We present thermodynamic measurements of various physical observables of the two-dimensional
S=1/2 isotropic quantum Heisenberg antiferromagnet on a square lattice, obtained by quantum Monte Carlo methods. The results are in excellent agreement with field-theoretical predictions. The issue of the existence of a crossover from quantum critical to renormalized classical regime is clarified.
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.
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.
We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.
We study the spin excitation spectra of the two-dimensional spin-
21 Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions
J1 and
J2, and the tuning parameter
g is defined as
J2/J1. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When
g is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when
g is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.
Motivated by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice, we reinvestigate the phase transition of this model induced by dimerization using first-principle Monte Carlo simulations. We focus on studying the finite-size scaling of ${\ensuremath{\rho}}_{s1}2L$ and ${\ensuremath{\rho}}_{s2}2L$, where
L stands for the spatial box size used in the simulations and ${\ensuremath{\rho}}_{si}$, with $i\ensuremath{\in}{1,2}$, is the spin-stiffness in the
i-direction. Remarkably, while we observe a large correction to scaling for the observable ${\ensuremath{\rho}}_{s1}2L$, the data for ${\ensuremath{\rho}}_{s2}2L$ exhibit a good scaling behavior without any indication of a large correction. As a consequence, we are able to obtain a numerical value for the critical exponent $\ensuremath{\nu}$, which is consistent with the known
O(3) result with moderate computational effort. Further, we additionally carry out an unconventional finite-size scaling analysis with which we assume that the ratio of the spatial winding numbers squared is fixed through all simulations. The theoretical correctness of our idea is argued and its validity is confirmed. Using this unconventional finite-size scaling method, even from ${\ensuremath{\rho}}_{s1}L$, which receives the most serious correction among the observables considered in this study, we are able to arrive at a value for $\ensuremath{\nu}$ consistent with the expected
O(3) value. A detailed investigation to compare these two finite-size scaling methods should be performed.