Fragility of $\mathcal{Z}_2$ topological invariant characterizing triplet excitations in a bilayer kagome magnet

TL;DR

This paper investigates the fragility of $$2 topological invariants in triplet excitations of a bilayer kagome magnet, showing that realistic anisotropies and magnetic fields can destroy or alter the topological properties.

Contribution

It demonstrates that exchange anisotropies and magnetic fields can break the $$2 topological invariants in triplet excitations, revealing their intrinsic fragility compared to electronic topological insulators.

Findings

Exchange anisotropies destroy $$2 topology.

Magnetic fields induce a stable triplet Chern insulator.

Characterization of Berry curvature and measurable spin Nernst and thermal Hall signals.

Abstract

The discovery by Kane and Mele of a model of spinful electrons characterized by a $\mathcal{Z}_2$ topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the band-like excitations of magnetic insulators, and recently models supporting $\mathcal{Z}_2$ topological invariants have been proposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)] excitations. In both cases, magnetic excitations form time--reversal (TR) partners, which mimic the Kramers pairs of electrons in the Kane-Mele model but do not enjoy the same type of symmetry protection. In this paper, we revisit this problem in the context of the triplet excitations of a spin model on the bilayer kagome lattice. Here the triplet excitations provide a faithful analog of the Kane-Mele model as long as the Hamiltonian preserves the TR$\times$U(1) symmetry. We find that exchange anisotropies, allowed by the point group and typical in realistic models, break the required TR$\times$U(1) symmetry and instantly destroy the $\mathcal{Z}_2$ band topology. We further consider the effects of TR breaking by an applied magnetic field. In this case, the lifting of spin-degeneracy leads to a triplet Chern insulator, which is stable against the breaking of TR$\times$U(1) symmetry. Kagome bands realize both a quadratic and a linear band touching, and we provide a thorough characterization of the Berry curvature associated with both cases. We also calculate the triplet-mediated spin Nernst and thermal Hall signals which could be measured in experiments. These results suggest that the $\mathcal{Z}_2$ topology of band-like excitations in magnets may be intrinsically fragile compared to their electronic counterparts.