Wandering randomly among Feynman diagrams

Feynman diagrams is the standard tool of theoretical physics which is usually associated with the analytic approach (never mind that the final integrals are done on computers).  I will argue that diagrammatic expansions are also an ideal numerical tool with enormous and yet to be explored potential for solving interacting many-body systems. To introduce the Diagrammatic Monte Carlo (DiagMC) approach I will first explain how DiagMC  works for polaron type models (particle coupled to its environment), starting with the standard electron-phonon polaron  (Bose environment) and proceeding with the properties of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles (Fermi environment). The current scheme is based on direct simulation of Feynman diagrams for the proper self-energy up to some high order.  Though the original series based on bare propagators are sign-alternating and often divergent one can determine the answer behind them by using two strategies (separately or together): (i) using proper series re-summation techniques, and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e. making the entire scheme self-consistent.  In the last part of the talk I will argue that the DiagMC approach is a generic sign-problem tolerant method for exact numerical solution of the interacting many-body Hamiltonians and present our first results for the Fermi-Hubbard model.