A note on topological invariants in condensed matter

TL;DR

This paper explores the classification of topological phases in condensed matter, questioning whether only two phases exist under time-reversal symmetry or if more distinct phases with different invariants are possible.

Contribution

It provides a mathematical perspective on the potential existence of additional topological phases beyond the well-known two, expanding the understanding of topological invariants.

Findings

Analysis suggests more than two topological phases may exist.

Mathematical framework indicates possible subclasses of topological phases.

Discussion highlights the complexity of classifying topological states.

Abstract

We discuss some aspects of topological invariants that classify topological states of matter with emphasis on topological insulators. The main aspect addressed is if there are only two topological phases to Bloch Hamiltonian that are time reversal invariant or if there are more phases that has different topological invariants. From a mathematical point of view may exist more topological phases of matter as a subclass of one well established phase.