Dynamical mean-field theory

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