Cluster Extensions to the Dynamical Mean-Field Theory

Transcription

1 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods?

2 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co.

3 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA

4 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA 4. Summary

5 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA 4. Summary Collaborators: Th. Maier, M. Jarrell

6 Standard model for e.g. TMO: One band Hubbard model

7 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice

8 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site

9 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site Hopping between nearest t

10 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site t Hopping between nearest t and next-nearest neighbors

11 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site t U Hopping between nearest t and next-nearest neighbors Coulomb correlations

12 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site t U Hopping between nearest t and next-nearest neighbors Coulomb correlations H = ij,σ t ij c iσ c jσ + U i n i n i Dispersion: ɛ k = ɛ 0 2t(cos k x + cos k y ) 4t (cos k x cos k y 1) Typical parameters: t 0.25eV, t /t 0.2

13 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site t U Hopping between nearest t and next-nearest neighbors Coulomb correlations H = ij,σ t ij c iσ c jσ + U i n i n i W/2 Dispersion: ɛ k = ɛ 0 2t(cos k x + cos k y ) 4t (cos k x cos k y 1) E F -W/2 Typical parameters: t 0.25eV, t /t 0.2 Γ M X M Γ Γ X

14 Standard model for e.g. TMO: One band Hubbard model Simplest possible lattice Only one relevant orbital per site t U Hopping between nearest t and next-nearest neighbors Coulomb correlations H = ij,σ t ij c iσ c jσ + U i n i n i W/2 Dispersion: ɛ k = ɛ 0 2t(cos k x + cos k y ) 4t (cos k x cos k y 1) E F -W/2 Typical parameters: t 0.25eV, t /t 0.2, U/t 8 W Γ M X M Γ Intermediate coupling Regime Γ X

15 Standard techniques Exact Diagonalization Quantum Monte-Carlo Properties of finite systems (ED: N < 20, QMC: N < 100) Density-Matrix RenormalizationGroup Dynamical Mean-Field Theory Ground state and dynamics for D = 1 Approach to local properties RenormalizationGroup Low-energy properties Perturbation Theory Resummation of sub-classes of diagrams (FLEX) Variational wave functions Ground state properties

16 Successful approach for qualitative properties: DMFT

17 Successful approach for qualitative properties: DMFT Metal-insulator transition for n = 1 Georges et al., RMP 96, Bulla et al.,prl 99 & PRB U<U c U>U c T/W metal DOS ω insulator U/W

18 Successful approach for qualitative properties: DMFT Metal-insulator transition for n = 1 Magnetism (AFM & FM) PSfrag replacements Zitzler et al., EPJ 02 U/(W+U) 1 0,8 0,6 0,4 0,2 FM AFM (??) AFM(PS) PM 0 0% 10% δ = 1 n 20% 30%

19 Successful approach for qualitative properties: DMFT Metal-insulator transition for n = 1 TP et al., PRB QP Magnetism (AFM & FM) DOS 0.4 LHB UHB Correlated metal for n < ω Problems: No dependency on dimensionality of system wrong for D = 1 Fermi liquid ubiquitous No phases with non-local order parameter e.g. d-wave sc

20 Idea: Try to combine ED, QMC and (D)MFT

21 Idea: Try to combine ED, QMC and (D)MFT FiniteSystemSimulations Numerical exact Local & non-local dynamics Thermodynamic limit

22 Idea: Try to combine ED, QMC and (D)MFT FiniteSystemSimulations DynamicalMeanFieldTheory Numerical exact Local & non-local dynamics Thermodynamic limit Thermodynamic limit Local dynamics Non-local dynamics

23 Idea: Try to combine ED, QMC and (D)MFT FiniteSystemSimulations DynamicalMeanFieldTheory Numerical exact Local & non-local dynamics Thermodynamic limit Thermodynamic limit Local dynamics Non-local dynamics Combination: Cluster MFT Hettler, TP et al., PRB 58, 7475( 98) Lichtenstein et al., PRB 62, R9283 ( 00) Kotliar et al. PRL 87, ( 01) Maier et al., RMP ( 05).

24 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA 4. Summary

25 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. Maier et al., RMP 05

26 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y k x Maier et al., RMP 05

27 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) (π, π) Choose N c cluster points K: (0, 0) (π, 0) k x Maier et al., RMP 05

28 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) (π, π) Choose N c cluster points K: (0, 0) (π, 0) k x K Maier et al., RMP 05

29 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) k (π, π) Choose N c cluster points K: Σ( k, z) (0, 0) (π, 0) k x K Maier et al., RMP 05

30 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) k (π, π) Choose N c cluster points K: Σ( k, z) = Σ( K + k, z) k K (0, 0) (π, 0) k x K Maier et al., RMP 05

31 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) k (π, π) Choose N c cluster points K: Σ( k, z) = Σ( K + k, z) Σ( K, z) k K (0, 0) (π, 0) k x K Maier et al., RMP 05

32 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) k (π, π) Choose N c cluster points K: Σ( k, z) = Σ( K + k, z) Σ( K, z) k (0, 0) K (π, 0) k x coarse graining: Ḡ( K, z) = N c G( K N + k, z) k K Maier et al., RMP 05

33 General scheme in all cluster MFT: Take into account short-ranged correlations exactly Neglect long-ranged correlations Example DCA: Reduce k-space resolution to K = 2π/L. k y (0, π) k (π, π) Choose N c cluster points K: Σ( k, z) = Σ( K + k, z) Σ( K, z) k (0, 0) K (π, 0) k x coarse graining: Ḡ( K, z) = N c G( K N + k, z) k K effective periodic cluster model Maier et al., RMP 05

34 Practical implementation: Initial guess for Σ(K)

35 Practical implementation: Initial guess for Σ(K) Ḡ(K) = N c N 1 k ω ɛ K+k + µ Σ(K)

36 Practical implementation: G 1 (K) = Ḡ 1 (K) + Σ(K) Initial guess for Σ(K) Ḡ(K) = N c N k 1 ω ɛ K+k + µ Σ(K)

37 Practical implementation: Cluster Solver G(K) G c (K) G 1 (K) = Ḡ 1 (K) + Σ(K) Initial guess for Σ(K) Ḡ(K) = N c N k 1 ω ɛ K+k + µ Σ(K)

38 Practical implementation: Cluster Solver G(K) G c (K) G 1 (K) = Ḡ 1 (K) + Σ(K) Σ(K) = G 1 (K) G c (K) 1 Ḡ(K) = N c N k 1 ω ɛ K+k + µ Σ(K)

39 Practical implementation: Cluster Solver G(K) G c (K) G 1 (K) = Ḡ 1 (K) + Σ(K) Σ(K) = G 1 (K) G c (K) 1 Ḡ(K) = N c N k 1 ω ɛ K+k + µ Σ(K) Exact limits: N c = 1 DMFT, N c = N exact

40 Other realizations: Define cluster in real space Lichtenstein et al., PRB 00; Kotliar et al. PRL 01 Neglect self-consistency cluster perturbation-theory Gros & Valenti, Ann. der Physik 94; Sénéchal et al., PRL 00 Unifying framework: Self-energy functional theory Potthoff, EPJ 03; Dahnken et al., 03 General problem: Reconstruct full k-dependence from coarse-grained self-energy e.g. DCA: interpolate Σ(K, z) Σ(k, z)

41 Schematic structure of effective cluster:

42 Schematic structure of effective cluster: Dynamical mean-field: infinitely many degrees of freedom (noninteracting)

43 Schematic structure of effective cluster: Dynamical mean-field: infinitely many degrees of freedom (noninteracting) Method of choice: Hirsch-Fye QMC

44 Schematic structure of effective cluster: Dynamical mean-field: infinitely many degrees of freedom (noninteracting) Method of choice: Hirsch-Fye QMC

45 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA 4. Summary

46 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0)

47 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N(ω) t ω/t

48 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T N = 0.34t N(ω) t ω/t ω/t

49 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.34t N(ω) t ω/t ω/t

50 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.34t T=0.31t N(ω) t ω/t ω/t N c = 1: No precursor of AF

51 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.34t T=0.31t N c = 4, U = W/2, t = 0 T=0.80t T N = 0.21t N(ω) t ω/t ω/t ω/t N c = 1: No precursor of AF

52 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.34t T=0.31t N c = 4, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.21t N(ω) t ω/t ω/t ω/t N c = 1: No precursor of AF

53 Maier, TP et al., EPJ B 13 ( 00) 2D Hubbard Modell: N c = 1 vs. N c > 1 (t = 0) 0.2 U=0 N c = 1, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.34t T=0.31t N c = 4, U = W/2, t = 0 T=0.80t T=0.40t T N = 0.21t T=0.17t N(ω) t ω/t ω/t ω/t N c = 1: No precursor of AF N c = 4: pseudo gap in paramagnet

54 Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV

55 M Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV ImΣ(ω+iδ) n = 0.80, T = 370K k a k b ω M X k b (c) Γ M k a (a) X Γ (b)

56 M Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV ImΣ(ω+iδ) n = 0.80, T = 370K k a k b ω M X k b (c) Γ M k a (a) X Γ (b) well defined quasi particles weak k-dependence of Σ( k, ω)

57 M Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV n = 0.80, T = 370K well defined quasi particles weak k-dependence of Σ( k, ω) reduced quasi-particle bandwidth

58 M Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV ImΣ(ω+iδ) n = 0.80, T = 370K k a k b ω M X k b (c) Γ M k a (a) X Γ (b) well defined quasi particles weak k-dependence of Σ( k, ω) reduced quasi-particle bandwidth

59 M Q Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV n = 0.80, T = 370K M X k b (c) n = 0.95, T = 370K M X k b (c) ImΣ(ω+iδ) k a -1.5 ImΣ(ω+iδ) k b ω k a k b ω Γ M k a (a) X Γ (b) Γ M k a (a) X Γ (b) well defined quasi particles weak k-dependence of Σ( k, ω) reduced quasi-particle bandwidth overdamping of structures near X

60 M Q Γ X Spectral functions Maier, TP et al., PRB 66 ( 02) N c = 16, U = W = 8t, t = 0.2t, t = 0.25eV n = 0.80, T = 370K M X k b (c) n = 0.95, T = 370K M X k b (c) ImΣ(ω+iδ) k a -1.5 ImΣ(ω+iδ) k b ω k a k b ω Γ M k a (a) X Γ (b) Γ M k a (a) X Γ (b) well defined quasi particles weak k-dependence of Σ( k, ω) reduced quasi-particle bandwidth overdamping of structures near X strong k-dependence of Σ( k, ω) non-fl Σ( k, ω) near X?

61 Fermi surface Maier, TP et al., PRB 66 ( 02) N c = 16, U = W, t = 0.2, T = 370K n = 0.95 n = 0.90 n = 0.85 n = 0.80 n 0.9 Small FS for n 1 Hole pockets? n 0.9 Large FS for n < 0.9 free-electron like FS

62 Thomas Pruschke Institut für Theoretische Physik Universität Göttingen Cluster Extensions to the Dynamical Mean-Field Theory 1. Why cluster methods? 2. Cluster extensions DCA, CDMFT and Co. 3. Spectral functions from the DCA 4. Summary

63 Aspects of cluster MFT Interpolation between finite system simulations and DMFT Thermodynamic limit for dynamics Systematic inclusion of short- and mid-ranged correlations Sensible results Reduction of transition temperatures Fluctuation induced precursors of order in spectra Nontrivial k-dependent renormalization of single-particle properties