Topological phases of fermions in one dimension

TL;DR

This paper explores how interactions alter the classification of topological phases in one-dimensional fermionic systems, specifically in the TR-invariant Majorana chain, revealing an 8-fold classification and physical interpretations.

Contribution

It demonstrates that the interacting phases form an 8-fold classification, extending the band theory classification, with a physical interpretation based on entanglement properties.

Findings

Eight distinct topological phases identified.

Interaction effects reduce classification from infinite to eight classes.

Matrix product state methods validate the classification.

Abstract

In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of 1D systems. We focus on the TR-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index $k$, it is known that phases characterized by values of $k$ in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half-chains. We generalize these results to the classification of all one dimensional gapped phases of fermionic systems with possible anti-unitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.