Periodic table for topological insulators and superconductors

TL;DR

This paper provides a comprehensive classification scheme for topological insulators and superconductors based on symmetry, dimension, and algebraic structures, revealing universal topological invariants and their physical implications.

Contribution

It introduces a unified classification framework using Clifford algebras and K-theory, extending understanding of topological phases beyond previous models.

Findings

Classification based on Bott periodicity and Clifford algebras.

Topological invariants are elements of Abelian groups like Z or Z_2.

Robustness of topological phases against disorder and interactions.

Abstract

Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.