Topological phases of fermions in one dimension

✍ Scribed by Fidkowski, Lukasz; Kitaev, Alexei

• Book ID: 120067556
• Publisher: The American Physical Society
• Year: 2011
• Tongue: English
• Weight: 302 KB
• Volume: 83
• Category: Article
• ISSN: 1098-0121
• DOI: 10.1103/PhysRevB.83.075103

✦ Synopsis

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.