Zero- and longitudinal-field muon spin relaxation (ZF-, LF-μSR) measurements were carried out on the spin-1/2 kagome lattice antiferromagnet (Rb1-xCsx)2Cu3SnF12 with x = 0.53 and 0.6, where xc = 0.53 is the quantum critical point between the antiferromagnetically ordered phase and the valence-bond-glass phase. In the case of xc = 0.53, the muon spin relaxation rate at LF 200 gauss increases monotonically with decreasing temperature and shows no evidence for a magnetic order. The LF dependence of the muon spin relaxation rate, which corresponds to the fluctuating frequency spectrum, shows "white" spectrum behavior down to 0.3 K. These results indicate that internal magnetic fields fluctuate and are consistent with reported results of magnetic susceptibility measurement for xc = 0.53. In the case of x = 0.60, which is adjacent to the quantum critical point, the partial missing asymmetry in ZF-μSR time spectra is observed from a higher temperature than the antiferromagnetic phase transition temperature. It is suggested that large spin fluctuations peculiar to the quantum critical region appear.
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