$Z_N$ symmetry in $SU(N)$ gauge theories

We study $Z_N$ symmetry in $SU(N)$ gauge theories in the presence of matter fields in the fundamental representation, by restricting the lattice partition function integration to matter fields which are uniform in spatial directions and gauge fields with vanishing spatial components. In this approximation the gauge matter field interaction effectively reduces to a 1-dimensional gauged chain. This makes analytical calculations of the matter field contribution to the Polyakov loop free energy possible. We show that in the limit of large number of temporal sites the explicit breaking of $Z_N$ symmetry in this free energy vanishes, driven by dominance of the density of states. We argue that the spatial links as well as the spatial modes of the matter fields determine the boundaries separating regions where $Z_N$ symmetry is realised from rest of the phase diagram.