Quark confinement in a finite-temperature $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ gauge theory is formulated as the realization of a global $Z_N$ symmetry. Spontaneous breakdown corresponds to a transition to a nonconfining, plasma phase. The free energy of a single quark is an order parameter which probes the phase structure, and it may be calculated in the Euclidean theory in terms of a "Wilson line" running the length of the system along the (periodic) time axis. We present results of a Monte Carlo calculation in the SU(2) lattice theory which confirm the transition at a critical temperature computed in terms of the zero-temperature string tension; data for the quark-antiquark potential are presented as well. We discuss the implications of the finite-temperature transition for efforts to calculate zero-temperature quantities on finite-size lattices. Finally, we note that restoration of $Z_N$ symmetry as the temperature is lowered may be understood as a condensation of instantons and other topological objects.

• Received 2 March 1981

DOI:https://doi.org/10.1103/PhysRevD.24.450