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 two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.
Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.
The ground-state parameters of the two-dimensional
S=1/2 antiferromagnetic Heisenberg model are calculated using the stochastic series expansion quantum Monte Carlo method for $L\ifmmode\times\else\texttimes\fi{}L$ lattices with
L up to
16. The finite-size results for the energy
E, the sublattice magnetization
M, the long-wavelength susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(q=2\ensuremath{\pi}/L)$, and the spin stiffness ${\ensuremath{\rho}}_{s},$ are extrapolated to the thermodynamic limit using fits to polynomials in
1/L, constrained by scaling forms previously obtained from renormalization-group calculations for the nonlinear $\ensuremath{\sigma}$ model and chiral perturbation theory. The results are fully consistent with the predicted leading finite-size corrections, and are of sufficient accuracy for extracting also subleading terms. The subleading energy correction $(\ensuremath{\sim}{1/L}^{4})$ agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground- state energy per spin is $E=\ensuremath{-}0.669437(5)$. The result from previous Green's function Monte Carlo (GFMC) calculations is slightly higher than this value, most likely due to a small systematic error originating from ``population control'' bias in GFMC. The other extrapolated parameters are
M=0.3070(3), ${\ensuremath{\rho}}_{s}=0.175(2)$, ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0625(9)$, and the spin-wave velocity
c=1.673(7). The statistical errors are comparable with those of previous estimates obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both
M and ${\ensuremath{\rho}}_{s}$ obtained from the finite-
T data are, however, a few error bars higher than the present estimates. It is argued that the
T=0 extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
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.