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

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

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

Feynman digrams Positive definite in momentum-imaginary time representation

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

Monte Carlo (Metropolis) cycle: Diagramsuggest a change Accept with probability 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: e.g. Solution: Example: for Frohlich polaron

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

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

M II I I M

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

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

NOT EASY BUTTON