Classification of magnon thermal Hall systems based on U(1) to non-Abelian gauge fields

Magnon thermal Hall effect in insulating magnets is the manifestation of Berry curvature in magnon bands, which is formulated using the emergent gauge fields that act on magnons as a fictitious magnetic field. In ferromagnets, it is commonly accepted as the outcome of U(1) gauge fields generated by Dzyaloshinskii-Moriya interactions and spin textures, but this mechanism is often suppressed by symmetry-enforced cancellations in many lattice geometries, known as a no-go rule. As a result, antiferromagnetic insulators have long been considered as unfavorable platforms for the effect. We show that antiferromagnets with multiple magnetic sublattices naturally host non-Abelian SU(N) gauge fields in magnon band structures, providing a robust rule-to-go mechanism. The noncommutativity of these gauge fields prevents Berry-curvature cancellation and guarantees a nonvanishing thermal Hall response. As a minimal realization, we demonstrate that a coplanar 120 • antiferromagnet with Dzyaloshinskii-Moriya interactions constitutes a canonical SU(3) platform for the magnon thermal Hall effect. We provide a table of so-far-known two-dimensional lattice geometries and variants of magnetic structures, along with the corresponding gauge fields, providing a unified guideline for identifying magnetic materials, including antiferromagnets and altermagnets, that host thermal Hall transport.