  Phase transition in U(n)xU(n) models is investigated for arbitrary flavor
number n. We present a nonperturbative, 3+1 dimensional finite temperature
treatment of obtaining the effective potential, based on a chiral invariant
expansion of the functional renormalization group flows. The obtained tower of
equations is similar but not identical to that of the Dyson-Schwinger hierarchy
and has to be truncated for practical purposes. We investigate the finite
temperature behavior of the system in an expansive set of the parameter space
for n = 2, 3, 4 and also perform a large-n analysis. Our method is capable of
recovering the one-loop beta functions of the coupling constants of the epsilon
expansion; furthermore, it shows direct evidence that regardless of the actual
flavor number, within our approximation, the system undergoes a fluctuation
induced first order phase transition.
