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,
Zc=1.10(3). Finite-size effects in the extraction of
Zc 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.
We present the first exact results for the T=0 dynamical structure factor of finite-sized two-dimensional square-lattice spin-(1/2 antiferromagnetic Heisenberg systems with up to 16 spins. Our results are compared to a recently proposed Schwinger boson mean-field theory. We find that the mean-field theory represents an excellent approximation to the exact spin excitation spectra at all allowed wave vectors and for all system sizes studied. We also determine the mean-field value for the T=0 staggered magnetization,
mMF=0.306, to be in excellent agreement with published Monte Carlo results.
We present a variational wave function for the ground state of the two-dimensional spin-1/2 Heisenberg antiferromagnet. It is a modification of a Monte Carlo variational wave function proposed by Huse and Elser which gives excellent results for the ground-state energy but breaks the rotational symmetry of the squared staggered magnetization. In their case the x component of the squared staggered magnetization (${\mathit{m}}_{\mathit{x}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$
)2 is much larger than (${\mathit{m}}_{\mathit{y}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$
)2 and (${\mathit{m}}_{\mathit{z}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$
)2, which are the y and z components, respectively. In this paper we show how to restore the symmetry between ${\mathit{m}}_{\mathit{x}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and ${\mathit{m}}_{\mathit{y}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$. However, ${\mathit{m}}_{\mathit{z}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ is still much smaller than ${\mathit{m}}_{\mathit{x}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$. We also study the enforcement of the full rotational symmetry in our trial wave function using only two spin interaction terms. Finally, we discuss how errors in Monte Carlo data affect the efficiency of algorithms which determine more systematically the best variational parameters for the ground-state wave function.
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.
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.
Conventional spin-wave approach relies on an expansion around a relevant classical (unentangled) ground state or equivalently the ground state in the mean field approximation for the quantum model.However, for systems which may admit valence bond ground states, such as the staggered J-J Heisenberg antiferromagnet, single site mean field approximation is obviously an incorrect zeroth order approximation.In this paper, we introduce a mean field approximation for clusters consisting of two spins connected by a strong bond in the aforementioned model.We identify the quantum critical point and calculate the ground state magnetization within this cluster mean field approximation in one and two dimensions.Finally, we derive an effective dimer Hamiltonian, in the standard basis operator formalism, which may be a basis for further spin-wave like expansions around the cluster mean field approximation ground state.