Non-Linear Thermoelectric Response Induced by Berry Curvature Quadrupole in Systems With Broken Time-Reversal Symmetry

Recent theoretical work has shown that higher-order moments of the Berry curvature, e.g., Berry curvature quadrupole or hexapole moments can produce the leading order non-linear anomalous Hall response (NLAH) in systems with special magnetic point group symmetry. Recent experimental work has reported the observation of the Berry curvature quadrupole-induced third-order NLAH (i.e., Hall voltage proportional to the third power of the external electric field) from cryogenic conditions to room temperature in an epitaxially grown material platform with broken time-reversal symmetry. In this paper, using semiclassical Boltzmann formalism in the relaxation time approximation, we compute the Berry curvature quadrupole-induced non-linear anomalous thermal Hall and Nernst coefficients in time reversal broken systems. In systems where Berry curvature monopole and dipole moments vanish by symmetry, our results predict the behavior of the leading order transverse anomalous thermoelectric coefficients proportional to the third power of the applied longitudinal temperature gradient, and are guaranteed to exist in systems that have already exhibited the third order non-linear anomalous Hall effect in recent experiments.