Homotopy invariant in time-reversal and twofold rotation symmetric systems

TL;DR

This paper introduces a new homotopy invariant for 2D systems with twofold rotation and time-reversal symmetry, linking it to K theory and demonstrating its effectiveness through examples and a tight-binding model.

Contribution

It develops a novel homotopy invariant based on Wilson loop lifting, connecting it to K theory, and explores topological distinctions beyond Wilson loop spectra.

Findings

New homotopy invariant matches K theory predictions

Identifies obstructions in Wilson loop unwinding for multiple bands

Provides a tight-binding model for a non-trivial phase

Abstract

The primary goal of this paper is to study topological invariants in two dimensional twofold rotation and time-reversal symmetric spinful systems. In this paper, firstly we build a new homotopy invariant based on the lifting of the Wilson loop to the universal covering group of the special orthogonal group. Furthermore, we prove that the invariant we built agrees with the K theory invariant. We go beyond the previous understanding of the Wilson loop unwinding in more than two occupied bands by finding an obstruction of such unwinding. Then, within this formalism, we show two examples that have the same Wilson loop spectrum but belong to different topological classes. Finally, we present a tight binding model realizing the non-trivial phase.