DIAGRAMMATIC MONTE CARLO From polarons to pathintegrals with

DIAGRAMMATIC MONTE CARLO: From polarons to path-integrals (with worm, of course) Nikolay Prokofiev, Umass, Amherst Many thanks to collaborators on major algorithm developments Boris Svistunov, Umass, Amherst Igor Tupitsyn, PITP Vladimir Kashurnikov, MEPI, Moscow Evgeni Burovski, Umass, Amherst Andrei Mishchenko, AIST, Tsukuba NASA Les Houches, June 2006, Lecture 2

Let … Diagram order Same-order diagrams Contribution to the answer or the diagram weight (positive definite, please) Integration variables ENTER

Polaron problem: electron phonons el. -ph. interaction Green function: + Sum of all Feynman diagrams + …

Feynman digrams Positive definite in momentum-imaginary time representation

Diagrams for: there also diagrams for optical conductivity, etc.

Monte Carlo (Metropolis) cycle: Accept with probability Diagram suggest a change Same order diagrams: Business as usual Updating the diagram order: Ooops

Balance Equation: If the desired probability density distribution of diagrams in the stochastic sum is (in most cases it is the same as the diagram weight ) then the MC process of updating diagrams should be stationary with respect to (equilibrium condition) Flux out of Flux to Is the probability density of “making” new variables, if any Detailed Balance: solve it for each pair of updates separately.

Equation: Solution: Example: e. g. for Frohlich polaron

imaginary time Lattice path-integrals for bosons and spins are “diagrams” of closed loops! + +

Diagrams for imaginary time Diagrams for lattice site The rest is conventional worm algorithm in continuous time

I I M M I I

Path-integrals in continuous space are “diagrams” of closed loops too! P 2 1 P

Not necessarily for closed loops! Feynman (space-time) diagrams for fermions with contact interaction (attractive) Pair correlation function connect vortexes with sum over all possible and connections Rubtsov ’ 03 Burovski et al. ’ 03 NOT EASY BUTTON

NOT EASY BUTTON