Thermal transport and low-temperature specific heat in 4Hb-TaS$_2$

We investigate the low-energy excitation spectrum of the van der Waals heterostructure superconductor 4Hb-TaS$_2$ using ultra-low-temperature specific-heat and thermal-conductivity measurements with magnetic fields applied parallel and perpendicular to the crystallographic $c$ axis. The specific heat is broadly consistent with a nodeless superconducting gap, but retains a finite residual linear contribution, indicating a small residual low-energy density of states in the superconducting state. In addition, a pronounced upturn appears below approximately 0.3K. Its weak magnetic-field dependence, together with the absence of a corresponding feature in thermal transport, supports an interpretation in terms of localized degrees of freedom, most likely a nuclear Schottky contribution. In contrast to the finite residual thermodynamic density of states, the thermal conductivity extrapolates to a vanishing zero-field electronic linear term within experimental uncertainty for both field orientations. Thus, the residual low-energy states do not form a detectable itinerant heat-conduction channel. In finite magnetic field, the electronic heat transport grows rapidly. For out-of-plane fields, this response is broadly consistent with previous thermal-conductivity measurements and with the behavior commonly associated with multigap nodeless superconductivity. The even steeper increase observed for in-plane fields suggests that the field-induced quasiparticle response of 4Hb-TaS$_2$ is more complicated than the standard multigap picture alone.