Topological phases of fermions in one dimension

By Lukasz Fidkowski, Alexei Y. Kitaev. 2010

591 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/b3a30c12f37fc7f27c5a9cb7bd692dffd6efbdae

How researchers see this paper

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In Atomic and Molecular Physics, and Optics

1. Ashvin Vishwanath
2. Ryan Thorngren
3. E. Witten
4. D. Stanford
5. D. Gaiotto
6. T. Senthil
7. S. Shenker
8. R. Verresen

Citations in the corpus

• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states (2012, background): “[57] Lukasz Fidkowski and Alexei Kitaev, Topological Phases of Fermions in One Dimension, Phys.”
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology (2014, result): “This agrees with the group cohomology classification [13, 14].”
• Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes (2016, background): “However, for spinless fermions one may introduce an artificial antiunitary symmetry T that squares to +1 [57].”
• Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model (2013, unknown): “[7] Lukasz Fidkowski and Alexei Kitaev.”
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface (2015, background): “[28] Lukasz Fidkowski and Alexei Kitaev, “Topological phases of fermions in one dimension,” Phys.”
• Average Symmetry-Protected Topological Phases (2022, unknown): “[33] Lukasz Fidkowski and Alexei Kitaev, “Topological”
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter (2016, background): “The general purpose of this paper is to explore the properties of spin-topological quantum field theories in 2 + 1 dimensions [1] and their relation to fermionic gapped phases of matter [2, 3].”
• Gapless Symmetry-Protected Topological Order (2017, background): “[24] Lukasz Fidkowski and Alexei Kitaev, “Topological phases of fermions in one dimension,” Phys.”
• Spin TQFTs and fermionic phases of matter (2015, background): “(6): the violation of the Leibniz rule for ∪a equals the symmetrization of ∪a−1.”
• Black holes and random matrices (2016, methodology): “The theory also has a particle-hole symmetry under the operator [39, 32, 31]”