Diagrammatic extensions of (E)DMFT: Dual boson

Transcription

1 Diagrammatic extensions of (E)DMFT: Dual boson IPhT, CEA Saclay, France ISSP, June 25, 2014

2 Collaborators Mikhail Katsnelson (University of Nijmegen, The Netherlands) Alexander Lichtenstein (University of Hamburg, Germany) Erik van Loon (University of Nijmegen, The Netherlands) Junya Otsuki (Tohoku University, Japan) Olivier Parcollet (CEA, Saclay, France) Alexey Rubtsov (Moscow State University, Russia)

3 Outline Motivation: Diagrammatic extensions of DMFT Overview Existing approaches Underlying idea Dual fermion: convergence properties Collective excitations: dual boson approach EDMFT Formalism Impurity Solver Collective excitations: DMFT / EDMFT

4 Motivation include momentum dependence beyond DMFT quantum criticality long-range Coulomb interaction collective excitations Feasible / accurate many body approaches for real materials...

5 Diagrammatic extensions of (extended) DMFT Diagrammatic extension of DMFT [H. Kusunose J. Phys. Soc. of Japan 75, (2006)] Multiscale approach [C. Slezak, M. Jarrell, T. Maier, and J. Deisz, arxiv: (2006)] DΓA [A. Toschi, A. A. Katanin, and K. Held, Phys. Rev. B 75, (2007)] Dual Fermion [A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 77, (2008)] Dual Boson [A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Ann. Phys. 327, 1320 (2012)] One-particle irreducible approach [G. Rohringer, A. Toschi, H. Hafermann, K. Held, V. I. Anisimov, and A. A. Katanin, Phys. Rev. B 88, (2013)]

6 Diagrammatic extensions of DMFT: underlying idea U (τ τ ) impurity model = Σ imp, vertex γ imp ω utilize effective two-particle interaction of the impurity model Γ = γ + γ Γ eh + Γ v ee Γ Σ(ω, k) = 1 γ Γ 2 γ = Γ eh + Γ + v 2 γ ee γ include spatial correlations through diagrammatic corrections: second-order, FLEX, Parquet, diagrammatic MC (?)...

7 Complementarity to cluster approaches Clusters Diagrammatic extensions Control parameter: cluster size Large clusters Rigorous summation of all diagrams on the cluster Diagrams chosen on physical considerations limited cluster size, difficult to converge in practice Truncation of diagrammatic series and fermion interaction

8 Illustration: dual fermion paramagnetic 2D Hubbard model, U/t = 8, T /t = DMFT Γ E f M M Γ X Dual Fermion Σ d = Q = (π, π) X ǫ k+q = ǫ k Γ Captures short-range dynamical AF correlations Critical U c /t 6.5 in good agreement with cluster methods [S. Brener, HH, A. N. Rubtsov, M. I. Katsnelson, A. I. Lichtenstein PRB 77, (2008)]

9 Convergence Properties I Small parameter? weak coupling limit (U 0): γ (4) U, γ (6) U 2,... strong coupling limit (t h k 0); atomic limit ( 0): Gω d 0 [ (k) = g ω gω + ( h k ) 1] 1 gω g ω h k g ω

10 Convergence Properties II λ (d) max T/t = DMFT DF a b DF a d DF a f LDFA U/t T Γ irr, m N ωω Ω=0 G ω (k)g ω (k+q) φ ω = λφ ω ω k a c e f b d Γ irr, G m ω +Ω(k + q) ωω Ω DF: Γ irr = γ (4), G = G d DMFT: Γ irr = γ irr imp, G = G DMFT [HH, G. Li, A. N. Rubtsov, M. I. Katsnelson, A. I. Lichtenstein, H. Monien, Phys. Rev. Lett. 102, (2009)]

11 Model Extended Hubbard model Hamiltonian H = t ( ) c rσc r δσ + c r δσ c rσ rδσ + U r n r n r rr V (r r )n r n r. r: discrete lattice site positions; half bandwidth D = 1 V q = 0 local interaction Hubbard model (2D) V q = V q 2, (screened) Coulomb interaction (3D) V q = 2V (cos q x +cos q y ), nearest-neighbor interaction (2D)

12 Extended dynamical mean-field theory (EDMFT) Treatment of models with long-range interaction V q Mapping to impurity model with retarded interaction W ω S imp [c, c] = σ c σ[ı + µ σ ]c σ + U ω n ω n ω + 1 n ω W ω n ω. 2 ω Self-consistency G 1 Xω 1 (k) = (g imp ) 1 + ɛ k (q) = (χ imp ) 1 + W ω V q g imp χ imp ω ω = 1 G (k) N k = 1 X ω (q) N q [Q. Si and J. L. Smith, Phys. Rev. Lett. 77, 3391 (1996)] [H. Kajueter, Ph.D. thesis, Rutgers University (1996)] [J. L. Smith and Q. Si, Phys. Rev. B 61, 5184 (2000)] [R. Chitra and G. Kotliar, Phys. Rev. B 63, (2001)]

13 dual bosons: basic idea S lat [c, c] = iσ c iσ[ı + µ]c iσ + U qω n qω n q, ω + ε k ckσ c kσ kσ Introduce impurity problem at each lattice site V q n qω n q ω. qω = S lat [c, c] = i S imp [c iσ, c iσ ] kσ c kσ ( ɛ k )c kσ ωq n ωq (W ω V q )n ωq decouple non-local terms through Hubbard-Stratonovich transformations, integrate out original fermions [A.N. Rubtsov, M.I. Katsnelson, A.I. Lichtenstein, Ann. Phys. 327, 1320 (2012)]

14 Transformation to Dual Fermions e c kσ ( ɛ k)c kσ = det [ g 1 D[f, f ]e f kσ g 1 ( ɛ k )g 1 ] 1 ( ɛ k )g 1 f kσ fkσ g 1 c c g 1 f kσ e 1 2 n ωq(w ω V q)n ωq = det [ χ 1 ω (W ω V q )χ 1 ω ] 1 2 D[φ] (2π) N e φωqχ 1 ω (W ω V q)χ 1 ω φ ωq φ ωqχ 1 ω n ωq n ωqχ 1 ω φ ωq dual action S[f, f ] = kσ fkσ [G d 0 (k)] 1 f kσ + i V [f i, f i ; φ i ] V [f, f ; φ] = 1 4 γ 1234f 1 f 2 f 3 f 4 + λ 123 f 1 f 2 φ G d 0 (k) = [ g 1 + ( h k ) ] 1 g Xω d 0 (k) = [ χ 1 ω + (W ω V q ) ] 1 χω

15 Dual perturbation theory Evaluate fermionic and bosonic self-energies in dual perturbation theory Σ, Π (a) G (0) k (b) X qω (0) + ω γ (c) γ imp ω + ω + ω (d) λ imp ω ω

16 Polarization corrections Σ = Π = 0 corresponds to (extended) DMFT: G 1 k Xqω 1 Define physical polarization: = g 1 (1 + Σ k g ) 1 + ɛ k = χ 1 ω (1 + Π qω χ ω ) 1 + W ω V q X 1 qω = Π 1 qω V q Π generates polarization corrections to EDMFT polarization This talk: Σ = 0, Π = Π 1 qω = χ ω (1 + Π qω χ ω ) 1 + W ω = λ Λ Λ λ Γ λ + Γ = γ + γ Γ

17 Impurity solver Hybridization expansion CTQMC with retarded interaction U ω and improved estimators n (τ ; τ) µscr 1 0 β K(τ τ ) Uscr 0 n (τ ; τ) Z = Z at k=0 dτ whyb (τ )w at (τ )w ret (τ ) w at (τ ) = e 1 2 w ret (τ ) = e 1 2 ij U ij l ij e µ i l ii α i α j ij 2k i,j α i,α j >0 sα i sα j K(τα i τα j ) e 2K (0 + ) i j ij l ij +K (0 + ) i l ii, K (τ) = U(τ), K(0) = K(β) = 0 [P. Werner et al., PRL 97, (2006)] [P. Werner et al., PRL 104, (2010)]

18 Improved estimators Σ a (iω) = F a(iω) G a (iω) ; F st a (τ τ ) = j F = F st +F ret nj (τ)u ja c a (τ)c a (τ ) F ret ab (τ τ ) = nj( τ) U( τ τ) a b τ τ β Fa ret (τ τ ) = d τ 0 i ni ( τ)u ret ( τ τ)c a (τ)c a (τ ) β d τn j ( τ)u ret ( τ τα) e j 0 = 2K (0 + ) s βj K (τ βj τα) e j β j Im Σ(ın) n Im Σ(ın) n n V = 0.00 V = 0.40 V = 0.60 V = 0.72 [HH, Phys. Rev. B 89, (2014)]

19 Improved estimators for vertex γ (4) + G (3),con ab (ı, ıω) = i i G a (i)f (3) ab (ı, ıω) F a(ı)g (3) ab (ı, ıω) + ω + ω + ω + ω } } {{ } } {{ (4) con G ω GG+ωχωλω { }} { γ (3) + ω } {{ } (3) con G ω ω (4) disc G ω ω + + ω } {{ } (3) disc G ω n λ a (ı, ıω) = 1 χ(ıω) ( b G 1 a (ı)g 1 (ı + ıω) a ) G (3),con ab (ı, ıω) 1 Re λn,ωm Re λn,ωm m = 0 m = 4 V = 0.00 V = 0.40 V = 0.60 V = n [HH, Phys. Rev. B 89, (2014)]

20 Hubbard model (2D) Collective charge excitations: DMFT (V q = 0) Exactly equivalent to DMFT susceptibility a) Γ = Γ irr + Γ irr Γ b) χ = + Γ zero sound mode in correlated metallic state

21 Simpler approximations = = ω ω 4 U = Γ X M Γ 4 U = Γ X M Γ approximations violate Ward identity (continuity equation) q F µ Γ µ (k, q) = G 1 (k) G 1 (k + q) improper treatment of vertex corrections leads to qualitatively wrong results

22 DMFT susceptibility DMFT is conserving! Luttinger-Ward functional Φ[G i j ] = i Φ[G i i ] = i φ imp [G i i ] Σ ij = δφ[g i j ] = δφ[g i i ] δg ji δg ii δ ji Γ irr ijkl = δ2 Φ[G i j ] = δ2 φ[g l l ] δg ji δg lk δg 2 δ li δ lj δ lk ll Σ = Σ imp, Γ irr = Γ irr,imp [G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961)][G. Baym, Phys. Rev. 127, 1391 (1962)]

23 Ward identity Γ µ;,ω (k, q) =γ µ (k, q) T k Γ irr ω G σ (k )G +ω(k + q)γ µ;,ω(k, q). Λ µ = χ Γ Γ irr µ = + Γ γ µ µ γ µ q F µ Γ µ; (k, q) =q F µ γ µ (k, q) T k Γ irr ω G (k )G +ω(k + q)[q F µ Γ µ; (k, q)] Σ+ω Σ, T Γirr ω [g +ω g ] Σ+ω Σ, T Γirr ω [g +ω g ] 10 m = 1 m = 1 15 m = 3 m = 3 m = 5 m = 5 m = 7 m = n n Σ U = U = ω Σ ω = T Γ irr [g +ω g ] ω ω

24 Collective charge excitations: EDMFT (V q = V /q 2, 3D) EDMFT: X ω (q) = ω Γ 1 [(χ imp ω ) 1 + W ω ] V q RPA: EDMFT X ω (q) = Π RPA ω 1 (q) V q 1 + V Π ω(q) /q 2 = V Π ω(q) (q)/q 2 = 0 Π ω 1/ω α = ω 1/q 2/α Π RPA ω (q) = χ 0 ω(q) g q 2 f (ω)

25 Collective charge excitations: (V q = V /q 2, 3D) = + ω Γ dual boson Πωm (q) m = 0 m = 1 m = 3 m = 5 m = 7 m = 9 dual boson polarization Π ω (q) q 2 (ω > 0) plasma frequency: ω 2 p = e 2 a 2 tv kσ 2 cos(k z a) n kσ Imǫ 1 ω (q 0) qx[π/a] V = 0.25 V = 0.50 V = ω

26 Extended Hubbard model: U V phase diagram (2D) transition to charge-ordered (CO) insulator V V U CO divergence of X qω at q = (π, π), ω = 0 dual boson diagrams: 0.3 EDMFT 0.2 DB, 2nd ord 0.1 MET DB, ladder RPA 0.0 V = U/ U large corrections to EDMFT even for small U RPA is reproduced non-trivially for U 0!

27 Summary DMFT susceptibility provides conserving description of the collective excitations of strongly correlated electrons dual boson approach yields conserving description of collective charge excitations in correlated systems (zero sound / plasmons) non-local vertex corrections are essential to fulfill charge conservation frequency dependent self-energy requires frequency dependent irreducible vertex Thank you for your attention!