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

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

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

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

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

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

Further advantages of the diagrammatic technique Calculate irreducible diagrams for,, … to get,, …. from Dyson equations Dyson Equation: Make the entire scheme self-consistent, i.e. all internal lines in,, … are “bold” = skeleton graphs or 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: quasiparticle E.g. Electrons in semiconducting crystals (electron-phonon polarons) e e electron phonons el.-ph. interaction

Fermi-polaron problem: Universal physics ( independent)

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. Sign problem! self-consistent and self-consistent only = Confirmed by ENS

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

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

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

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

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

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

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

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

Universal function F 0 Single shot data Andre Schirotzek, Ariel Sommer, Mark Ku, and Martin Zwierlein

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.

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

term order different terms of of the same order Integration variables Contribution to the answer Monte Carlo (Metropolis-Rosenbluth-Teller) cycle: Diagramsuggest a change Accept with probability Collect statistics: sign problem and potential trouble!, but …

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

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