A continuous time algorithm for quantum impurity models

Transcription

1 ISSP, Aug. 6 p.1 A continuou time algorithm for quantum impurity model Philipp Werner Department of Phyic, Columbia Univerity cond-mat/512727

2 ISSP, Aug. 6 p.2 Outline Introduction Dynamical mean field theory Exiting impurity olver New approach Diagrammatic expanion in the impurity-bath hybridization Monte Carlo ampling Scaling with temperature and interaction trength Application Temperature- and doping dependent Mott tranition Free energy calculation Generalization Matrix formalim Collaborator A. J. Milli, M. Troyer

3 ISSP, Aug. 6 p.3 Introduction Dynamical mean field theory Metzner & Vollhardt (1989), George & Kotliar (1992) Replace lattice problem by a ingle ite effective problem Neglect patial, keep dynamical fluctuation Become exact in the infinite coordination number limit

4 ISSP, Aug. 6 p.3 Introduction Dynamical mean field theory Metzner & Vollhardt (1989), George & Kotliar (1992) Lattice model (Denity of tate D(ɛ), Self energy Σ lat (iω n, k)) H lat = µ i (n i + n i ) + U i n i n i + t i,j,σ c i,σ c j,σ t Single impurity (Hybridization V k, Self energy Σ imp (iω n )) H = µ(n + n ) + Un n + k ɛbath k,σ nbath k,σ + k,σ (V kc σa bath k,σ + h.c.) V

5 ISSP, Aug. 6 p.3 Introduction Dynamical mean field theory Metzner & Vollhardt (1989), George & Kotliar (1992) Lattice model (Denity of tate D(ɛ), Self energy Σ lat (iω n, k)) H lat = µ i (n i + n i ) + U i n i n i + t i,j,σ c i,σ c j,σ t Effective Action (Hybridization F(τ), Self energy Σ imp (iω)) S = dτ( µ(n + n ) + Un n ) σ dτdτ c σ (τ)f σ (τ τ )c σ(τ ) F( τ τ )

6 Introduction Dynamical mean field theory Metzner & Vollhardt (1989), George & Kotliar (1992) Lattice model (Denity of tate D(ɛ), Self energy Σ lat (iω n, k)) H lat = µ i (n i + n i ) + U i n i n i + t i,j,σ c i,σ c j,σ t Effective Action (Hybridization F(τ), Self energy Σ imp (iω)) S = dτ( µ(n + n ) + Un n ) σ dτdτ c σ (τ)f σ (τ τ )c σ(τ ) Self-conitency F( τ τ ) S Σ lat (iω n ) Σ imp (iω n ) G lat (iω n ) = dɛ D(ɛ) iω n +µ ɛ Σ lat (iω n ) S ISSP, Aug. 6 p.3

7 Previou QMC approache Hirch-Fye olver Hirch & Fye (1986) Hubbard model: Z = TrT τ e S with action S = S F + S loc S F = β σ dτdτ c σ (τ)f σ (τ τ )c σ(τ ) S loc = µ β dτ(n + n ) + U β dτn n Dicretize imaginary time into N equal lice τ Decouple Un n uing dicrete Hubbard-Stratonovich tranformation e τu(n n +1/2(n +n )) = 1 2 =±1 eλ(n +n ), λ = coh(e τu/2 ) Perform Gauian integral Z = i det G 1, ( 1,..., N )G 1, ( 1,..., N ) MC ampling of auxiliary Iing pin Initial drop of Green function e Uτ/2 Matrix ize: N 5βU Low temperature not acceible ~exp( U τ/2) ISSP, Aug. 6 p.4

8 Previou QMC approache Rubtov olver Rubtov et al. (25) Hubbard model: Z = TrT τ e S with action S = S + S U S = µ β dτ(n +n ) β σ dτdτ c σ (τ)f σ (τ τ )c σ(τ ) S U = U β dτn n Continuou-time olver baed on a diagrammatic expanion of Z Prokof ev et al. (1996) Treat quadratic part S a unperturbed action and expand e U R dτn n Z = k dτ1...dτ k Tr [ T τ e S n (τ 1 )n (τ 1 )...n (τ k )n (τ k ) ] ( U) k k! Perform Gauian integral Z = ( U) k k k! dτ1...dτ k det G, (τ 1,..., τ k )G, (τ 1,..., τ k ) MC ampling of vertice {n (τ i )n (τ i )} i=1,2,...,k Matrix ize: k.5βu 1 + G G,, + U U U +... ISSP, Aug. 6 p.5

9 ISSP, Aug. 6 p.6 New impurity olver Expanion in the impurity-bath hybridization F (cond-mat/512727) Non-interacting model: Z = TrT τ exp( β dτdτ c(τ)f(τ τ )c (τ )) Expand exponential, evaluate in the occupation number bai {, 1 } Z = 1! Tr ! TrT τ + 1 2! TrT τ +... dτ 1 dτ1 e c(τ1)f(τ e 1 e τ1)c (τ1) dτ 1 dτ1dτ e 2dτ 2 e c(τ1)f(τ e 1 e τ1)c (τ1) c(τ2)f(τ e 2 e τ2)c (τ2) + β τ F( τ τ ) 1 e 1 1 τ 1 e + β τ 2 τ 1 e τ 1 τ 2 e β + τ 2 e e τ3 τ 1 τ 2 τ3 τ 1 e β +...

10 New impurity olver Expanion in the impurity-bath hybridization F (cond-mat/512727) Non-interacting model: Z = TrT τ exp( β dτdτ c(τ)f(τ τ )c (τ )) Expand exponential, evaluate in the occupation number bai {, 1 } Z = 1! Tr ! TrT τ + 1 2! TrT τ +... dτ 1 dτ1 e c(τ1)f(τ e 1 e τ1)c (τ1) dτ 1 dτ1dτ e 2dτ 2 e c(τ1)f(τ e 1 e τ1)c (τ1) c(τ2)f(τ e 2 e τ2)c (τ2) + β τ F( τ τ ) 1 e 1 1 τ 1 e + β τ 2 τ 1 e τ 1 τ 2 e β + τ 2 e e τ3 τ 1 τ 2 τ3 τ 1 e β +... Some diagram have negative weight ISSP, Aug. 6 p.6

11 ISSP, Aug. 6 p.7 New impurity olver Expanion in the impurity-bath hybridization F (cond-mat/512727) Non-interacting model: Z = TrT τ exp( β dτdτ c(τ)f(τ τ )c (τ )) Collect the k! diagram with the ame {c(τ i ), c (τ e i )} i=1...k into a determinant Z k (τ 1, τ e 1; τ 2, τ e 2;...;τ k, τe k ) = detf (k) δ τe k τ 1 F (k) m,n = F(τ e m τ n) reum huge number of diagram eliminate the ign problem c c c c c c τ τ e τ τ e τ τ e β F( τe τ ) 1 1 Z k can be viualized a a configuration of k egment on a circle Z = 2 + β k=1 dτ 1... β τk 1 e dτ k τ 1 τ k dτ e k Z k (τ 1, τ e 1; τ 2, τ e 2;...;τ k, τe k )

12 Sign problem Expanion in the impurity-bath hybridization F (cond-mat/512727) Example: F(τ) = c, β-antiperiodic F(iω n ) = c iω n Diagram: p(k) k! (cβ2 ) k (2k)! Determinant: p(k) 2 k (cβ 2 ) k (2k)! c 3 probability 1 1e-5 1e-1 diagram cβ 2 =1 cβ 2 =1 cβ 2 =1 cβ 2 =1 c 3 1e-15 determinant c 3 c 3 1e order c 3 3 c ISSP, Aug. 6 p.8

13 ISSP, Aug. 6 p.9 Monte Carlo ampling Expanion in the impurity-bath hybridization F (cond-mat/512727) Sampling of Z through local update (i) inertion/removal of egment (ii) inertion/removal of anti-egment (iii) hift of the egment end point β local update (i) (ii) (iii) Detailed balance k k+1 = k + p in ( ) p rem ( ) = Z k+1( k+1 ) Z k ( k ) βl max k + 1 e lµ Store and update the matrix M = F 1 acce to determinant ratio efficient computation of G G(τ) = i,j M j,i (τ, τi e τ j ) 1 β

14 Monte Carlo ampling Expanion in the impurity-bath hybridization F (cond-mat/512727) Hubbard model (U ): Segment configuration for pin up/down pin up pin down δl overlap β β Acceptance rate for MC move now alo depend on egment overlap Detailed balance p( ) p( ) = Z k( ) e( l l)µ Uδl overlap Z k () Obviouly: E pot = U l total overlap n σ = G σ (β) = β 1 lσ total... ISSP, Aug. 6 p.1

15 ISSP, Aug. 6 p.11 Number of egment Expanion in the impurity-bath hybridization F (cond-mat/512727) Efficiency of the algorithm depend on the matrix ize k Computational effort O(k 3 ) Order of diagram k β k decreae with increaing U p(k) U/t= U/t=3 U/t=4 U/t=5 βt=1 mall matrice work even at very low T order k Method Hirch-Fye Expanion in U Expanion in F Matrix ize k β β β βt = 1, U/t = βt = 1, U/t = βt = 1, U/t =

16 ISSP, Aug. 6 p.12 Reult Semi-circular denity of tate (Bethe lattice) with band width 4t Enforce paramagnetic phae Self-conitency condition: F(τ) = t 2 G( τ) Phae diagram (ketch) T/t U c1.5 I.1 M U c U/t

17 Reult - Green function Can map out teep drop of G(τ) with almot perfect reolution Different large-τ behavior for metallic/inulating Green function U/t=4.95 βt=2 βt=3 βt=2 βt=4.1 U/t=4.95 βt=2 βt=3 βt=2 βt=4 -G(τ).2 -G(τ) τ/β τ/β.5 T/t.4.5 -G(τ).3.2 U/t=4.95 βt=2 βt=3 βt=2 βt=4 I.1.1 M τ/β U/t ISSP, Aug. 6 p.13

18 ISSP, Aug. 6 p.14 Reult - Kinetic energy Low temperature phyic near Mott tranition eaily acceible Hirch-Fye: ytematic error due to inufficient time-reolution.53 -E kin /t exact diagonalization continuou-time Hirch-Fye U/t=4 -E kin /t exact diagonalization U/t=4.95 U/t=5.3 U/t= (T/t) 2 T/t (T/t) 2 E kin = 2t 2 dτg(τ) 2.5 I ED method: M. Capone et al., cond-mat/ M U/t

19 Reult - Free energy Acce to low temperature allow to compute the entropy C met (T) = de met /dt S met (T) = T dt C met (T ) T S in (T) = ln(2) Croing of free energy curve yield firt order tranition point.3.28 E kin E met -TS E in -TS E met E/t e-5.1 (T/t) 2 U/t=5.3 (left cale) T/t ISSP, Aug. 6 p.15

20 Reult - Doping dependence Algorithm work away from half filling compute n(µ) for U U c2. Doping dependent Mott tranition (for T > ) i firt order.5-n βt=4 U/t=6.5 U/t=6. U/t=5.8 U/t=5.7 U/t=5.6 (.5-n) 1/ βt=4 U/t=6.5 U/t=6. U/t=5.8 U/t=5.7 U/t= µ/(u/2) µ/(u/2).5 T/t.4 µ=1.u/2 µ=.7u/2 µ=.5u/2 µ=.3u/2 U/t=6.5 βt=4.5 -G(τ).3.2 µ /t I.1.1 M τ/β U/t ISSP, Aug. 6 p.16

21 ISSP, Aug. 6 p.17 Reult - Doping dependence Doping dependent Mott tranition remain firt order down to T = Change in F = E TS µn yield precie location of the tranition (-(n-.5)) 1/2.3 U/t=6. βt=4.25 βt=2.2 βt= µ/(u/2) (F in -F met ) / t T/t µ/(u/2) βt=4 including S met βt=2 including S met βt=1.5 µ /t I.1 M U/t

22 ISSP, Aug. 6 p.18 General formalim Expanion in the impurity-bath hybridization function General model: Z = TrT τ e S with action S = S F + S loc S F = a β dτdτ ψ a (τ)f a (τ τ )ψ a(τ ) S loc = β dτ(ψ Qψ T + U abcd ψ aψ b ψ cψ d ) }{{} H loc Expand in F a, reum diagram into determinant Z = TrT τ e S loc a k a = dτ a1... dτa e det(f (k a) ka a ({τ a })) ψ a(τ a 1 )ψ a (τ e a 1 )...ψ a(τ a k a )ψ a(τ e a k a ) Configuration conit of k a creation (annihilation) operator at time τ a 1 <... < τ a k a (τe a 1 <... < τ e a k a ) β

23 ISSP, Aug. 6 p.19 General formalim Expanion in the impurity-bath hybridization function General model: Z = TrT τ e S with action S = S F + S loc S F = a β dτdτ ψ a (τ)f a (τ τ )ψ a(τ ) S loc = β dτ(ψ Qψ T + U abcd ψ aψ b ψ cψ d ) }{{} H loc Weight of... i (K(τ) = e Hlocτ ) β w f det(f (k f) f )Tr[K(β τ e a k a )ψ a(τ e a k a )K(τe a k a τ b kb )ψ b (τ b kb )......ψ b (τ e b 1 )K(τ e b 1 τ e c 1 )ψ c (τ e c 1 )K(τ e c 1 τ a 1 )ψ a(τ a 1 )K(τ a 1 )] K, ψ, ψ are n n matrice Ue eigenbai of K K(τ) = diag(e α 1τ, e α 2τ,...,e α nτ )

24 ISSP, Aug. 6 p.2 Concluion Strong-coupling continuou-time impurity olver, baed on a diagrammatic expanion in the impurity-bath hybridization Matrix ize k decreae with increaing U Allow acce to low T, even at large U No detectable ign problem Can be generalized to model with exchange Conference talk matrix ize Hirch-Fye 5βU Rubtov.5βU trong-coupling U/t On-going and future project Multi-orbital model and cluter Small (or truncated) ytem uing matrix formalim Large ytem uing double expanion in F and J