Continuous time QMC methods

Transcription

1 Continuous time QMC methods Matthias Troyer (ETH Zürich) Philipp Werner (Columbia ETHZ) Emanuel Gull (ETHZ Columbia) Andy J. Millis (Columbia) Olivier Parcollet (Paris) Sebastian Fuchs, Thomas Pruschke (Göttingen) Evgeny Kozik, Lode Pollet, Brigitte Surer (ETHZ) Nikolay Prokof ev and Boris Svistunov (UMass Amherst) P. Werner, A. Comanac, L. De Medici, A.J. Millis and M. Troyer, PRL 97, (2006) E. Gull, P. Werner, O. Parcollet and M. Troyer, EPL 82, (2008)

2 Band structure of La CuO Strong correlations and novel phases High temperature superconductors Undoped material: half-filled band and metal according to DFT but antiferromagnetic insulator in experiment! Band structure calculation breaks down! Doped material: a high-temperature superconductor not yet theoretically understood after more than two decades

3 The Mott transition As interactions are increased the conduction band splits into two subbands and the half-filled band is insulating We need to go beyond density functional theory and try to solve a simplified model accurately, the Hubbard model H = t i,j,σ (c i,σ c j,σ + c j,σ c i,σ) +U i n i, n i, strong repulsion Image: Kotliar & Vollhardt, Physics Today (2004)

4 Path integral QMC Use Trotter-Suzuki or a a simple low-order formula gives a mapping to a (d+1)-dimensional classical model imaginary time Z = Tre βh = Tr e MΔτH = Tr e ΔτH i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 ( ) M = Tr 1 ΔτH = i 1 1 ΔτH i 2 i 2 1 ΔτH i 3 i M 1 ΔτH i 1 {(i 1...i M )} space direction ( ) M + O(βΔτ) place particles for Hamiltonians conserving particle number we get world lines partition function of quantum system is sum over classical world lines

5 Why discretize an integral? Deterministic method: Riemann sum b! f (x)dx = b " a N a N ' i =1 # f a + i b " a % $ N & + O(1/N) Monte Carlo sampling draw random samples from the continuum no discretization needed f = " f ( x! )d x! d x!! "! f! 1 M M " i =1 f (! x i )

6 the limit Δτ 0 can be taken in the algorithm [Prokof'ev et al., Pis'ma v Zh.Eks. Teor. Fiz. 64, 853 (1996)] imaginary time i 1 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i 1 The continuous time limit space direction space direction discrete time: store configuration at all time steps continuous time: store times at which configuration changes imaginary time τ 5 τ 6 τ 1 τ 4 τ 2 τ 3

7 Advantages of continuous time No need to extrapolate in time step a single simulation is sufficient no additional errors from extrapolation Less memory and CPU time required Instead of a time step Δτ << t we only have to store changes in the configuration happening at mean distances t Speedup of 1 / Δτ 10 for bosons and 1 / Δτ for fermions Conceptual advantage we directly sample a diagrammatic perturbation expansion

8 Bose-Einstein condensation in cold atomic gases At close to zero temperatures, a macroscopic fraction of all atoms in a Bose gas occupy the same quantum state A diverging occupation of the zero momentum state Momentum distribution function

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10 How do we detect these quantum gases? release the atoms faster atoms fly farther the image reflects the momentum distribution

11 Optical lattices formed by standing waves from three pairs of laser beams realize quantum lattice models of fermions or bosons: the bosonic and fermionic Hubbard models are reliable microscopic models and exact ab initio simulations are possible

12 Ab-initio validation of experiments Ab-initio QMC simulation if ultracold bosonic gases on 8 million lattice site Only 10 hours on a PC Excellent agreement with experiment validates the control achieved in experiments

13 Fermionic continuous time General recipe: Split Hamiltonian into two parts: H = H 1 + H 2 Use interaction representation in which O(τ) = e τh 1 Oe τh 1 Write partition function as time-ordered exponential, expand in powers of H 2 Z = T r e βh 1 T e R β 0 dτh 2(τ) = k diagrammatic QMC β 0 dτ 1... β 0 dτ k ( 1) k k! T r e βh 1 T H 2 (τ 1 )... H 2 (τ k ) Weak-coupling expansion: Rombouts et al., (1999), Rubtsov et al. (2005), Gull et al. (2008) expand in interactions, treat quadratic terms exactly

14 Diagrammatic expansion in interaction too many to draw all p! 2 possible topologies but easy to sum all of them! (about 5000! 2 )

15 Diagrammatic expansion in interaction A.N. Rubtsov & A.I. Lichtenstein, Pis'ma v JETP 80, 67 (2004) A.N. Rubtsov, V.V. Savkin, A.I. Lichtenstein, Phys. Rev. B 72, (2005) 1 3 The sum of all p! 2 diagrams for a given vertex configuration is a determinant squared 2 9 Sign-problem free for attractive interactions U<0 and balanced population of up and down spins

16 Critical temperature of the resonant Fermi gas Phase diagram of dilute Fermi gas weak attraction: BCS of Cooper pairs??? BEC strong attraction: BEC of molecules BCS Universal behavior at resonance (unitary point) where 2-particle bound state crosses lower band edge

17 Previous results for the unitary point 0.5 M Holland, SJJMF Kokkelmans, ML Chiofalo, and R. Walser J Kinhast, A Turlapov, JE Thomas, Q Chen, J Stajic, K Levin V.K. Akkineni, N. Trivedi, D. Ceperley P Nozieres, S Schmitt-Rink XJ Liu, H Ho P. Nikolic, S Sachdev A Bulgac, JE Drut, P Magierski R. Haussmann, W. Rantner S. cerrito, W. Zwerger M Wingate BEC limit

18 V.K. Akkineni, N. Trivedi, D. Ceperley (fixed node) A Bulgac, JE Drut, P Magierski Visual inspection estimate from E(T) A. Sewer et al. T.A. Maier et al.

19 Dynamical Mean Field Theory is an approximative but successful method for describing strongly interacting (repulsive) fermions in high dimensions E. Müller-Hartmann, Z. Phys. B (1989). M. Metzner and D. Vollhard, PRL 62, 324 (1989). A. Georges and G. Kotliar Phys. Rev. B 45, 6479 (1992). A. Georges et al., Rev. Mod. Phys. 68, 13 (1996). solves a few-site problem in the presence of a self-consistent bath provided by the rest of the system with an effective nonlocal action S = β 0 β 0 dτdτ σ c σ(τ)f (τ τ )c σ (τ ) + β 0 dτun (τ)n (τ)

20 Mean-field theory for Ising Model Lattice model (nearest neighbor coupling J, coordination number z) H latt = J i,j S is j J Single site model (m i = S i, h eff = J 0,i m i = zjm) H 0 = h eff S 0 h eff Self-consistency condition m = m 0 H0 ( = tanh(βh eff ) = tanh(βzjm) )

21 Dynamical mean field theory Lattice model H latt = U i n i n i t i,j,σ c iσ c jσ Metzner & Vollhardt, PRL (1989) Georges & Kotliar, PRB (1992) t Quantum impurity model H imp = Un n k,σ (t kc σa bath k,σ + h.c.) + H bath t k

22 Analogy to DFT G. Kotliar et al., Rev. Mod. Phys. 78, (2006) DFT is an (in principle) exact theory based on a functional of the density LDA approximation approximates the exchange-correlation part of the functional by that of a uniform electron gas works well when correlations are weak DMFT is based on an (in principle) exact theory using a functional of the local Green s function The DMFT approximation uses the energy of a quantum impurity problem with the same local Green s function works well near the Mott transition, where physics is local

23 Dynamical mean field theory Self-consistency loop lattice model Metzner & Vollhardt, PRL (1989) Georges & Kotliar, PRB (1992) impurity model t t k G latt self-consistency condition G latt G imp H imp dk 1 iω n +µ k Σ latt Σ latt DMFT approximation Σ latt Σ imp impurity solver G imp, Σ imp Computationally expensive step: solution of the impurity model

24 Continuous time solvers Standard QMC solver was discrete time method with local updates Hirsch-Fye algorithm (PRL 86) New continuous time solvers have been developed recently continuous time expansion in U Rubtsov, Savkin and Lichtenstein, PRB (2005) continuous time expansion in hybridization Werner, Comanac, De Medici, Millis and Troyer, PRL (2006) continuous time version of Hirsch-Fye Gull, Werner, Parcollet and Troyer, EPL (2008) offer substantial speedup can do on a laptop what used to require a supercomputer

25 General recipe: Split Hamiltonian into two parts: H = H 1 + H 2 Use interaction representation in which O(τ) = e τh 1 Oe τh 1 Write partition function as time-ordered exponential, expand in powers of H 2 Z = T r e βh 1 T e R β 0 dτh 2(τ) = k Diagrammatic QMC β 0 dτ 1... β 0 dτ k ( 1) k k! T r e βh 1 T H 2 (τ 1 )... H 2 (τ k ) Weak-coupling expansion: Rombouts et al., (1999), Rubtsov et al. (2005), Gull et al. (2008) expand in interactions, treat quadratic terms exactly Hybridization expansion: Werner et al., (2006), Werner & Millis (2006), Haule (2007) expand in hybridizations, treat local terms exactly

26 CT-AUX auxiliary field QMC Rombouts et al., PRL (1999) Gull et al., EPL (2008) Impurity model given by H = H 0 + H U H 0 = K/β (µ U/2)(n + n ) + H hyb + H bath H U = U(n n (n + n )/2) K/β Expand partition function into powers of the interaction term Z = ( 1) k dτ 1... dτ k T r T e βh 0 H U (τ 1 )... H U (τ k ) k! k Decouple the interaction terms using Rombouts et al., PRL (1999) H U = K 2β s=±1 e γs(n n ), cosh(γ) = 1 + βu 2K then integrate over fermionic Gaussian integrals

27 CT-AUX auxiliary field QMC Rombouts et al., PRL (1999) Gull et al., EPL (2008) Configuration space: all possible time-ordered spin configurations Weight: w(τ 1, s 1 ;... ; τ k, s k ) = Kdτ 2β k N 1 σ = e Γ σ G 0σ e Γ σ 1 σ det N 1 σ ({τ i, s i }) e Γ σ = diag(e γσs 1,..., e γσs k ) Monte Carlo updates: random insertion/removal of a spin 10 x smaller matrix than in discrete time Hirsch Fye algorithm Closely related to weak-coupling solver by Rubtsvov (2005)

28 Cluster versions of DMFT: DCA DMFT: momentum independent self energy cluster DMFT methods: approximate momentum-dependence of the self-energy Σ(p, ω) = a φ a (p)σ a (ω) Dynamical cluster approximation (DCA): ``tiling of the Brillouin zone

29 Convergence of DCA For small U/t we can also directly sample a fermionic diagrammatic perturbative expansion (DiagMC) and compare: E.Kozik et al EPL 90, (2010) DCA results converge with cluster size! Continuous time solvers now allow clusters with up to 144 k-points reliable thermodynamics for Hubbard models and cold atmic gases s r T =0.4 t T =0.5 t T =0.6 t T =1t T =2t T =4t S. Fuchs et al arxiv:

30 M-I transition in the 2D Hubbard model Doping the insulator produces electron/hole pockets 8-site cluster has a ``tile at the expected position of the pockets 8-site DCA-result at U/t=7: first 8% of dopants go into the B sector t=40, B t=40, C t=20, B t=20, C n B C B C !/t

31 M-I transition in the 2D Hubbard model Doping the insulator produces electron/hole pockets 8-site cluster has a ``tile at the expected position of the pockets 8-site DCA-result at U/t=7: first 8% of dopants go into the B sector Assuming an ellipsoidal shape for the pocket, we can estimate the aspect ratio b a 1 10 a b

32 CT-HYB: Hybridization expansion Werner et al., PRL (2006) Werner & Millis, PRB (2006) Haule, PRB (2007) Impurity model given by H = H loc + H bath + H hyb H loc = Un n µ(n + n ) H hyb = p,σ t σ p c σa p,σ + h.c. Expand partition function into powers of the hybridization term Z = 1 dτ 1... dτ 2k T r T e β(h loc+h bath ) H hyb (τ 1 )... H hyb (τ 2k ) 2k! k Trace over bath degrees of freedom yields determinant of hybridization functions F T r bath [...] = σ det Mσ 1, Mσ 1 (i, j) = F σ (τ (c) i τ (c ) j ) t σ p 2 F σ ( iω n ) = p iω n p

33 Hybridization expansion Werner et al., PRL (2006) Werner & Millis, PRB (2006) Haule, PRB (2007) Monte Carlo configurations consist of segments for spin up and down Monte Carlo updates: random insertion/removal of (anti-)segments Weight of a segment configuration: w τ σ(c) 1, τ σ(c ) 1 ;... ; τ σ(c) k σ, τ σ(c ) k σ = e Ul overlap +µ(l +l ) T r imp [...] Determinant of a k x k matrix resums k! diagrams F det σ (τ (c) 1 τ (c ) 1 ) F σ (τ (c) 1 τ (c ) 2 ) F σ (τ (c) 2 τ (c ) 1 ) F σ (τ (c) 2 τ (c ) 2 ) Eliminates sign problem σ det Mσ 1 dτ 2k σ T r bath [...]

34 Solver Comparison - Matrix Sizes Weak CT-AUX Coupling Algorithm Hybridization CT-HYB Expansion Hirsch-Fye Matrix Size βt U/t=4

35 Matrix Sizes - coupling depenence 100 CT-AUX CT-HYB Weak Coupling Algorithm Hybridization Expansion Matrix Size Typical region of interest: U/t βt=30

36 Generalizations Multi-orbital and cluster problems with full Coulomb interaction P. Werner and A.J. Millis, Phys. Rev. B 74, (2006) K. Haule, Phys. Rev. B 75, (2007) Phonons can be added at no cost P. Werner and A.J. Millis, Phys. Rev. Lett. 99, (2007) Frequency dependent interactions P. Werner and A.J. Millis, Phys. Rev. Lett. 104, (2010)

37 Transition metal oxides Full Coulomb interaction can be used instead of just density-density Statistics of atomic states accessible Example: NiO density-density interactions fu" Coulomb interaction Probability sector statistics S = Sectors Probability sector statistics S = 1 Sectors Spin rotation symmetry preserved!

38 Second example: CoO 0.3 sector statistics 0.25 Probability S=3/2 quadruplet Sectors

39 Spin freezing transition in a 3-orbital model 1 site, 3 degenerate orbitals (semi-circular DOS, bandwidth 4t) H loc = α,σ µn α,σ + α Un α, n α, + α>β,σ U n α,σ n 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.) Captures essential physics of SrRuO3 Similar models for other transition metal oxides, actinides, Fe based superconductors,...

40 Spin freezing transition in a 3-orbital model Phase diagram for U = U = 2J, J/U = 1/6, βt = U/t 8 U/t µ/t Mott insulator ( t=50) n Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons

41 Spin freezing transition in a 3-orbital model U/t Phase diagram 0.25 for U = U = 2J, J/U = 1/6, βt = <n 0 (0)n 1 ( )>, <S z (0)S z ( )> µ/t glass transition Mott insulating ``lobes with 1, 2, 3, (4, 5) electrons U/t Fermi liquid glass transition Mott insulator ( t=50) t n=1.21 n=1.75 n=2.23 n=2.62 n=2.97 n frozen moment In the metallic phase: transition from Fermi liquid to ``spin glass

42 Spin freezing transition in a 3-orbital model U/t Phase diagram for Im /t U = U = 2J, J/U = 1/6, βt = 50 n=2.62 n=2.35 n=1.99 n=1.79 n=1.60 U/t Fermi liquid frozen moment µ/t glass transition Critical exponents 0 associated 0.5 with the 1 transition 1.5 can be seen 2 in a wide quantum critical regime ( n /t) 0.5 e. g. non Fermi-liquid self-energy glass transition Mott insulator ( t=50) ImΣ/t (iω n /t) α, α 0.5 n

43 From effective models to real materials Single-orbital DMFT can now be solved on a laptop We can consider more complex problems, such as strongly correlated metals with d and f orbitals interfaces... The current challenge: Seven f-orbitals with full Coulomb interaction Low temperatures Frequency dependent interactions

44 Continuous time QMC methods (CT-QMC) Diagrammatic continuous-time QMC methods enable huge systems of millions of sites for spins and bosons enable hundreds of attractive fermions but sign problem for repulsively interacting fermions CT-QMC for DMFT impurity solvers enable efficient DMFT simulations of fermionic lattice models Weak-coupling solver CT-AUX ideal for large clusters Hybridization expansion solver CT-HYB allows to treat multiorbital models with full Coulomb interactions Applications Accurate simulations of single-band Hubbard models by DCA LDA+DMFT studies of transition metal oxides, f-electron materials, superconductivity in multi-band models,...