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.
Ground state properties of the randomly diluted two-dimensional Heisenberg antiferromagnet are discussed. Quantum Monte Carlo (stochastic series expansion) data for L×L lattices, with L up to 64, are extrapolated to infinite size for dilution fractions p between 0 (clean system) and p* (the percolation threshold). The sublattice magnetization is calculated using a decomposition into a classical and a quantum mechanical factor, which are evaluated separately. The quantum mechanical factor (the sublattice magnetization of the largest connected cluster of magnetic sites) remains finite at the percolation threshold, implying that the dilution driven order-disorder transition is a classical percolation transition. The spin stiffness is calculated using the winding number fluctuations.
To understand the role of local sublattice imbalance in low-energy spectra of
s=21 quantum antiferromagnets, we study the
s=21 quantum nearest neighbor Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform many-body calculations using an implementation of the density matrix renormalization group (DMRG) technique for generic tree graphs. We discover that the bond-centered Cayley tree has a quasidegenerate set of a low-lying tower of states and an ``anomalous'' singlet-triplet finite-size gap scaling. For understanding the construction of the first excited state from the many-body ground state, we consider a wave function ansatz given by the single-mode approximation, which yields a high overlap with the DMRG wave function. Observing the ground-state entanglement spectrum leads us to a picture of the low-energy degrees of freedom being ``giant spins'' arising out of sublattice imbalance, which helps us analytically understand the scaling of the finite-size spin gap. The Schwinger-boson mean-field theory has been generalized to nonuniform lattices, and ground states have been found which are spatially inhomogeneous in the mean-field parameters.
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.