Intrinsic Paramagnetism of Optical Phonons and Thermal Hall Effect in Quantum Paraelectrics

Abstract

We propose a microscopic theory of the phonon-mediated thermal Hall effect in quantum paraelectrics, driven by the intrinsic paramagnetism due to quantum fluctuations of the (nearly) critical optical phonon modes. The effective magnetization due to temporal fluctuations of the assosciated dipoles couple to an external magnetic field resulting the magnetoelectric term $\mathbf{H} \cdot (\mathbf{P} \times \dot{\mathbf{P}})$ in the effective Lagrangian density of the polarisation field $\mathbf{P}$. This coupling splits the transverse optical phonon spectrum into right- and left-circularly polarized chiral branches. Using the chiral phonon Green's functions and the Kubo formalism, we derive the thermal Hall conductivity $κ_{xy}$ from the heat-current correlation function. We observe that $κ_{xy}$ is proportional to the difference in spectral weights of the two chiral modes, $κ_{xy} \propto A_+^2 - A_-^2$, establishing a direct microscopic link between phonon chirality and transverse heat transport. Our theory predicts three key observations: (i) a linear-in-magnetic-field response $κ_{xy} \propto H$ at low fields, evolving into nonlinear behavior at higher fields (ii) a non-monotonic temperature dependence of $κ_{xy}$ with a peak arising from the competition between thermal population and phonon scattering and (iii) strong enhancement of $κ_{xy}$ near the quantum critical point $ω_0 \to 0$, it exhibits power-law scaling $κ_{xy} \propto T^α$ with $α\approx 0.41$. Our findings provide an understanding of thermal Hall transport in quantum paraelectrics due to intrinsic paramagnetism and is aligned with the observed behaviour in SrTiO$_3$ and related materials.