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We review the Extended Mean Field Theory (EMFT) approximation and apply it tocomplex, scalar $\phi^4$-theory on the lattice. We study the criticalproperties of the Bose condensation driven by a nonzero chemical potential$\mu$ at both zero and nonzero temperature and determine the $(T,\mu)$ phasediagram. The results are in very good agreement with recent Monte Carlo datafor all parameter values considered. EMFT can be formulated directly in thethermodynamic limit which allows us to study lattice spacings for which MonteCarlo studies are not feasible with present techniques. We find that the EMFTapproximation accurately reproduces many known phenomena of the exact solution,like the "Silver Blaze" behavior at zero temperature and dimensional reductionat finite temperature.

Extended Mean Field study of complex $\phi^4$-theory at finite density and temperature