Phase transitions in $Z_N$ gauge theory and twisted $Z_N$ topological phases

We find a series of non-Abelian topological phases that are separated from the deconfined phase of $Z_N$ gauge theory by a continuous quantum phase transition. These non-Abelian states, which we refer to as the “twisted” $Z_N$ states, are described by a recently studied $U\left(1\right)\times U\left(1\right)⋊Z_2$ Chern-Simons (CS) field theory. The $U\left(1\right)\times U\left(1\right)⋊Z_2$ CS theory provides a way of gauging the global $Z_2$ electric-magnetic symmetry of the Abelian $Z_N$ phases, yielding the twisted $Z_N$ states. We introduce a parton construction to describe the Abelian $Z_N$ phases in terms of integer quantum Hall states, which then allows us to obtain the non-Abelian states from a theory of $Z_2$ fractionalization. The non-Abelian twisted $Z_N$ states do not have topologically protected gapless edge modes and, for $N>2$, break time-reversal symmetry.