# The "not-A", RSPT and Potts phases in an $S_3$-invariant chain

## TL;DR

This paper explores an $S_3$-invariant quantum chain, revealing four distinct phases, including topological and RSPT orders, with exact ground states and phase transitions described by the three-state Potts model.

## Contribution

It provides a detailed analysis of an $S_3$-invariant chain, identifying new phases and exact ground states, and connects lattice models to Potts conformal field theory.

## Key findings

- Identification of four proximate gapped phases with distinct orders
- Exact frustration-free ground states for each phase
- Phase transitions in the universality class of the three-state Potts model

## Abstract

We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in the region of a U(1)-invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Also nearby is a phase with "representation" symmetry-protected topological (RSPT) order. Its dual exhibits an unusual "not-A" order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact RSPT ground state is similar to that of Affleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-A phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. A field-theory analysis shows that all transitions are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.