On the K-Theoretic Classification of Topological Phases of Matter

Abstract

We present a rigorous and fully consistent $K$-theoretic framework for studying gapped topological phases of free fermions such as topological insulators. It utilises and profits from powerful techniques in operator $K$-theory. From the point of view of symmetries, especially those of time reversal, charge conjugation, and magnetic translations, operator $K$-theory is more general and natural than the commutative topological theory. Our approach is model-independent, and only the symmetry data of the dynamics, which may include information about disorder, is required. This data is completely encoded in a suitable $C^*$-superalgebra. From a representation-theoretic point of view, symmetry-compatible gapped phases are classified by the super-representation group of this symmetry algebra. Contrary to existing literature, we do not use $K$-theory to classify phases in an absolute sense, but only relative to some arbitrary reference. $K$-theory groups are better thought of as groups of obstructions between homotopy classes of gapped phases. Besides rectifying various inconsistencies in the existing literature on $K$-theory classification schemes, our treatment has conceptual simplicity in its treatment of all symmetries equally. The Periodic Table of Kitaev is exhibited as a special case within our framework, and we prove that the phenomena of periodicity and dimension shifts are robust against disorder and magnetic fields.