Topological phases of fermions in one dimension

TLDR

Band theory predicts an integer topological index \(k\), but phases with \(k\) in the same equivalence class modulo 8 can be adiabatically connected by adding interactions. The paper demonstrates how interactions modify the classification of one‑dimensional topological phases and extends this result to all 1D gapped fermionic systems with antiunitary symmetries using a central‑extension algebraic framework. The authors analyze the time‑reversal‑invariant Majorana chain (BDI class) with matrix‑product‑state techniques and employ a central‑extension algebraic framework to derive the classification. They prove that the eight equivalence classes are distinct and exhaustive, provide a physical interpretation of the modulo‑8 invariant, and show that each phase corresponds to a different Altland‑Zirnbauer class of the reduced density matrix for an entanglement bipartition.

Abstract

In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-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 antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.