Universal scaling of the specific heat in $S=1/2$ quantum kagome antiferromagnet herbertsmithite

The antiferromagnetic spin-1/2 Heisenberg model on a kagome lattice is one of the most paradigmatic models in the context of quantum spin liquids (QSLs). However, the exact understanding of the kagome QSL nature has suffered from the lack of fundamental information, in particular thermodynamic properties of kagome layers. Through the specific heat and thermal conductivity measurements in magnetic fields with high resolution, here we investigated the intrinsic thermodynamic properties of single-crystal herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$, a canonical candidate for bearing a QSL on a perfect kagome lattice. It is striking that the intrinsic magnetic specific heat contribution arising from the kagome layers exhibits excellent scaling collapse as a function of $T/H$ (temperature/magnetic field). In addition, no residual linear term in the thermal conductivity $\kappa/T(T\rightarrow 0)$ is observed in zero and applied magnetic fields, indicating the absence of itinerant gapless excitations. These results capture a new essential feature of the QSL state of the kagome layers; localized orphan spins induced by exchange bond randomness, surrounded by a non-itinerant QSL.