Weak-Coupling CT-QMC: projective schemes and applications to retarded interactions.

Transcription

1 Weak-Couplig CT-QMC: projective schemes ad applicatios to retarded iteractios. F.F. Assaad (IPAM Jauary 26-3, 29) Outlie Weak couplig CT-QMC ( Rubtsov et al. PRB 5 ). Projective schemes. Retarded iteractios: phoo degrees of freedom. Applicatio to the 1D quarter filled Holstei model. Coclusios. F.F. Assaad ad T.C. Lag Phys. Rev. B 76, (27), Phys. Rev. B (28)

2 I. The method. Aderso Model (Rubtsov et. al PRB (5)) S = dτdτ ' dσ ( τ) G ( τ τ ') dσ ( τ ') + U dτ d () τ d () τ d () τ d () τ S () τ β Dyso. Expasio aroud U=. β H Tr e β H = Tr e β τ 1 dτ dτ ( U) ( τ ) ( τ ) ( τ ) ( τ ) Wick =1 G 1 1 Udet Udet M1( τ1) G G ( τ1, τ1) U G ( τ, τ ) = G ( τ, τ ) = Tdˆ ( τ ) dˆ ( τ ) σ σ 2 σ 1 = ( ) 2 = U det M2 τ1, τ 2

3 βh β τ 1 Tr e τ τ ( τ τ ) β = d 1 d ( U) det M 1,, H Tr e Sum with Mote Carlo Weight Weight / Sig. U K HU ( [ s ])( [ s ]) e 2 s=± 1 2β s d d 1/2 δ 1/2 + δ = d d α ( ) s K 2 1 = Uβδ ( 1/4), cosh( α) 1 =, 2 2( δ 1/ 4) δ > 1/2 New dyamical variable s. Exact mappig oto CT-Hirsch-Fye (K. Mikelsos et al. preprit) (Rombouts et al. PRL 99, Gull et. al EPL 8) Sig problem behaves as i Hirsch-Fye. (Abset for oe-dimesioal chais, particle-hole symmetry, impurity models)

4 βh β τ 1 Tr e τ τ ( τ τ ) β = d 1 d ( U) det M 1,, H Tr e Sum with Mote Carlo Weight Weight / Sig. U d d K H ( [ 1/2 sδ] )( [ 1/2 + sδ ]) = e 2 s=± 1 2β s U d d α ( ) s K 2 1 = Uβδ ( 1/4), cosh( α) 1 =, 2 2( δ 1/ 4) δ > 1/2 ( [ ]) [ ] ( ) d d HU U 1/2 1 + U ( d d = δ / 2 + δ δ ) Absorb i H Particle-Hole symmetry δ = ad oly eve powers of occur i expasio.

5 βh β τ 1 e τ τ U τ τ β d 1 d det M 1, s1,, s H e 2 s1 s Tr = ( ) Tr Sum with Mote Carlo Weight Samplig. Cofiguratio C: set of -vertices at imagiary times [ τ, s ] [ τ, s ], [ τ, s ] Add τ τ τ τ [ τ 1 ] [ 2][ ] [ 4 1, s1 τ 2, s2 τ 3, s3 τ 4, s4] [ τ1, τs11][ τ 2 τ, s 2 2][ τ 3, s 3] Remove [ τ, s ] [ τ, s ]

6 βh β τ 1 e τ τ U τ τ β d 1 d det M 1, s1,, s H e 2 s1 s Tr = ( ) Tr Sum with Mote Carlo Weight Measuremets. [ ] [ ] ˆ σ TH [ τ, s] H [ τ, s] TH τ, s H τ, s d ( τ) dˆ ( τ') G G G M G 1 ( ) U U σ σ σ σ σ σ C ( τ, τ') = ( τ, τ') ( τ, τα) αβ ( τβ, τ') U 1 1 U αβ, = 1 Wick theorem applies for each cofiguratio C of vertices. Direct calculatio of Matsubara Gree fuctios. 1 ( ) σ σ σ iωτ m α σ σ C m = m m αβ αβ, = 1 G ( iω ) G ( iω ) G ( iω ) e M G ( τ,) β

7 βh β τ 1 e τ τ U τ τ β d 1 d det M 1, s1,, s H e 2 s1 s Tr = ( ) Tr Sum with Mote Carlo Weight Average Expasio parameter. d d 2 ( 1/2)( 1/2) δ = β U CPU time scales as <> 3 same scalig as Hirsch-Fye. <> is miimal at particle-hole symmetric poit, δ = Histogram of expasio parameter.

8 Examples. a) Particle-hole symmetric Aderso Model, U/t=4. G() τ β t = 4 = 27 Hirsch-Fye: LTrot = 4 /.2 ( Δ τ t =.2) Speedup: ( 2 / 27) 3 4 b) Off particle-hole Symmetry, U/t=4 βt=4. G(τ) = 65 -Im(G(iω m )) A( ) Gi ( ω ω m) = d ω i ω ω m Speedup ( 2 / 65) 3 3 Direct calculatio of is possible. G(iω m )

9 Weak-Couplig CT-QMC: projective schemes ad applicatios to retarded iteractios. F.F. Assaad (IPAM Jauary 26-3, 29) Weak couplig CT-QMC ( Rubtsov et al. PRB 5 ). Projective schemes. (M. Feldbacher, K Held, FFA PRL 24) Retarded iteractios: phoo degrees of freedom. Applicatio to the 1D quarter filled Holstei model. Coclusios. F.F. Assaad ad T.C. Lag Phys. Rev. B 76, (27), Phys. Rev. B (28)

10 II) Projective Schemes O = Groud state lim θ Ψ T e Ψ θ θ H H 2 2 T e O e θh Ψ T Ψ T Fiite temperature O = Tr e Tr e βh O βh No thermal fluctuatios Thermal fluctuatios Formulatio. ψ T ψ e T θh ψ T ψ T θ τ 1 τ τ U = d 1 d det M ( τ1, s1, τ, s) 2 s s 1 Replace G ( τ, τ ) = Tdˆ ( τ ) dˆ ( τ ) by σ σ 2 σ 1 ˆ+ ψ Td ( τ ) dˆ ( τ ) ψ T σ T 2 σ 1 ψ ψ T T i determiat. Note: ψ T has to be a Slater determiat!

11 PQMC+DMFT DMFT: self-cosistecy cycle. N( ε ) Σ( iω) Dyso G( iω) = dε i ω ε + μ Σ( i ω) G( τ ) PQMC G ( τ ) G ( iω) = G ( iω) Σ( iω) 1 1 Questio: [ ] iωmτ G( τ) for τ θm, θm, G( iωm) = dτ e G( τ)? MaxEt. G( τ ) = dω K( τ, ω) A( ω) A( ) Gi ( ω ω m) = d ω i ω ω m MaxEt as a fittig procedure.

12 Example: Mott trasitio. Hamiltoia: Hˆ = t cˆ cˆ + U ˆ ˆ i, j i, σ j, σ i, i, i PQMC vs. QMC: Bethe-DOS: badwidth W=4 Phase diagram: Bulla NRG: Simulatios at θt 2 3 give good estimate of groud state properties.

13 Weak-Couplig CT-QMC: projective schemes ad applicatios to retarded iteractios. F.F. Assaad (IPAM Jauary 26-3, 29) Weak couplig CT-QMC ( Rubtsov et al. PRB 5 ). Projective schemes. (M. Feldbacher, K Held, FFA PRL 24) Retarded iteractios: phoo degrees of freedom. (Hybridizatio expasio Werer & Millis PRL 99 7) Applicatio to the 1D quarter filled Holstei model. Coclusios. F.F. Assaad ad T.C. Lag Phys. Rev. B 76, (27), Phys. Rev. B (28)

14 III) Phoos. Itegrate out phoos i favor of a retarded iteractio. Hubbard-Holstei: ˆ Hˆ t cˆ cˆ U ˆ ˆ ˆ P k = + + g Q + + Q + i, j i, σ j, σ i, i, i, j, σ i Itegrate out the phoos i 2 i ˆ 2 ( ˆ i i 1) i i 2M 2 β β β + Z = dc dc e x p S U dτ ( τ ) ( τ ) + dτ dτ' [ i( τ) 1] D ( i j, τ τ') [ j( τ') 1] i i i, j 2 g D ( i j, τ τ ') = δi, j P( τ τ ') 2k ω τω ( β τ ) ω P( τ) = e + e, ω = k / M βω 2(1 e ) Attractive, retarded iteractio (time scale 1/ω ). Atiadiabatic limit: lim P( ) ( ) Attractive Hubbard. ω τ = δ τ

15 III) Phoos. Itegrate out phoos i favor of a retarded iteractio. β β β + Z = dc dc e x p S U dτ ( τ ) ( τ ) + dτ dτ' [ i( τ) 1] D ( i j, τ τ') [ j( τ') 1] i i i, j DDQMC. Expad both i Hubbard ad retarded phoo iteractio. Vertices: Hubbard. Phoo. D ( i j, τ τ ') i, τ j, τ ' i, τ j, τ ' U i, τ i, τ i, τ j, τ '

16 Weak-Couplig CT-QMC: projective schemes ad applicatios to retarded iteractios. F.F. Assaad (IPAM Jauary 26-3, 29) Weak couplig CT-QMC ( Rubtsov et al. PRB 5 ). Projective schemes. (M. Feldbacher, K Held, FFA PRL 24) Retarded iteractios: phoo degrees of freedom. (Hybridizatio expasio Werer & Millis PRL 99 7) Applicatio to the 1D quarter filled Holstei model. Coclusios. F.F. Assaad ad T.C. Lag Phys. Rev. B 76, (27), Phys. Rev. B (28)

17 Oe-dimesioal quarter filled Holstei model. ˆ ˆ P k H = k cˆ cˆ + g Q ˆ + + Q 2 + ˆ i ˆ 2 ε ( ) k, σ k, σ i( i 1) i k, σ i i 2M 2 ¼ Filled Holstei ω =.1t Luttiger Liquid. Peierls phase. Bipolaroic CDW. Gapless spi Gapless charge. λ c Gapfull spi Gapless charge. Luther-Emery. λ Flat bad width W Σ ( iω m ) = Obtaied from: 2 * g 2 = 1 + λ, λ = 2k W Static ad dyamical spi ad charge structure factors, ad optical coductivity (Lattice simulatios; L=2, 28, T/t=1/4). Temperature depedece of the sigle particle spectral fuctio (CDMFT, L c =8-12). m m

18 Static properties. Lattice simulatios. ω =.1t iqr Charge N ( q) = e ˆ( r) ˆ() r Domiat 2k F charge correlatios, at λ ~.35 2k F 4k F

19 Static properties. Lattice simulatios. ω =.1t Pairig P( r) = Δˆ ( r) Δˆ (), Δ ˆ ( r) = cˆ cˆ r r,, Short raged pairig correlatios grow Two electros with opposite spi share the same potetial well (Bipolaros). Log rage pairig correlatios drop Bipolaros ted to localize.

20 Static properties. Lattice simulatios. ω =.1t Spi iqr S( q) = ˆ ( ) ˆ e Sz r Sz() r Pairig suppresses spi respose. 2k F Luttiger liquid. ~.3 Peierls phase. Bipolaroic CDW. λ Isulator or metal?

21 Charge dyamical structure factor. Lattice simulatios. 1 β E 2 N( q, ω ) = e m ˆ ( q) m δ( E Em ω) Z m, βt ω =.1t = 4, ρ =.5 Luttiger liquid. Peierls. λ = λ =.15 λ =.35 π /2 q π Charge couples to phoo. Phoo modes are apparet i charge respose. q q Pilig up of spectral weight at 2k f. Slow dyamics of the CDW.

22 Charge dyamical structure factor. Lattice simulatios. 1 β E 2 N( q, ω ) = e m ˆ ( q) m δ( E Em ω) Z m, βt ω =.1t = 4, ρ =.5 Luttiger liquid. Peierls. λ = λ =.15 λ =.35 Charge velocity? π /2 q π Charge couples to phoo. Phoo modes are apparet i charge respose. q q Pilig up of spectral weight at 2k f. Slow dyamics of the CDW.

23 Optical Coductivity. Cotiuity equatio: Log wavelegth limit: ω βω σ '( q, ω) = 2 ( 1 e ) N( q, ω) q ( ) N( q, ω) N( q) δ vcq ω with N( q) αq σ '( ω) = lim σ '(, ω) α δ( ω) q q v c at T=. λ=.35 is a metallic state!

24 Spi dyamical structure factor. DDQMC lattice simulatios. 1 2 β E S( q, ω ) = m ˆ e Sz( q) m δ( E Em ω) Z m, βt = 4, ρ =.5 ω =.1t Luttiger liquid. Peierls. λ = λ =.15 λ =.35 π /2 q π q q Bipolaro formatio suppressio of low eergy spectral weight. Iterpretatio: At λ=.35, Δ sp ~.2t, Δ c =

25 Two particle data is cosistet with: Luttiger Liquid. Peierls phase. Bipolaroic CDW. Gapless spi Gapless charge. ~.3 Gapfull spi Gapless charge. Luther-Emery. λ Cofirmatio with sigle particle spectral fuctio.

26 b) Sigle particle spectral fuctio. Lattice model. Luttiger Liquid phase. CDMFT L c =8. T =.2t > ω λ =.25, ω =.1 t, ρ =.5 T =.125t << ω T =.67t k F k F k F

27 b) Sigle particle spectral fuctio. Lattice model. Luttiger Liquid phase. CDMFT L c =8. QMC T =.2t > ω λ =.25, ω =.1 t, ρ =.5 Self-cosistet Bor Approximatio. k F Egelsberg, Schrieffer Phys. Rev Σ ( iω m ) =

28 b) Sigle particle spectral fuctio. Lattice model. Luttiger Liquid phase. CDMFT L c =8. QMC λ =.25, ω =.1 t, ρ =.5 T =.125t << ω Luttiger Liquid approach/bosoizatio. Mede, Schöhammer, Guarso, PRB 94. Lˆk, σ Rˆk, σ k F Electro-Phoo iteractio. Phoo creatio operator.

29 b) Sigle particle spectral fuctio. Lattice model. Luttiger Liquid phase. CDMFT L c =8. QMC λ =.25, ω =.1 t, ρ =.5 T =.125t << ω Luttiger Liquid approach/bosoizatio. Mede, Schöhammer, Guarso, PRB 94. Polaro Gapped Charge Spi k F σ : Spi desity (boso), decouples. ρ ˆq ˆq : Charge desity (boso), mixes with phoo. Bogoliubov trasformatio.

30 b) Sigle particle spectral fuctio. Lattice model. Luttiger Liquid phase. CDMFT L c =8. QMC λ =.25, ω =.1 t, ρ =.5 T =.125t << ω Luttiger Liquid approach/bosoizatio. Mede, Schöhammer, Guarso, PRB 94. Polaro Gapped Charge Spi k F

31 b) Sigle particle spectral fuctio. Peierls phase isulatig phase. CDMFT L c =12. QMC T =.5t < ω λ =.35, ω =.1 t, ρ =.5 QMC T=.25t < ω Iterpretatio: Breakig of a bipolaro. Eergy cost is the spi gap: Δ sp ~.2t~2Δ qp Cosistet with Luther-Emery liquid. k F k F

32 Luttiger Liquid, λ=.15,.25 Luther-Emery Liquid, λ=.35 Domiat 2k f charge Kρ<1 2k F 4k F Pairig correlatios fall of quicker tha 1/r 2 Kρ<1/2

33 Summary. Weak-couplig CT-QMC. Simple ad flexible method. Perfectly suited for cluster methods (DCA, CDMFT) Allows to acces large clusters. Projective schemes Geeralizatio to iclude phoos. ¼ Filled Holstei model. Luttiger Liquid. Peierls phase. Bipolaroic CDW. Gapless spi Gapless charge. λ c Gapfull spi Gapless charge. Luther-Emery. λ Charge, spi ad sigle particle spectral fuctios, ad temperature depedece thereof.