We study the effects of bond randomness in the spin-
1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ triangular Heisenberg model using exact diagonalization and density-matrix renormalization group. With increasing bond randomness, we identify a randomness-induced spin-liquid-like phase without any magnetic order, dimer order, spin-glass order, or valence-bond-glass order. The finite-size scaling of gaps suggests the gapless nature of both spin triplet and singlet excitations, which is further supported by the broad continuum of the dynamical spin structure factor. By studying the bipartite entanglement spectrum of the system on cylinder geometry, we identify the features of the low-lying entanglement spectrum in the spin-liquid-like phase, which may distinguish this randomness-induced spin-liquid-like phase and the intrinsic spin-liquid phase in the clean ${J}_{1}\ensuremath{-}{J}_{2}$ triangular Heisenberg model. We further discuss the nature of this spin-liquid-like phase and the indication of our results for understanding spin-liquid-like materials with triangular-lattice structure.
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.