Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model

The ground-state parameters of the two-dimensional $S=1/2\mathrm{}$ antiferromagnetic Heisenberg model are calculated using the stochastic series expansion quantum Monte Carlo method for $L\times L$ lattices with $L$ up to $16$. The finite-size results for the energy $E$, the sublattice magnetization $M$, the long-wavelength susceptibility ${\chi }_{\perp }(q=2\mathrm{}\pi /L)$, and the spin stiffness ${\rho }_s,$ 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 $\sigma$ 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 $\left(\sim {1/L}^4\right)$ 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 $E=-0.669437\left(5\right)\mathrm{}$. 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 $M=0.3070\mathrm{}\left(3\right)\mathrm{}$, ${\rho }_s=0.175\mathrm{}\left(2\right)\mathrm{}$, ${\chi }_{\perp }=0.0625\mathrm{}\left(9\right)\mathrm{}$, and the spin-wave velocity $c=1.673\mathrm{}\left(7\right)\mathrm{}$. 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 ${\rho }_s$ obtained from the finite-$T$ data are, however, a few error bars higher than the present estimates. It is argued that the $T=0\mathrm{}$ extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.