  A quantum spin liquid (QSL) is an exotic state of matter characterized by
quantum entanglement and the absence of any broken symmetry. A long-standing
open problem, which is a key for fundamental understanding the mysterious QSL
states, is how the quantum fluctuations respond to randomness due to quenched
disorder. Transition metal dichalcogenide 1T-TaS$_2$ is a candidate material
that hosts a QSL ground state with spin-1/2 on the two-dimensional perfect
triangular lattice. Here, we performed systematic studies of low-temperature
heat capacity and thermal conductivity on pure, Se-substituted and electron
irradiated crystals of 1T-TaS$_2$. In pure 1T-TaS$_2$, the linear temperature
term of the heat capacity $\gamma T$ and the finite residual linear term of the
thermal conductivity in the zero-temperature limit
$\kappa_{0}/T\equiv\kappa/T(T\rightarrow0)$ are clearly resolved, consistent
with the presence of gapless spinons with a Fermi surface. Moreover, while the
strong magnetic field slightly enhances $\kappa_0/T$, it strongly suppresses
$\gamma$. These unusual contrasting responses to magnetic field imply the
coexistence of two types of gapless excitations with itinerant and localized
characters. Introduction of additional weak random exchange disorder in
1T-Ta(S$_{1-x}$Se$_x$)$_2$ leads to vanishing of $\kappa_0/T$, indicating that
the itinerant gapless excitations are sensitive to the disorder. On the other
hand, in both pure and Se-substituted systems, the magnetic contribution of the
heat capacity obeys a universal scaling relation, which is consistent with a
theory that assumes the presence of localized orphan spins forming random
singlets. Electron irradiation in pure 1T-TaS$_2$ largely enhances $\gamma$ and
changes the scaling function dramatically, suggesting a possible new state of
spin liquid.
