In magnetic insulators, magnons (spin waves) and phonons (lattice vibrations) scatter under time-reversal-broken conditions, generating a transverse heat current — the thermal Hall (Righi-Leduc) effect — without any electrical current.
Magnon & Phonon Dispersion with Hybridization
Thermal Hall Conductivity κ_xy
κ_xy arises from the Berry curvature Ω(k) of hybridized magnon-phonon bands: κ_xy = -(k_B²T/ℏV)Σ_n ∫ c₂(n̄)Ω_n(k) d²k, where c₂ is the Bose-Einstein distribution function and n̄ is the occupation.