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We report the magnetic-field induced Bose glass$-$BEC transition of magnonsin Tl$_{1-x}$K$_x$CuCl$_3$ and its critical behavior investigated throughspecific heat and ESR measurements. The field dependence of the BEC transitiontemperature $T_{\rm N}$ can be described by the power law $\left[H\,{-}\,H_{\rmc}\right]\,{\propto}\,T_{\rm N}^{\phi}$ near the quantum critical point $H_{\rmc}\,{\sim}\,3.5$ T. The critical exponent ${\phi}$ tends to reach a valuesmaller than 1/2 with decreasing fitting window in contrast with${\phi}\rightarrow 3/2$ for the standard BEC in pure system. At sufficientlylow temperatures, the ESR line shape for $H\,{\simeq}\,H_{\rm c}$ isintermediate between Gaussian and Lorentzian. This indicates the localizationof magnons for $H\,{<}H_{\rm c}$ at T=0.

Bose Glass-BEC Transition of Magnons in Tl$_{1-x}$K$_x$CuCl$_3$