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The kagome Heisenberg antiferromagnet formed by frustrated spins arranged ina lattice of corner-sharing triangles is a prime candidate for hosting aquantum spin liquid (QSL) ground state consisting of entangled spin singlets.But the existence of various competing states makes a convincing theoreticalprediction of the QSL ground state difficult, calling for experimental cluesfrom model materials. The kagome lattice materials Zn-barlowiteZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do notexhibit long range order, and they are considered the best realizations of thekagome Heisenberg antiferromagnet known to date. Here we use $^{63}$Cu nuclearquadrupole resonance combined with the inverse Laplace transform (ILT) to probelocally the inhomogeneity of delicate quantum ground states affected bydisorder. We present direct evidence for the gradual emergence of spin singletswith spatially varying excitation gaps, but even at temperatures far below thesuper-exchange energy scale their fraction is limited to approximately 60\% ofthe total spins. Theoretical models need to incorporate the role of disorder toaccount for the observed inhomogeneously gapped behaviour.

Emergence of spin singlets with inhomogeneous gaps in the kagome Heisenberg antiferromagnets Zn-barlowite and herbertsmithite