Nematicphases, breaking spontaneously rotational symmetry, provide for ubiquitously observed states of matter in both classical and quantum systems. These nematic states may be further classified by their $N$-fold rotational invariance described by cyclic groups $C_N$ in 2 + 1D. Starting from the space groups of underlying 2D crystals, we present a general classification scheme incorporating $C_N$ nematic phases that arise from dislocation-mediated melting and discuss the conventional tensor order parameters. By coupling the $O\left(2\right)$ matter fields to the $Z_N$ lattice gauge theory, a unified $O\left(2\right)/Z_N$ lattice gauge theory is constructed in order to describe all these nematic phases. This lattice gauge theory is shown to reproduce the $C_N$ nematic-isotropic liquid phase transitions and contains an additional deconfined phase. Finally, using our $O\left(2\right)/Z_N$ gauge-theory framework, we discuss phase transitions between different $C_N$ nematics.

• Received 15 May 2014
• Revised 10 December 2014

DOI:https://doi.org/10.1103/PhysRevB.91.075103