Classification of topological quantum matter with symmetries

TL;DR

The work surveys a unifying framework to classify topological quantum matter using symmetry principles. It centers on the ten-fold AZ classification for noninteracting fermions, extends to topological crystalline insulators, and then to gapless topological materials, with a detailed treatment of bulk–boundary and bulk–defect correspondences. It also discusses how interactions collapse or modify these classifications, presenting group cohomology and cobordism as tools for organizing interacting SPT phases, and highlights experimental platforms and future directions for realizing and probing these phases. Overall, the paper provides a comprehensive, physics-oriented guide to topology in quantum matter across gapped, gapless, and interacting regimes. The synthesis of algebraic, geometric, and field-theoretic methods offers a coherent path from abstract invariants to observable boundary phenomena and defect states across condensed-matter systems.

Abstract

Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. We consider both fully gapped and gapless topological materials and their classification in terms of nonspatial symmetries, such as time-reversal, as well as spatial symmetries, such as reflection. Furthermore, we survey the classification of gapless modes localized on topological defects. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K-theory, and non-linear sigma models describing the Anderson (de-)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, we also provide a brief overview of recent results and open questions concerning the topological classifications of interacting systems.

Figures (14)

• Figure 1: The 8 real symmetry classes that involve the antiunitary symmetries $T$ (time reversal) and/or $C$ (particle-hole) are specified by the values of $T^2 = \pm 1$ and $C^2=\pm 1$. They can be visualized on an eight-hour "clock". Adapted from Teo:2010fk.
• Figure 2: Topological defects characterized by a $D$ parameter family of $d$-dimensional Bloch-BdG Hamiltonians. Line defects correspond to $d-D=2$, while point defects correspond to $d-D=1$. Temporal cycles for point defects correspond to $d-D=0$. Adapted from Teo:2010fk.
• Figure 3: (a) Spatial configuration of the $\mathbf{d}$-vector around a half-quantum vortex of a $p+ip$ SC. (b) Zero energy Majorana modes of a half-quantum vortex (HQV) and a full quantum vortex (FQV).
• Figure 4: (a) Dislocation on a square lattice. (b) Two inequivalent $\Omega=-\pi/2$ disclinations. (c) A $\Omega=\pm\pi/2$ disclination dipole.
• Figure 5: Zero-energy MBSs (yellow dots) in heterostructures: (a) superconductor (SC) - magnet (M) domain wall along a QSH edge or a Chern insulator interface; (b) a flux vortex across a superconducting interface between a 3d topological (TI) and trivial insulator (I).