High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square lattice

TL;DR

This work delivers highly precise ground-state benchmarks for the two-dimensional S=1/2 Heisenberg antiferromagnet on the square lattice using sign-free stochastic series expansion QMC, computing e0, m_s, ρ_s, χ_⊥, c, and χ_s/L^2 across large L and extrapolating to the thermodynamic limit. It rigorously tests finite-size scaling forms predicted by chiral perturbation theory, confirming leading and subleading corrections and identifying a multiplicative log correction to the L^{-2} term in m_s^2 with γ ≈ 0.82, while providing consistent estimates for ρ_s, χ_⊥, and c. The paper also extends data to open and cylindrical boundaries, revealing edge-induced distortions that decay as stretched exponentials and offering valuable benchmarks for non-periodic boundary implementations. Overall, these results supply the most precise numerical values to date for the model and reinforce the validity of the theoretical finite-size corrections, with broad relevance for benchmarking emerging numerical and variational methods.

Abstract

Several ground state properties of the square-lattice $S=1/2$ Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility, and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures $T$ sufficiently low to realize the $T \to 0$ limit. Results for periodic $L\times L$ lattices with $L \in [6,96]$ are tabulated versus $L$ and extrapolations to infinite system size are carried out. The extrapolated ground state energy density is $e_0=-0.669441857(7)$, which represents an improvement in precision of three orders of magnitude over the previously best result. The leading and subleading finite-size corrections to $e_0$ are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is $m_s=0.307447(2)$, which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in $L^{-1}$ correction to $m^2_s$ agrees with the value from chiral perturbation theory and the presence of a factor $\ln^γ(L)$ in the second-order correction is also confirmed, with the previously not known value of the exponent being $γ= 0.82(4)$. The finite-size corrections to the staggered susceptibility point to logarithmic corrections also in this quantity. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.

Figures (10)

• Figure 1: Uniform susceptibility versus the system size for three values of the aspect ratio $\beta/L$. The dashed lines show predictions based on a quantum rotor tower with the infinite-size value of the $T=0$ susceptibility, $\chi_u(\infty)=0.04379$, while the solid lines show results for $\chi_u(L)$ given by an fourth-order polynomial fitted to the finite-size data in Table \ref{['ltable']}.
• Figure 2: Probability vs the lattice size of the system being thermally excited ($P_0$ being the ground-state probability) under the valid assumption that only the rotor states contribute. The results for $\beta=32$ (black), $64$ (blue) and $128$ (red) were computed using the interpolated $\chi_\perp(L)$ function in the rotor spectrum Eq. (\ref{['erotor']}), the same as used for the solid curves in Fig. \ref{['xu_beta']}. The range of $L \in [6,96]$ corresponds to that for which SSE results are presented here.
• Figure 3: Temperature dependent $q\to 0$ susceptibility computed in two different ways in simulations with $\beta=2L^{1/2}$; with $q=0$ in $\chi_u$ and $q=2\pi/L$ in $\chi_1$. The dashed line is the $T=0$ value obtained by extrapolating the $\chi_\perp$ data in Table \ref{['ltable']} to the thermodynamic limit and adjusted for the factor $3/2$ in Eq. (\ref{['qsusc']}). The black solid line is a linear fit to the $\chi_u$ data. The other dashed lines are guides to the eye.
• Figure 4: Scatter plot of the coefficients of the cubic and quartic corrections of the energy based on 2000 fits with Gaussian noise added to the energy data in Table. \ref{['ltable']}, with the minimum size being $L_{\rm min}=6$ (green dots), $8$ (red dots), and $10$ (blue dots). The horizontal and vertical pairs of dashed lines indicate the expected values based on Eq. (\ref{['fitforms']}) with $\rho_s$ and $\chi_\perp$ taking the previously extrapolated values, plus and minus two standard deviations in each case. The two horizontal lines almost coincide on the scale used here.
• Figure 5: The squared sublattice magnetization with the fitted terms of Eq. (\ref{['form_m']}) gradually removed from the data set and the corresponding remaining functions. No higher-order terms were included. In (a), only the constant $a=m^2_s(\infty)$ has been removed. The solid red curve shows the remaining part of the fitted function and the blue dashed line represents the term $b/L$, where the coefficient $b$ is given by Eq. (\ref{['form_m']}) with the previously infinite-size extrapolated values of $\rho_s$ and $\chi_\perp$. In (b) also the linear term has been removed from both the data set and the fitted function. In (c) the remaining $\mu\ln^\gamma(L/\xi)L^{-2}$ term has been removed. There are no sign of higher-order corrections within the statistical errors.