$Z_3$ and $(\times Z_3)^3$ symmetry protected topological paramagnets

Abstract

We identify 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 study microscopic models for the gapless edge and discuss the corresponding conformal field theories and their central charges. These are the edge states of s = 1 paramagnets protected by Z 3 ×Z 3 ×Z 3 and Z 3 symmetries, in analogy with the free fermion XX model edge discussed by Levin and Gu for the spin-1/2 Z 2 Ising paramagnet. We show that in spin-1 magnets, there are numerous options to obtain self-dual Hamiltonians and gapless edge modes. These models form the basis for the realization of a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.