Nikolay Prokofiev, Umass, Amherst

1. DIAGRAMMATIC MONTE CARLO: WHAT HAPPENS TO THE SIGN-PROBLEM Nikolay Prokofiev, Umass, Amherst work done in collaboration with Matthias Troyer ETH, Zurich Lode Pollet Harvard EvgenyKozik ETH, Zurich Kris van Houcke UMass, Amherst Univ. Gent Boris Svistunov UMass, Amherst Emanuel Gull Columbia RPMBT15, Ohio 2009

2. Outline Feynman Diagrams vsIsing spins: Is there any difference from the Monte Carlo perspective? Acceptable solution to the sign problem? Yes! (so far …) Polarons in Fermi systems Many-body implementation for (i) the Fermi-Hubbard model in the Fermi liquid regime and (ii) the resonant Fermi gas

3. Feynman Diagrams: graphical representation for the high-order perturbation theory explicit graphical representation for all terms and easy rules to convert graphs to math Diagrammatic technique: explicit summation of geometric series “on-the-go” with self-consistent re-formulation of the diagrams

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

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

6. Standard Monte Carlo setup: (depends on the model and it’s representation) - configuration space - each cnf. has a weight factor - quantity of interest Monte Carlo configurations generated from the prob. distribution

7. 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

8. 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”

9. 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/divergent or convergent series • Good news: BCS theory is non-analytic at U 0 , • and yet this is accounted for within • the lowest-order diagrams!

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

11. Fermi-polaron problem: particle dressed by interactions with the Fermi sea: cold Fermi gases with population strong imbalance, orthogonalitycatastrophe, X-ray singularities, heavy fermions, quantum diffusion in metals, ions in He-3, etc. Extra fermion BEC BCS Universal physics ( independent) Non-interacting Fermi sea Bosonicquasiparticle (molecule) Fermionicquasiparticle (polaron)

12. 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!

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

14. 3D Resonant Fermi gas at unitarity : Bridging the gap between different limits

15. 2D Fermi-Hubbard model in the Fermi-liquid regime Fermi –liquid regime was reached Bare series convergence: yes, after order 4

16. 2D Fermi-Hubbard model in the Fermi-liquid regime Comparing DiagMC with cluster DMFT (DCA implementation) !

17. 2D Fermi-Hubbard model in the Fermi-liquid regime Momentum dependence of self-energy along

18. 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

19. 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.

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

22. 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 …

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

24. 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: