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Nikolay Prokofiev, Umass, Amherst

DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS. Nikolay Prokofiev, Umass, Amherst. work done in collaboration with . Lode Pollet Harvard. Matthias Troyer ETH. Evgeny Kozik ETH. Emanuel Gull Columbia. Boris Svistunov UMass. Kris van Houcke UMass Univ. Gent. Felix Werner

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Nikolay Prokofiev, Umass, Amherst

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  1. DIAGRAMMATIC MONTE CARLO FOR CORRELATED FERMIONS Nikolay Prokofiev, Umass, Amherst work done in collaboration with Lode Pollet Harvard Matthias Troyer ETH EvgenyKozik ETH Emanuel Gull Columbia Boris Svistunov UMass Kris van Houcke UMass Univ. Gent Felix Werner UMass PITP, Dec. 4, 2009

  2. 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

  3. fifth order term: momentum representation: Elements of the diagrammatic expansion: Fermi-Hubbard model:

  4. = + + + + … + + + + Why not sample the diagrams by Monte Carlo? The full Green’s Function: Configuration space = (diagram order, topology and types of lines, internal variables)

  5. Diagram order MC update MC update MC update Diagram topology This is NOT: write diagram after diagram, compute its value, sum

  6. Sign-problem Variational methods Determinant MC Cluster DMFT / DCA methods Diagrammatic MC + universal - often reliable only at T=0 - systematic errors - finite-size extrapolation + universal - diagram-order extrapolation + universal - cluster size extrapolation + “solves” case - CPU expensive - not universal - finite-size extrapolation Computational complexity Is exponential : Cluster DMFT Diagrammatic MC for irreducible diagrams diagram order linear size

  7. Further advantages of the diagrammatic technique Calculate irreducible diagrams for , , … to get , , …. from Dyson equations Dyson Equation: or 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”

  8. 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!

  9. quasiparticle Polaron problem: E.g. Electrons in semiconducting crystals (electron-phonon polarons) electron phonons e e el.-ph. interaction

  10. Fermi-polaron problem: Universal physics ( independent)

  11. Examples: Electron-phonon polarons (e.g. Frohlich model) = particle in the bosonic environment. Too “simple”, no sign problem, Fermi –polarons (polarized resonant Fermi gas = particle in the fermionic environment. = self-consistent and self-consistent only Sign problem! Confirmed by ENS

  12. “Exact” solution: Polaron Molecule Enter sure, press Updates:

  13. 2D Fermi-Hubbard model in the Fermi-liquid regime Fermi –liquid regime was reached Bare series convergence: yes, after order 4

  14. 2D Fermi-Hubbard model in the Fermi-liquid regime Comparing DiagMC with cluster DMFT (DCA implementation) !

  15. 2D Fermi-Hubbard model in the Fermi-liquid regime Momentum dependence of self-energy along

  16. 3D Fermi-Hubbard model in the Fermi-liquid regime DiagMCvs high-T expansion in t/T (up to 10-th order) 10 8 8 2 Unbiased high-T expansion in t/T fails at T/t>1 before the FL regime sets in 10

  17. 3D Resonant Fermi gas at unitarity : Bridging the gap between different limits

  18. S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon ENS data DiagMC Seatlle’s Det. MC

  19. Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein DiagMC fit

  20. Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein Universal function F0 Single shot data

  21. 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 ( version) and the continuous • electron gas or jellium model (screening version). • Next step: Effects of disorder, broken symmetry phases, additional • correlation functions, etc.

  22. Configuration space = (diagram order, topology and types of lines, internal variables)

  23. 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 suggest a change Collect statistics: sign problem and potential trouble!, but …

  24. Fermi-Hubbard model: Self-consistency in the form of Dyson, RPA Extrapolate to the limit.

  25. Large quantum system Large classical system state described by numbers state described by numbers Possible to simulate “as is” (e.g. molecular dynamics) Impossible to simulate “as is” Math. mapping for quantum statistical predictions approximate Feynman Diagrams:

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