Periodic table for topological insulators and superconductors

TL;DR

The paper addresses unifying the classification of gapped free-fermion phases across dimensions and symmetry classes using $K$-theory and Bott periodicity. It develops a stable-equivalence framework with Clifford-algebra representations, connecting physical mass terms to classifying spaces $C_q$ and $R_q$ to categorize continuous, band, and discrete systems via real and complex $K$-theory and $K$-homology. A central result is the Bott-periodic, eightfold pattern for the real classes (and a twofold pattern for complex classes), organizing a diagonal table that predicts phase distinctions as dimension and symmetry vary. It further clarifies the robustness of these classifications to disorder and the nuanced role of interactions, including concrete counterexamples in 1D where interactions can alter free-fermion classifications. Overall, the work provides a mathematically rigorous, unified framework for predicting and understanding topological phases of free electrons and their boundary phenomena.

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.