Green's Function Monte Carlo Research Articles

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

The ground-state parameters of the two-dimensional $S=1/2$ 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.

The Use of Parameter-Free Correlation Functions in Green's Function Monte Carlo Simulations of Small Electronic Systems

Abstract Using an old formulation for correlation functions with correct cusp-behaviour, the Schrödinger equation transforms to a new differential equation which provides a very simple expression for the local electronic energy with limited range. This, together with the simplicity of the formulation promises a high performance in Green's function Monte Carlo (GFMC) simulations of small electronic systems. The behaviour of the local energy is studied on a few simple examples because the variance of this function determines the quality of the results in the GFMC methods. Calculations for one-and two-electron systems are presented and compared with results from well-known functions. The form of the function is then extended to systems with more than two electrons. Results for the Be atom are given and the extension to larger electronic systems is discussed.

Visualization of the local contribution to the nodal surface of a many-fermion wave function.

An essential part of variational Monte Carlo or Green's function Monte Carlo (GFMC) algorithms is the trial wave function. In the case of particles obeying Fermi statistics, this wave function is antisymmetric and cannot be interpreted directly as a probability distribution, thereby making calculations difficult. Some progress can be made in GFMC algorithms, however, by requiring the trial wave function to have the same ``fixed'' nodes as the variational function. The sensitivity of the energy to changes in the nodal surface has remained a significant unresolved issue for two reasons: (a) the many-dimensional nodal surface is hard to quantify or visualize and (b) it has been difficult to vary this surface in a controlled fashion. As a first step toward gaining further information, we have developed a method of quantifying the many-dimensional nodal surface by creating and visually characterizing a related surface. The related surface is the surface of the local volume in which any given particle can range without changing the overall sign of the wave function, and this surface is mapped for different configurations and wave functions. A comparison of these surfaces allows us to characterize the local contributions to the nodal surface of a particular wave function and thereby visually differentiate between different types of wave functions. We show that introducing backflow correlations into the trial wave function results in a constrained volume with the undesired effect of significantly slowing down GFMC calculations. This study demonstrates that advanced visualization methods can serve a useful role in the process of algorithm development rather than just in the presentation of results. \textcopyright{} 1996 The American Physical Society.

Scattering in the Framework of the Green's Function Monte-Carlo Method

A method based on the Green's Function Monte-Carlo approach is used to solve the integral equation for the scattering wave function. The algorithm employed is explained for two-body scattering and tested for cases where the exact solution is known.

Calculation of the α-Particle Ground State

The correlated hyperspherical harmonic expansion method is used to calculate α-particle properties with a realistic Hamiltonian consisting of the Argonne V14 two-nucleon and Urbana model VIII three-nucleon potentials. The calculated binding energy, mass radius and wave percentages are close to the corresponding quantities obtained with Green's-function Monte-Carlo and Faddeev-Yakubovsky techniques.

Trial shadow wave function for the ground state of 4He.

A trial shadow wave function is introduced to describe the ground state of $^{4}\mathrm{He}$ in the solid and liquid phases. We have used Monte Carlo integration to optimize the parameters of this function, and have carried out a thorough analysis of the shadow description of the system. This shows improved pair correlations, an improved condensate fraction, substantially reduced variational energies, and a good equation of state. We have explored the melting and freezing transition, and find the transition densities to be in good agreement with the exact results of Green's function Monte Carlo (GFMC) calculations. We introduce a second shadow wave function in which a basis set expansion is used to optimize the two-particle correlations. This shadow wave function yields pair-correlation functions in excellent agreement with GFMC, as well as a substantial reduction in the variational energies at all densities.

Quantum simulation of the electronic structure of diatomic molecules

The domain Green’s function Monte Carlo (GFMC) method with the fixed-node approximation is used to calculate the binding energies and correlation energies of most of the first row diatomic molecules. Our results show that the many-electron correlations are significant and that GFMC corrections to Hartree–Fock wave functions produce 80% to 90% of the correlation energy. Our GFMC calculations have an accuracy comparable to or better than standard methods in quantum chemistry.

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

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.

Quantum Monte Carlo calculation of the long-range order in the Heisenberg antiferromagnet.

A Green's-function Monte Carlo (GFMC) method is used to compute the staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ in the two-dimensional, spin-1/2 Heisenberg antiferromagnet on L\ifmmode\times\else\texttimes\fi{}L square lattices, up to L=12. Unlike previous GFMC calculations, the present method, which uses the forward-walking algorithm is unbiased and projects out the exact, rotationally invariant ground state. These calculations provide confirmation of the existence of long-range antiferromagnetic order in the ground state. A known relationship between ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and the leading finite-size correction, coupled with high-precision ground-state-energy calculations, is used to reduce the error in extrapolating to the thermodynamic limit. The data extrapolate to ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.3075\ifmmode\pm\else\textpm\fi{}0.0025, only slightly different from the spin-wave-theory result, 0.3034. Several perfect singlet trial wave functions used to reduce the statistical error are discussed. A possible explanation as to why exact-diagonalization extrapolations tend to yield low values of ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ is presented.

Aziz fermions and liquid 3He.

Starting from the well-known quantum-field-theory low-density expansion for the ground-state energy of a strongly interacting many-fermion assembly, we rearrange the series into a quantum-thermodynamic perturbation theory that is then applied to liquid $^{3}\mathrm{He}$. Extrapolation to physical densities is implemented order by order via Pad\'e and related approximants. We employ both the hard-core square-well potential and the Aziz pair potential with the Barker-Henderson (BH) and Weeks-Chandler-Andersen (WCA) potential-decomposition schemes familiar from classical fluid studies. The binding energy, saturation density, and sound velocity are calculated, with no adjustable parameters, and compared with Green-function Monte Carlo (GFMC) computer simulations, experimental data, and recent variational calculations. Our binding-energy and saturation-density results for the Aziz-WCA case, and the sound velocity result for the Aziz-BH case, are found to be within 2% of the GFMC data.

Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study.

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 ${\mathit{E}}_{0}$/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 ${\mathit{Z}}_{\mathit{c}}$==c/${\mathit{c}}_{\mathit{s}}$=1.14(5).

Numerical study of ground state and low lying excitations of quantum antiferromagnets

We have studied via Green Function Monte Carlo (GFMC) the S=1/2 Heisenberg quantum antiferromagnet in two dimensions on a square lattice. We obtain the ground state energy with only statistical errors E0/J= -0.6692(2), the staggered magnetization m†=0.31(2), and from the long wave length behavior of the structure factor, the spin wave velocity c/cO=1.14(5). We also show that the ground state wave function has long range pair correlations arising from the zero point motion of spin waves.

Green-function Monte Carlo study of quantum antiferromagnets.

We have studied via Green-function Monte Carlo (GFMC) technique the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions on a square lattice. GFMC is a T=0 stochastic method that projects out the component of the ground state in a given variational wave function. From studies on lattices up to 12\ifmmode\times\else\texttimes\fi{}12, we find the ground-state energy per site ${E}_{0}$/J=-0.6692(2). We include the zero-point motion of the elementary excitations in the ground state and show that it produces long-range correlations in the wave function. We obtain a staggered magnetization ${m}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) in units in which the classical N\'eel state is 0.5. The structure factor at long wavelengths is S(q)\ensuremath{\sim}q and from the slope we deduce the spin-wave velocity.

Fermion Monte Carlo algorithms and liquid 3He.

Variational Monte Carlo and several many-fermion Green's-function Monte Carlo (GFMC) algorithms are used to study the ground state of liquid $^{3}\mathrm{He}$. We report the first mirror potential GFMC calculations in a many-fermion problem, comparing them with transient estimation and fixed-node studies to illustrate the strengths and weaknesses of each. GFMC results with the Aziz HFDHE2 interaction are in good agreement with experiment, yielding energies within approximately 0.1 K per particle. In addition, each of these calculations predict a kinetic energy per particle of between 12 and 12.5 K.

Model ground state calculations with two-variable integro-differential equations for 16O

The accuracy of an integro-differential equation, incorporating all two-body correlations, is tested against the variational and Green function Monte Carlo (GFMC), and the hypernetted chain methods for 16 bosons and fermions, with the Malfiiet-Tjon V potential. Good agreement is obtained with all the other methods or fermions, except for the GFMC, at a minute fraction of the numerical effort.

Microscopic calculation of a model 16O nucleus

We present variational calculations for a hypothetical system of 16 nucleons interacting via the Malfliet-Tjon potential and compare our results with the Green-function Monte Carlo (GFMC) data of Helmbrecht et al. For a system of 16 bosons, we find that the (approximate) ground-state energy of our optimized hypernetted-chain calculation agrees with the GFMC data within 1%. For fermions, we find that the optimized Fermi hypernetted chain calculation leads to a ground state energy between the GFMC data and the variational Monte Carlo result with non-optimized correlations; the one-body density predicted by the FHNC theory agrees well with the one predicted by the GFMC calculation.

Monte Carlo calculations of atoms and molecules

The variational and Green's function Monte Carlo (GFMC) methods can treat many interesting atomic and molecular problems. These methods can give chemical accuracy for up to 10 or so electrons. The various implementations of the GFMC method, including the domain Green's function method and the short-time approximation, are discussed. Results are presented for several representative atoms and molecules.

The statistical error of green's function Monte Carlo

The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached withN identical molecules, the computer time needed to reach a given error per molecule increases asN h where 0.5 <b < 1.5 and as the nuclear chargeZ of a system is increased the computer time necessary to reach a given error grows asZ 5.5. Thus GFMC simulations will be most useful for calculating the properties of lowZ elements. The implications for choosing the optimal trial function from a series of trial functions is also discussed.

Application of the Green's-function Monte Carlo method to the compact Abelian lattice gauge theory

We have applied the Green's-function Monte Carlo (GFMC) method to the Hamiltonian formulation of the compact U(1) lattice gauge theory in three and two (space) dimensions on small lattices, 3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3 and 5\ifmmode\times\else\texttimes\fi{}5. The GFMC method is a Monte Carlo method of finding the ground state of a quantum-mechanical system with many degrees of freedom, by iteration of an integral operator of which the ground state is an eigenstate. An interesting aspect of this method is an importance-sampling technique that makes use of a trial wave function to accelerate convergence of the Monte Carlo estimates. We used two importance functions in these calculations, which were designed to be accurate in the small- and large-coupling limits. These importance functions were optimized by the variational principle; the results of the variational calculations are interesting in their own right. Our Monte Carlo results exhibit evidence of the phase transition of the three-dimensional compact U(1) lattice gauge theory, and indicate the nonexistence of a phase transition in the two-dimensional theory.

Three-body correlations in liquidHe4

We report results of variational calculations of liquid $^{4}\mathrm{He}$ with wave functions containing optimized two-body and three-body correlations. The hypernetted-chain (HNC) summation method is used, and the elementary and Abe diagrams are calculated with the scaling approximation. Comparisons with the existing Monte Carlo calculations suggest that this HNC-scaling method is almost exact. The logarithm of the three-body correlation of ${f}_{\mathrm{ijk}}$ contains terms having ${P}_{l}({\stackrel{^}{r}}_{\mathrm{ij}}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{^}{r}}_{\mathrm{ik}})$, $l=0, 1, \mathrm{and} 2$. As expected on theoretical grounds, the $l=1$ term of $\mathrm{ln}{f}_{\mathrm{ijk}}$ dominates, while the $l=0 \mathrm{and} 2$ terms give rather small changes in the binding energy. The ${f}_{\mathrm{ijk}}$ makes up \ensuremath{\sim}85% of the difference between the Jastrow and presumably exact Green's-function Monte Carlo (GFMC) energies. The best variational energies obtained with the HFDHE2 potential of Aziz et al. are within (2 \ifmmode\pm\else\textpm\fi{} 1)% of the GFMC and experimental results. The liquid structure function $S(k)$ is also well explained by the variational wave function.