Phononic-magnetic dichotomy of the thermal Hall effect in the Kitaev-Heisenberg candidate material Na$_2$Co$_2$TeO$_6$

Majorana fermions as emergent excitations of the Kitaev quantum spin liquid ground state constitute a promising concept in fault tolerant quantum computation. Experimentally, the recently reported topological half-quantized thermal Hall effect in the Kitaev material $\alpha$-RuCl$_3$ seems to confirm the Majorana nature of the material's magnetic excitations. It has been argued, however, that the thermal Hall signal in $\alpha$-RuCl$_3$ rather stems from phonons or topological magnons than from Majorana fermions. Here we investigate the thermal Hall effect of the closely related Kitaev quantum material Na$_2$Co$_2$TeO$_6$, and we show that the thermal Hall signal emerges from at least two components, phonons and magnetic excitations. This dichotomy results from our discovery that the transversal heat conductivity $\kappa_{xy}$ carries clear signatures of the phononic $\kappa_{xx}$, but changes sign upon entering the low-temperature, magnetically ordered phase. We systematically resolve the two components by considering the detailed temperature and field dependence of both $\kappa_{xy}$ and $\kappa_{xx}$. Our results demonstrate that uncovering a genuinely quantized magnetic thermal Hall effect in a Kitaev topological quantum spin liquid requires to disentangle phonon vs. magnetic contributions where the latter include potentially fractionalized excitations such as the expected Majorana fermions.