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DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS Nikolay Prokofiev, Umass, Amherst work done in collaboration with Boris Svistunov Kris van Houcke Felix Werner UMass UMass UMass Univ. Gent Evgeny Kozik Lode Pollet Emanuel Gull Matthias Troyer Harvard ETH ETH Columbia PITP, Dec. 4, 2009 Outline Feynman Diagrams: An acceptable solution to the sign problem? (i) proven case of fermi-polarons Many-body implementation for (ii) the Fermi-Hubbard model in the Fermi liquid regime and (iii) the resonant Fermi gas H t Fermi-Hubbard model: ij , t ai a †j U ni ni ni i , i U momentum representation: H ( k )ak† ak k , kpq , ' U q ak† q a †p q ' a p ' ak ' Elements of the diagrammatic expansion: fifth order term: G (k , 2 1 ) Tr a,k ( 2 ) a,k ( 1 ) e ( 0) , U q ( 1 2k) H0 /T G(0) (k , 3 4 ) 1 k U q ( 1 2 ) 1 2 k q G(0) ( p, 5 6 ) q p 2 pq G, ( p, 2 1 ) Tr a, p ( 2 ) a, p ( 1 ) e H / T The full Green’s Function: G0 ( p, ) = + G0 ( p, ) + + + + + + +… Why not sample the diagrams by Monte Carlo? Configuration space = (diagram order, topology and types of lines, internal variables) Diagram order {qi , i , pi } Diagram topology This is NOT: write diagram after diagram, compute its value, sum 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 Further advantages of the diagrammatic technique Calculate irreducible diagrams for , , … to get G , U , …. from Dyson equations G ( p, ) + Dyson Equation: G0 ( p, ) + ( p,1 2 ) + (0) + + ... or U U + + G (0) U + GG G (0)G (0) T G Make the entire scheme self-consistent, i.e. all internal lines in , , … are “bold” = skeleton graphs Every analytic solution or insight into the problem can be “built in” -Series expansion in U is often divergent, or, even worse, asymptotic. Does it makes sense to have more terms calculated? Yes! (i) Unbiased resummation techniques (ii) there are interesting cases with convergent series (Hubbard model at finite-T, resonant fermions) - It is an unsolved problem whether skeleton diagrams form asymptotic or convergent series Good news: BCS theory is non-analytic at U 0 , and yet this is accounted for within the lowest-order diagrams! Polaron problem: H H particle H environment H coupling quasiparticle E ( p 0), m* , G( p, t ), ... E.g. Electrons in semiconducting crystals (electron-phonon polarons) H ( p)a p a p electron p e e ( p)(b b q q 1/ 2) phonons q V a q pq p q a p bq h.c. el.-ph. interaction 2 p Fermi-polaron problem: H H Fermi sea V (r r ') n(r ') dr ' 2m V (r ) (r ) k F aS ~ 1 r0 r k F r0 0 Universal physics ( V ( r ) independent) n1/ 3 ~ k F / Examples: Electron-phonon polarons (e.g. Frohlich model) = particle in the bosonic environment. Too “simple”, no sign problem, N 10 2 G Fermi –polarons (polarized resonant Fermi gas = particle in the fermionic environment. G self-consistent G and ( k F a ) = self-consistent G only 0.615 Confirmed by ENS Sign problem! N max 11 “Exact” solution: Polaron Molecule sure, press Updates: Enter 2D Fermi-Hubbard model in the Fermi-liquid regime U /t 4 / t 1.5 n 0.6 T / t 0.025 EF /100 Fermi –liquid regime was reached T Bare series convergence: E (T ) E (0) ( F F' EF ) yes, after order 4 6 2 n(T ) n(0) F' 2T 2 6 2 2D Fermi-Hubbard model in the Fermi-liquid regime 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 p y U /t 4 ( nU ) / t 1.5 n 1.35 3D Fermi-Hubbard model in the Fermi-liquid regime T / t 0.1 EF / 50 DiagMC vs high-T expansion in t/T (up to 10-th order) 10 8 8 Unbiased high-T expansion in t/T fails at T/t>1 before the FL regime sets in 2 10 3D Resonant Fermi gas at unitarity (k F a ) : Bridging the gap between different limits T / EF 0.3 TC / EF 0.152(7) S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon ENS data DiagMC P / P0 Seatlle’s Det. MC e /T Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein 0 0 2.5 2.0 Optical Density 0 0 0 1.5 DiagMC fit 1.0 0.5 0.0 0 0 50 100 150 Pixel [1.2 m] 0 500 600 700 800 900 1000 200 250 300 Andre Schirotzek, Ariel Sommer, MarkF Ku, and Martin Zwierlein Universal function 0 Single shot data 60 50 3 T F0 n 40 30 20 10 0 -3 -2 -1 0 1 2 3 Conclusions/perspectives • Bold-line Diagrammatic series can be efficiently simulated. - combine analytic and numeric tools - thermodynamic-limit results - sign-problem tolerant (small configuration space) • Work in progress: bold-line implementation for the Hubbard model and the resonant Fermi-gas ( G version) and the continuous electron gas or jellium model (screening version). • Next step: Effects of disorder, broken symmetry phases, additional correlation functions, etc. H ( k )a ak k , † k U a q kpq , † k q a † p q ' a p ' ak ' ' U q ( 1 2 ) G(0) (k , 3 4 ) G(0) ( p, 5 6 ) Configuration space = (diagram order, topology and types of lines, internal variables) A y d x1d x 2 n 0 d x n Wn ; x1 , x 2 , x n , y W Integration variables term order Contribution to the answer different terms of of the same order Monte Carlo (Metropolis-Rosenbluth-Teller) cycle: Accept with probability Diagram Collect statistics: suggest a change Racc W ' 1 W Pv v ' (new {x}) Acounter y Acounter y sign( ) sign problem and potential trouble!, but … Fermi-Hubbard model: H t ij , ai a †j U ni ni ni i , i G ( p, ) U (q) G ( p, ) Self-consistency in the form of Dyson, RPA G G (0) Extrapolate to the + N limit. U U + Large classical system state described by #N numbers Large quantum system state described by Possible to simulate “as is” (e.g. molecular dynamics) e# N numbers Impossible to simulate “as is” approximate Math. mapping for quantum statistical predictions Feynman Diagrams: