Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

TL;DR

This paper classifies one-dimensional interacting fermion phases using entanglement, revealing a finite set of eight or sixteen distinct phases depending on symmetries, contrasting with the infinite non-interacting classification.

Contribution

It introduces a framework based on entanglement to classify interacting fermion phases, establishing a finite $ ext{Z}_8$ group structure and identifying bulk invariants.

Findings

Eight distinct phases with unique entanglement invariants

Presence of translational symmetry doubles the phases to sixteen

Interactions reduce the classification from infinite to finite groups

Abstract

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find that in the presence of interactions there are only eight distinct phases, which obey a $\mathbb{Z}_8$ group structure. This is in contrast to the $\mathbb{Z}$ classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.