  Quantum fluctuations are expected to lead to highly entangled spin-liquid
states in certain two-dimensional spin-1/2 compounds. We have synthesized and
measured thermodynamic properties and muon spin relaxation rates in the
copper-based two-dimensional triangular-lattice spin liquids
Lu$_3$Cu$_2$Sb$_3$O$_{14}$ and Lu$_3$CuZnSb$_3$O$_{14}$. The former is the
least disordered of this kind discovered to date. Magnetic entropy generation
at high temperatures has been ruled out after carefully correcting for the
lattice specific heat. Surprisingly, roughly half of the magnetic entropy is
missing down to temperatures of O(10$^{-3}$) the exchange energy, independent
of magnetic field up to $g\mu_B H \gtrsim k_B\Theta_W$, where $\Theta_W$ is the
Weiss temperature. The magnetic specific heat divided by temperature $C_M(T)/T$
and muon spin relaxation rate $\lambda(T)$ are both temperature-independent at
low temperatures, followed by logarithmic decreases with increasing
temperature. This behavior can be simply characterized by scale-invariant
time-dependent fluctuations with a single parameter. Since no cooperative
effects due to impurities are observed, the measured properties are intrinsic.
They are evidence that in Lu$_3$Cu$_2$Sb$_3$O$_{14}$ massive quantum
fluctuations lead to either a gigantic specific heat peak from singlet
excitations at very low temperatures or, perhaps less likely, an extensively
degenerate possibly topological singlet ground state.
