Topological phases of fermions in one dimension

TL;DR

This work shows that interactions in one-dimensional fermionic systems qualitatively modify the band-structure classification, reducing the Z to Z8 for the TR-invariant Majorana chain. Using matrix product states and entanglement spectra, it constructs interacting invariants tied to projective symmetry representations on the entanglement space, identifying eight distinct phases corresponding to k mod 8. It then formulates a general 1D framework based on central extensions and Wall invariants to classify all gapped 1D fermionic phases with unitary and anti-unitary symmetries, connecting these invariants to Altland-Zirnbauer classes. The results unify edge-state physics, entanglement properties, and algebraic classification into a coherent picture with potential extensions to higher dimensions.

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.

Figures (3)

• Figure 1: Schematic representation of the Hamiltonians $H_0$ and $H_1$. The dots denote Majorana fermions, and the edges the quadratic couplings in the Hamiltonian.
• Figure 2: Schematic representation of the Hamiltonian $H_k$. The dashed vertical line can represent a physical cut or an entanglement bipartition. Note that on the left side of the cut we have $k$ unpaired Majorana fermions, $c_2, \ldots, c_{2k}$.
• Figure 3: Schematic representation of the Hamiltonian $H_{2m+1}$.