${\mathbb Z}_2$ Topological Invariant for Magnon Spin Hall Systems

TL;DR

This paper introduces a ${\mathbb Z}_2$ topological invariant for magnon spin Hall systems, enabling the characterization of edge states in bosonic analogs of topological insulators through a new formalism.

Contribution

It defines a ${\mathbb Z}_2$ index for magnon systems using bosonic Berry connections, extending topological classification to bosonic non-interacting systems.

Findings

The ${\mathbb Z}_2$ index accurately predicts helical edge states.

Explicit magnon spin Hall models demonstrate the index's effectiveness.

The formalism applies to various non-interacting bosonic systems.

Abstract

We propose a definition of a ${\mathbb Z}_2$ topological invariant for magnon spin Hall systems which are the bosonic analog of two-dimensional topological insulators in class AII. The existence of "Kramers pairs" in these systems is guaranteed by pseudo-time-reversal symmetry which is the same as time-reversal symmetry up to some unitary transformation. The ${\mathbb Z}_2$ index of each Kramers pair of bands is expressed in terms of the bosonic counterparts of the Berry connection and curvature. We construct explicit examples of magnon spin Hall systems and demonstrate that our ${\mathbb Z}_2$ index precisely characterizes the presence or absence of helical edge states. The proposed ${\mathbb Z}_2$ index and the formalism developed can be applied not only to magnonic systems but also to other non-interacting bosonic systems.