Homotopy of periodic two by two matrices

TL;DR

This paper classifies the homotopy classes of 2x2 periodic matrices with symmetries, relating to gapped Bloch Hamiltonians in 1D, using elementary topological methods involving closed curves in three dimensions.

Contribution

It provides a complete description of homotopy classes of symmetric 2x2 matrices relevant to topological phases in condensed matter physics.

Findings

Classifies homotopy classes of symmetric 2x2 matrices

Relates matrix homotopy to closed curves in three dimensions

Connects mathematical classification to physical models in 1D

Abstract

We describe the homotopy classes of 2 by 2 periodic simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in three dimensions. The matrices represent gapped Bloch Hamiltonians in 1D with a two-dimensional Hilbert space per unit cell.