Dynamical mean-field theory for bosons

We discuss the recently developed bosonic dynamical mean-field (B-DMFT) framework, which maps a bosonic lattice model onto the selfconsistent solution of a bosonic impurity model with coupling to a reservoir of normal and condensed bosons. The effective impurity action is derived in several ways: (i) as an approximation to the kinetic energy functional of the lattice problem, (ii) using a cavity approach, and (iii) by using an effective medium approach based on adding a one-loop correction to the selfconsistently defined condensate. To solve the impurity problem, we use a continuous-time Monte Carlo algorithm based on a sampling of a perturbation expansion in the hybridization functions and the condensate wave function. As applications of the formalism we present finite temperature B-DMFT phase diagrams for the bosonic Hubbard model on a 3d cubic and 2d square lattice, the condensate order parameter as a function of chemical potential, critical exponents for the condensate, the approach to the weakly interacting Bose gas regime for weak repulsions, and the kinetic energy as a function of temperature.