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Topological phases of fermions in one dimension

Fidkowski, Lukasz and Kitaev, Alexei (2011) Topological phases of fermions in one dimension. Physical Review B, 83 (7). Art. No. 075103. ISSN 1098-0121. doi:10.1103/PhysRevB.83.075103.

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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.

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Additional Information:© 2011 American Physical Society. Received 13 September 2010; published 8 February 2011. We would like to acknowledge useful discussions with Jason Alicea, Matthew Hastings, Netanel Lindner, John Preskill, Gil Refael, Ari Turner, and Dan Freed. A.K. is grateful to the Aspen Center for Physics for hospitality. This work was supported in part by the Institute for Quantum Information under National Science Foundation Grant No. PHY-0803371.
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NSF Institute for Quantum InformationPHY-0803371
Issue or Number:7
Classification Code:PACS: 71.10.Pm
Record Number:CaltechAUTHORS:20110308-123056073
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Official Citation:Topological phases of fermions in one dimension Lukasz Fidkowski and Alexei Kitaev Published 8 February 2011 (13 pages) 075103
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:22711
Deposited By: Ruth Sustaita
Deposited On:10 Mar 2011 16:11
Last Modified:09 Nov 2021 16:07

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