Muon Spin Relaxation Study of Spin Dynamics on a Kitaev honeycomb material H$_3$LiIr$_2$O$_6$

The vacancy effect in quantum spin liquid (QSL) has been extensively studied. A finite density of random vacancies in the Kitaev model can lead to a pileup of low-energy density of states (DOS), which is generally experimentally determined by a scaling behavior of thermodynamic or magnetization quantities. Here, we report detailed muon spin relaxation ($\mu$SR) results of H$_3$LiIr$_2$O$_6$, a Kitaev QSL candidate with vacancies. The absence of magnetic order is confirmed down to 80 mK, and the spin fluctuations are found to be persistent at low temperatures. Intriguingly, the time-field scaling law of longitudinal-field (LF)-$\mu$SR polarization is observed down to 0.1 K. This indicates a dynamical scaling, whose critical exponent 0.46 is excellently consistent with the scaling behavior of specific heat and magnetization data. All the observations point to the finite DOS with the form $N(E) \sim E^{-0.5}$ , which is expected for the Kitaev QSL in the presence of vacancies. Our {\mu}SR study provides a dynamical fingerprint of the power-law low-energy DOS, and introduces a crucial new insight into the vacancy effect in QSL.