Green Function Monte Carlo Calculations Research Articles

Quantum Monte Carlo calculations of light nuclei using chiral potentials.

We present the first Green's function MonteCarlo calculations of light nuclei with nuclear interactions derived from chiral effective field theory up to next-to-next-to-leading order. Up to this order, the interactions can be constructed in a local form and are therefore amenable to quantum MonteCarlo calculations. We demonstrate a systematic improvement with each order for the binding energies of A=3 and A=4 systems. We also carry out the first few-body tests to study perturbative expansions of chiral potentials at different orders, finding that higher-order corrections are more perturbative for softer interactions. Our results confirm the necessity of a three-body force for correct reproduction of experimental binding energies and radii, and pave the way for studying few- and many-nucleon systems using quantum MonteCarlo methods with chiral interactions.

Nuclear quantum Monte Carlo: Expanding into the continuum

I discuss calculations of nuclear properties using the variational Monte-Carlo (VMC) and Green’s function Monte-Carlo (GFMC) methods in combination with realistic two- and three-nucleon interactions. Work to date has emphasized energies of bound and narrow states, and the results agree well with measured masses and energies. We are now directing more effort toward other nuclear properties, including weak decay rates and nuclear charge radii. The main new direction is the computation of low-energy scattering and reaction properties. We have incorporated R-matrix-like boundary conditions into GFMC calculations, and our first calculation of this kind has yielded phase shifts for neutron scattering on 4He up to a few MeV. We find strong dependence of the results on the three-nucleon interaction used.

Nonlinear Schrödinger equation for a superfluid Fermi gas in the BCS-BEC crossover

We introduce a quasi-analytic nonlinear Schr\odinger equation with beyond mean-field corrections to describe the dynamics of a zero-temperature dilute superfluid Fermi gas in the crossover from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime, where $k_F|a| \ll 1$ with $a$ the s-wave scattering length and $k_F$ the Fermi momentum, through the unitarity limit, $k_Fa \to \pm \infty$, to the Bose regime where $k_Fa >0$. The energy of our model is parametrized using the known asymptotic behavior in the BCS, Bose, and the unitarity limits and is in excellent agreement with accurate Green function Monte Carlo calculations. The model generates good results for frequencies of collective breathing oscillation of a trapped Fermi superfluid.

Quantum Monte Carlo for light nuclei

Quantum Monte Carlo calculations using realistic two- and three-nucleon interactions are presented for nuclei with up to nine nucleons. Our Greens function Monte Carlo calculations are accurate to ∼1% for the binding energy. We have constructed Hamiltonians using the Argonne v 18 NN interaction and reasonable three-nucleon interactions that reproduce the energies of these nuclear states with only ∼500 keV rms error. Other predictions, such as form factors, decay rates, and spectroscopic factors also agree well with data.

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.

Proof for an upper bound in fixed-node Monte Carlo for lattice fermions.

We justify a recently proposed prescription for performing Green function Monte Carlo calculations on systems of lattice fermions, by which one is able to avoid the sign problem. We generalize the prescription such that it can also be used for problems with hopping terms of different signs. We prove that the effective Hamiltonian, used in this method, leads to an upper bound for the ground-state energy of the real Hamiltonian, and we illustrate the effectiveness of the method on small systems.

Monte-Carlo calculation of bound-state properties

We elaborate on the role of the Green's function Monte-Carlo method in connection with quantum few-body systems and their computational treatment algorithms for calculation of bound-state properties are presented. We report results obtained from Green's function Monte-Carlo calculations both of ground-state properties of the mesic moleculesdtμ,ddμ, andpdμ as well as of excited states of the three-dimensional anharmonic oscillator and thedtμ molecule.

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.

Quantum Monte Carlo study of polyexcitons in semiconductors.

A variational Monte Carlo method was used to calculate the binding energies and ground-state wave functions of two-, three-, and four-exciton complexes in indirect-band-gap semiconductors within the spherical-effective-mass approximation. The results for two-exciton complexes (biexcitons) have been compared with those of Green's-function Monte Carlo calculations which produce exact ground-state energies within statistical error; the variational results recover about 90% of the binding energy of these systems. For three- and four-exciton complexes in Si, we predict that the binding energy with respect to the smaller complex with one less exciton is about 2.2 meV, in fair agreement with the experimental value of 2.5 meV.

Ground state of the two-dimensional electron gas.

Variational and fixed-node Green's-function Monte Carlo calculations have been performed to find the ground-state properties of the two-dimensional electron gas in the density range 1\ensuremath{\le}${r}_{s}$\ensuremath{\le}100. Our calculations predict a Wigner crystallization at the density ${r}_{s}$\ensuremath{\simeq}37\ifmmode\pm\else\textpm\fi{}5. The electron system is found to be in the normal- (paramagnetic) fluid state below the transition density, but the fully polarized state is very close in energy. We have tabulated the values of pair distribution function g(r), the static structure factor S(k), and the momentum distribution n(k) at several densities of interest both in the normal and the polarized phases. An estimate of the spin susceptibility \ensuremath{\chi} is also given.

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.

Superconductivity in the dilute electron gas.

We determine the effective interaction of quasiparticles on the Fermi surface for the electron gas in a rigid positive background. The quasiparticle scattering amplitude is calculated from the two coupled Bethe-Salpeter equations for the two-particle vertex functions in the particle-hole (p-h) channels for densities from r/sub s/ = 1 to r/sub s/ = 37. The density and spin-density mean fields are fitted to the compressibility and spin susceptibility of Green's-function Monte Carlo calculations and the local-field factors G(q) of microscopic models. We find p-wave superconductivity for 1035.

Condensate fraction and momentum distribution in the ground state of liquid 4He.

Calculations of the condensate fraction and momentum distribution of liquid $^{4}\mathrm{He}$ are carried out using variational ground-state wave functions containing two-body (Jastrow) and three-body correlations. These wave functions give a satisfactory description of the equation of state of the liquid at T=0. The calculations are performed within the scheme of hypernetted-chain equations, using the scaling approximation for evaluating the contributions of the elementary diagrams. Results are reported at densities \ensuremath{\rho}=0.365${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$, 0.401${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$, and 0.438${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$, and compared with momentum distributions obtained from experimental data and Green's-function Monte Carlo calculations.

Green's-function Monte Carlo calculations on the SU(2) and U(1) lattice gauge theories

An application of the Green's-function Monte Carlo method to the Hamiltonian formulation of the SU(2) and U(1) lattice gauge theories is described. The Green's function is that of a diffusion process in the gauge group space. A small-step approximation of the diffusion distribution is used in actual calculations. Also, a variance reduction technique is implemented, importance sampling with a disordered trial wave function optimized by the variational principle. The results of computations are reported for a 3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3 spatial lattice. The quantities computed are the ground-state energy and the expectation value of the magnetic energy, as a function of the gauge coupling constant. The results are compared to variational estimates and to weak-coupling perturbation theory.