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 $S=1/2$ 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 $\kappa {\rm -(ET)_2 Cu_2 (CN)_3}$and ${\rm EtMe_3 Sb[Pd(dmit)_2]_2}$ 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, $S({\bf q})$ and $S({\bf q},\omega)$, by means of an exact diagonalization method. In both triangular and kagome models, the computed $S({\bf q},\omega)$ in the random-singlet state depends on the wavevector ${\bf q}$ only weakly, robustly exhibiting gapless behaviors accompnied by the broad distribution extending to higher energy $\omega$. Especially in the strongly random kagome model, $S({\bf q},\omega)$ hardly depends on ${\bf q}$, 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.