Intrinsic (valley) thermal Hall conductivity of phonons in graphene for integer quantum Hall phases

RESUMO

We calculate the intrinsic contribution to the (valley) thermal Hall conductivity of phonons in (Semenoff-gapped) graphene in the quantum Hall regime where bulk electron thermal transport is suppressed. The (valley) thermal Hall transport of phonons considered here originates from a phonon (valley) Hall viscosity induced by electron (valley) Hall conductivity and viscosity via geometric electron-phonon coupling. The so generated phonon (valley) Hall viscosity is identified with the emergent (valley) Hall viscosity recently introduced in \textit{Phys. Lett. A} \textbf{595}, 132096 (2026). While our calculations refer to graphene, they may very well be generalized to other Dirac materials like group-VI transition metal dichalcogenides. We discuss the prospect of measuring the phonon thermal Hall conductivity in graphene-based heterostructures and estimate the corresponding valley response induced by electrons in the inversion symmetry broken phase.