Topological phases of fermions in one dimension

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.

Figure 1

Schematic representation of the Hamiltonians $H_0$ and $H_1$. The dots denote Majorana fermions, and the edges are 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, $c_2,\dots ,c_{2k}$.

Figure 3

Schematic representation of the Hamiltonian $H_{2m+1}$.