Low-Energy Dynamics of the Two-Dimensional $S=1/2$ Heisenberg Antiferromagnet on Percolating Clusters

We investigate the quantum dynamics of site diluted $S=1/2$ Heisenberg antiferromagnetic clusters at the 2D percolation threshold. We use Lanczos diagonalization to calculate the lowest excitation gap $\Delta$ and, to reach larger sizes, use quantum Monte Carlo simulations to study an upper bound for $\Delta$ obtained from sum rules involving the staggered structure factor and susceptibility. Scaling the gap distribution with the cluster length $L$, $\Delta \sim L^{-z}$, we obtain a dynamic exponent $z\approx 2D_f$, where $D_f=91/48$ is the fractal dimensionality of the percolating cluster. This is in contrast with previous expectations of $z=D_f$. We argue that the low-energy excitations are due to weakly coupled effective moments formed due to local imbalance in sublattice occupation.

Figure 1

(a) Singlet-triplet gap distribution for clusters of size $n=16$, 18, and 20. Representative clusters corresponding to the two peaks are shown. (b) Distribution of the upper bound ${\Delta }^{*}$ for the gap for $n=16$, 32, and 64.

Figure 2

Finite-size scaling of the average of the gap $\Delta$ and the upper bound ${\Delta }^{*}$. The slope of the line fitted to $⟨{\Delta }^{*}⟩$ is $z/D_f=1.99$.

Figure 3

Local gaps ${\Delta }_i$ for $n=128$ clusters, shown as a color coded mapping of $-\ln ({\Delta }_i)$ to $[0,1]$ (for each cluster separately). The smallest gaps are ${\Delta }_{\min }/J=0.027$ (left), 0.014 (middle), and 0.012 (right). The largest gaps are ${\Delta }_{\max }/J=0.9$ (left and right) and 0.22 (middle).

Figure 4

(a) Distribution of the scaled gap ${ϵ}_i={\Delta }_i{L}^a$, with $a=2.84$. The line corresponds to a gap exponent $\omega =1$. (b) Distribution of the upper bound ${\Delta }^{*}$ scaled with dynamic exponent $z=3.6$. The line shows the asymptotic small-gap behavior with $\omega =1$. The dashed curve is the Frechet distribution, Eq. (9).