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Recently, neutron scattering data on powder samples of Zn-paratacamite,Zn$_x$Cu$_{4-x}$(OH)$_6$Cl$_{2}$, with small Zn concentration has beeninterpreted as evidence for valence bond solid and N\'eel ordering [1]. Westudy the classical and quantum Heisenberg models on the distorted kagomelattice appropriate for Zn-paratacamite at low Zn doping. Our theory naturallyleads to the emergence of the valence bond solid and collinear magnetic orderat zero temperature. Implications of our results to the existing experimentsare discussed. We also suggest future inelastic neutron and X-ray scatteringexperiments that can test our predictions.

Theory of N\'eel and Valence-Bond-Solid Phases on the Kagome Lattice of Zn-paratacamite