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Nematic phases, breaking spontaneously rotational symmetry, provide forubiquitously observed states of matter in both classical and quantum systems.These nematic states may be further classified by their $N$--fold rotationalinvariance described by cyclic groups $C_N$ in 2+1D. Starting from the spacegroups of underlying $2d$ crystals, we present a general classification schemeincorporating $C_N$ nematic phases that arise from dislocation-mediated meltingand discuss the conventional tensor order parameters. By coupling the $O(2)$matter fields to the $Z_N$ lattice gauge theory, an unified $O(2)/Z_N$ latticegauge theory is constructed in order to describe all these nematic phases. Thislattice gauge theory is shown to reproduce the $C_N$ nematic-isotropic liquidphase transitions and contains an additional deconfined phase. Finally, usingour $O(2)/Z_N$ gauge theory framework, we discuss phase transitions betweendifferent $C_N$ nematics.

Classification of nematic order in 2+1D: Dislocation melting and $O(2)/Z_N$ lattice gauge theory