Origin of the large phonon thermal Hall effect in cuprates

In hole-doped cuprates inside the pseudogap phase and in the Mott insulators La2CuO4 and Sr2CuO2Cl2, a large negative $\kappa_{xy}$, present for heat flow both in the CuO2 planes and along $c$, is attributed to phonons (Grissonnanche et al. 2019, 2020); a planar thermal Hall response with $B$ in the plane was reported in 2024 (Chen et al., Physical Review X 14, 041011). Identify what couples phonons to $B$ (spin fluctuations of the antiferromagnetic background, defects, or charge-order domains) and predict the doping, field-direction and temperature dependence.

A microscopic coupling that predicts doping, field-direction and temperature dependence, checked across hole- and electron-doped cuprates and across defect densities.