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Interacting fermionic topological insulators/superconductors in three dimensions

Chong Wang, T. Senthil

TL;DR

The paper classifies three-dimensional interacting electronic SPT phases across the ten symmetry classes, clarifying which free-fermion classifications survive interactions and when bosonic SPTs contribute additional phases. A central method is gauging symmetries to invoke theta terms and monopole physics, enabling a bulk diagnostic via surface terminations and monopole quantum numbers. The authors establish complete interacting classifications for symmetry groups with a normal U(1), yielding results such as Z8×Z2 for U(1)×Z2^T and Z2^5 for CII, and they map a wide range of surface states, from symmetry-breaking surfaces to symmetry-preserving topological orders, including non-Abelian cases and symmetry-enforced gaplessness. The work connects bulk SPT order to surface dynamics through theta terms, Witten effects, and Hopf terms, with implications for materials and theoretical QFT, including spin liquids and anomaly constraints in lower dimensions.

Abstract

Symmetry Protected Topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper we describe the classification and properties of such phases for three dimensional(3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1) subgroup we show that this classification is complete. We describe the non-trivial surface and bulk properties of these states. In particular we discuss interesting correlated surface states that are not captured in a free fermion description. We show that in many, but not all cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.

Interacting fermionic topological insulators/superconductors in three dimensions

TL;DR

The paper classifies three-dimensional interacting electronic SPT phases across the ten symmetry classes, clarifying which free-fermion classifications survive interactions and when bosonic SPTs contribute additional phases. A central method is gauging symmetries to invoke theta terms and monopole physics, enabling a bulk diagnostic via surface terminations and monopole quantum numbers. The authors establish complete interacting classifications for symmetry groups with a normal U(1), yielding results such as Z8×Z2 for U(1)×Z2^T and Z2^5 for CII, and they map a wide range of surface states, from symmetry-breaking surfaces to symmetry-preserving topological orders, including non-Abelian cases and symmetry-enforced gaplessness. The work connects bulk SPT order to surface dynamics through theta terms, Witten effects, and Hopf terms, with implications for materials and theoretical QFT, including spin liquids and anomaly constraints in lower dimensions.

Abstract

Symmetry Protected Topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper we describe the classification and properties of such phases for three dimensional(3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1) subgroup we show that this classification is complete. We describe the non-trivial surface and bulk properties of these states. In particular we discuss interesting correlated surface states that are not captured in a free fermion description. We show that in many, but not all cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.

Paper Structure

This paper contains 27 sections, 31 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Generalities
  3. Surface terminations
  4. Symmetry broken surface
  5. Symmetry preserving surface topological order
  6. Gauging the symmetry: $\theta$ terms
  7. Gauging the symmetry: bulk monopoles and surface states
  8. $U(1)\times\mathbb{Z}_2^T$: AIII class
  9. 8 Dirac cones: triviality
  10. 4 Dirac cones: boson SPT
  11. 2 Dirac cones: Kramers monopole
  12. Surface topological order
  13. 1 Dirac cone: $\theta=\pi$
  14. Non-Abelian Surface topological order
  15. $\mathbb{Z}_8\times\mathbb{Z}_2$ classification
  16. ...and 12 more sections