Z3 and (×Z3)3 symmetry protected topological paramagnets

A bstract We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (× Z 3 ) 3 ≡ Z 3 × Z 3 × Z 3 symmetry and smaller Z 3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SU k (3)/ SU k (2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.