On the K-theoretic classification of topological phases of matter

• Published in: arXiv.org 2014-11
• Main Author: Guo Chuan Thiang
• Format: Article
• Language: English
• Subjects: Charge reversal; Classification; Coding; Conjugation; Fermions; Group theory; Obstructions; Operators (mathematics); Periodic table; Periodic variations; Phases; Representations; Symmetry; Topological insulators; Topology; Translations
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1406.7366

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.