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

TLDR

We compute ground‑state parameters of the 2D S=1/2 antiferromagnetic Heisenberg model on L×L lattices up to L=16 using stochastic series expansion quantum Monte Carlo, then extrapolate energy, sublattice magnetization, transverse susceptibility, and spin stiffness to the thermodynamic limit with polynomial fits constrained by renormalization‑group and chiral‑perturbation‑theory scaling forms. The extrapolated values—E=−0.669437(5), M=0.3070(3), ρ_s=0.175(2), χ_⊥=0.0625(9), and c=1.673(7)—agree with leading finite‑size corrections and confirm subleading 1/L^4 energy corrections to within a few percent, while previous GFMC estimates are slightly higher due to population‑control bias.

Abstract

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.