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

Nematic phases, 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 $O(2)$ matter fields to the $Z_N$ lattice gauge theory, an unified $O(2)/Z_N$ 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 $O(2)/Z_N$ gauge theory framework, we discuss phase transitions between different $C_N$ nematics.