From Gutzwiller Wave Functions to Dynamical Mean-Field Theory

Transcription

1 From utzwiller Wave Functions to Dynamical Mean-Field Theory Dieter Vollhardt Autumn School on Correlated Electrons DMFT at 25: Infinite Dimensions Forschungszentrum Jülich, September 15, 2014 Supported by FOR 1346

2 Outline: Electronic correlations Approximation schemes: Mean-field theories, variational wave functions utzwiller wave functions, utzwiller approximation Derivations of the utzwiller approximation From one to infinite dimensions Simplification of many-body perturbation theory in Dynamical mean-field theory (DMFT) d = Applications: LDA+DMFT for correlated electron materials

3 Correlations in fermionic matter < 1950s 1950s - Mott insulators/metal-insulator transitions - Ferromagnetism in 3d transition metals Liquid Helium s 1970s Kondo effect (1964) Heavy fermion systems Hubbard model (1963) 1980s 1990s 2000s - Fractional quantum Hall effect - High-T c superconductivity Colossal magnetoresistance Cold fermionic atoms in optical lattices etc.

4 material realistic model modeling Þ Þ maximal simplification?

5 Hubbard model utzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time ˆ = t ˆ ˆ U ˆ nˆ σ σ + i j i i ij,, σ i H c c n Dimension of Hilbert space L: # lattice sites L O(4 ) nˆ nˆ nˆ nˆ i i i i Static (Hartree-Fock-type) mean-field theories generally insufficient Computational time for N 2 molecule: ca. 1 year with compute nodes Purely numerical approaches (d=2,3): hopeless!

6 AND Reliable, widely applicable, non-perturbative approximation scheme for correlated electron models PROFESSORSHIP Best-known approaches: Mean-field theories Variational wave functions

7 Mean-field theory (MFT) 1) Construction by decoupling/factorization e.g., spins: AB A B SS i j Si Sj Weiss molecular field theory (1907) Prototypical single site MFT mean field.

8 Mean-field theory (MFT) 2) Construction by continuum/infinity limit Spin S Degeneracy N Dimension d /coordination number Z Weiss MFT for spin models }

9 Variational Wave Functions One-particle wave function Variational parameters Applications, e.g.: - Quantum liquids 3 He, 4 He - Nuclear physics - Correlated electrons - Heavy fermions - FQHE - High-T c superconductivity Correlation operator reduces energetically unfavorable configurations in Expectation value of some operator E exact Energy expectation value E λ var 0 i λ * i = λ * i Minimization determines variational parameters

10 Martin utzwiller ( ) utzwiller Wave Functions

11 utzwiller wave functions Hˆ = Hˆ + Hˆ i, j, σ kin int ˆ ˆ ijciσcjσ U i i = t + utzwiller variational wave functions nˆ nˆ i Dˆ i Dˆ ψ Op. of local double occupation Op. of total double occupation = e λhˆ int Cˆ F utzwiller (1963) One-particle (product) wave function

12 utzwiller wave function Hˆ = Hˆ + Hˆ i, j, σ kin int ˆ ˆ ijciσcjσ U i i = t + utzwiller variational wave functions λu e g Hˆ UDˆ Dˆ λ int λ e = e = g g = 1 U = 0 g = 0 U = i i nˆ nˆ i Dˆ i Dˆ ψ, 0 g 1 Op. of local double occupation Op. of total double occupation = e λhˆ int Cˆ F Fermi gas utzwiller (1963) lobal reduction of configurations with too many doubly occupied sites utzwiller projection

13 utzwiller wave function with Electronic configuration with D doubly occupied sites Set of all different electron configurations with given D Probability of the electron configuration i D How to calculate?

14 utzwiller Approximation Approximates quantum mechanical expectation values by counting classical spin configurations

15 utzwiller approximation with... Simplest approximation:? Neglect correlations between electrons: count classical configurations of, electrons all probabilities equal = const Probability for a configuration of electrons with spin σ N D : # electron configurations with given D

16 utzwiller approximation L: # lattice sites N σ : # electrons with spin σ D: # doubly occupied sites : # singly occupied sites with spin σ : # empty sites : # spin configurations with given D Thermodynamic limit: Sum over D determined by the largest term: d dd g N = 2D ( D) 0 D D / L d d( g) Physical meaning?

17 utzwiller approximation = [...][...] [...][...] Law of mass action for the chemical reaction 2 g + + Quasi-chemical equilibrium: Typical mean-field result! 2D g N D Norm, D= Dg ( )

18 utzwiller approximation Similarly: Kinetic energy ˆ ˆ ψ H kin ψ 0 H kin = Ekin = qd (, n) Ekin ψ ψ n 2 Interaction energy round state energy L ε 2 2(1 δ 2 d)( d + δ + d ) 2, δ =1- n = n = q= n 1 δ Interactions multiplicative renormalization of kinetic energy UDˆ H = ψ ψ E = UD ˆ U ψ ψ 1 ˆ E ( d( g), n) H = q( d, n) ε0 + Ud L L E = 0 gd ( *) In the following: d* d d d * Study origin of band ferromagnetism utzwiller (1965) 0 int

19 utzwiller-brinkman-rice theory Brinkman, Rice (1970) n = 1 d q 1 = 1 U U 4 c U = 1 U c 2 Uc = 8 ε 0 E L = ε 0 1 U U c 2 U U : E 0 and E 0 Electrons localize c kin int no charge current utzwiller approximation describes a correlation induced ("Mott") metal-insulator transition as in V 2 O 3 Applicable to normal liquid 3 He? Anderson, Brinkman (1975)

20 Properties of normal liquid 3 He Strongly renormalized Fermi gas reywall (1983) DV (1984) Spin susceptibility Why does χ s become so large? o χs Paramagnon theory: χs =, U 1 UN F Fermi liquid theory: N 1 F Effective mass m*/m Wilson ratio Anderson, Brinkman (1975) 3 He close to ferromagnetic instability 3 He close to Mott (localization) transition?

21 utzwiller approximation and Fermi liquid theory Change of the ground state energy E L δ E = qε + Ud = ( q δε + δ q ε ) L d * Brinkman, Rice (1970) DV (1984) Correspondence with Landau Fermi liquid theory: δ E FL change of: particle energy distribution of particles... l < 1 m* m m m* N 0 0 *(0) = N (0), εp = ε σ p σ bare q = m m *

22 utzwiller approximation and Fermi liquid theory Fermi liquid relations Spin susceptibility Compressibility utzwiller approximation U U U c DV (1984) F ( U) = F ( U) a s = 0 0

23 utzwiller approximation and Fermi liquid theory Fermi liquid relations Spin susceptibility Compressibility utzwiller approximation U = U U c a U U 3 3 F c 0 p = 4 4 const DV (1984) U U Wilson ratio c 4 = const No divergence!

24 utzwiller approximation and Fermi liquid theory DV (1984) utzwiller approximation utzwiller approximation 3 4 Conclusion: no magnetic instability Spin susceptibility χ s becomes large because m*/m becomes large! Liquid 3 He: "almost localized Fermi liquid" (close to Mott transition at U c )

25 utzwiller approximation and Fermi liquid theory DV (1984) utzwiller approximation utzwiller approximation 3 4 Conclusion: no magnetic instability Spin susceptibility χ s becomes large because m*/m becomes large!

26 utzwiller approximation: 1) Quasi-classical, non-perturbative approximation scheme for correlated fermions 2) Describes a: - correlated Fermi liquid with a - Mott metal-insulator transition for n=1 at a critical U c 3) Shows mean-field behavior Open question: Systematic derivation by more conventional methods of quantum many-body theory?

27 Slave-boson mean-field theory Lecture. Kotliar Hubbard model Introduce local Bose fields ei pi pi di Local constraints zi σ : renormalization factors (operators) Saddle point approximation results of utzwiller approximation

28 utzwiller wave function Diagrammatic evaluation of, 0 g 1 E = ψ ψ Hˆ ψ ψ Metzner, DV (1987, 1988) c ( ) m n 1) Hˆ = U Dˆ = LUd( gn, ) int i 2 2 m 1 = LUg ( g 1) m= 1 c m ( n) Diagrammatic representation of c ( ) m n Lines correspond to FT 0 0 ij, σ = iσ jσ 0 k σ g c c n (probability amplitude/ momentum distribution) Diagrams for the Hubbard interaction (φ 4 theory)

29 utzwiller wave function Diagrammatic evaluation of E = ψ ψ Hˆ ψ ψ Metzner, DV (1987, 1988), 0 g 1 fm σ ( k) 2) Hˆ kin i = ε ( k) nˆ kσ kσ nkσ 2 (, ) ( 1) m k σ = + m ( k) m= 2 fm σ ( k) n gn g f σ Diagrammatic representation of Lines correspond to FT 0 0 ij, σ = iσ jσ 0 k σ g c c n (probability amplitude/ momentum distribution) Diagrams for the kinetic energy

30 utzwiller wave function Diagrammatic evaluation of E = ψ ψ Hˆ ψ ψ Metzner, DV (1987, 1988) c ( ) m n fm σ ( k) Diagrams for the kinetic energy d=1: analytic calculation of all diagrams possible Diagrams for the Hubbard interaction

31 1) Hˆ U ˆ LUd(, int = D = gn) density of doubly occupied sites dgn (, ) 2) Hˆ kin = ε ( k) nˆ kσ kσ n kσ momentum distribution Hˆ + Hˆ E kin int Minimization of E (g,d(g),n,u): U ε 0 : E 4 = π 2 2 t U 1 ln( U / ε ) 0

32 Correlation functions ebhard, DV (1987, 1988) Spin, density, empty sites, doubly occupied sites, local Cooper pairs d=1: analytic calculation of all diagrams possible E. g., spin-spin correlations: U = n = 1 Excellent agreement with exact result for antiferromagnetic Heisenberg chain (j=1,2) One of 4 sets of diagrams Exact result for S=1/2 antiferromagnetic Heisenberg chain with exchange coupling Haldane(1988), Shastry (1988) utzwiller wave function = Anderson s RVB state in 1D Very interesting wave function! d=2,3?

33 utzwiller wave function Diagrammatic evaluation of c ( ) m n E = ψ ψ Hˆ ψ ψ Metzner, DV (1988) fm σ ( k) Diagrams for the kinetic energy Diagrams for the Hubbard interaction Analytic evaluation of all diagrams not possible in d=2,3

34 utzwiller wave function Diagrammatic evaluation of c ( ) m n E = ψ ψ Hˆ ψ ψ Metzner, DV (1988) Calculation of individual diagrams: Numerical check by Monte-Carlo integration d : 3 n 4 i j Example: Diagrams for the Hubbard interaction 2 n 3 2 = 4 n, 2 3, d = 1 d = dimension Each diagram takes simple value n = n n = 2 d reat simplifications for : internal momenta become independent # lines

35 utzwiller wave function Diagrammatic evaluation of E = ψ ψ Hˆ ψ ψ Metzner, DV (1988) d : Conclusion: utzwiller approximation gives the exact result for expectation values calculated with the utzwiller wave function in d Calculation of individual diagrams: Numerical check by Monte-Carlo integration i j d : 3 n 4 dimension Diagrammatic derivation of utzwiller approximation

36 Optimal form of utzwiller wave functions in d = D Ψ = g Ψ0 ˆ arbitrary one-particle wave function In spite of diagrammatic collapse: Remaining diagrams have to be calculated with difficult Ψ 0 Re-write by gauge transformation µ i σ nˆ iσ iσ Ψ = g Ψ 0 0 chose local chemical potentials such that all Hartree bubbles disappear in d = ebhard (1990) For arbitrary Ψ 0 : Hˆ = t q q g + U d d = ij, σ renormalization factors iσ jσ 0 ij, σ i i qd (, n n ), n = Ψ nˆ Ψ i σ σ σ 0 σ 0 d = Diagrammatic derivation in slave boson saddle point solution of the Kotliar-Ruckenstein

37 d limit for lattice fermions

38 Hˆ kin 0 i, σ j( NN i) Z ˆ ˆ σ = t ci cjσ 0 g 0 ij, σ 1 Probability amplitude for hopping Z j NN i Amplitude for hopping j NN i 2 1 = Probability for hopping j NN i Amplitude for hopping j NN i 0 g ij, σ = 1 1 or Z d Z In general:

39 Hˆ kin 0 Z or d = t cˆ cˆ i j σ σ 0 i, σ j( NN i) 0 1 Z g ij, σ Z Collapse of all connected, irreducible diagrams in position space Example: 1 Z 1 Z 1 Z 1 Z Z or d reat simplification of many-body perturbation theory, e.g., self-energy diagram purely local δ ij Metzner (1989)

40 Hˆ kin 0 Z or d = t cˆ cˆ i j σ σ 0 i, σ j( NN i) 0 1 Z g ij, σ Z Collapse of all connected, irreducible diagrams in position space Example: 1 Z 1 Z 1 Z 1 Z Z or d δ ij Valid for probability amplitude/momentum distribution and the full, time-dependent propagator, since FT 0 0 ij, σ = iσ jσ 0 k σ g c c n

41 Hˆ kin 0 Z or d = t cˆ cˆ i j σ σ 0 i, σ j( NN i) 0 1 Z g ij, σ Z Collapse of all connected, irreducible diagrams in position space Example: correlation energy of Hubbard model 1 Z 1 Z 1 Z 1 Z Z or d δ ij 2 E 2/U Excellent approximation for d=3

42 Z Hˆ kin 0 or d t 1 Z J t * J t= = Z Z Classical Quantum * cˆ cˆ Scaling σ σ 0 NN ) rescaling = i j i, σ j( i Z 1 Z Example: correlation energy of Hubbard model 1 Z 1 Z 1 Z 1 Z Z or d δ ij 2 E 2/U Excellent approximation for d=3

43 ω Σ( k, ) Fermi liquid Only Hubbard interaction remains dynamical

44 Solution of the Hubbard model in d : Towards dynamical mean-field theory

45 Self-consistent DMFT equations: eneralization of coherent potential approximation (CPA) to interacting lattice fermions

46 eneralization of coherent potential approximation (CPA) to interacting lattice fermions ω ε + µ Σ( ω) = 0 ( ω Σ( ω)) free electrons in a dynamic potential Σ(ω) Dynamical (single-site) mean-field theory Numerical solution?

47 Preprint May 6, 1991 Solution of the Hubbard model in d : = Mapping to a single-impurity Anderson model + self-consistency condition Lecture A. eorges Jarrell (1992)

48 Self-consistent DMFT equations (i) Effective impurity problem: local propagator..., single-site ("impurity") action A (ii) k-integrated Dyson equation ( lattice reen function : lattice enters) ω ε + µ Σ( ω) Impurity solver QMC Hirsch-Fye (1986) Jarrell (1992) ED Caffarel, Krauth (1994), Si et al. (1994) NR Bulla (1999)

49 d The solution of the Hubbard model in provides a dynamical mean-field theory (DMFT) of correlated electrons DMFT: local theory with full many-body dynamics Σ( k, ω) Σ( ω) Kotliar, DV (2004) Spectral function Mott-Hubbard metal-insulator transition U Review: eorges, Kotliar, Krauth, Rozenberg (RMP, 1996)

50 LDA+DMFT for Correlated Electron Materials Lecture A. Lichtenstein

51 Held (2004) How to combine? Held (2004) time

52 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, A,...) or W + Local electronic correlations (Many-body theory: DMFT) X=LDA, A; W, X+DMFT Anisimov, Poteryaev, Korotin, Anokhin, Kotliar(1997) Lichtenstein, Katsnelson (1998)

53 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, A,...) or W + Local electronic correlations (Many-body theory: DMFT) LDA+DMFT Anisimov, Poteryaev, Korotin, Anokhin, Kotliar(1997) Lichtenstein, Katsnelson (1998)

54 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, A,...) or W + Local electronic correlations (Many-body theory: DMFT) LDA+DMFT Held, Nekrasov, Keller, Eyert, Blümer, McMahan, Scalettar, Pruschke, Anisimov, DV (Psi-k 2003)

55 Conclusions and Outlook Quantum many-body perturbation theory for lattice fermions greatly simplifies in d Calculations with utzwiller wave function in = utzwiller approximation d DMFT: canonical mean-field theory for correlated electrons oal: Development of LDA+DMFT into a computational scheme for correlated materials with predictive power