Classification of Topological Insulators and Superconductors

TL;DR

This paper provides a comprehensive classification of topological insulators and superconductors across dimensions, identifying five distinct classes characterized by topological invariants, and relates them to symmetry classes of Hamiltonians.

Contribution

It introduces a systematic classification scheme reducing the problem to Anderson localization at boundaries, revealing five classes in each dimension and linking them to symmetry classes.

Findings

Five topological classes in each spatial dimension.

Identification of topological invariants as winding numbers or Z_2 quantities.

Connection between topological phases and symmetry classes of Hamiltonians.

Abstract

An exhaustive classification scheme of topological insulators and superconductors is presented. The key property of topological insulators (superconductors) is the appearance of gapless degrees of freedom at the interface/boundary between a topologically trivial and a topologically non-trivial state. Our approach consists in reducing the problem of classifying topological insulators (superconductors) in d spatial dimensions to the problem of Anderson localization at a (d-1) dimensional boundary of the system. We find that in each spatial dimension there are precisely five distinct classes of topological insulators (superconductors). The different topological sectors within a given topological insulator (superconductor) can be labeled by an integer winding number or a Z_2 quantity. One of the five topological insulators is the 'quantum spin Hall' (or: Z_2 topological) insulator in d=2, and its generalization in d=3 dimensions. For each dimension d, the five topological insulators correspond to a certain subset of five of the ten generic symmetry classes of Hamiltonians introduced more than a decade ago by Altland and Zirnbauer in the context of disordered systems (which generalizes the three well known "Wigner-Dyson'' symmetry classes).