Complex $\varphi^4$ Theory at Finite Temperature and Density via Extended Mean Field Theory

We review the Extended Mean Field (EMFT) approximation and apply it to complex, scalar $\varphi^4$ theory on the lattice. We determine the ($T,\mu$) phase diagram and study the critical properties of the Bose condensation transition, at zero and finite temperature. In both cases we obtain results which agree very well with recent Monte Carlo studies. Within our approximation we can reach the thermodynamic limit and can thus obtain results at lattice spacings unreachable by present Monte Carlo simulations. We find that our approximation reproduces accurately many phenomena of the model, like the "Silver Blaze" behavior at zero temperature and dimensional reduction at finite temperature.