Parafermionic phases with symmetry-breaking and topological order

TL;DR

This paper introduces a broader concept of topological order in 1D parafermionic chains, demonstrating robust groundstate properties, new boundary conditions, and symmetry-breaking phases with parafermionic order parameters.

Contribution

It generalizes topological order and edge modes in 1D parafermionic systems, extending previous models and introducing twisted boundary conditions and parafermionic condensates.

Findings

Groundstates are mutually indistinguishable by local symmetric probes.

Topological robustness of edge modes is established.

New twisted boundary conditions ensure topological groundstates.

Abstract

Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in 1D open chains, which generalizes the seminal work by Fendley [J. Stat. Mech., P11020 (2012)]. The first essential property is that the groundstates are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the groundstates. These two properties are shown to be topologically robust, and applicable to a wider family of topologically-ordered Hamiltonians than has been previously considered. An an application of these edge modes, we formulate a new notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain groundstate is topological, i.e., it originates from the topological manifold of degenerate states on the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalizes Cooper pairing in superconductors. The stability of these condensates are investigated on both open and closed chains.