Inspired by the recent theoretical suggestion that the random-bond S=1/2 antiferromagnetic Heisenberg model on the triangular and the kagome lattices might exhibit a randomness-induced quantum spin liquid (QSL) behavior when the strength of the randomness exceeds a critical value, and that this ``random-singlet state'' might be relevant to the QSL behaviors experimentally observed in triangular organic salts $\ensuremath{\kappa}\text{\ensuremath{-}}{(\mathrm{ET})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{CN})}_{3}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{\mathrm{EtMe}}_{3}\mathrm{Sb}{[\mathrm{Pd}{(\mathrm{dmit})}_{2}]}_{2}$ and in kagome herbertsmithite ZnCu3(OH)6Cl2, we further investigate the nature of the static and the dynamical spin correlations of these models. We compute the static and the dynamical spin structure factors, S(q) and $S(\mathbf{q},\ensuremath{\omega})$, by means of an exact diagonalization method. In both triangular and kagome models, the computed $S(\mathbf{q},\ensuremath{\omega})$ in the random-singlet state depends on the wave vector q only weakly, robustly exhibiting gapless behaviors accompanied by the broad distribution extending to higher energy $\ensuremath{\omega}$. Especially in the strongly random kagome model, $S(\mathbf{q},\ensuremath{\omega})$ hardly depends on q, and exhibits an almost flat distribution for a wide range of $\ensuremath{\omega}$, together with a $\ensuremath{\omega}=0$ peak. These features agree semiquantitatively with the recent neutron-scattering data on a single-crystal herbertsmithite. Furthermore, the computed magnetization curve agrees almost quantitatively with the experimental one recently measured on a single-crystal herbertsmithite. These results suggest that the QSL state observed in herbertsmithite might indeed be the randomness-induced QSL state, i.e., the random-singlet state.
We have realized a thorough quantum mechanical analysis of an extensive number of low lying levels of the Heisenberg Hamiltonian on small periodic samples. Striking qualitative differences appear between the spectra of the triangular lattice and those of the Kagomé lattice. Symmetries and finite size scaling of the low lying levels of the triangular antiferromagnetic point to an à la Néel symmetry breaking, and the dynamical structure factors corroborate this conclusion. The Kagomé antiferromagnet spectra are qualitatively different: no signature of a symmetry breaking has been exhibited and the dynamical structure factors of the low lying levels display broad bands. The Heisenberg antiferromagnet on a Kagomé lattice is a serious candidate for a spin liquid.
Through exact diagonalization study of the spin - 1/2 Heisenberg model on Kagome lattice with ring-exchange coupling Jr, we find the pure Heisenberg model with Jr=0 stands as a quantum critical point, as evidenced by avoided level crossing and divergence of the second derivative of the ground state energy with respect to Jr at Jr=0. The pure Heisenberg model should thus be gapless in the thermodynamic limit, contrary to common beliefs. At the same time, the ring exchange coupling is found to drive the system into a state with more short ranged spin correlation and with a local spin correlation pattern equivalent to that of the antiferromagnetic Heisenberg model on triangular lattice(the peak of spin structure factor moves to the momentum q=(34π,0)). The resemblance of state with the Marston-Zeng spin Pereils solid state(in terms of dimer-dimer correlation) is also much enhanced by the ring exchange coupling, although it is unclear if such correlation would solidify into static order in the thermodynamic limit.