Diagrammatic Monte Carlo Method for the Fermi Hubbard Model Youjin Deng Univ. of Sci. & Tech. of China (USTC) Adjunct: Umass, Amherst Nikolay Prokof’ev UMass Boris Svistunov UMass ANZMAP 2012, Lorne Outline • • • • Fermi-Hubbard Model Diagrammatic Monte Carlo sampling Preliminary results Discussion Fermi-Hubbard model Hamiltonian H t ij , ai a †j U ni ni ni i , i momentum representation: H ( k )ak† ak k , Rich Physics: U kpq , q ak†q a †p q ' a p 'ak ' Ferromagnetism Anti-ferromagnetism Metal-insulator transition Superconductivity ? Many important questions still remain open. t U Feynman’s diagrammatic expansion Quantity to be calculated: G, ( p, 2 1 ) Tr a, p ( 2 ) a, p ( 1 ) e H / T Feynman diagrammatic expansion: The full Green’s function: The bare Green’s function : G(0,) (k , 2 1 ) Tr a,k ( 2 ) a,k ( 1 ) e H 0 / T 1 k 2 k The bare interaction vertex : 1 k q q U q ( 1 2 ) p 2 pq Uq A fifth order example: G(0) (k , 2 3 ) G(0) ( p, 4 5 ) Full Green’s function is expanded as : G0 ( p, ) = + G0 ( p, ) + + + + + + +… Boldification: Calculate irreducible diagrams for to get G ( p, ) Dyson Equation G0 ( p, ) G ( p,1 2 ) + G G0 G0 G : Calculate irreducible diagrams for The bare Ladder The bold Ladder + + ... + to get 0 U U 0 0 : 0 0 0 + U : 0 + 0 0 Two-line irreducible Diagrams: Self-consistent iteration G, Diagrammatic expansion Dyson’s equation , Monte Carlo sampling Diagrammatic expansion Why not sample the diagrams by Monte Carlo? Configuration space = (diagram order, topology and types of lines, internal variables) Standard Monte Carlo setup: - configuration space - each cnf. has a weight factor Wcnf A W W cnf - quantity of interest A cnf cnf cnf cnf Monte Carlo MC Acnf cnf configurations generated from the prob. distribution Wcnf Diagram order {qi , i , pi } Diagram topology This is NOT: write diagram after diagram, compute its value, sum Preliminary results 2D Fermi-Hubbard model in the Fermi-liquid regime U /t 4 / t 1.5 n 0.6 T / t 0.025 EF /100 N: cutoff for diagram order Series converge fast Fermi –liquid regime was reached E (T ) E (0) ( F F' EF ) n(T ) n(0) F' 2T 2 6 2T 2 6 U /t 4 / t 3.1 n 1.2 T / t 0.4 EF /10 Comparing DiagMC with cluster DMFT (DCA implementation) ! 2D Fermi-Hubbard model in the Fermi-liquid regime U /t 4 / t 3.1 n 1.2 T / t 0.4 EF /10 Momentum dependence of self-energy 0 T , p along px py Discussion • Absence of large parameter The ladder interaction: + U (t ) (t ) Trick to suppress statistical fluctuation 0 1 + Define a “fake” function: + • Does the general idea work? Skeleton diagrams up to high-order: do they make sense for g 1 ? NO Diverge for large g even if are convergent for small g. Dyson: Expansion in powers of g is asymptotic if for some (e.g. complex) g one finds pathological behavior. Electron gas: Bosons: Math. Statement: # of skeleton graphs 2n n3/2 n ! asymptotic series with zero conv. radius (n! beats any power) e ie U U [collapse to infinite density] AN g 1 Asymptotic series for with zero convergence radius 1/ N Skeleton diagrams up to high-order: do they make sense for g 1 ? YES Divergent series outside of finite convergence radius can be re-summed. Dyson: - Does not apply to the resonant Fermi gas and the Fermi-Hubbard model at finite T. - not known if it applies to skeleton graphs which are NOT series in bare coupling g: recall the BCS answer (one lowest-order diagram) # of graphs is 2n n3/2 n ! but due to sign-blessing they may compensate each other to accuracy better then 1/ n! leading to finite conv. radius e 1/ g - Regularization techniques From strong coupling theories based on one lowest-order diagram To accurate unbiased theories based on millions of diagrams and limit N • Proven examples Resonant Fermi gas: Nature Phys. 8, 366 (2012) Universal results in the zero-range, kF r0 0, and thermodynamic limit Square and Triangular lattice spin-1/2 Heisenberg model test: arXiv:1211.3631 Square lattice (“exact”=lattice PIMC) T J TMF Triangular lattice (ED=exact diagonalization) T 1.25J • Computational complexity Sign-problem Variational methods Determinant MC + “solves” ni ni 1 case + universal - often reliable only at T=0 - systematic errors - finite-size extrapolation - CPU expensive - not universal - finite-size extrapolation Cluster DMFT /MC DCA methods Diagrammatic + universal - diagram-order cluster size extrapolation extrapolation Computational complexity Is exponential : exp{# } Cluster DMFT F T D L linear size Diagrammatic MC N diagram order for irreducible diagrams Thank You! Key elements of DiagMC resummation technique Example: A cn 3 9 / 2 9 81/ 4 ... ? n 0 Define a function such that: f n, N f n, N Nb 1 for n N 1 f n, N e f n, N e n ln( n ) f n , N 0 for n N Na n Construct sums n2 / N AN AN cn f n , N and extrapolate Nlim to get A n 0 AN ln 4 1/ N (Gauss) (Lindeloef) Key elements of DiagMC self-consistent formulation Calculate irreducible diagrams for , , … to getG, U , …. from Dyson equations G ( p, ) Dyson Equation: + G0 ( p, ) ( p,1 2 ) + Screening: + + ... + U U G Irreducible 3-point vertex: 3 1 3 More tools: (naturally incorporating Dynamic mean-field theory solutions) Ladders: (contact potential) (0) + U U What is DiagMC = + + + + + + MC sampling Feyman Diagrammatic series: • Use MC to do integration • Use MC to sample diagrams of different order and/or different topology What is the purpose? • Solve strongly correlated quantum system(Fermion, spin and Boson, Popov-Fedotov trick) +…