Anisotropic thermal conductivity of proton fluctuation-induced quantum spin liquid $\kappa $-H$_{\mathrm{3}}$(Cat-EDT-TTF)$_{\mathrm{2}}$

We report the thermal transport properties of a quantum spin liquid candidate $\kappa $-H$_{\mathrm{3}}$(Cat-EDT-TTF)$_{\mathrm{2}}$ (H-CAT) with a two-dimensional nearly isotropic triangular lattice. Above 1.0 K, thermal conductivity of H-CAT is substantially smaller than that of a deuterated non-magnetic sample (D-CAT) despite no spin thermal conductivity in D-CAT. In the zero-temperature limit, a finite $T$-linear term of the thermal conductivity of H-CAT is clearly observed when the heat current is parallel to $c$-axis, while it is almost zero when the heat current is parallel to $b$-axis. These features would be attributed to anisotropic proton fluctuations present in H-CAT.