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We studied the field dependent thermal conductivity ($\kappa$) ofNa$_2$Co$_2$TeO$_6$, a compound considered as the manifestation of the Kitaevmodel based on the high-spin $d^7$ Co$^{2+}$ ions. We found that in-planemagnetic fields beyond a critical value $B_c \approx$~10 T are able todrastically enhance $\kappa$ at low temperatures, resulting in a double-peakstructure of $\kappa(T)$ that closely resembles the behavior of$\alpha$-RuCl$_3$. This result suggests that heat transport inNa$_2$Co$_2$TeO$_6$ is primarily phononic, and it is strongly affected byscattering from magnetic excitations that are highly tunable by externalfields. Interestingly, for magnetic fields $B // a$ (i.e., along the zigzagdirection of the Co-Co bonds), there is an extended field range which separatesthe long-range magnetic order for $B\leq B_c\approx10$ T and the partiallyspin-polarized gapped high-field phase for $B\gtrsim 12$ T. The low-energyphonon scattering is particularly strong in this field range, consistent withthe notion that the system becomes a quantum spin liquid with prominent spinfluctuations down to energies of no more than 2 meV.

Unusual heat transport of the Kitaev material Na$_2$Co$_2$TeO$_6$: putative quantum spin liquid and low-energy spin excitations