MC sampling of skeleton Feynman diagrams Road to

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

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

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 …

Feynman Diagrams 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?

Skeleton diagrams up to high-order: do they make sense for ? NO Diverge for large even if are convergent for small. 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 Math. Statement: # of skeleton graphs asymptotic series with zero conv. radius (n! beats any power)

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

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

Configuration space = (diagram order, topology and types of lines, internal variables) Diagram order te pda u C M e t da up Diagram topology This is NOT: write diagram after diagram, compute its value, sum Computational complexity is factorial :

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

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

resummation and extrapolation for density controls contributing diagram orders

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)

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

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

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 …

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