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Physical review. B, Condensed matter

Finite-size study of the ground-state energy, susceptibility, and spin-wave velocity for the Heisenberg antiferromagnet

Karl J. Runge

Published Jun 1, 1992• 58 citations
Publisher

Abstract

The Green's-function Monte Carlo (GFMC) method is used to calculate very accurate ground-state energies of the two-dimensional, spin-1/2 Heisenberg antiferromagnet. The computations are performed on L\ifmmode\times\else\texttimes\fi{}L square lattices up to L=16 with varying uniform magnetization, which allows the extraction of the perpendicular susceptibility (\ensuremath{\chi}) and spin-wave velocity (c). These two quantities are the lattice- or cutoff-dependent parameters that allow one to map the long-wavelength properties of the antiferromagnet onto the nonlinear \ensuremath{\sigma} model and so are of general interest. Systematic errors present in previous GFMC calculations are addressed and corrected to yield results in excellent agreement with other numerical methods. I find, for the ground-state energy per site, -0.669 34(3); the susceptibility renormalization factor, ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$=0.535(5); and the spin-wave velocity renormalization factor, ${\mathit{Z}}_{\mathit{c}}$=1.10(3). Finite-size effects in the extraction of ${\mathit{Z}}_{\mathit{c}}$ and ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ are discussed. The value of ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ computed here is in agreement with the series-expansion results of Singh and of Zheng, Oitmaa, and Hamer, thereby clearing up a previous inconsistency between the series-expansion and quantum Monte Carlo predictions.

DOI
10.1103/physrevb.45.12292

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The Green's-function Monte Carlo (GFMC) method is used to calculate very accurate ground-state energies of the two-dimensional, spin-1/2 Heisenberg antiferromagnet. The computations are performed on L\ifmmode\times\else\texttimes\fi{}L square lattices up to L=16 with varying uniform magnetization, which allows the extraction of the perpendicular susceptibility (\ensuremath{\chi}) and spin-wave velocity (c). These two quantities are the lattice- or cutoff-dependent parameters that allow one to map the long-wavelength properties of the antiferromagnet onto the nonlinear \ensuremath{\sigma} model and so are of general interest. Systematic errors present in previous GFMC calculations are addressed and corrected to yield results in excellent agreement with other numerical methods. I find, for the ground-state energy per site, -0.669 34(3); the susceptibility renormalization factor, Z\ensuremathχ{\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}Z\ensuremathχ​=0.535(5); and the spin-wave velocity renormalization factor, Zc{\mathit{Z}}_{\mathit{c}}Zc​=1.10(3). Finite-size effects in the extraction of Zc{\mathit{Z}}_{\mathit{c}}Zc​ and Z\ensuremathχ{\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}Z\ensuremathχ​ are discussed. The value of Z\ensuremathχ{\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}Z\ensuremathχ​ computed here is in agreement with the series-expansion results of Singh and of Zheng, Oitmaa, and Hamer, thereby clearing up a previous inconsistency between the series-expansion and quantum Monte Carlo predictions.
The ground-state parameters of the two-dimensional S=1/2S=1/2S=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 LLL up to 161616. The finite-size results for the energy EEE, the sublattice magnetization MMM, 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/L1/L1/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)M=0.3070(3)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)c=1.673(7)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 MMM and ${\ensuremath{\rho}}_{s}$ obtained from the finite-TTT data are, however, a few error bars higher than the present estimates. It is argued that the T=0T=0T=0 extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.
We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site E0{\mathit{E}}_{0}E0​/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be Zc{\mathit{Z}}_{\mathit{c}}Zc​==c/cs{\mathit{c}}_{\mathit{s}}cs​=1.14(5).
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, 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 → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous 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 = 0.307 447 ( 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 s 2 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 unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. 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.
We study the spin-1/2 Heisenberg antiferromagnet on a square lattice using variational Monte Carlo calculations. We construct a variational wave function which gives accurate ground-state energy and is consistent with sum rules for the dynamical structure function. The spin-wave velocity and the spin-stiffness constant can be calculated using sum rules. We obtain Zc{Z}_{c}Zc​==c/c0{c}_{0}c0​, where c0{c}_{0}c0​ is the classical spin-wave velocity. This and other ground-state quantities are in good agreement with spin-wave theory and Monte Carlo calculations.
We describe in detail a simple and efficient Green function Monte Carlo technique for computing both the ground state energy and the ground state properties by the ``forward walking'' scheme. The simplicity of our reconfiguration process, used to maintain the walker population constant, allows us to control any source of systematic error in a rigorous and systematic way. We apply this method to the Heisenberg model and obtain accurate and reliable estimates of the ground state energy, the order parameter, and the static spin structure factor S(q)S(q)S(q) for several momenta. For the latter quantity we also find very good agreement with available experimental data on the La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ antiferromagnet.
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.