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Classification and analysis of two dimensional abelian fractional topological insulators

Michael Levin, Ady Stern

TL;DR

This work develops a bulk Chern-Simons framework for time-reversal invariant 2D abelian insulators, characterized by $(\mathcal{K}, \boldsymbol{\tau}, T, \boldsymbol{\chi})$, and derives a sharp edge-stability criterion: edge modes are protected if and only if $\frac{1}{e^*}\, \boldsymbol{\chi}^T \mathcal{K}^{-1} \boldsymbol{\tau}$ is odd, where $e^*$ is the smallest charge. It introduces local Kramers degeneracy and a local Kramers theorem, and proves the edge protection via microscopic edge analysis and a generalized flux-insertion argument that preserves the topological sector. The results unify prior fractional topological insulator constructions, provide a general classification, and suggest extensions to bosonic systems and higher dimensions. The framework paves the way for systematic exploration of interacting, TR-invariant topological phases with abelian statistics and their protected edge phenomena.

Abstract

We present a general framework for analyzing fractionalized, time reversal invariant electronic insulators in two dimensions. The framework applies to all insulators whose quasiparticles have abelian braiding statistics. First, we construct the most general Chern-Simons theories that can describe these states. We then derive a criterion for when these systems have protected gapless edge modes -- that is, edge modes that cannot be gapped out without breaking time reversal or charge conservation symmetry. The systems with protected edge modes can be regarded as fractionalized analogues of topological insulators. We show that previous examples of 2D fractional topological insulators are special cases of this general construction. As part of our derivation, we define the concept of "local Kramers degeneracy" and prove a local version of Kramers theorem.

Classification and analysis of two dimensional abelian fractional topological insulators

TL;DR

This work develops a bulk Chern-Simons framework for time-reversal invariant 2D abelian insulators, characterized by , and derives a sharp edge-stability criterion: edge modes are protected if and only if is odd, where is the smallest charge. It introduces local Kramers degeneracy and a local Kramers theorem, and proves the edge protection via microscopic edge analysis and a generalized flux-insertion argument that preserves the topological sector. The results unify prior fractional topological insulator constructions, provide a general classification, and suggest extensions to bosonic systems and higher dimensions. The framework paves the way for systematic exploration of interacting, TR-invariant topological phases with abelian statistics and their protected edge phenomena.

Abstract

We present a general framework for analyzing fractionalized, time reversal invariant electronic insulators in two dimensions. The framework applies to all insulators whose quasiparticles have abelian braiding statistics. First, we construct the most general Chern-Simons theories that can describe these states. We then derive a criterion for when these systems have protected gapless edge modes -- that is, edge modes that cannot be gapped out without breaking time reversal or charge conservation symmetry. The systems with protected edge modes can be regarded as fractionalized analogues of topological insulators. We show that previous examples of 2D fractional topological insulators are special cases of this general construction. As part of our derivation, we define the concept of "local Kramers degeneracy" and prove a local version of Kramers theorem.

Paper Structure

This paper contains 25 sections, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Chern-Simons theories for time reversal invariant insulators
  3. The time-reversal breaking case
  4. Including time reversal symmetry
  5. Explicit parameterization of time reversal invariant abelian Chern-Simons theories
  6. Examples and physical realizations
  7. Stability of the edge: microscopic analysis
  8. Basic setup
  9. Example 1: Laughlin quantum spin Hall state
  10. Example 2: Two component fractional quantum spin Hall state
  11. Microscopic analysis in general case
  12. Stability of the edge modes: flux insertion argument
  13. Integer case
  14. Topological order and ground state degeneracy on a cylinder
  15. General case: outline of the argument
  16. ...and 10 more sections

Figures (2)

  • Figure 1: The flux insertion argument for the $\nu = k$ toy model: we start with the ground state $\Psi_0$ and adiabatically inset $\pm \Phi_0/2$ flux through the cylinder, obtaining two states $\Psi_1, \Psi_2$. If $k$ is odd, then $\Psi_1$ has a Kramers degeneracy at the two ends of the cylinder, and is therefore degenerate in energy with three other states, one of which is $\Psi_2$. If we start in one of these three degenerate states and then adiabatically reduce the flux to $0$, we obtain an excited state $\Psi_{ex}$ whose energy gap $\Delta$ vanishes in the thermodynamic limit.
  • Figure 2: (a) A schematic portrait of the ground state $\Psi_0$ of the $\nu = 1$ toy model. Working in the Landau gauge, the single particle states consist of spin-up and spin-down Landau orbitals. In the time reversal invariant ground state, the spin-up and spin-down orbitals are equally occupied. After inserting $\Phi_0/2$ flux, the spin-up and spin-down orbitals shift in opposite directions, resulting in the state $\Psi_1$. (b) The state $\Psi_1$ has a Kramers degeneracy on both ends of the cylinder and is therefore degenerate with its time reversed partner (c) $\Psi_2$, as well as two other states, (d) $\Psi_3$ and (e) $\Psi_4$.