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Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

Ari M. Turner, Frank Pollmann, Erez Berg

TL;DR

The paper tackles the problem of how interactions modify the 1D topological classification of spinless fermions with time-reversal symmetry and fermion parity. It develops a bulk entanglement framework that analyzes the symmetry properties of the Schmidt eigenstates at a bipartition, showing that phases are distinguished by projective representations of the symmetry on the entanglement ends. The main finding is that interacting 1D fermions realize eight distinct phases forming a $\mathbb{Z}_8$ group, labeled by invariants $(\mu,\phi,\kappa) \in \{0,\pi\}^3$, with characteristic degeneracies in the entanglement spectrum; translational symmetry further doubles the count to 16 via a $\theta$ invariant. This approach provides a bulk, wavefunction-based understanding of interaction effects on topological classifications, consistent with Fidkowski and Kitaev, and suggests avenues to extend the framework to other symmetry classes and higher dimensions.

Abstract

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find that in the presence of interactions there are only eight distinct phases, which obey a $\mathbb{Z}_8$ group structure. This is in contrast to the $\mathbb{Z}$ classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

TL;DR

The paper tackles the problem of how interactions modify the 1D topological classification of spinless fermions with time-reversal symmetry and fermion parity. It develops a bulk entanglement framework that analyzes the symmetry properties of the Schmidt eigenstates at a bipartition, showing that phases are distinguished by projective representations of the symmetry on the entanglement ends. The main finding is that interacting 1D fermions realize eight distinct phases forming a group, labeled by invariants , with characteristic degeneracies in the entanglement spectrum; translational symmetry further doubles the count to 16 via a invariant. This approach provides a bulk, wavefunction-based understanding of interaction effects on topological classifications, consistent with Fidkowski and Kitaev, and suggests avenues to extend the framework to other symmetry classes and higher dimensions.

Abstract

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find that in the presence of interactions there are only eight distinct phases, which obey a group structure. This is in contrast to the classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

Paper Structure

This paper contains 11 sections, 29 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Fermionic models with pairing terms
  3. Classifying phases by symmetry properties of the entanglement eigenstates
  4. Fermion parity
  5. Time Reversal Symmetry
  6. Addition of Phases
  7. Translational Invariance and $\theta$
  8. Summary and outlook
  9. Low Energy Operators and the Ends of the Chain
  10. Factoring antiunitary operators
  11. Parity of the ground state on a periodic chain

Figures (1)

  • Figure 1: Illustration of a bipartition of a 1D chain into a segment (S) of length L and a surrounding environment (E). The operators $\mathcal{O}_A$ and $\mathcal{O}_B$ act on the edges of the segment.