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

Recently, neutron scattering data on powder samples of Zn-paratacamite, Zn${}_x$Cu${}_{4-x}$(OH)${}_6$Cl${}_2$, with small Zn concentration has been interpreted as evidence for valence bond solid and N'eel ordering [1]. We study the classical and quantum Heisenberg models on the distorted kagome lattice appropriate for Zn-paratacamite at low Zn doping. Our theory naturally leads to the emergence of the valence bond solid and collinear magnetic order at zero temperature. Implications of our results to the existing experiments are discussed. We also suggest future inelastic neutron and X-ray scattering experiments that can test our predictions.