Static and dynamical spin correlations of the $S=\frac{1}{2}$ random-bond antiferromagnetic Heisenberg model on the triangular and 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 $\kappa \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3\mathrm{and}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ and in kagome herbertsmithite ${\mathrm{ZnCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2$, 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, $S\left(\mathbf{q}\right)$ 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 wave vector $\mathbf{q}$ only weakly, robustly exhibiting gapless behaviors accompanied 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 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.

Figure 1

The temperature dependence of the rescaled sublattice magnetization per spin ${\tilde{m}}_{\mathrm{s}}$ of the triangular-lattice Heisenberg antiferromagnet for the randomness $\mathrm{\Delta }=0,0.7$, and $1.0$ and for the sizes $N=12$ and 18. The definition of ${\tilde{m}}_{\mathrm{s}}$ is given in the text.

Figure 2

Intensity plots of the static spin structure factors $S\left(\mathbf{q}\right)$ of the triangular-lattice Heisenberg antiferromagnet in the wave vector $\left(q_x,q_y\right)$ plane for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.3 (b), 0.6 (c), and 1.0 (d). The lattice constant $a=1$ is the nearest-neighbor distance of the triangular lattice. The system size is $N=30$. The solid black line depicts the first Brillouin zone of triangular lattice. Each small hexagon corresponds to the resolution unit, and the center of each distorted hexagon is the wave-vector point we can treat in $N=30$ system. The black point in (a) represents the $K$ point, while the purple point the $M$ point: see the text for details.

Figure 3

The $\omega$ dependence of the dynamical structure factors of the triangular-lattice Heisenberg antiferromagnet taken at the $K$ point, $\mathbf{q}=(\pm 2\pi /3,\pm 2\pi /\sqrt{3})$ and $(\pm 4\pi /3,0)$, for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.3 (b), 0.6 (c), and 1.0 (d). The critical randomness separating the AF state and the random-singlet state is ${\mathrm{\Delta }}_{\mathrm{c}}\simeq 0.6$. Note the difference in the ordinate scale between (a) and (b)–(d). In the inset of (a) and (b), the $\omega$ value of the dominant peak is plotted vs the inverse system size $1/N$.

Figure 4

The $\omega$ dependence of the dynamical structure factors of the triangular-lattice Heisenberg antiferromagnet taken at the $M$ point, $\mathbf{q}=(0,\pm 2\pi /\sqrt{3})$ and $(\pm \pi ,\pm \pi /\sqrt{3})$, for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.3 (b), 0.6 (c), and 1.0 (d). The critical randomness separating the AF state and the random-singlet state is estimated to be ${\mathrm{\Delta }}_{\mathrm{c}}\simeq 0.6$. Note the difference in the ordinate scale between (a) and (b)–(d), and between this figure and Fig. 3.

Figure 5

The temperature dependence of the rescaled sublattice magnetization per spin, ${\tilde{m}}_{\mathrm{s}}$, associated with the $q=0$ and $\sqrt{3}\times \sqrt{3}$ orders of the kagome-lattice Heisenberg antiferromagnet for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.4 (b), 0.7 (c), and 1.0 (d).

Figure 6

Intensity plots of the static spin structure factors $S\left(\mathbf{q}\right)$ of the kagome-lattice Heisenberg antiferromagnet in the wave-vector $\left(q_x,q_y\right)$ plane for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.4 (b), 0.7 (c), and 1.0 (d). The lattice constant $a=1$ is the nearest-neighbor distance of the kagome lattice. The system size is $N=30$. The solid black line depicts the zone boundary of the extended BZ. The black square in (a) represents the $\mathrm{\Gamma }$ point, while the blue triangle represents the $M$ point.

Figure 7

The $\omega$ dependence of the dynamical spin structure factor $S(\mathbf{q},\omega )$ of the kagome-lattice Heisenberg antiferromagnet taken at the $\mathrm{\Gamma }$ point, $\mathbf{q}=(0,\pm 2\pi /\sqrt{3})$ and $(\pm \pi ,\pm \pi /\sqrt{3})$, for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.3 (b), 0.6 (c), and 1.0 (d). The critical randomness separating the randomness-irrelevant QSL state and the random-singlet state is estimated to be ${\mathrm{\Delta }}_{\mathrm{c}}\simeq 0.4$. The insets are magnified views of the low-$\omega$ region in units of meV, where $J=17$ meV is assumed with herbertsmithite in mind. Note the difference in the ordinate scale between (a) and (b)–(d).

Figure 8

The $\omega$ dependence of the dynamical spin structure factor $S(\mathbf{q},\omega )$ of the kagome-lattice Heisenberg antiferromagnet taken at the $M$ point, $\mathbf{q}=(0,\pm \pi /\sqrt{3})$ and $(\pm \pi /2,\pm \pi /2\sqrt{3})$, for several values of the randomness $\mathrm{\Delta }=0$ (a), 0.3 (b), 0.6 (c), and 1.0 (d). The critical randomness separating the randomness-irrelevant QSL state and the random-singlet state is estimated to be ${\mathrm{\Delta }}_{\mathrm{c}}\simeq 0.4$. The insets are magnified views of the low-$\omega$ region in units of meV, where $J=17$ meV is assumed with herbertsmithite in mind. Note the difference in the ordinate scale between (a) and (b)–(d).

Figure 9

The $T=0$ magnetization curve for the randomness of $\mathrm{\Delta }=0.7$ for $N=18$ and 24, in the wider field range (inset) and in the lower field range (main panel) corresponding to the region indicated by the arrow in the inset. With herbertsmithite in mind, the magnetization $M$ is normalized per formula unit of herbertsmithite, i.e., the saturation value taken to be 3, while an applied field is given in units of tesla with assuming the experimental $g$ factor $\left(g\simeq 2.2\right)$ and $J$ value $\left(J\simeq 200\mathrm{K}\right)$ of herbertsmithite [65].

Figure 10

The rescaled $T=0$ squared sublattice magnetization per spin, $m_{\mathrm{s}}^2$, of the square-lattice Heisenberg antiferromagnet plotted vs $1/\sqrt{N}$ ($N$ the lattice size) for several values of the randomness $\mathrm{\Delta }$.

Figure 11

The temperature dependence of the rescaled sublattice magnetization per spin, ${\tilde{m}}_{\mathrm{s}}$, of the square-lattice Heisenberg antiferromagnet for the size $N=36$, and for the randomness $\mathrm{\Delta }=0$ and $1.0$.

Figure 12

Intensity plots of the $T=0$ static spin structure factor $S\left(\mathbf{q}\right)$ of the square-lattice Heisenberg antiferromagnet in the wave-vector $\left(q_x,q_y\right)$ plane for the randomness $\mathrm{\Delta }=0$ (a), and 1 (b). The solid black line represents the zone boundary of the first BZ of the square lattice.

Figure 13

The $\omega$ dependence of the $T=0$ dynamical spin structure factor $S(\mathbf{q},\omega )$ of the square-lattice antiferromagnet taken at the wave vector $\mathbf{q}=(\pi ,\pi )$ for the randomness $\mathrm{\Delta }=0$ (a) and 1 (b).

Figure 14

The lattice shapes used in the exact diagonalization calculation of the triangular-lattice model for various $N$. Periodic boundary conditions are applied in all directions.

Figure 15

The lattice shapes used in the exact diagonalization calculation of the kagome-lattice model for various $N$. Periodic boundary conditions are applied in all directions.

Figure 16

The lattice shapes used in the exact diagonalization calculation of the square-lattice model for various $N$. Periodic boundary conditions are applied in all directions.