Topological magnons in a kagome lattice spin system with $XXZ$ and Dzyaloshinskii-Moriya interactions

TL;DR

This paper investigates topological magnon excitations in a Kagome lattice spin system with $XXZ$ and Dzyaloshinskii-Moriya interactions, revealing multiple topological phases distinguished by Chern numbers and thermal Hall effects.

Contribution

It identifies four distinct topological phases in a classical spin system with identical ground states, based on magnon band topology and edge states.

Findings

Four topological phases with different Chern numbers identified.

Edge states localized on system boundaries depend on topological phase.

Thermal Hall conductivity varies with temperature, distinguishing phases.

Abstract

We study the phases of a spin system on the Kagome lattice with nearest-neighbor $XXZ$ interactions with anisotropy ratio $\Delta$ and Dzyaloshinsky-Moriya interactions with strength $D$. In the classical limit where the spin $S$ at each site is very large, we find a rich phase diagram of the ground state as a function of $\Delta$ and $D$. There are five distinct phases which correspond to different ground state spin configurations in the classical limit. We use spin wave theory to find the bulk energy bands of the magnons in some of these phases. We also study a strip of the system which has infinite length and finite width; we find modes which are localized on one of the edges of the strip with energies which lie in the gaps of the bulk modes. In the ferromagnetic phase in which all the spins point along the $+ \hat z$ or $- \hat z$ direction, the bulk bands are separated from each other by finite energy gaps. This makes it possible to calculate the Berry curvature at all momenta, and hence the Chern numbers for every band; the number of edge states is related to the Chern numbers. Interestingly, we find that there are four different regions in this phase where the Chern numbers are different. Hence there are four distinct topological phases even though the ground state spin configuration is identical in all these phases. We calculate the thermal Hall conductivity of the magnons as a function of the temperature in the above ferromagnetic phase; we find that this can distinguish between the various topological phases. These results are valid for all values of $S$.In the other phases, there are no gaps between the different bands; hence the edge states are not topologically protected.