 1 ) r r important for strongly localized 3d, 4d, 4f,... electrons Hubbard model: e.g. 1 band, onlyu=v iiii H Hubbard = ijσ t ij c + iσ c jσ + U i n i n i lectures
4 Dynamical mean-field theory limit of large coordination numberzor large dimensiond: scaling: t=t / Z with Z d Green function: G ij (ω) d R i R j /2 self energy: Σ ij (ω) =δ ij Σ(ω) local! mapping onto single-site problem: self-energyσ ii [G ii ] same as for dynamical single-site problem e.g. Anderson impurity model numerical methods Dynamical mean-field theory Metzner & Vollhardt 89; Müller-Hartmann 89; Georges & Kotliar 92; Georges et al. RMP 96,...
5 Part I Introduction 1. Green functions Spectral representations Self-energy Path-integral formulation 2. Useful concepts Quasiparticles Hubbard bands Mott-Hubbard transition
6 1. Green functions imaginary-time-ordered fermionic Green functiong αβ (τ): e.g., Negele & Orland { G αβ (τ)= T τ c α (τ)c + β (0) = cα (τ)c + β (0) τ> 0 c + β (0)c α(τ) τ 0 = G αβ (τ+β) for β<τ< 0 with Heisenberg operatorsa(τ)=e Hτ Ae Hτ Matsubara Green function: G αβ (τ)=t G αβ (iω n )= + n= β 0 G αβ (iω n )e iω nτ dτg αβ (τ)e iω nτ with fermionic Matsubara frequenciesiω n = 2πT(n+ 1 2 )
7 Spectral representations spectral function: G αβ (iω n )= dω A αβ(ω) iω n ω A αβ (ω)= 1 π ImG αβ(ω+i0) }{{} retarded Green function = n c + β m m c α n e βe m e βe n n,m Z δ(e n E m ω) local Green function: G iiσ (iω n )=G σ (iω n )= A iiσ (ω)=a σ (ω)= 1 π Im 1 L 1 L G kσ (iω n ) k G kσ (ω+i0) = interacting density of states k
8 Free particles free particles: H µn= kσ(ǫ k µ)c + kσ c kσ G (0) kσ (iω n)= 1 iω n +µ ǫ k local Green function: G σ (iω n )= 1 L A σ (ω)= 1 L k 1 iω n +µ ǫ k = dǫ δ(ω+µ ǫ k ) =ρ(ω+µ) k ρ(ǫ) iω n +µ ǫ with free density of states (which characterizesǫ k ) ρ(ω)= k δ(ω ǫ k )
9 Self-energy self-energy Σ k (iω n ): G kσ (iω n )= 1 iω n +µ ǫ k Σ kσ (iω n ) G kσ (iω n ) 1 =G (0) kσ (iω n) 1 Σ kσ (iω n ) Dyson equation matrix notation: G ijσ (iω n )=(G) ij,σ,n G 1 =G (0) 1 Σ or G =G (0) +G (0) ΣG diagrammatic notation: = + Σ
10 Path-integral formulation partition function for fermionic HamiltonianH({c + α},{c α }) : Z= Tre β(h µn) = D(φ α (τ),φ α(τ)) exp(a) φ α (β)= φ α (0) = functional integral over Grassmann variablesφ α (τ) action: A= β 0 dτ [ ] φ α( τ µ)φ α +H({φ α},{φ α }) α e.g., Negele & Orland imaginary-time-ordered fermionic Green function: G αβ (τ)= 1 Z D(φ φ) φ α (τ)φ β (0) exp(a) APBC
11 Part I Introduction 1. Green functions Spectral representations Self-energy Path-integral formulation 2. Useful concepts Quasiparticles Hubbard bands Mott-Hubbard transition
12 2. Useful concepts. Quasiparticles spectral function: describes single-particle excitations A k (ω)= 1 π ImΣ k (ω) [ω ǫ k +µ ReΣ k (ω)] 2 +[ImΣ k (ω)] 2 real part vanishes if ω=ǫ k µ+ ReΣ k (ω) solutions ω=e k
13 2. Useful concepts. Quasiparticles spectral function: describes single-particle excitations A k (ω)= 1 π ImΣ k (ω) [ω ǫ k +µ ReΣ k (ω)] 2 +[ImΣ k (ω)] 2 real part vanishes if ω=ǫ k µ+ ReΣ k (ω) solutions ω=e k forω E k : LSCO Zhou et al. (2006) G k (ω) Z k (E k ) ω E k +iτ k (E k ) 1 Z k (ω)=1/[1 ω ReΣ k(ω)] τ k (ω)=1/[ Z k ImΣ(ω)] Fermi liquid: coherent quasiparticles for sufficiently smallω
14 Hubbard bands, Mott transition atomic limit: H at = i[un i n i µ(n i +n i )] G at σ(iω n )= n σ iω n +µ U + 1 n σ iω n +µ spectral function: ω peaks broaden fort ij 0 Hubbard bands Hubbard 63 Hubbard bands merge for large enough t ij quasiparticle bands develops gaps for large enoughu (non-magnetic) Mott-Hubbard transition atu=u c andn=1 Mott 46
15 Part II Fermions in infinite dimensions 1. Free fermions Scaling of hopping amplitudes Density of states 2. Many-body theory Diagrammatic expansions Power-counting in 1/d Simplifications ind=
16 1. Free fermions crystal lattices ind=3: simple cubic lattice (Z= 8) face-centered cubic lattice (Z= 12)... generalized lattices for any (large) dimensiond? easy for hypercubic lattice: inddimensions: e 1 =(1, 0, 0,... ) d=1 d=2 d=3 e 2 =(0, 1, 0,... )...=... e d =(0, 0, 0,...,1)
17 Next-neighbor hopping kinetic energy: H kin = ijσ t ij c + iσ c jσ = kσ ǫ k c + kσ c kσ NN hopping: t ij =t(r i R j )= { t if Ri R j =±e n 0 else dispersion: ǫ k = 2t d i=1 cosk i nontrivial limitd? density of states: ρ(ǫ)= 1 L δ(ǫ ǫ k ) k L = d d k (2π) dδ(ǫ ǫ k)
18 Scaling of hopping amplitudes elegant answer: Metzner & Vollhardt 89 random variablesx i = 2 cosk i (mean=0, variance=1) X d := 1 d d i=1 cosk i central limit theorem: ford : X d in law Gaussian r.v. (mean=0, variance=1) density of states: distribution function of 2dtX d ρ(ǫ)= 1 2π t e ǫ 2 2t 2 for t= t 2d
19 Density of states ρ(ω) Vollhardt 93
20 Part II Fermions in infinite dimensions 1. Free fermions Scaling of hopping amplitudes Density of states Generalized lattices 2. Many-body theory Diagrammatic expansions Power-counting in 1/d Simplifications ind=
21 2. Many-body theory Feynman diagrams for Green functions: = non-interacting Green function lineg (0) = interaction vertex = full (interacting) Green function lineg perturbation expansion: =
22 Self-energy proper self-energy diagrams: external vertex amputated cannot be cut in two pieces proper proper not proper proper self-energy: Σ =
23 Skeleton expansion so far: Σ[G (0) ] now: omit self-energy insertions, etc. skeleton expansionσ[g] Σ = avoid double counting should be equivalent when summing all diagrams not equivalent when summing some diagrams
24 Power counting in 1/d d dependence ofg ijσ (ω) ford? hopping amplitudes: t ij =t ij d 1 2 R i R j kinetic energy: E kin,σ = ij t ij c + iσ c jσ = dω t ij 2πi G ijσ(ω)e iω0+ =O(d 0 ) ij }{{} O(d R i R j ) Green function: G ijσ (ω)=o(d 1 2 R i R j ), G iiσ (ω)=o(d 0 ) simplifications for Feynman diagrams!
25 Diagrammatic simplifications Hugenholtz diagrams: (Hubbard model: no exchange diagrams) i,σ i, σ = Un i n i = skeleton expansion: Σ = (1) consider fixed i: compare j i with j = i i j
26 Collapse of position space diagrams skeleton expansion: 3 independent paths fromitoj Green function lines: O(d 3 2 R i R j ) summation overj: O(d R i R j ) j skeleton diagram iso(d 1 2 R i R j ) i ind= : all vertices inσ[g] have the same site label! self-energy is local! Σ ijσ (ω)=δ ij Σ iiσ (ω)=δ ij Σ σ (ω) Σ kσ (ω)=σ σ (ω) independent ofk!
27 Consequences of local self-energy simplekdependence: G kσ (iω n )= 1 iω n +µ ǫ k Σ σ (iω n ) local Green function: G σ (iω n )= d d k 1 (2π) d iω n +µ ǫ k Σ σ (iω n ) Dyson equation = dǫ ρ(ǫ) iω n +µ Σ σ (iω n ) ǫ Hilbert transform (later: self-consistency equation )
28 Part III Dynamical mean-field theory 1. Mapping onto impurity models 2. A solvable example 3. Impurity solvers
29 1. Mapping onto impurity models Effective single-site action: A=A 1 +A 2 Kotliar & Georges 92, Jarrell 92 A 1 = β 0 dτ β 0dτ σ c σ(τ)g 1 σ (τ,τ )c σ (τ ) = n,σc σ (iω n)g σ (iω n ) 1 c σ (iω n ) A 2 = U β 0 dτc (τ)c (τ)c (τ)c (τ) local Hubbard interaction Weiss fieldg: (G 1 ) τ,τ =G 1 σ (τ,τ ) Green function: G σ (iω n )= c σ (iω n )c σ(iω n ) A[G]
30 Dynamical mean-field theory in generala 1 is not due to a single-site Hamiltonian G is a dynamical mean field only single-site Hamiltonian H at forg 1 = τ +µ define impurity self-energy Σ via G= [ G 1 Σ] 1 impurity Dyson equation skeleton expansion: Σ[G] = one site only! = Σ[G] same as for Hubbard model ind=!
31 Dynamical mean-field equations lattice Dyson equation: G σ (iω n )= together with = d d k 1 (2π) d iω n +µ ǫ k Σ σ (iω n ) dǫ ρ(ǫ) iω n +µ Σ σ (iω n ) ǫ self-consistency (1) G σ (iω n )= [ G σ (iω n ) 1 Σ σ (iω n )] 1 (2) G σ (iω n )= c σ (iω n )c σ(iω n ) A[G] (solve numerically) (3) three equations for unknownsg, G, Σ
32 Some simple limits non-interacting case,u= 0: Σ σ (iω n )=0 (1) G σ (iω n )=G (0) σ (iω n )= 1 L k G (0) k (iω n) (2) G σ (iω n )=G σ (iω n ) (3) atomic limit,t ij = 0,ǫ k = 0: ρ(ǫ)=δ(ǫ) (1) G σ (iω n )= 1 iω n +µ Σ σ (iω n ) (2) G σ (iω n ) 1 =iω n +µ G 1 σ (τ)= τ +µ (3)
33 2. A solvable example Falicov-Kimball model: hopping only fordspin species H= ij t ij d + i d j +E f f + i f i +U i d + i d i f+ i f i i d electrons hop on background off electrons f configuration optimizes the free energy half-filling, bipartitie lattice,d 2: checkerboard phase foru> 0 andt c >T > 0 Lieb 86 DMFT exactly solvable Brandt & Mielsch 89, van Dongen 90, Si et al. 92, Freericks & Zlatic 03
34 DMFT equations self-consistency forf electrons: G 1 f = τ +µ DMFT action: A= + β dτ 0 β 0 β 0 dτ d (τ)g 1 d (τ,τ )d(τ ) dτf (τ)( τ µ+e f )f(τ) U β 0 dτd (τ)d(τ)f (τ)f(τ) integrate outf electrons: (atomic limit!) G d (iω n )= d(iω n )d (iω n ) A = n f G d (iω n ) 1 U + 1 n f G d (iω n ) 1
35 DMFT solution self-consistency equations: G d (iω n )= dǫρ d (ǫ) iω n +µ Σ d (iω n ) ǫ G d (iω n ) 1 =G d (iω n ) 1 Σ d (iω n ) determinesg d (iω n ) for any density of statesρ d (ǫ) skeleton functionalσ d [G d ]: Σ d (iω n )= U 2 1 2G d (iω n ) ± ( ) U Un f 2G d (iω n ) G d (iω n ) involves all orders inu
36 Spectral function of itinerant electrons Bethe lattice, homogeneous phase,n d =n f = 1 2,U= 0.5, 1.0, Freericks & Zlatic 03 Mott metal-insulator transition atu= 2 non-fermi-liquid spectrumt independent in homogeneous phase
37 3. Impurity solvers representation ofg via Anderson impurity model: H= lσ ǫ l a + lσ a lσ + lσ V l (a + lσ c σ+c + σa lσ )+Uc + c c+ c integrate out host degrees of freedom actionawith G 1 σ (iω n )=iω n +µ l =iω n +µ 1 π V 2 l iω n ǫ l dω (ω) iω n ω (ω)=π l V 2 l δ(ω ǫ l) hybridization function
38 3. Impurity solvers representation ofg via Anderson impurity model: H= lσ ǫ l a + lσ a lσ + lσ V l (a + lσ c σ+c + σa lσ )+Uc + c c+ c integrate out host degrees of freedom actionawith G 1 σ (iω n )=iω n +µ l =iω n +µ 1 π V 2 l iω n ǫ l dω (ω) iω n ω (ω)=π l V 2 l δ(ω ǫ l) hybridization function
39 Numerical methods Hirsch-Fye QMC Trotter decomposition of imaginary-time action Continuous-time QMC Expansion in hopping or interaction ED exact diagonalization for small number of host sites NRG logarithmic discretization of host spectrum, sites added successively DMRG blocks with varying number of sites, dynamical quantities available lectures
40 Metal-insulator transition Hubbard model, Bethe lattice, homogeneous phase,n=1, DMFT(NRG) 1.5 U/W=1.0 U/W=1.42 U/W= A(ω)*W ω/w Bulla 99
41 Summary DMFT: exact ford numerical solution of local dynamical many-body problem input: kinetic energy, interactions, band-filling (materials!) lectures Outlook: multiband systems, real materials, LDA+DMFT numerical methods spatial fluctuations: cluster theories, dual fermions, DΓA,... lectures
5 Introduction to Dynamical Mean-Field Theory
5 Introduction to Dynamical Mean-Field Theory Marcus Kollar Center for Electronic Correlations and Magnetism University of Augsburg Contents 1 Introduction 2 2 Fermions in infinite dimensions 6 3 Simplifications
More informationElectronic correlations in models and materials. Jan Kuneš
Electronic correlations in models and materials Jan Kuneš Outline Dynamical-mean field theory Implementation (impurity problem) Single-band Hubbard model MnO under pressure moment collapse metal-insulator
More informationDiagrammatic Monte Carlo methods for Fermions
Diagrammatic Monte Carlo methods for Fermions Philipp Werner Department of Physics, Columbia University PRL 97, 7645 (26) PRB 74, 15517 (26) PRB 75, 8518 (27) PRB 76, 235123 (27) PRL 99, 12645 (27) PRL
More informationDynamical Mean Field Theory and Numerical Renormalization Group at Finite Temperature: Prospects and Challenges
Dynamical Mean Field Theory and Numerical Renormalization Group at Finite Temperature: Prospects and Challenges Frithjof B. Anders Institut für Theoretische Physik Universität Bremen Göttingen, December
More informationFrom Gutzwiller Wave Functions to Dynamical Mean-Field Theory
From utzwiller Wave Functions to Dynamical Mean-Field Theory Dieter Vollhardt Autumn School on Correlated Electrons DMFT at 25: Infinite Dimensions Forschungszentrum Jülich, September 15, 2014 Supported
More informationIntroduction to DMFT
Introduction to DMFT Lecture 2 : DMFT formalism 1 Toulouse, May 25th 2007 O. Parcollet 1. Derivation of the DMFT equations 2. Impurity solvers. 1 Derivation of DMFT equations 2 Cavity method. Large dimension
More informationMagnetic Moment Collapse drives Mott transition in MnO
Magnetic Moment Collapse drives Mott transition in MnO J. Kuneš Institute of Physics, Uni. Augsburg in collaboration with: V. I. Anisimov, A. V. Lukoyanov, W. E. Pickett, R. T. Scalettar, D. Vollhardt,
More information1 From Gutzwiller Wave Functions to Dynamical Mean-Field Theory
1 From Gutzwiller Wave Functions to Dynamical Mean-Field Theory Dieter Vollhardt Center for Electronic Correlations and Magnetism University of Augsburg Contents 1 Introduction 2 1.1 Modeling of correlated
More informationO. Parcollet CEA-Saclay FRANCE
Cluster Dynamical Mean Field Analysis of the Mott transition O. Parcollet CEA-Saclay FRANCE Dynamical Breakup of the Fermi Surface in a doped Mott Insulator M. Civelli, M. Capone, S. S. Kancharla, O.P.,
More information9 Making Use of Self-Energy Functionals: The Variational Cluster Approximation
9 Making Use of Self-Energy Functionals: The Variational Cluster Approximation Contents Michael Potthoff I. Institut für Theoretische Physik Universität Hamburg Motivation 2 2 The cluster approach 4 2.
More informationInelastic light scattering and the correlated metal-insulator transition
Inelastic light scattering and the correlated metal-insulator transition Jim Freericks (Georgetown University) Tom Devereaux (University of Waterloo) Ralf Bulla (University of Augsburg) Funding: National
More informationRole of Hund Coupling in Two-Orbital Systems
Role of Hund Coupling in Two-Orbital Systems Gun Sang Jeon Ewha Womans University 2013-08-30 NCTS Workshop on Quantum Condensation (QC13) collaboration with A. J. Kim, M.Y. Choi (SNU) Mott-Hubbard Transition
More informationCluster Extensions to the Dynamical Mean-Field Theory
Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? Thomas Pruschke Institut für Theoretische Physik Universität
More informationMott transition : beyond Dynamical Mean Field Theory
Mott transition : beyond Dynamical Mean Field Theory O. Parcollet 1. Cluster methods. 2. CDMFT 3. Mott transition in frustrated systems : hot-cold spots. Coll: G. Biroli (SPhT), G. Kotliar (Rutgers) Ref:
More informationLocal moment approach to the multi - orbital single impurity Anderson and Hubbard models
Local moment approach to the multi - orbital single impurity Anderson and Hubbard models Anna Kauch Institute of Theoretical Physics Warsaw University PIPT/Les Houches Summer School on Quantum Magnetism
More information16 Dynamical Mean-Field Approximation and Cluster Methods for Correlated Electron Systems
16 Dynamical Mean-Field Approximation and Cluster Methods for Correlated Electron Systems Thomas Pruschke Institute for Theoretical Physics, University of Göttingen, 37077 Göttingen, Germany pruschke@theorie.physik.uni-goettingen.de
More informationFerromagnetism and Metal-Insulator Transition in Hubbard Model with Alloy Disorder
Ferromagnetism and Metal-Insulator Transition in Hubbard Model with Alloy Disorder Krzysztof Byczuk Institute of Physics, Augsburg University Institute of Theoretical Physics, Warsaw University October
More informationDMFT for correlated bosons and boson-fermion mixtures
DMFT for correlated bosons and boson-fermion mixtures Workshop on Recent developments in dynamical mean-field theory ETH ürich, September 29, 2009 Dieter Vollhardt Supported by Deutsche Forschungsgemeinschaft
More informationDiagrammatic Green s Functions Approach to the Bose-Hubbard Model
Diagrammatic Green s Functions Approach to the Bose-Hubbard Model Matthias Ohliger Institut für Theoretische Physik Freie Universität Berlin 22nd of January 2008 Content OVERVIEW CONSIDERED SYSTEM BASIC
More informationarxiv:cond-mat/ v1 [cond-mat.str-el] 21 Mar 2006
Non-Fermi-liquid phases in the two-band Hubbard model: Finite-temperature exact diagonalization study of Hund s rule coupling A. Liebsch and T. A. Costi Institut für Festkörperforschung, Forschungszentrum
More informationw2dynamics : operation and applications
w2dynamics : operation and applications Giorgio Sangiovanni ERC Kick-off Meeting, 2.9.2013 Hackers Nico Parragh (Uni Wü) Markus Wallerberger (TU) Patrik Gunacker (TU) Andreas Hausoel (Uni Wü) A solver
More informationThe frustrated Hubbard model on the Bethe lattice - an investigation using the self-energy functional approach
The frustrated Hubbard model on the Bethe lattice - an investigation using the self-energy functional approach von Martin Eckstein Diplomarbeit im Fach Physik vorgelegt der Mathematisch-Naturwissenschaftlichen
More informationDiagrammatic Monte Carlo simulation of quantum impurity models
Diagrammatic Monte Carlo simulation of quantum impurity models Philipp Werner ETH Zurich IPAM, UCLA, Jan. 2009 Outline Continuous-time auxiliary field method (CT-AUX) Weak coupling expansion and auxiliary
More informationFrom Materials to Models and Back. Dieter Vollhardt
From Materials to Models and Back Dieter Vollhardt 28 th Edgar Lüscher Seminar, Klosters; February 8, 2017 From Materials to Models and Back - The Need for Models in Condensed Matter Physics - Outline:
More informationA continuous time algorithm for quantum impurity models
ISSP, Aug. 6 p.1 A continuou time algorithm for quantum impurity model Philipp Werner Department of Phyic, Columbia Univerity cond-mat/512727 ISSP, Aug. 6 p.2 Outline Introduction Dynamical mean field
More informationPhase Diagram of the Multi-Orbital Hubbard Model
Phase Diagram of the Multi-Orbital Hubbard Model Submitted by Bence Temesi BACHELOR THESIS Faculty of Physics at Ludwig-Maximilians-Universität München Supervisor: Prof. Dr. Jan von Delft Munich, August
More informationQuantum Cluster Methods (CPT/CDMFT)
Quantum Cluster Methods (CPT/CDMFT) David Sénéchal Département de physique Université de Sherbrooke Sherbrooke (Québec) Canada Autumn School on Correlated Electrons Forschungszentrum Jülich, Sept. 24,
More informationLocal moment approach to multi-orbital Anderson and Hubbard models
Local moment approach to multi-orbital Anderson and Hubbard models Anna Kauch 1 and Krzysztof Byczuk,1 1 Institute of Theoretical Physics, Warsaw University, ul. Hoża 69, PL--681 Warszawa, Poland Theoretical
More informationSimulation of the electron-phonon interaction in infinite dimensions
1 Simulation of the electron-phonon interaction in infinite dimensions J. K. Freericks Department of Physics, University of California, Davis, CA 95616 M. Jarrell Department of Physics, University of Cincinnati,
More informationThe Mott Metal-Insulator Transition
Florian Gebhard The Mott Metal-Insulator Transition Models and Methods With 38 Figures Springer 1. Metal Insulator Transitions 1 1.1 Classification of Metals and Insulators 2 1.1.1 Definition of Metal
More informationDYNAMICAL MEAN-FIELD THEORY FOR CORRELATED LATTICE FERMIONS
1 DYNAMICAL MEAN-FIELD THEORY FOR CORRELATED LATTICE FERMIONS K. BYCZUK (1) Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute for Physics, University of Augsburg, 86135
More informationSurprising Effects of Electronic Correlations in Solids
Center for Electronic Correlations and Magnetism University of Augsburg Surprising Effects of Electronic Correlations in Solids Dieter Vollhardt Supported by TRR 80 FOR 1346 University of Florida, Gainesville;
More information6. Auxiliary field continuous time quantum Monte Carlo
6. Auxiliary field continuous time quantum Monte Carlo The purpose of the auxiliary field continuous time quantum Monte Carlo method 1 is to calculate the full Greens function of the Anderson impurity
More informationQuantum Impurity Solvers for DMFT
Quantum Impurity Solvers for DMFT Vijay B. Shenoy (shenoy@physics.iisc.ernet.in) Centre for Condensed Matter Theory Department of Physics Indian Institute of Science, Bangalore VBS Quantum Impurity Solvers
More informationPHYSICAL REVIEW B VOLUME 57, NUMBER 19 ARTICLES. Charge-transfer metal-insulator transitions in the spin- 1 2
PHYSICAL REVIEW B VOLUME 57, NUMBER 19 15 MAY 1998-I ARTICLES Charge-transfer metal-insulator transitions in the spin- 1 2 Falicov-Kimball model Woonki Chung and J. K. Freericks Department of Physics,
More informationUbiquity of Linear Resistivity at Intermediate Temperature in Bad Metals
Ubiquity of Linear Resistivity at Intermediate Temperature in Bad Metals G. R. Boyd V. Zlatić Institute of Physics, Zagreb POB 304, Croatia and J. K. Freericks (Dated: October 3, 2018) arxiv:1404.2859v1
More informationDiagrammatic extensions of (E)DMFT: Dual boson
Diagrammatic extensions of (E)DMFT: Dual boson IPhT, CEA Saclay, France ISSP, June 25, 2014 Collaborators Mikhail Katsnelson (University of Nijmegen, The Netherlands) Alexander Lichtenstein (University
More information561 F 2005 Lecture 14 1
56 F 2005 Lecture 4 56 Fall 2005 Lecture 4 Green s Functions at Finite Temperature: Matsubara Formalism Following Mahan Ch. 3. Recall from T=0 formalism In terms of the exact hamiltonian H the Green s
More informationAn efficient impurity-solver for the dynamical mean field theory algorithm
Papers in Physics, vol. 9, art. 95 (217) www.papersinphysics.org Received: 31 March 217, Accepted: 6 June 217 Edited by: D. Domínguez Reviewed by: A. Feiguin, Northeastern University, Boston, United States.
More informationarxiv:cond-mat/ v1 [cond-mat.str-el] 11 Aug 2003
Phase diagram of the frustrated Hubbard model R. Zitzler, N. ong, h. Pruschke, and R. Bulla Center for electronic correlations and magnetism, heoretical Physics III, Institute of Physics, University of
More informationGreen s functions: calculation methods
M. A. Gusmão IF-UFRGS 1 FIP161 Text 13 Green s functions: calculation methods Equations of motion One of the usual methods for evaluating Green s functions starts by writing an equation of motion for the
More informationDynamical mean field approach to correlated lattice systems in and out of equilibrium
Dynamical mean field approach to correlated lattice systems in and out of equilibrium Philipp Werner University of Fribourg, Switzerland Kyoto, December 2013 Overview Dynamical mean field approximation
More informationLecture 1: The Equilibrium Green Function Method
Lecture 1: The Equilibrium Green Function Method Mark Jarrell April 27, 2011 Contents 1 Why Green functions? 2 2 Different types of Green functions 4 2.1 Retarded, advanced, time ordered and Matsubara
More informationField-Driven Quantum Systems, From Transient to Steady State
Field-Driven Quantum Systems, From Transient to Steady State Herbert F. Fotso The Ames Laboratory Collaborators: Lex Kemper Karlis Mikelsons Jim Freericks Device Miniaturization Ultracold Atoms Optical
More informationDynamical Mean Field within Iterative Perturbation Theory
Vol. 111 (2007) ACTA PHYSICA POLONICA A No. 5 Proceedings of the XII National School Correlated Electron Systems..., Ustroń 2006 Dynamical Mean Field within Iterative Perturbation Theory B. Radzimirski
More informationThe mapping problem in nonequilibrium dynamical mean-field theory
The mapping problem in nonequilibrium dynamical mean-field theory Masterarbeit im Fach Physik vorgelegt der Mathematisch-Naturwissenschaflichen Fakultät der Universität Augsburg von Christian Gramsch Januar
More informationPerurbation theory. d 3 r ρ(r)ρ(r )V C (r r ) (1)
Perurbation theory One way to go beyond the static mean field approximation is to explore the higher order terms in perturbation theory with respect to Coulomb interaction. This is not an optimal way of
More informationSolution of the Anderson impurity model via the functional renormalization group
Solution of the Anderson impurity model via the functional renormalization group Simon Streib, Aldo Isidori, and Peter Kopietz Institut für Theoretische Physik, Goethe-Universität Frankfurt Meeting DFG-Forschergruppe
More informationLecture 6: Fluctuation-Dissipation theorem and introduction to systems of interest
Lecture 6: Fluctuation-Dissipation theorem and introduction to systems of interest In the last lecture, we have discussed how one can describe the response of a well-equilibriated macroscopic system to
More informationIMPACT ionization and thermalization in photo-doped Mott insulators
IMPACT ionization and thermalization in photo-doped Mott insulators Philipp Werner (Fribourg) in collaboration with Martin Eckstein (Hamburg) Karsten Held (Vienna) Cargese, September 16 Motivation Photo-doping:
More informationSurprises in Correlated Electron Physics
Vol. 111 (2007) ACTA PHYSICA POLONICA A No. 4 Proceedings of the XII National School Correlated Electron Systems..., Ustroń 2006 Surprises in Correlated Electron Physics K. Byczuk a,b, W. Hofstetter c,
More informationDynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions Antoine Georges Laboratoire de Physique Théorique de l Ecole Normale Supérieure, 24, rue Lhomond,
More informationStatic and dynamic variational principles for strongly correlated electron systems
Static and dynamic variational principles for strongly correlated electron systems Michael Potthoff Citation: AIP Conf. Proc. 1419, 199 (2011); doi: 10.1063/1.3667325 View online: http://dx.doi.org/10.1063/1.3667325
More informationBose-Hubbard Model (BHM) at Finite Temperature
Bose-Hubbard Model (BHM) at Finite Temperature - a Layman s (Sebastian Schmidt) proposal - pick up Diploma work at FU-Berlin with PD Dr. Axel Pelster (Uni Duisburg-Essen) ~ Diagrammatic techniques, high-order,
More informationDynamical Mean Field Theory of inhomogeneous correlated systems
Dynamical Mean Field Theory of inhomogeneous correlated systems Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt
More informationarxiv:cond-mat/ v2 [cond-mat.str-el] 24 Feb 2006
Applications of Cluster Perturbation Theory Using Quantum Monte Carlo Data arxiv:cond-mat/0512406v2 [cond-mat.str-el] 24 Feb 2006 Fei Lin, Erik S. Sørensen, Catherine Kallin and A. John Berlinsky Department
More informationarxiv:cond-mat/ v1 [cond-mat.str-el] 4 Jul 2000
EPJ manuscript No. (will be inserted by the editor) arxiv:cond-mat/742v1 [cond-mat.str-el] 4 Jul 2 Phase diagram of the three-dimensional Hubbard model at half filling R. Staudt 1, M. Dzierzawa 1, and
More informationExact dynamical mean-field theory of the Falicov-Kimball model
Exact dynamical mean-field theory of the Falicov-Kimball model J. K. Freericks* Department of Physics, Georgetown University, Washington, D.C. 257, USA V. Zlatić Institute of Physics, Bijenička cesta 46,
More informationDynamical Mean-Field Theory for Correlated Electron Materials Dieter Vollhardt
Center for Electronic Correlations and Magnetism University of Augsburg Dynamical Mean-Field Theory for Correlated Electron Materials Dieter Vollhardt XXIII Latin American Symposium on Solid State Physics
More informationDynamical cluster approximation: Nonlocal dynamics of correlated electron systems
PHYSICAL REVIEW B VOLUME 61, NUMBER 19 15 MAY 2-I Dynamical cluster approximation: Nonlocal dynamics of correlated electron systems M. H. Hettler* Department of Physics, University of Cincinnati, Cincinnati,
More informationContinuous Time Monte Carlo methods for fermions
Continuous Time Monte Carlo methods for fermions Alexander Lichtenstein University of Hamburg In collaboration with A. Rubtsov (Moscow University) P. Werner (ETH Zurich) Outline Calculation of Path Integral
More information2. The interaction between an electron gas and the potential field is given by
68A. Solutions to Exercises II. April 5. Carry out a Hubbard Stratonovich transformation for the following interaction Hamiltonians, factorizing the terms in brackets, and write one or two sentences interpreting
More informationSurprising Effects of the Interaction between Electrons in Solids. Dieter Vollhardt
Surprising Effects of the Interaction between Electrons in Solids Dieter Vollhardt Shanghai Jiao Tong University; April 6, 2016 Augsburg: founded in 15 BC Roman Emperor Augustus 63 BC 14 AD Center founded
More informationarxiv: v2 [cond-mat.str-el] 11 Nov 2010
Multi-orbital simplified parquet equations for strongly correlated electrons Pavel Augustinský and Václav Janiš Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-822 Praha
More informationarxiv:cond-mat/ v1 [cond-mat.str-el] 6 Jun 1997
Insulating Phases of the d = Hubbard model David E Logan, Michael P Eastwood and Michael A Tusch Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ (U.K.)
More informationFluctuating exchange theory of dynamical electron correlations and magnetism
Fluctuating exchange theory of dynamical electron correlations and magnetism Václav Drchal Institute of Physics ASCR, Praha, Czech Republic Grant Agency of ASCR: project IAA11616 Workshop Frontiers in
More informationMasterthesis: Continuous Time - Quantum Monte Carlo impurity solver for the Hubbard model
Masterthesis: Continuous Time - Quantum Monte Carlo impurity solver for the Hubbard model Version: July 2, 24 Masterarbeit im Fach Physik vorgelegt der Mathematisch-Naturwissenschaflichen Fakultät der
More informationThe Gutzwiller Density Functional Theory
The Gutzwiller Density Functional Theory Jörg Bünemann, BTU Cottbus I) Introduction 1. Model for an H 2 -molecule 2. Transition metals and their compounds II) Gutzwiller variational theory 1. Gutzwiller
More informationContinuous time QMC methods
Continuous time QMC methods Matthias Troyer (ETH Zürich) Philipp Werner (Columbia ETHZ) Emanuel Gull (ETHZ Columbia) Andy J. Millis (Columbia) Olivier Parcollet (Paris) Sebastian Fuchs, Thomas Pruschke
More informationFunctional renormalization group approach to interacting Fermi systems DMFT as a booster rocket
Functional renormalization group approach to interacting Fermi systems DMFT as a booster rocket Walter Metzner, Max-Planck-Institute for Solid State Research 1. Introduction 2. Functional RG for Fermi
More informationLinearized dynamical mean-field theory for the Mott-Hubbard transition
Eur. Phys. J. B 13, 257 264 (2000) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Springer-Verlag 2000 Linearized dynamical mean-field theory for the Mott-Hubbard transition
More informationarxiv: v2 [cond-mat.str-el] 17 Jan 2011
vtex Efficient treatment of two-particle vertices in dynamical mean-field theory Jan Kuneš Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická, 6 53 Praha 6, Czech Republic (Dated:
More information9 Hirsch-Fye Quantum Monte Carlo Method for Dynamical Mean-Field Theory
9 Hirsch-Fye Quantum Monte Carlo Method for Dynamical Mean-Field Theory Nils Blümer Institut für Physik Johannes Gutenberg-Universität Mainz Contents 1 Introduction 2 1.1 Dynamical mean-field theory...........................
More informationFinite Temperature Field Theory
Finite Temperature Field Theory Dietrich Bödeker, Universität Bielefeld 1. Thermodynamics (better: thermo-statics) (a) Imaginary time formalism (b) free energy: scalar particles, resummation i. pedestrian
More information4 Development of the LDA+DMFT Approach
4 Development of the LDA+DMFT Approach Alexander Lichtenstein I. Institut für Theoretische Physik Universität Hamburg Contents 1 Introduction 2 2 Functional approach: from DFT to DMFT 4 3 Local correlations:
More informationTight-binding treatment of the Hubbard model in infinite dimensions
PHYSICAL REVIEW B VOLUME 54, NUMBER 3 15 JULY 1996-I Tight-binding treatment of the Hubbard model in infinite dimensions L. Craco Instituto de Física, Universidade Federal do Rio Grande do Sul, 9151-97
More informationSurprises in correlated electron physics
Surprises in correlated electron physics K. Byczuk 1,, W. Hofstetter 3, M. Kollar 1, and D. Vollhardt 1 (1) Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute for Physics,
More information5 Projectors, Hubbard U, Charge Self-Consistency, and Double-Counting
5 Projectors, Hubbard U, Charge Self-Consistency, and Double-Counting Tim Wehling Institute for Theoretical Physics Bremen Center for Computational Material Sciences University of Bremen Contents 1 Introduction
More informationNumerical Methods in Quantum Many-body Theory. Gun Sang Jeon Pyeong-chang Summer Institute 2014
Numerical Methods in Quantum Many-body Theory Gun Sang Jeon 2014-08-25 Pyeong-chang Summer Institute 2014 Contents Introduction to Computational Physics Monte Carlo Methods: Basics and Applications Numerical
More informationRealistic Modeling of Materials with Strongly Correlated Electrons
Realistic Modeling of Materials with Strongly Correlated Electrons G. Keller 1, K. Held 2, V. Eyert 1, V. I. Anisimov 3, K. Byczuk 1, M. Kollar 1, I. Leonov 1, X. Ren 1, and D. Vollhardt 1 1 Theoretical
More informationAn introduction to the dynamical mean-field theory. L. V. Pourovskii
An introduction to the dynamical mean-field theory L. V. Pourovskii Nordita school on Photon-Matter interaction, Stockholm, 06.10.2016 OUTLINE The standard density-functional-theory (DFT) framework An
More informationA. Field theoretical methods for the partition function, perturbative expansions
IV. QUANTUM FIELD THEORY, GREEN S FUNCTION, AND PERTURBATION THEORY Markus Holzmann LPMMC, Maison de Magistère, Grenoble, and LPTMC, Jussieu, Paris markus@lptl.jussieu.fr http://www.lptl.jussieu.fr/users/markus
More informationarxiv:cond-mat/ v1 [cond-mat.str-el] 10 Jan 2003
Self-energy-functional approach to systems of correlated electrons Michael Potthoff * Lehrstuhl Festkörpertheorie, Institut für Physik, Humboldt-Universität zu Berlin, 10115 Berlin, Germany arxiv:cond-mat/0301137v1
More informationTowards exploiting non-abelian symmetries in the Dynamical Mean-Field Theory using the Numerical Renormalization Group
master s sis chair of oretical solid state physics faculty of physics ludwig-maximilians-universität münchen Towards exploiting non-abelian symmetries in Dynamical Mean-Field Theory using Numerical Renormalization
More informationIntroduction to SDFunctional and C-DMFT
Introduction to SDFunctional and C-DMFT A. Lichtenstein University of Hamburg In collaborations with: M. Katsnelson, V. Savkin, L. Chioncel, L. Pourovskii (Nijmegen) A. Poteryaev, S. Biermann, M. Rozenberg,
More informationwhere β = 1, H = H µn (3.3) If the Hamiltonian has no explicit time dependence, the Greens functions are homogeneous in time:
3. Greens functions 3.1 Matsubara method In the solid state theory class, Greens functions were introduced as response functions; they can be used to determine the quasiparticle density of states. They
More informationPHYSICAL REVIEW B 80,
PHYSICAL REVIEW B 8, 6526 29 Finite-temperature exact diagonalization cluster dynamical mean-field study of the two-dimensional Hubbard model: Pseudogap, non-fermi-liquid behavior, and particle-hole asymmetry
More informationMOTTNESS AND STRONG COUPLING
MOTTNESS AND STRONG COUPLING ROB LEIGH UNIVERSITY OF ILLINOIS Rutgers University April 2008 based on various papers with Philip Phillips and Ting-Pong Choy PRL 99 (2007) 046404 PRB 77 (2008) 014512 PRB
More informationVertex-corrected perturbation theory for the electron-phonon problem with nonconstant density of states
PHYSICAL REVIEW B VOLUME 58, NUMBER 17 1 NOVEMBER 1998-I Vertex-corrected perturbation theory for the electron-phonon problem with nonconstant density of states J. K. Freericks Department of Physics, Georgetown
More informationMany body physics issues with the core-hole propagator and resonant inelastic X-ray scattering
Many body physics issues with the core-hole propagator and resonant inelastic X-ray scattering Jim Freericks (Georgetown University) Funding: Civilian Research and Development Foundation In collaboration
More information(r) 2.0 E N 1.0
The Numerical Renormalization Group Ralf Bulla Institut für Theoretische Physik Universität zu Köln 4.0 3.0 Q=0, S=1/2 Q=1, S=0 Q=1, S=1 E N 2.0 1.0 Contents 1. introduction to basic rg concepts 2. introduction
More informationarxiv:cond-mat/ v2 [cond-mat.str-el] 21 Aug 1997
Interpolating self-energy of the infinite-dimensional Hubbard model: Modifying the iterative perturbation theory arxiv:cond-mat/9704005v2 [cond-mat.str-el] 21 Aug 1997 M. Potthoff, T. Wegner and W. Nolting
More informationThe Hubbard model out of equilibrium - Insights from DMFT -
The Hubbard model out of equilibrium - Insights from DMFT - t U Philipp Werner University of Fribourg, Switzerland KITP, October 212 The Hubbard model out of equilibrium - Insights from DMFT - In collaboration
More informationGreen's Function in. Condensed Matter Physics. Wang Huaiyu. Alpha Science International Ltd. SCIENCE PRESS 2 Beijing \S7 Oxford, U.K.
Green's Function in Condensed Matter Physics Wang Huaiyu SCIENCE PRESS 2 Beijing \S7 Oxford, U.K. Alpha Science International Ltd. CONTENTS Part I Green's Functions in Mathematical Physics Chapter 1 Time-Independent
More informationNiO - hole doping and bandstructure of charge transfer insulator
NiO - hole doping and bandstructure of charge transfer insulator Jan Kuneš Institute for Physics, Uni. Augsburg Collaboration: V. I. Anisimov S. L. Skornyakov A. V. Lukoyanov D. Vollhardt Outline NiO -
More informationMOMENTUM DISTRIBUTION OF ITINERANT ELECTRONS IN THE ONE-DIMENSIONAL FALICOV KIMBALL MODEL
International Journal of Modern Physics B Vol. 17, No. 27 (23) 4897 4911 c World Scientific Publishing Company MOMENTUM DISTRIBUTION OF ITINERANT EECTRONS IN THE ONE-DIMENSIONA FAICOV KIMBA MODE PAVO FARKAŠOVSKÝ
More informationNumerical Studies of the 2D Hubbard Model
arxiv:cond-mat/0610710v1 [cond-mat.str-el] 25 Oct 2006 Numerical Studies of the 2D Hubbard Model D.J. Scalapino Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Abstract
More informationQUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS. Wolfgang Belzig
QUANTUM FIELD THEORY IN CONDENSED MATTER PHYSICS Wolfgang Belzig FORMAL MATTERS Wahlpflichtfach 4h lecture + 2h exercise Lecture: Mon & Thu, 10-12, P603 Tutorial: Mon, 14-16 (P912) or 16-18 (P712) 50%
More informationarxiv: v2 [cond-mat.mes-hall] 5 Sep 2015
Non-equilibrium AC Stark effect and long-lived exciton-polariton states in semiconductor Mie resonators arxiv:154.949v2 [cond-mat.mes-hall] 5 Sep 215 Andreas Lubatsch 1 and Regine Frank 2,3,4 1 Georg-Simon-Ohm
More informationTransport Coefficients of the Anderson Model via the numerical renormalization group
Transport Coefficients of the Anderson Model via the numerical renormalization group T. A. Costi 1, A. C. Hewson 1 and V. Zlatić 2 1 Department of Mathematics, Imperial College, London SW7 2BZ, UK 2 Institute
More information