Longitudinal-field muon spin-relaxation (LF-μSR) measurements have been performed in a bond-randomness introduced quantum spin system, Tl1-xKx CuCl3 with x = 0.40, in which the appearance of a Bose glass (BG) phase adjacent to the Bose-Einstein condensation (BEC) phase of excited triplets is observed. In the zero- field limit below 500 G, LF-μSR time spectra shows an exponential-like decay, and the muon spin-relaxation rate does not have a significant temperature dependence down to 25 mK. This indicates that spins fluctuate in time and in space, and that the spin system is not into a spin gapped state. It is suggested that the theoretically predicted gapped Mott insulating phase adjacent to the BG phase disappears in this case. This result obtained by microprobe LF-μSR measurements is consistent with an indication deduced from the results of macroscopic magnetization measurements.
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