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

The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in a lattice of corner-sharing triangles is prime candidate for hosting quantum spin liquid (QSL) ground state consisting entangled singlets. But the existence various competing states makes convincing theoretical prediction QSL difficult, calling experimental clues from model materials. materials Zn-barlowite ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not exhibit long range order, they are considered best realizations known to date. Here we use $^{63}$Cu nuclear quadrupole resonance combined with inverse Laplace transform (ILT) probe locally inhomogeneity delicate affected disorder. We present direct evidence gradual emergence singlets spatially varying excitation gaps, but even at temperatures far below super-exchange energy scale their fraction limited approximately 60\% total spins. Theoretical models need incorporate role disorder account observed inhomogeneously gapped behaviour.