1. INT, 05/18/2011 MC sampling of skeleton Feynman diagrams: Road to solution for interacting fermions/spins? Nikolay Prokofiev, Umass, Amherst Boris Svistunov UMass work done in collaboration with Kris van Houcke UMass, U. Gent Evgeny Kozik ETH Felix Werner UMass, ENS + proof from Nature MIT group: Mrtin Zwierlein, Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek

2. Feynman diagrams have become our everyday’s language. “Particle A scatters off particle B by exchanging a particle C … “ Feynman Diagrams: graphical representation for the high-order perturbation theory

3. Diagrammatic technique: admits partial resummation and self-consistent formulation Calculate irreducible diagrams for,, … to get,, …. from Dyson equations Dyson Equation: Screening: Ladders: (contact potential) More tools: (naturally incorporating Dynamic mean-field theory solutions) Higher “level”: diagrams based on effective objects (ladders), irreducible 3-point vertex …

4. Physics of strongly correlated many-body systems, i.e. no small parameters: Are they useful in higher orders? And if they are, how one can handle billions of skeleton graphs? Feynman Diagrams

5. Skeleton diagrams up to high-order: do they make sense for ? NO Diverge for large even if are convergent for small. Math. Statement: # of skeleton graphs asymptotic series with zero conv. radius (n! beats any power) Dyson: Expansion in powers of g is asymptotic if for some (e.g. complex) g one finds pathological behavior. Electron gas: Bosons: [collapse to infinite density] Asymptotic series for with zero convergence radius

6. Skeleton diagrams up to high-order: do they make sense for ? YES # of graphs is but due to sign-blessing they may compensate each other to accuracy better then leading to finite conv. radius 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 : e.g. the BCS theory answer (lowest-order diagrams) - Regularization techniques are available. Divergent series far outside of convergence radius can be re-summed. From strong coupling theories based on one lowest-order diagram To accurate unbiased theories based on billions of diagrams and limit

7. Define a function such that: Construct sums and extrapolate to get Example: бред какой то Re-summation of divergent series with finite convergence radius. Lindeloef Gauss

8. Diagram order Diagram topology MC update This is NOT: write diagram after diagram, compute its value, sum Configuration space = (diagram order, topology and types of lines, internal variables) Computational complexity is factorial :

9. Resonant Fermions: Universal results in the zero-range,, and thermodynamic limit Unitary gas:.

10. Useful ‘bold’ relations: all ladder diagrams Skeleton graphs based on

12. controls contributing diagram orders resummation and extrapolation for density

13. Unitary gas EOS (full story in previous talks) (in the universal & thermodynamic limit with quantifiable error bars) Goulko, Wingate ‘10 (calculated independently and cross-checked for universality)

14. Mean-field behavior: Critical point from pair distribution function Criticality: from

15. Burovski et. al ’06, Kozik et. al ‘08Goulko & Wingate ‘10

16. Conclusions/perspectives Diag.MC for skeleton graphs works all the way to the critical point Phase diagrams for strongly correlated states can be done, generically Res. Fermions: population imbalance, mass imbalance, etc Fermi-Hubbard model (any filling) Coulomb gas Frustrated magnetism …

17. Cut one line – interpret the rest as self-energy for this line: