Variational Monte Carlo Results Research Articles

Neural network ansatz for periodic wave functions and the homogeneous electron gas

The authors introduce here WAPNET, a periodic neural network (NN) variational ansatz for solving the ground state of a homogeneous electron gas with high accuracy over a broad range of the density coupling constant ${r}_{s}$. Going beyond recent work for molecules, this contribution establishes NN models as flexible and powerful ansatz for electronic structure calculations in extended systems. In all density regimes, WAPNET-based variational Monte Carlo results are comparable to or better than state-of-the-art benchmarks obtained by diffusion Monte Carlo with iterative backflow.

Neutrinoless double- β decay: Combining quantum Monte Carlo and the nuclear shell model with the generalized contact formalism

We devise a framework based on the generalized contact formalism that combines the nuclear shell model and quantum Monte Carlo methods and compute the neutrinoless double-beta decay of experimentally relevant nuclei, including $^{76}$Ge, $^{130}$Te, and $^{136}$Xe. In light nuclei, we validate our nuclear matrix element calculations by comparing against accurate variational Monte Carlo results. Due to additional correlations captured by quantum Monte Carlo and introduced within the generalized contact formalism, in heavier systems, we obtain long-range nuclear matrix elements that are about 30% smaller than previous shell-model results. We also evaluate the recently recognized short-range nuclear matrix element estimating its coupling by the charge-independence-breaking term of the Argonne $v_{18}$ potential used in the Monte Carlo calculations. Our results indicate an enhancement of the total nuclear matrix element by around 30%.

Space-warp coordinate transformation for efficient ionic force calculations in quantum Monte Carlo.

Ab initio quantum Monte Carlo (QMC) methods are a state-of-the-art computational approach to obtaining highly accurate many-body wave functions. Although QMC methods are widely used in physics and chemistry to compute ground-state energies, calculation of atomic forces is still under technical/algorithmic development. Very recently, force evaluation has started to become of paramount importance for the generation of machine-learning force-field potentials. Nevertheless, there is no consensus regarding whether an efficient algorithm is available for the QMC force evaluation, namely, one that scales well with the number of electrons and the atomic numbers. In this study, we benchmark the accuracy of all-electron variational Monte Carlo (VMC) and lattice-regularized diffusion Monte Carlo (LRDMC) forces for various mono- and heteronuclear dimers (1 ≤ Z ≤ 35, where Z is the atomic number). The VMC and LRDMC forces were calculated with and without the so-called space-warp coordinate transformation (SWCT) and appropriate regularization techniques to remove the infinite variance problem. The LRDMC forces were computed with the Reynolds (RE) and variational-drift (VD) approximations. The potential energy surfaces obtained from the LRDMC energies give equilibrium bond lengths (req) and harmonic frequencies (ω) very close to the experimental values for all dimers, improving the corresponding VMC results. The LRDMC forces with the RE approximation improve the VMC forces, implying that it is worth computing the DMC forces beyond VMC despite the higher computational cost. The LRDMC forces with the VD approximations also show improvement, which unfortunately comes at a much higher computational cost in all-electron calculations. We find that the ratio of computational costs between QMC energy and forces scales as Z∼2.5 without the SWCT. In contrast, the application of the SWCT makes the ratio independent of Z. As such, the accessible QMC system size is not affected by the evaluation of ionic forces but governed by the same scaling as the total energy one.

Plasma Confined Ground and Excited State Helium Atom: A Comparison Theorem Study Using Variational Monte Carlo and Lagrange Mesh Method

The energy eigenvalues of the ground state helium atom and lowest two excited states corresponding to the configurations 1s2s embedded in the plasma environment using Hulthén, Debye–Hückel and exponential cosine screened Coulomb model potentials are investigated within the variational Monte Carlo method, starting with the ultracompact trial wave functions in the form of generalized Hylleraas–Kinoshita functions and Guevara–Harris–Turbiner functions. The Lagrange mesh method calculations of energy are reported for the He atom in the ground and excited 1S and 3S states, which are in excellent agreement with the variational Monte Carlo results. Interesting relative ordering of eigenvalues are reported corresponding to the different screened Coulomb potentials in the He ground and excited electronic states, which are rationalized in terms of the comparison theorem of quantum mechanics.

Unitary Dynamics of Strongly Interacting Bose Gases with the Time-Dependent Variational Monte Carlo Method in Continuous Space

We introduce time-dependent variational Monte Carlo for continuous-space Bose gases. Our approach is based on the systematic expansion of the many-body wave-function in terms of multi-body correlations and is essentially exact up to adaptive truncation. The method is benchmarked by comparison to exact Bethe-ansatz or existing numerical results for the integrable Lieb-Liniger model. We first show that the many-body wave-function achieves high precision for ground-state properties, including energy and first-order as well as second-order correlation functions. Then, we study the out-of-equilibrium, unitary dynamics induced by a quantum quench in the interaction strength. Our time-dependent variational Monte Carlo results are benchmarked by comparison to exact Bethe ansatz results available for a small number of particles, and also compared to quench action results available for non-interacting initial states. Moreover, our approach allows us to study large particle numbers and general quench protocols, previously inaccessible beyond the mean-field level. Our results suggest that it is possible to find correlated initial states for which the long-term dynamics of local density fluctuations is close to the predictions of a simple Boltzmann ensemble.

Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor Hamiltonians

In a recent contribution [Phys. Rev. B 81, 165104 (2010)] fermionic Projected Entangled-Pair States (PEPS) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t$-$J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions, and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest-neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.

Improved variational many-body wave function in light nuclei

We propose and implement a simple method for improving the variational wave function of a many-body system. We have obtained a significant improvement in the binding energies, wave functions, and variance for the light nuclei {sup 3}H, {sup 4}He, and {sup 6}Li, using the fully realistic Argonne (AV{sub 18}) two-body and Urbana-IX (UIX) three-body interactions. The energy of {sup 4}He was improved by about 0.2 MeV and the {sup 6}Li binding energy was increased by {approx_equal}1.7 MeV compared to earlier variational Monte Carlo results. The latter result demonstrates the significant progress achieved by our method, and detailed analyses of the improved results are given. With central interactions the results are found to be in agreement with the 'exact' calculations. Our study shows that the relative error in the many-body wave functions, compared to two-body pair correlations, increases rapidly at least proportionally to the number of pairs in the system. However, this error does not increase indefinitely since the pair interactions saturate owing to convergence of cluster expansion.

Spontaneous breaking of the Fermi-surface symmetry in thet−Jmodel: A numerical study

We present a variational Monte Carlo (VMC) study of spontaneous Fermi-surface symmetry breaking in the $t\text{\ensuremath{-}}J$ model. We find that the variational energy of a Gutzwiller projected Fermi sea is lowered by allowing for a finite asymmetry between the $x$ and the $y$ directions. However, the best variational state remains a pure superconducting state with $d$-wave symmetry, as long as the underlying lattice is isotropic. Our VMC results are in good overall agreement with slave boson mean field theory (SBMFT) and renormalized mean field theory (RMFT), although apparent discrepancies do show up in the half-filled limit, revealing some limitations of mean field theories. VMC and complementary RMFT calculations also confirm the SBMFT predictions that many-body interactions can enhance any anisotropy in the underlying crystal lattice. Thus, our results may be of consequence for the description of strongly correlated superconductors with an anisotropic lattice structure.

Variational Monte Carlo Study of Flat Band Ferromagnetism –Application to CeRh3B2–

A new mechanism for ferromagnetism in CeRh_3B_2 is proposed on the basis of variational Monte Carlo results. In a one-dimensional Anderson lattice where each 4f electron hybridizes with a ligand orbital between neighboring Ce sites, ferromagnetism is stabilized due to a nearly flat band which is a mixture of conduction and 4f electron states. Because of the strong spin-orbit interaction in 4f electron states, and of considerable amount of hybridization in the nearly flat band, the magnetic moments from 4f and conduction electrons tend to cancel each other. The resultant ferromagnetic moment becomes smaller as compared with the local 4f moment, and the Fermi surface in the ferromagnetic ground state is hardly affected by the presence of 4f electrons. These theoretical results are consistent with experimental observations in CeRh_3B_2 by neutron scattering and dHvA effects.

Isotopic Effects in Solid LiH and LiD at Very Low Temperature

The ground state of the ionic solids LiD and LiH is theoretically studied using the Variational Monte Carlo (VMC) method. Our main focus has been the calculation of relevant properties of the H− and D− ions using a fully quantum approach. In particular, we report results on their kinetic energies, mean-squared displacements, Einstein frequencies, radial distribution functions, and density profiles around the sites. The microscopic results obtained for both isotopes show corrections beyond trivial isotopic effects due to their quantum behavior. Finally, the VMC results are compared with predictions from self consistent average phonon (SCAP) theory at T = 0 K.

A semi-empirical approach to electron correlations based on a non-local formalism

We present a new correlation function based on variational Monte Carlo results. The function is, for ease of implementation, semi-local, spin-independent, and semi-empirical. This approach is applied to semiconductor systems and the quantity of interest is the fundamental gap energy. We found that except for InN, these gap energies improved remarkably when compared to local density approximation results.

One-body density matrix, natural orbits, and quasihole states in16Oand40Ca

The one body density matrix, momentum distribution, natural orbits and quasi hole states of 16O and 40Ca are analyzed in the framework of the correlated basis function theory using state dependent correlations with central and tensor components. Fermi hypernetted chain integral equations and single operator chain approximation are employed to sum cluster diagrams at all orders. The optimal trial wave function is determined by means of the variational principle and the realistic Argonne v8' two-nucleon and Urbana IX three-nucleon interactions. The correlated momentum distributions are in good agreement with the available variational Monte Carlo results and show the well known enhancement at large momentum values with respect to the independent particle model. Diagonalization of the density matrix provides the natural orbits and their occupation numbers. Correlations deplete the occupation number of the first natural orbitals by more than 10%. The first following ones result instead occupied by a few percent. Jastrow correlations lower the spectroscopic factors of the valence states by a few percent (~1-3%) and an additional ~8-12% depletion is provided by tensor correlations. It is confirmed that short range correlations do not explain the spectroscopic factors extracted from (e,e'p) experiments. 2h-1p perturbative corrections in the correlated basis are expected to provide most of the remaining strength, as in nuclear matter.

Three-Body Correlations on the Antiferromagnetic State - An Extension of Local Approximation -

We propose an approximate version of the three-body scattering theory as an extension of local approximation. We solve the three-body problem by limiting the number of three-body states, instead of solving the Faddeev equation, in the calculation of the self-energy. Applying this simple extension to the antiferromagnetic state at half-filling in the two-dimensional Hubbard model, we demonstrate that it gives a reasonable spectral function for the single-particle Green's function in comparison with the results of the exact diagonalization for finite clusters and the quantum Monte Carlo method. We also show that it gives the ground state energy and the sublattice magnetization comparable to previous variational Monte Carlo results.

Variationally processed Monte Carlo transport theory

Monte Carlo methods are widely used in neutron and reactor studies. They commonly are accelerated by a variety of variance reduction techniques, including adjoint weighting. We propose a new use of the adjoint — in a variational principle to process the Monte Carlo results. In many cases the estimation of the adjoint function is available from the Monte Carlo simulation without further sampling. In this paper we explore a simple model, chosen for known exact results, to compare the benefit of such a variational processing over straight analogue Monte Carlo simulation. The variational Monte Carlo results, for this system, are shown to yield improvement in the accuracy for the same computational cost. Despite the fact that variational results when used during Monte Carlo runs can be biased, they are associated with large reductions in variance. Comparing the ways in which variational expressions are processed, the most gain in accuracy is obtained by completing the Monte Carlo simulation before post-processing with the variational principle.

Transient-estimate Monte Carlo in the two-dimensional electron gas.

Energies of the ground state and low-lying excited states of the two-dimensional electron gas have been calculated by a transient-estimate Monte Carlo method. This is an exact fermion quantum Monte Carlo method that systematically improves upon the results of a variational energy without imposing nodal constraints. We focus upon the density r s51, where our previous variational Monte Carlo calculation found qualitative differences in the effective mass from other theoretical approaches. Starting from a wave function with backflow and two-body correlations, the best trial function in our previous variational study, we find a ground-state energy only very slightly lower than the previously reported backflow fixed-node energy, reinforcing the conclusion that backflow wave functions are quite accurate. The effective mass derived from excitation energies does not differ significantly from the variational Monte Carlo results, giving a value of m*/m50.9360.01, so we conclude that the effective mass is indeed less than bare electron mass for a range of densities around r s51.

Ground-state properties of the half-filled Kondo lattice model: A numerical study.

Ground-state properties of the one-dimensional Kondo lattice model are investigated using the quantum Monte Carlo (QMC) simulation method. To reduce the difficulty of the negative sign problem, we employ the variationally optimized singlet wave function. Since the improvement of Monte Carlo sampling efficiency is achieved in the half-filling case, we can perform the simulations on large systems (up to 26-site system). The QMC data on the local spin correlations are closely compared with those of the variational Monte Carlo (VMC) calculations as well as with the exact diagonalization data of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) lattice model. We find that the local spin correlation decays exponentially with the correlation length ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}2.54 in the QMC data, while ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}1.08 in the VMC results (J/t=1). This implies the absence of the long-range order (the existence of the spin gap) and the importance of intersite correlation effects which are not taken into account in the VMC results. On the other hand, the system size dependence is almost absent in the Kondo lattice model results, which contrasts the behavior in the RKKY data.

Variational and diffusion Monte Carlo techniques for quantum clusters.

The H[sub 2]-[sup 4]He dimer and small [sup 4]He clusters are studied using Monte Carlo sampling techniques. We consider alternative wave-function forms in order to obtain high accuracy efficiently. For the smaller systems, both guided and unguided Metropolis walks are used and the effciencies are studied. Of particular concern is accurate sampling at small particle separations and the behavior of the local energy in this regime. As a final step, we compute exact energies by a diffusion Monte Carlo method. We obtain converged energies significantly below the Green's-function Monte Carlo values, which employed an earlier He-He potential with a slightly shallower well. For He[sub 3] and He[sub 20], the Green's-function Monte Carlo energies are reproduced when employing the same potential. However, for the 112-atom cluster, our converged energy lies below the Green's-function Monte Carlo value. Second-order estimates of the exact density profiles and particle separation distributions, [ital p], are also determined. For the 14- and 20-atom clusters, second-order estimates of [ital p] show enhanced structure in comparison to variational Monte Carlo results. Statistically meaningful oscillations in the second-order estimates of the exact density profiles are not observed.

Brinkman-Rice transition in the three-band Hubbard model

The ${\mathit{U}}_{\mathit{d}}$=\ensuremath{\infty} three-band Hubbard model on a two-dimensional ${\mathrm{CuO}}_{2}$ network is studied based on a Gutzwiller wave function by use of the variational Monte Carlo (VMC) method, where ${\mathit{U}}_{\mathit{d}}$ is the Coulomb repulsion between holes on the Cu site (hole picture). The VMC results are compared with those of the Rice-Ueda-type Gutzwiller approximation (GA) in a half-filled band, where the average number of holes per Cu site is n=1. In the GA the Brinkman-Rice (BR) transition occurs at a finite value of ${\mathrm{\ensuremath{\Delta}}}_{0}$=(${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{p}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{d}}$)/2, and then the number of d holes becomes ${\mathit{n}}_{\mathit{d}}$=n=1. Whether the BR transition really occurs or not within the Gutzwiller wave function is discussed in detail. In finite systems the absence of the BR transition is shown analytically. The thermodynamic limit is examined assuming the asymptotic behavior of ${\mathit{n}}_{\mathit{d}}$ is given by a scaling function. The results of the finite-size scaling analysis suggest the absence of the BR transition as that in the single-band Hubbard model.

Quark models of nuclear matter: (I). Basic models and ground-state properties

Nuclear matter is modeled directly in the constituent quark coordinates. A many-body string flip potential is used which confines quarks, allows hadrons to separate and is symmetric in all quark coordinates. Variational Monte Carlo results are presented for the energy, wave function and quark correlation function of nuclear and quark matter for various densities. These show how quark clustering decreases with density. A percolation phase transition from nuclear to quark matter is observed which is characterized by a dramatic rearrangement of strings.

Antiferromagnetism and superconductivity in a two-dimensional Heisenberg model: Further studies on possible coexistence

The question of possible coexistence of antiferromagnetism (AF) and superconductivity (SC) is reexamined within the t-J model by using an improved mean-field treatment. The renormalization of the effective Hamiltonian and the symmetry of the order parameters are properly taken into account. Our calculations reconfirm the possible coexistence of AF and SC below critical doping concentrations ${\ensuremath{\delta}}_{c}$=0.04 and 0.06 for t/J=5 and 2.5, respectively, in good agreement with recent variational Monte Carlo results. This conclusion is consistent with the muon-spin-rotation data. Among pairings of various symmetries, the d-wave pairing has the lowest energy.