  The thermal Hall effect, defined as a heat current response transversal to an
applied temperature gradient, is a central experimental probe of exotic
electrically insulating phases of matter. A key question is how the interplay
between magnetic and structural degrees of freedom gives rise to a nonzero
thermal Hall conductivity (THC). Here, we present evidence for an intrinsic
thermal Hall effect in the Heisenberg-Kitaev antiferromagnet and spin-liquid
candidate Na$_2$Co$_2$TeO$_6$ brought about by the quantum-geometric Berry
curvature of so-called magnon polarons, resulting from magnon-phonon
hybridization. At low temperatures, our field- and temperature-dependent
measurements show a negative THC for magnetic fields below 10 T and a sign
change to positive THC above. Theoretically, the sign and the order of
magnitude of the THC cannot be solely explained with magnetic excitations. We
demonstrate that, by incorporating spin-lattice coupling into our theoretical
calculations, the Berry curvature of magnon polarons counteracts the purely
magnonic contribution, reverses the overall sign of the THC, and increases its
magnitude, which significantly improves agreement with experimental data. Our
work highlights the crucial role of spin-lattice coupling in the thermal Hall
effect.
