Diagrammatic Monte Carlo methods for Fermions

Transcription

1 Diagrammatic Monte Carlo methods for Fermions Philipp Werner Department of Physics, Columbia University PRL 97, 7645 (26) PRB 74, (26) PRB 75, 8518 (27) PRB 76, (27) PRL 99, (27) PRL 99, (27) Support: NSF-DMR Urbana, June 8 p.1

2 Urbana, June 8 p.2 Outline Motivation Dynamical mean field theory for fermionic lattice models impurity models Recent advances im methodology Diagrammatic Monte Carlo approach weak-coupling expansion expansion in hybridization Application Metal-insulator transition in the Hubbard model "Spin glass" transition in a 3-orbital model Collaborators A. J. Millis, E. Gull, M. Troyer

3 Urbana, June 8 p.3 Introduction Theoretical Physics Experimental Physics Computational Physics Computer science template <class model, class lattice> class simulation{... }; Algorithms Hardware

4 Introduction Theoretical Physics Hubbard model Experimental Physics correlation driven metal-insulator transition Computational Physics H = U i n i n i t i,j,σ c i,σ c j,σ U t not analytically solvable McWhan et al., (1973) Urbana, June 8 p.3

5 Urbana, June 8 p.4 Introduction Simulation of correlated lattice models H Hubbard = U i n i n i t i,j,σ c i,σ c j,σ U t Exact diagonalization: up to 2 sites Monte Carlo: fermion sign problem Simulation of 2D, 3D lattice models not possible Need new methods / approximate descriptions e. g. Dynamical Mean Field Theory (DMFT)

6 Urbana, June 8 p.5 Motivation Dynamical mean field theory Metzner & Vollhardt (1989), Georges & Kotliar (1992) Lattice model H latt = U i n i n i t i,j,σ c i,σ c j,σ t Quantum impurity model H imp = Un n k,σ (t kc σa bath k,σ + h.c.) + H bath t k

7 Motivation Dynamical mean field theory Metzner & Vollhardt (1989), Georges & Kotliar (1992) Lattice model H latt = U i n i n i t i,j,σ c i,σ c j,σ t Effective action (hybridization function F(τ)) S = U dτn (τ)n (τ) σ dτdτ c σ (τ)f σ (τ τ )c σ(τ ) F( τ τ ) Self-consistency condition G loc latt (τ) = G imp(τ) Urbana, June 8 p.5

8 Urbana, June 8 p.6 Motivation Dynamical mean field theory Metzner & Vollhardt (1989), Georges & Kotliar (1992) Self-consistency loop couples the impurity to the lattice Lattice model Impurity model t F( τ τ ) Hilbert transform G loc latt = G imp S imp [F(τ)] impurity solver Σ latt = Σ imp G imp (τ) Computationally expensive step: solution of the impurity problem

9 Urbana, June 8 p.7 Example: Hubbard model Correlation driven metal-insulator (Mott) transition Phasediagram for V 2 O 3 McWhan et al., (1973) paramagnetic DMFT solution 1-band Hubbard model Georges & Krauth (1993), Blümer (22) T/4t.1 Mott insulator metal t/U More realistic multi-band simulation requires powerful impurity solvers

10 Urbana, June 8 p.8 Diagrammatic Monte Carlo Weak coupling vs. strong coupling approach Diagrammatic QMC = stochastic sampling of Feynman diagrams Hubbard model: Z = TrT τ e S with action S = β σ dτdτ c σ (τ)f σ (τ τ )c σ(τ β ) +U }{{} dτn n }{{} S F Weak-coupling expansion Rombouts et al., PRL (1999); Rubtsov et al., PRB (25); Gull et al., EPL (28) Treat quadratic part (S F ) exactly, expand Z in powers of S U Hybridization expansion Werner et al., PRL (26); Werner & Millis, PRB (26); Haule, PRB (27); Werner & Millis, PRL (27) Treat local part (S U ) exactly, expand Z in powers of S F S U

11 Urbana, June 8 p.9 Diagrammatic Monte Carlo Expansion in U + auxiliary field decomposition Rombouts et al., PRL (1999), Gull et al., EPL (28) Expand Z in powers of K/β U(n n (n + n )/2) τ τ τ β τ 4 Decouple "interaction vertices" using Rombouts et al., PRL (1999) K/β U(n n (n + n )/2) = (K/2β) s= 1,1 cosh(γ) = 1 + (βu/2k) e γs(n n ) β

12 Diagrammatic Monte Carlo Expansion in U + auxiliary field decomposition Rombouts et al., PRL (1999), Gull et al., EPL (28) Weight of the configuration (Γ σ = diag(γσs 1,...), (G ) ij = g (τ i τ j )) w({s i, τ i }) = ( Kdτ 2β ) n det σ ( ) e Γ σ G σ (e Γ σ I) Local updates: insertion/removal of an auxiliary spin β β Advantage: less spins than Hirsch-Fye method faster updates, shorter autocorrelation (thermalization) times Urbana, June 8 p.1

13 Urbana, June 8 p.11 Diagrammatic Monte Carlo Expansion in the impurity-bath hybridization F Werner et al., PRL (26) Non-interacting model: Z = TrT τ exp [ β dτdτ c(τ)f(τ τ )c (τ ) ] Expand exponential in powers of F F( τ τ ) 1 e s 1 β + τ 1 s τ 1 e β + empty orbital τ 2 s τ 1 e τ 1 s τ 2 e occupied orbital β + τ 2 s e s e s τ3 τ 1 τ 2 τ3 τ 1 e β +... Some diagrams have negative weight sampling individual diagrams leads to a severe sign problem

14 Urbana, June 8 p.12 Diagrammatic Monte Carlo Expansion in the impurity-bath hybridization F Werner et al., PRL (26) Collect the diagrams with the same {c(τi s), c (τi e )} into a determinant F( τe τs ) 1 1 det F (F) m,n = F(τ e m τ s n) resums huge numbers of diagrams (1! = ) eliminates the sign problem Z = sum of all operator sequences c c c c c c τs τe τs τe τs τe β

15 Diagrammatic Monte Carlo Generalizations Arbitrary interactions: U αβγδ c αc β c γc δ, S c α σ α,β c β, S L,... Werner & Millis, PRB (26); Haule, PRB (27) [ ] w = Tr e H locτ 1 ψ e H loc(τ 2 τ 1 ) ψ e H loc(τ 3 τ 2 ) ψ... det F det F β local problem treated exactly flexible histogram of relevant states weight triplet Mott insulator Metal orbitally polarized insulator scales exponentially with.2 # sites, orbitals basis state Urbana, June 8 p.13

16 Urbana, June 8 p.14 Efficiency Scaling of the average perturbation order k Gull et al, PRB (27) Computational effort grows O(k 3 ) with size k of determinants Weak coupling expansion: k U perturbation order <k> weak coupling expansion hybridization expansion metal Mott insulator Hybridization expansion: k decreases with increasing U U/t In the strong correlation regime, speed-ups of

17 1-band Hubbard model Metal-insulator transition on the 2D square lattice (bandwidth = 8t) Gull et al, EPL (28) Single site DMFT: H loc = Un n "Mott" transition at U c 12t 4 site DMFT: H loc = k,σ ɛ kc k,σ c k,σ + i Un n "Slater" transition at U c 4t collapse into plaquette singlet state Probability U/t Urbana, June 8 p.15

18 3-orbital model Non-Fermi liquid behavior in multi-orbital models with Hund coupling Werner et al, arxiv:cond-mat/ H loc = α Un α, n α, + α β,σ U n α,σ n β, σ + α β,σ (U J)n α,σ n β,σ α β J(ψ α, ψ β, ψ β, ψ α, +ψ β, ψ β, ψ α, ψ α, +h.c.) α,σ µn α,σ Bethe lattice with bandwidth 4t, U = U 2J Phase diagram for J = U/6 (left) and self-energy at U/t = 8 (right) n=2.62 n=2.35 n=1.99 n=1.79 n=1.6 U/t Im Σ/t µ/t (ω n /t).5 Urbana, June 8 p.16

19 3-orbital model Non-Fermi liquid behavior in multi-orbital models with Hund coupling Werner et al, arxiv:cond-mat/ H loc = α Un α, n α, + α β,σ U n α,σ n β, σ + α β,σ (U J)n α,σ n β,σ α β J(ψ α, ψ β, ψ β, ψ α, +ψ β, ψ β, ψ α, ψ α, +h.c.) α,σ µn α,σ Transition to a phase with frozen moments Broad quantum critical regime ImΣ ω n σ(ω) 1/ Ω /6 2/6 3/ U/t 1 8 U/t 1 8 Fermi liquid frozen moment µ/t n=2 glass transition 4 2 βt=5 βt=1 Mott insulator (βt=5) n Urbana, June 8 p.17

20 Urbana, June 8 p.18 Conclusions & Outlook Diagrammatic MC simulation of impurity models: 1 weak coupling expansion Weak-coupling method for large impurity clusters "Strong-coupling" method for multi-orbital models perturbation order <k> hybridization expansion metal Mott insulator On-going projects: U/t LDA+DMFT simulation of transition metal oxides and actinide compounds Adaptation of the diagrammatic approach to real-time dynamics (non-equilibrium systems) Job openings: PhD and postdoc position at ETH Zürich