Impurities, which are inherently present in any real material, may play an important role in the magnetism of frustrated spin systems with spin-liquid ground states. We address the impurity issue in quantum kagome antiferromagnets by investigating ZnCu$_3$(OH)$_6$SO$_4$ (Zn-brochantite) by means of muon spin spectroscopy. We show that muons couple to the impurity magnetism, originating from Cu-Zn intersite disorder, and that the impurities are highly correlated with the kagome spins, allowing us to probe the intrinsic kagome physics via a Kondo-like effect. The low-temperature plateau in local susceptibility identifies the spin-liquid ground state as being gapless. The corresponding spin fluctuations exhibit an unconventional spectral density and an intriguing field dependence.
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