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It has been suggested that the nearest-neighbour (NN) antiferromagneticHeisenberg (HAF) model on the Kagome lattice may be a good starting point tounderstand the quantum spin-liquid (QSL) behaviour discovered inHerbertsmithite. We investigate possible QSL phases in the presence ofexperimentally relevant spin-rotation symmetry-breaking perturbations such asDzyaloshinskii-Moriya and Ising interactions, as well as second-neighbourantiferromagnetic Heisenberg interactions. We use the projective symmetry groupanalysis within the slave-fermion framework of QSL phases and systematicallyclassify possible QSLs in the presence of these perturbations. The dynamicalspin-structure factor for relevant QSLs is computed and their effects arestudied. Our calculations reveal dispersive features in the spin structurefactor embedded in a generally diffuse background due to the existence offractionalized S=1/2 excitations called spinons. For two of the previouslyproposed $Z_2$ states, the dispersive features are almost absent, and diffusescattering dominates over a large energy window throughout the Brillouin zone.This resembles the structure factor observed in recent inelastic neutronscattering experiments on singlet crystals of Herbertsmithite. Furthermore, oneof the $Z_2$ states with the spin structure factor with mostly diffusescattering is gapped, and it may be adiabatically connected to the gapped QSLstate observed in recent DMRG calculations for the NN HAF. The aboveperturbations are found to enhance the diffuse nature of the spin structurefactor and reduce the momentum dependencies of the spin gap. We also calculatethe electron spin resonance absorption spectra that further characterize therole of spin-rotation symmetry breaking perturbations, and can shed more lightinto the nature of the ground state in Herbertsmithite.

Quantum spin liquids in the absence of spin-rotation symmetry: application to Herbertsmithite