Vacuum condensate in (2+1)-dimensional gauge theories

The one-loop effective potential for a background field in (2+1)-dimensional $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theory is calculated at arbitrary temperature. The perturbative vacuum is found to be unstable against spontaneous formation of a gauge-field condensate at zero temperature, corresponding to a nontrivial minimum of the effective potential. The condensate is found to "evaporate" at a first-order phase transition—above a critical temperature $T_c$, the minimum of the free energy lies at a zero background field. The condensate also vanishes for a sufficiently large number of massless fermions. These properties of the gauge-field effective potential are shown to provide a mean-field description of interacting charges in 2+1 dimensions that exhibits linear confinement (to be compared with a logarithmic interaction in the purely classical theory), and a first-order ("deconfining") phase transition. Similar qualitative features have been found in the one-loop effective potential for (3+1)-dimensional gauge theories.