We study the low-energy dynamics of
S=1/2 antiferromagnetic Heisenberg clusters constructed by diluting a square lattice at vacancy concentration
p at and below the percolation threshold ${p}^{\ensuremath{\ast}}\ensuremath{\approx}0.407$. The finite-size scaling behavior of the average excitation gap, $⟨\ensuremath{\Delta}⟩\ensuremath{\sim}{L}^{\ensuremath{-}z}$, where
L is the cluster length, is obtained using quantum Monte Carlo results for an upper bound ${\ensuremath{\Delta}}^{\ensuremath{\ast}}$ to $\ensuremath{\Delta}$, derived from sum rules. At the percolation threshold, we obtain a dynamic exponent $z=3.6\ifmmode\pm\else\textpm\fi{}0.1\ensuremath{\approx}2{D}_{f}$ for clusters with singlet
(S=0) ground state. Here
Df=91/48 is the fractal dimensionality of the percolating cluster. We argue that this large dynamic exponent---roughly twice that expected for quantum-rotor excitations---is a consequence of weakly interacting localized effective magnetic moments, which form due to local sublattice imbalance. This picture is supported by an extremal-value analysis of local spectral gaps, which delivers an exponent relation (between
z and two exponents characterizing the local-gap distribution) reproduced by our simulation data. However, the average $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ over all clusters, which have mostly ground-state spin
S>0, scales with a smaller exponent than for the
S=0 clusters alone; $z\ensuremath{\approx}1.5{D}_{f}$. Lanczos exact diagonalization for small clusters show that typically, $S\ensuremath{\rightarrow}S\ensuremath{-}1$ in the lowest-energy excitations while the dominant spectral weight originates from $S\ensuremath{\rightarrow}S+1$ excitations. Thus, the scaling of $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ for clusters with ground state
S>0 does not reflect the lowest-energy excitations but the higher $S\ensuremath{\rightarrow}S+1$ excitations. This result can be understood within a valence bond picture. To further explore the scenario of localized moments, we introduce a classical dimer-monomer aggregation model to study the distribution of nearest-neighbor sites forming dimers (which are the objects used in mapping to the quantum-rotor model) and unpaired spins (monomers). The monomers are localized and, thus, effective magnetic moments should form in the spin system. We also study the lowest triplet excitation of
S=0 clusters using quantum Monte Carlo calculations in the valence bond basis. The triplet is concentrated at some of the classical monomer regions, confirming the mechanism of moment formation. The number of spins (and moment regions) affected by the excitation scales as a nontrivial power of the cluster size. For a dimer-diluted bilayer Heisenberg model with weak interlayer coupling (where the system remains N\'eel ordered), there is no sublattice imbalance. In this case we find $z\ensuremath{\approx}{D}_{f}$, consistent with quantum-rotor excitations. For a single layer at $p<{p}^{\ensuremath{\ast}}$ we find $z\ensuremath{\approx}2=D$, which indicates that the weakly interacting localized moment mechanism is valid only exactly at the percolation point. There is a crossover behavior close to the percolation point.
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 approximately L(-), we obtain a dynamic exponent z approximately 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.
We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.
The scaling behavior of the dynamical structure factor S(q,\ensuremath{\omega}) of percolating Heisenberg antiferromagnets is investigated in terms of a dynamic scaling argument and numerical calculations. It is found, based on the single-length-scaling postulate, that the asymptotic behavior of S(q,\ensuremath{\omega}) can be characterized by only two exponents: the dynamic exponent
za and a new exponent y. This theoretical prediction is confirmed by direct numerical calculations. The values of the exponents
za and y are determined for the case of d=2 percolating Heisenberg antiferromagnets.
The two-dimensional antiferromagnetic
S=1/2 Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number
Nc of spins on an infinite lattice at the percolation density are studied for
Nc up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality
d=91/48, have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.