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Gapped Boundary Phases of Topological Insulators via Weak Coupling

Nathan Seiberg, Edward Witten

TL;DR

Seiberg and Witten construct explicit, weakly coupled boundary models for a 3+1D topological insulator that reproduce the standard gapless boundary state and also realize gapped, topologically ordered phases. The approach hinges on coupling to an emergent boundary U(1) gauge field and carefully tracking spin/charge constraints on spin$_c$ manifolds, monopole zero-modes, and anomalies through anomaly inflow, APS index theory, and eta-invariants. The paper derives a long-distance topological field theory with a dual Abelian sector and an Ising-like non-Abelian sector, connected via a Z$_2$ quotient, and shows how it recovers known boundary states (including MKF-type theories) while enabling new configurations with nontrivial quasiparticle content and non-Abelian statistics. It further extends the framework to topological superconductors, elucidating how CT-time-reversal variants yield T-Pfaffian × SF-type boundaries and how ν_sc can be tuned by model parameters, thereby enriching the landscape of symmetry-preserving gapped boundaries. The results provide explicit, tractable models for exploring boundary anomalies, spin/charge constraints, and the interplay between bulk topological data and surface anyon content with potential experimental implications for TI and TSC boundary phenomenology.

Abstract

The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.

Gapped Boundary Phases of Topological Insulators via Weak Coupling

TL;DR

Seiberg and Witten construct explicit, weakly coupled boundary models for a 3+1D topological insulator that reproduce the standard gapless boundary state and also realize gapped, topologically ordered phases. The approach hinges on coupling to an emergent boundary U(1) gauge field and carefully tracking spin/charge constraints on spin manifolds, monopole zero-modes, and anomalies through anomaly inflow, APS index theory, and eta-invariants. The paper derives a long-distance topological field theory with a dual Abelian sector and an Ising-like non-Abelian sector, connected via a Z quotient, and shows how it recovers known boundary states (including MKF-type theories) while enabling new configurations with nontrivial quasiparticle content and non-Abelian statistics. It further extends the framework to topological superconductors, elucidating how CT-time-reversal variants yield T-Pfaffian × SF-type boundaries and how ν_sc can be tuned by model parameters, thereby enriching the landscape of symmetry-preserving gapped boundaries. The results provide explicit, tractable models for exploring boundary anomalies, spin/charge constraints, and the interplay between bulk topological data and surface anyon content with potential experimental implications for TI and TSC boundary phenomenology.

Abstract

The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.

Paper Structure

This paper contains 47 sections, 202 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Anomalies And Boundary States Of A Topological Insulator
  3. Coupling To Background Fields
  4. Outline
  5. Review Of The Standard Gapless Boundary State
  6. Action And Symmetries
  7. Anomalies
  8. The Spin/Charge Relation And Spin$_c$ Manifolds
  9. A Simple Class Of Boundary States
  10. Some Models
  11. Monopole Operators And The Anomaly
  12. Breaking A Symmetry
  13. The Vortex
  14. Quasiparticles
  15. Enumerating the Quasiparticles
  16. ...and 32 more sections

Figures (4)

  • Figure 1: We are interested in solving a complicated microscopic theory and finding its macroscopic description. Instead, we replace it with a simplified model that captures some degrees of freedom. This model is not supposed to be exact. This model is weakly coupled and allows us to find its long distance behavior easily. The hope is that it is in the same universality class as the original microscopic theory.
  • Figure 2: Vortex 1 is initially to the left of vortex 2. The two vortices are exchanged via a motion in the counterclockwise direction.
  • Figure 3: Two manifolds $X$ and $X'$ with the same boundary $W$ are glued together -- after reversing the orientation of $X'$ so that the orientations are compatible -- to make a compact oriented manifold $X^*$ without boundary. (With a view to later generalizations, $X$ and $X'$ are depicted as being topologically different.)
  • Figure 4: A magnetic monopole penetrates a topological insulator. Its world-line is represented by an 't Hooft line (the solid line). It penetrates the topological insulator at the red point. It continues as an 't Hooft line (the dashed line) inside the material, now with electric charge $\pm 1/2$, and leaves behind a quasiparticle described by the dotted line on the surface. Here time flows upward and we depicted only 2 out of the 3 dimensions of the bulk.