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

TL;DR

This paper explores 2D three-state Potts paramagnets with gapless edge modes protected by $(\times Z_3)^3$ symmetry, deriving models, analyzing their conformal properties, and connecting them to coset conformal field theories.

Contribution

It introduces microscopic models for symmetry-protected topological phases with gapless edges and analyzes their conformal field theory descriptions, including numerical and analytical methods.

Findings

Identification of gapless edge modes protected by $(\times Z_3)^3$ symmetry.

Derivation of models and analysis of their conformal properties.

Connection to $SU_k(3)/SU_k(2)$ coset conformal field theory with $k=2$.

Abstract

We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by $(\times Z_3)^3\equiv Z_3\times Z_3\times Z_3$ 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 is 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.