Phase structure and the effective potential at fixed charge

We calculate exactly the one-loop finite-temperature effective potential for a self-interacting theory with a fixed U(1) charge. We analyze in detail two limits of this exact expression: the very-low- and very-high-temperature regimes. In the former we reproduce many standard results of the nonrelativistic Bose gas. In the high-temperature limit, we describe the effects, which can be striking, of nonzero charge on vacuum structure and critical temperatures. Generally we find that in order to accommodate the fixed charge a system tends to remain longer in a spontaneously broken phase.