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Wire deconstructionism of two-dimensional topological phases

Titus Neupert, Claudio Chamon, Christopher Mudry, Ronny Thomale

TL;DR

This work develops a coupled-wire framework to classify and realize two-dimensional topological phases of fermions, spanning both short-range entangled (SRE) and long-range entangled (LRE) regimes. By arranging N wires with M channels and enforcing symmetry constraints via a generalized K-matrix formalism, the authors use the Haldane criterion to select inter-wire tunneling that gaps the bulk while leaving symmetry-protected edge modes. They reproduce the tenfold classification for noninteracting SRE phases, demonstrate SRE phases beyond the tenfold with independent charge and heat responses, and extend the construction to LRE phases with fractional excitations across symmetry classes A, AII, D, DIII, and C. The fractionalized extensions yield Abelian FQH-like states and symmetry-enriched variants, with Z or Z2 classifications depending on the class and parity constraints, highlighting a unifying wire-based path to both conventional and exotic topological order.

Abstract

A scheme is proposed to construct integer and fractional topological quantum states of fermions in two spatial dimensions. We devise models for such states by coupling wires of non-chiral Luttinger liquids of electrons, that are arranged in a periodic array. Which inter-wire couplings are allowed is dictated by symmetry and the compatibility criterion that they can simultaneously acquire a finite expectation value, opening a spectral gap between the ground state(s) and all excited states in the bulk. First, with these criteria at hand, we reproduce the tenfold classification table of integer topological insulators, where their stability against interactions becomes immediately transparent in the Luttinger liquid description. Second, we construct an example of a strongly interacting fermionic topological phase of matter with short-range entanglement that lies outside of the tenfold classification. Third, we expand the table to long-range entangled topological phases with intrinsic topological order and fractional excitations.

Wire deconstructionism of two-dimensional topological phases

TL;DR

This work develops a coupled-wire framework to classify and realize two-dimensional topological phases of fermions, spanning both short-range entangled (SRE) and long-range entangled (LRE) regimes. By arranging N wires with M channels and enforcing symmetry constraints via a generalized K-matrix formalism, the authors use the Haldane criterion to select inter-wire tunneling that gaps the bulk while leaving symmetry-protected edge modes. They reproduce the tenfold classification for noninteracting SRE phases, demonstrate SRE phases beyond the tenfold with independent charge and heat responses, and extend the construction to LRE phases with fractional excitations across symmetry classes A, AII, D, DIII, and C. The fractionalized extensions yield Abelian FQH-like states and symmetry-enriched variants, with Z or Z2 classifications depending on the class and parity constraints, highlighting a unifying wire-based path to both conventional and exotic topological order.

Abstract

A scheme is proposed to construct integer and fractional topological quantum states of fermions in two spatial dimensions. We devise models for such states by coupling wires of non-chiral Luttinger liquids of electrons, that are arranged in a periodic array. Which inter-wire couplings are allowed is dictated by symmetry and the compatibility criterion that they can simultaneously acquire a finite expectation value, opening a spectral gap between the ground state(s) and all excited states in the bulk. First, with these criteria at hand, we reproduce the tenfold classification table of integer topological insulators, where their stability against interactions becomes immediately transparent in the Luttinger liquid description. Second, we construct an example of a strongly interacting fermionic topological phase of matter with short-range entanglement that lies outside of the tenfold classification. Third, we expand the table to long-range entangled topological phases with intrinsic topological order and fractional excitations.

Paper Structure

This paper contains 27 sections, 135 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Strategy for constructing topological phases
  4. Representation of symmetries
  5. Particle-hole symmetry in interacting superconductors
  6. Conditions for a spectral gap
  7. Reproducing the tenfold way
  8. Symmetry class A
  9. SRE phases in the tenfold way
  10. SRE phases beyond the tenfold way
  11. Symmetry class AII
  12. Symmetry class D
  13. Symmetry classes DIII and C
  14. Summary
  15. Fractionalized phases
  16. ...and 12 more sections

Figures (1)

  • Figure 1: (Color online) The boundary conditions determine whether a topological phase has protected gapless modes or not. (a) With open boundary conditions, gapless modes exist near the wires $j=1$ and $j=N$, the scattering between them is forbidden by imposing locality in the limit $N\to\infty$. (b) Periodic boundary conditions allow the scattering vector $\mathcal{T}^{(0)}$ that gaps modes which were protected by locality before.