Static and dynamical spin correlations of the S=1/2 random-bond antiferromagnetic Heisenberg model on the triangular and the kagome lattices

Inspired by the recent theoretical suggestion that the random-bond $S=1\mathrm{/}2$ 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 $\kappa -(\mathrm{E}\mathrm{T}{)}_2\mathrm{C}{\mathrm{u}}_2(\mathrm{C}\mathrm{N}{)}_3$and $\mathrm{E}\mathrm{t}\mathrm{M}{\mathrm{e}}_3\mathrm{S}\mathrm{b}[\mathrm{P}\mathrm{d}(\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{t}{)}_2{]}_2$ and in kagome herbertsmithite ${\rm CuZn_3(OH)_6Cl_2}$, 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, $S(\mathbf{q})$ and $S(\mathbf{q},\omega )$, by means of an exact diagonalization method. In both triangular and kagome models, the computed $S(\mathbf{q},\omega )$ in the random-singlet state depends on the wavevector $\mathbf{q}$ only weakly, robustly exhibiting gapless behaviors accompnied by the broad distribution extending to higher energy $\omega$. Especially in the strongly random kagome model, $S(\mathbf{q},\omega )$ hardly depends on $\mathbf{q}$, and exhibits an almost flat distribution for a wide range of $\omega$, together with a $\omega =0$ peak. These features agree semi-quantitatively with the recent neutron-scattering data on a single-crystal herbertsmithite, suggesting that the QSL state observed in herbersmithite might indeed be the randomness-induced QSL state, {\it i.e./}, the random-singlet state.