Understanding anomalous magnetothermal transport via disentangling shear and compression phonons

Magnetothermal transport in various frustrated magnets exhibits striking field-dependent anomalies that deviate from conventional magnon or phonon transport. To understand such anomalies, we derive an effective spin-phonon Hamiltonian in which phonons with different polarizations couple selectively to distinct spin operators in the strong spin-orbit coupling limit, and show that symmetry-constrained spin-lattice coupling naturally leads to mode-selective spin-phonon interactions. As a result, compression and shear phonon modes contribute to heat current across different magnetic-field regimes. Using a Landauer transport framework combined with exact diagonalization of spin chains coupled to a phonon bath, we show that this mechanism produces a characteristic peak-dip-peak structure in the field dependence of heat current, providing a microscopic explanation for field-induced transport anomalies in spin-orbit-coupled Mott insulators.