Transition from Bose glass to a condensate of triplons in Tl<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>K<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>CuCl<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

We report the magnetic-field-induced Bose glass--Bose-Einstein condensate (BEC) transition of triplons in Tl${}_{1\ensuremath{-}x}$K${}_{x}$CuCl${}_{3}$ and its critical behavior, which were investigated through specific heat and electron spin resonance (ESR) measurements. The field dependence of the BEC transition temperature ${T}_{N}$ can be described by the power law $[H\ensuremath{-}{H}_{c}]\ensuremath{\propto}{T}_{N}^{\ensuremath{\phi}}$ near the quantum critical point ${H}_{c}~3$ T. The critical exponent $\ensuremath{\phi}$ tends to reach a value smaller than 1/2 with decreasing fitting window in contrast to $\ensuremath{\phi}\ensuremath{\rightarrow}3/2$ for the standard BEC in a pure system. At sufficiently low temperatures, the ESR line shape for $H\ensuremath{\simeq}{H}_{c}$ is intermediate between Gaussian and Lorentzian functions. This implies the localization of triplons for $H&lt;{H}_{c}$ at $T=0$.

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