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 applied to quantum field theory in collaboration with: O. Akerlund, P. de Forcrand, A. Georges Dynamical mean field theory for bosonic lattice systems in collaboration with: P. Anders, L. Pollet, M. Troyer Dynamical mean field theory for fermionic lattice systems in collaboration with: E. Gull, A. Millis Nonequilibrium extension of dynamical mean field theory in collaboration with: M. Eckstein, M. Kollar, N. Tsuji, T. Oka, H. Aoki Some applications: Hubbard model in strong electric fields in collaboration with: M. Eckstein, N. Tsuji, T. Oka, H. Aoki
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Lagrangian density L [ ϕ ( x )] = 1 2( ∂ µ ϕ ( x )) 2 − 1 0 ϕ ( x ) 2 − g 0 2 m 2 4! ϕ ( x ) 4 Spontaneous symmetry breaking for negative (renormalized) m 2 Z 2 After Wick rotation, discretization and variable transformation ! X X µ ϕ x + ϕ 2 x + λ ( ϕ 2 x − 1) 2 S = − 2 κ ϕ x + b x µ In the limit (Ising model) λ → ∞ , ϕ x = ± 1
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Mean field theory: mapping to a 0-dimensional effective model replace all interactions by an interaction with a constant background field v Partition function of the lattice model d ! Z Y X − ϕ 2 x − λ ( ϕ 2 x − 1) 2 +2 κ Z = D [ ϕ ] exp ϕ x ϕ x + b µ x µ =1 Z ∞ − ϕ 2 − λ ( ϕ 2 − 1) 2 + 2 κ (2 d ) v ϕ becomes � � Z MF = d ϕ exp −∞ Self-consistency condition Z ∞ 1 ⇣ � ϕ 2 � λ ( ϕ 2 � 1) 2 + 2 κ (2 d ) v ϕ ⌘ v ⌘ h ϕ i = ϕ ϕ exp Z MF −∞
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Mean field theory: Phase diagram (d=3+1) Monte Carlo, 32 4 0.14 Mean Field 0.13 0.12 symmetry-broken phase 0.11 � 0.1 0.09 symmetric phase 0.08 0.07 0 1 2 3 4 5 �
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field theory: mapping to a (0+1)-dimensional effective model explicitly treat fluctuations in one dimension, freeze fluctuations in the (d-1) other dimensions Dynamical dimension: (with conjugate momentum ) t ω Frozen dimensions: (conjugate momenta ) k 1 , . . . , k d − 1 x 1 , . . . , x d − 1 This convention allows to study finite-temperature behavior by varying the extent of the dynamical dimension Breaks Lorentz invariance, but let’s try it anyhow ...
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field theory: mapping to a (0+1)-dimensional effective model explicitly treat fluctuations in one dimension, freeze fluctuations in the (d-1) other dimensions Schematically: non-zero in the symmetry-broken ! Z X X X phase Z = D [ ϕ ] exp 2 κ V ( ϕ x ) ϕ x ϕ x + b µ − x µ x ! Z X X X K � 1 ( t − t 0 ) ϕ t ϕ t 0 − Z DMFT = D [ ϕ ] exp V ( ϕ t ) + h ϕ t − t,t 0 t t effect of the frozen dimensions on the local dynamics
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Cavity construction: separate action into x 6 = ( ~ x = ( ~ internal and external 0 , t ) 0 , t ) degrees of freedom t = S int + ∆ S + S ext S � S x h int ,t − 1) 2 i X − 2 κϕ int ,t +1 ϕ int ,t + ϕ 2 int ,t + λ ( ϕ 2 = S int t X X = − 2 κ ∆ S ϕ int ,t ϕ ext ,t t h int , ext i h x − 1) 2 i X X ⌫ ϕ x + ϕ 2 x + λ ( ϕ 2 = − 2 κ S ext ϕ x + b ⌫ x 6 =( ~ 0 ,t ) Expand and integrate out the external degrees of freedom exp( − ∆ S )
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Cavity construction: to allow for symmetry breaking, we write φ ext + δϕ ext ,t , h ϕ ext i = φ ext ϕ ext ,t = t φ int + δϕ int ,t , h ϕ int i = φ int ϕ int ,t = � S x ! X X 2( d − 1) φ † δϕ † = ext δϕ int ,t + + f ( ϕ ext ) ∆ S − κ int ,t δϕ ext ,t t h int , ext i X + + f ( ϕ ext ) S 1 δ S ≡ t can be included in S int can be included in S ext Expand exp( − δ S )
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Cavity construction: Terms up to second order yield Z t = D ϕ int exp( � S int � S 1 ) Z Z ext � S x h δ S ( t ) i ext + 1 ⇣ ⌘ X X h δ S ( t ) δ S ( t 0 ) i ext + . . . ⇥ 1 � 2 t t,t 0 First order term is zero because proportional to h δϕ ext i ext = 0 “hybridization function” Second order term is non-zero: h δ S ( t ) δ S ( t 0 ) i ext ⌘ δϕ † int ,t ∆ ( t � t 0 ) δϕ int ,t 0
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Cavity construction: After re-exponentiation and switching back from to δϕ ϕ we find the effective single-site action t � S x X X X ϕ t K � 1 ( ϕ 2 t − 1) 2 − h imp ,c ( t − t 0 ) ϕ t 0 + λ = S imp ϕ t t,t 0 t t Fourier transform of nn interaction in time e 1 − 2 κ cos( ω ) − e K � 1 imp ,c ( ω ) = ∆ ( ω ) because action written in terms of phi 2 φ ext (2 κ ( d − 1) − e = ∆ (0)) h We call it “impurity action” (condensed-matter convention)
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: Hybridization function and ∆ φ ext are fixed by self-consistency conditions t � S x X X X ϕ t K � 1 ( ϕ 2 t − 1) 2 − h imp ,c ( t − t 0 ) ϕ t 0 + λ = S imp ϕ t t,t 0 t t e 1 − 2 κ cos( ω ) − e K � 1 imp ,c ( ω ) = ∆ ( ω ) 2 φ ext (2 κ ( d − 1) − e = ∆ (0)) h
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: Hybridization function and ∆ φ ext are fixed by self-consistency conditions t � S x Local lattice Green’s function G loc ( ω ) = P DMFT approximation: identify lattice and e 1 impurity self-energy k e ( k, ω )+ e G − 1 Σ ( k, ω ) 0 0 ( k, ω ) = 1 − 2 κ P d e G − 1 i =1 cos( k i ) GF of the d-dimensional free theory Impurity Green’s function e 1 1 G imp ( ω ) = Σ imp ( ω ) = K − 1 e imp ,c ( ω )+ e 1 − 2 κ cos( ω ) − e ∆ ( ω )+ e Σ imp ( ω )
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 S int S ext Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: Hybridization function and ∆ φ ext are fixed by self-consistency conditions t � S x Local lattice Green’s function X 1 e G loc ( ω ) = ∆ ( ω ) − 2 κ P d − 1 e imp ( ω ) + e G − 1 i =1 cos k i Self-consistency equations: substitution yields an e e G imp ( ω ) = G loc ( ω ) implicit equation for the hybridization function h ϕ i S imp = φ ext
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field equations: Hybridization function and ∆ φ ext are fixed by self-consistency conditions Lattice model impurity model DMFT self-consistency G latt S [ ∆ , φ ext ] loc G latt loc ⌘ G imp , h ϕ i ⌘ φ ext impurity solver momentum average Σ latt ≡ Σ imp k = [ G − 1 G latt 0 ,k + Σ latt ] − 1 G imp , Σ imp , h ϕ i k k DMFT approximation
DMFT for quantum field theories Simple example: real, scalar quantum field theory ϕ 4 Akerlund, de Forcrand, Georges & Werner (2013) Dynamical mean field phase diagram (d=3+1): Monte Carlo, 32 4 0.14 Mean Field 0.13 DMFT, L=75 0.12 symmetry-broken 0.11 � phase 0.1 0.09 symmetric phase 0.08 0.07 0 1 2 3 4 5 �
DMFT for Bosons Simple example: Bose-Hubbard model Fisher, Grinstein, Weichmann & Fisher (1989) i b j + U b † X X X H = − t n i ( n i − 1) − µ n i 2 h i,j i i i 3U U t Mott-Insulator to superfluid transition at low temperature
DMFT for Bosons Simple example: Bose-Hubbard model Derivation of DMFT formalism analogous to the case ϕ 4 Anders, Pollet, Gull, Troyer & Werner (2011) Split action of the lattice model into S = S int + ∆ S + S ext Z β int b int + U h i − µb † = 2 n int ( n int − 1) S int d τ 0 Z β X ( b † int b ext + b † = ext b int ) ∆ S − t 0 h int,ext i Z β ext b ext + U h i − µb † = 2 n ext ( n ext − 1) S ext d τ 0 To allow for symmetry breaking (condensation), write b ext = φ ext + δ b ext , b int = φ int + δ b int
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