Twisted equivariant matter

TL;DR

This paper develops a mathematical framework using twisted equivariant K-theory to classify topological phases of quantum systems, generalizing classical symmetry classifications and providing new invariants for topological insulators.

Contribution

It introduces a generalized symmetry framework leading to twisted equivariant K-theory, enabling finer classification of topological phases in quantum systems.

Findings

Established a foundation for continuous families of quantum systems.

Defined invariants for topological phases of matter.

Connected twisted equivariant K-theory to topological insulators.

Abstract

We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical 3-fold way of real/complex/quaternionic representations as well as a corresponding 10-fold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.