Variational Monte Carlo Method Research Articles

Performance assessment of the effective core potentials under the fermionic neural network: First and second row elements.

The rapid development of deep learning techniques has driven the emergence of a neural network-based variational Monte Carlo (VMC) method (referred to as FermiNet), which has manifested high accuracy and strong predictive power in the electronic structure calculations of atoms, molecules, and some periodic systems. Recently, the implementation of the effective core potential (ECP) scheme has further facilitated more efficient calculations in practice. However, there is still a lack of comprehensive assessments of the ECP's performance under the FermiNet. In this work, we set sail to fill this gap by conducting extensive tests on the first two row elements regarding their atomic, spectral, and molecular properties. Our major finding is that, in general, the qualities of ECPs have been correctly reflected under FermiNet. Two recently built ECP tables, namely, correlation consistent ECP (ccECP) and energy consistent correlated electron pseudopotential (eCEPP), seem to prevail in terms of overall performance. In particular, ccECP performs slightly better on spectral precision and covers more elements, while eCEPP is more systematically built from both shape and energy consistency and better treats the core polarization. On the other hand, the high accuracy of the all-electron calculations is hindered by the absence of relativistic effects as well as the numerical instabilities in some heavier elements. Finally, with further in-depth discussions, we generate possible directions for developing and improving FermiNet in the near future.

Ensemble variational Monte Carlo for optimization of correlated excited state wave functions

Variational Monte Carlo methods have recently been applied to the calculation of excited states; however, it is still an open question what objective function is most effective. A promising approach is to optimize excited states using a penalty to minimize overlap with lower eigenstates, which has the drawback that states must be computed one at a time. We derive a general framework for constructing objective functions with minima at the the lowest N eigenstates of a many-body Hamiltonian. The objective function uses a weighted average of the energies and an overlap penalty, which must satisfy several conditions. We show this objective function has a minimum at the exact eigenstates for a finite penalty, and provide a few strategies to minimize the objective function. The method is demonstrated using ab initio variational Monte Carlo to calculate the degenerate first excited state of a CO molecule.

Towards a transferable fermionic neural wavefunction for molecules.

Deep neural networks have become a highly accurate and powerful wavefunction ansatz in combination with variational Monte Carlo methods for solving the electronic Schrödinger equation. However, despite their success and favorable scaling, these methods are still computationally too costly for wide adoption. A significant obstacle is the requirement to optimize the wavefunction from scratch for each new system, thus requiring long optimization. In this work, we propose a neural network ansatz, which effectively maps uncorrelated, computationally cheap Hartree-Fock orbitals, to correlated, high-accuracy neural network orbitals. This ansatz is inherently capable of learning a single wavefunction across multiple compounds and geometries, as we demonstrate by successfully transferring a wavefunction model pre-trained on smaller fragments to larger compounds. Furthermore, we provide ample experimental evidence to support the idea that extensive pre-training of such a generalized wavefunction model across different compounds and geometries could lead to a foundation wavefunction model. Such a model could yield high-accuracy ab-initio energies using only minimal computational effort for fine-tuning and evaluation of observables.

A DOMAIN-DECOMPOSED VAE METHOD FOR BAYESIAN INVERSE PROBLEMS

Bayesian inverse problems are often computationally challenging when the forward model is governed by complex partial differential equations (PDEs). This is typically caused by expensive forward model evaluations and highdimensional parameterization of priors. This paper proposes a domain-decomposed variational autoencoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. Through partitioning the global physical domain into small subdomains, the proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations. Gaussian process models with active learning address the domain decomposition interface conditions. Then inversions are conducted on each subdomain independently in parallel and in low-dimensional latent parameter spaces. The local inference solutions are postprocessed through the Poisson image blending procedure to result in an efficient global inference result. Numerical examples are provided to demonstrate the performance of the proposed method.

Neural network quantum states analysis of the Shastry-Sutherland model

We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

Compressed lithium atom under Debye screening

AbstractIn the present paper, we introduced two approximation methods to investigate the spherically compressed lithium atom under the influence of Debye plasma. Specifically, the diffusion and the variational Monte Carlo methods are used for the first time to study this case. We incorporated an exponential screening into the nuclear Coulomb potential to account for the plasma influence. We focused our investigation on analyzing the combined effect of compression and plasma environment on the behavior of the lithium atom in its ground state. The resulting outcomes exhibited consistent pattern across various confinement radii and Debye plasma lengths . Many of the results are novel contributions, as they have yet to be explored. Moreover, the findings from this study demonstrated reasonable agreement with the limited existing theoretical results.

The beryllium atom, ions, and iso-electronic ions in magnetic field using the variational Monte Carlo method

The variational Monte Carlo method is applied to investigate several properties of the beryllium atom, ions, and its iso-electronic ions. For the ground and the excited states of the beryllium atom and ions, the energy eigenvalues were evaluated freely and under the influence of magnetic field. Furthermore, the iso-electronic ions (B+ and C2+) are also investigated under the influence of the external magnetic field. Suitable trial wave functions including the spin and the correlated parts are used in these investigation. Some new excited states were included in the present work, such as the low-lying states (1s22s3s and 1s22s3p) and the core states (1s2s23s and 1s2s3s2). For the spin functions of these states, we used two different functions for the singlet and the triplet excited states. Moreover, the energies of the beryllium ions (Be+ and Be2+) were evaluated freely and in the presence of magnetic field. The obtained results are in good agreement with the corresponding results of other works.Graphical abstract

Comprehensive AbInitio Investigation of the Phase Diagram of Quasi-One-Dimensional Molecular Solids.

An abinitio investigation of the family of molecular compounds TM_{2}X is conducted, where TM is either TMTSF or TMTTF and X takes centrosymmetric monovalent anions. By deriving the extended Hubbard-type Hamiltonians from first-principles band calculations and evaluating not only the intermolecular transfer integrals but also the Coulomb parameters, we discuss their material dependence in the unified phase diagram. Furthermore, we apply the many-variable variational MonteCarlo method to accurately determine the symmetry-breaking phase transitions, and show the development of the charge and spin orderings. We show that the material-dependent parameter can be taken as the correlation effect, represented by the value of the screened on-site Coulomb interaction U relative to the intrachain transfer integrals, for the comprehensive understanding of the spin and charge ordering in this system.

Wigner crystallization in quasi-one-dimensional quantum wire

We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period 2r_text {s} a_0, where a_0 is the Bohr’s radius and r_text {s} is Wigner Seitz radius or electron-density parameter which corresponds to a k=4k_text {F} peak with k_text {F} being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as r_text {s} increases or as the wire width b decreases.

Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a nonintrusive generalization of the adaptive Galerkin finite element method with residual-based error estimation. It combines the nonintrusive character of a randomized least squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the nonintrusive adaptive algorithm, showing the expected convergence rates of single-level strategies.

Correlation energy of the paramagnetic electron gas at the thermodynamic limit

The variational and diffusion quantum Monte Carlo methods are used to calculate the correlation energy of the paramagnetic three-dimensional homogeneous electron gas at intermediate to high density. Ground-state energies in finite cells are determined using Slater-Jastrow-backflow trial wave functions, and finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Our correlation energies in the thermodynamic limit are more accurate than previous results. The present diffusion quantum Monte Carlo energies together with our recently reported [Phys. Rev. B 105, 245135 (2022)] results at low density, are used to parametrize the correlation energy of the electron gas using a functional form that satisfies the exact asymptotic behavior at high density.

Wire-width and electron-density dependence of the crossover in the peak of the static structure factor from 2kF → 4kF in one-dimensional paramagnetic electron gases

We use the variational quantum Monte Carlo (VMC) method to study the wire-width $(b)$ and electron-density $({r}_{\text{s}})$ dependences of the ground-state properties of quasi-one-dimensional paramagnetic electron fluids. The onset of a quasi-Wigner crystal phase is known to depend on electron density and the crossover occurs in the low density regime. We study the effect of wire width on the crossover of the dominant peak in the static structure factor from $k=2{k}_{\text{F}}$ to $k=4{k}_{\text{F}}$. It is found that, for a fixed electron density, in the charge structure factor the crossover from the dominant peak occurring at $2{k}_{\text{F}}$ to $4{k}_{\text{F}}$ occurs as the wire width decreases. Our study suggests that the crossover is due to the interplay of both ${r}_{\text{s}}$ and $b<{r}_{\text{s}}$. The finite wire-width correlation effect is reflected in the peak height of the charge and spin structure factors. We fit the dominant peaks of the charge and spin structure factors assuming fit functions based on our finite wire-width theory and clues from bosonization, resulting in a good fit of the VMC data. The pronounced peaks in the charge and spin structure factors at $4{k}_{\text{F}}$ and $2{k}_{\text{F}}$, respectively, indicate the complete decoupling of the charge and spin degrees of freedom. Furthermore, the wire-width dependence of the electron correlation energy and the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ is found to be significant.

Monte Carlo study of cuprate superconductors in a four-band model: role of orbital degrees of freedom

Understanding the various competing phases in cuprate superconductors is a long-standing challenging problem. Recent studies have shown that orbital degrees of freedom, both Cu e g orbitals and O p orbitals, are a key ingredient for a unified understanding of cuprate superconductors, including the material dependence. Here we investigate a four-band model derived from the first-principles calculations with the variational Monte Carlo method, which allows us to elucidate competing phases on an equal footing. The obtained results can consistently explain the doping dependence of superconductivity, antiferromagnetic and stripe phases, phase separation in the underdoped region, and also novel magnetism in the heavily-overdoped region. The presence of p orbitals is critical to the charge-stripe features, which induce two types of stripe phases with s)-wave and d-wave bond stripe. On the other hand, the presence of orbital is indispensable to material dependence of the superconducting transition temperature (), and enhances local magnetic moment as a source of novel magnetism in the heavily-overdoped region as well. These findings beyond one-band description could provide a major step toward a full explanation of unconventional normal state and high in cuprate supercondutors.

Solving the nuclear pairing model with neural network quantum states.

We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.

Bayesian geoacoustic parameter inversion based on dispersion characteristics of normal mode water wave and ground wave

Most of shallow water geoacoustic inversions based on modal dispersion cannot reliably estimate the deep geoacoustic parameters. Because these studies focused on the dispersions of water waves but ignored the dispersions of ground waves. Therefore, in this paper a Bayesian geoacoustic inversion is studied based on wideband modal dispersions of water waves and ground waves. Firstly, the modal dispersion curves with Airy phase components are discussed. Secondly, the Bayesian inversion theory and a novel sample-efficient inference algorithm, namely Variational Bayesian Monte Carlo, are introduced briefly. In the Bayesian inversion, the posterior probability densities of unknown parameters are inferred, which can provide the prediction closest to the observation data and the uncertainty of the prediction. Considering that the forward acoustic model is computationally intensive, the posterior analysis is carried out by using the Variational Bayesian Monte Carlo method. It is performed by finding the variational distribution closest to the target distribution and requires less computation time than the Markov chain Monto Carlo method. In the simulation study, a range-independent two-layer seabed, including the sediment layer and basement layer, is modeled, on the assumption that the water column is homogeneous. The function of shear wave in waveguide is ignored. The compressional sound speed of the sediment layer varies linearly from <i>c</i><sub>1U</sub> to <i>c</i><sub>1L</sub> between 0 and <i>h</i><sub>1</sub>, while other geoacoustic parameters are constant. By comparing the inversion results with and without the information of ground waves for different signal-to-noise ratios, it can be concluded that the deep geoacoustic parameters are more sensitive to the dispersions of ground waves. And then, a shallow-water experimental study is carried out in the Bohai Sea of China. The average water depth is about 20 m. The wideband pulse signals are recorded by a hydrophone with a sensitivity of –170 dB re 1 V/μPa. The received signals include well-defined Airy phase components, and the modal dispersion curves of water waves and ground waves are extracted accurately. The experimental results indicate that the Bayesian inversion combining water and ground wave dispersions can not only estimate the deep geoacoustic parameters reliably, but also reduce the inversion uncertainties of other model parameters, such as the shallow geoacoustic parameters, water depth, etc. The estimated source-receiver range and water sound speed are close to their measured values. The modal dispersion curves predicted by the posterior mean samples are in good consistence with those extracted from the experimental data in different ranges. In addition, the good forecast of transmission loss also demonstrates the reliability of the joint inversion.

Excited States Beyond Mott Gap in Half-Filled-Band Hubbard Model

In connection with recent experiments on excitation in which Mott insulators change to conductors, we study the properties of excited states beyond the Mott gap as quasi-stationary states for a two-dimensional Hubbard (t-t'-U) model at half filling. A variational Monte Carlo method is used with trial wave functions for paramagnetic or normal (PM), superconducting with dx2-y2-wave (d-SC), isotropic s-wave, and extended s-wave symmetries, and antiferromagnetic (AF) states. The excited states are generated by imposing a minimum number of doubly occupied sites (doublons) D_L on the lowest-energy states. For U>W (W: band width), d_L=D_L/Ns (Ns: number of sites) corresponds to the excitation intensity. It is found that the AF state is the most stable among the states we treated for d_L<0.14 and insulating. The PM and d-SC states become conductive over a threshold d_Lc, and the conduction is caused by unbound doublons and holons (empty sites). The PM state arises for d_L>0.14, but the d-SC state is always hidden by the AF state. The s-wave-type superconducting states are not stabilized for any parameter set.

Enhanced Spin–Orbit Coupling in a Correlated Metal

We investigate the correlation effects on spin-orbit coupling (SOC) in a two-orbital Hubbard model on a square lattice by applying the variational Monte Carlo method. We consider an effective SOC constant $\lambda_{\text{eff}}$ in the one-body part of the variational wave function and mainly discuss the cases of the electron number per site $n=1$, that is, quarter filling. We find that $\lambda_{\text{eff}}$ is proportional to the bare value $\lambda$ and depends on the electron-electron interactions through $(U'-J')$ in a relatively wide parameter range in the paramagnetic (PM) phase, where $U'$ is the interorbital Coulomb interaction and $J'$ is the pair hopping interaction. Increasing the electron-electron interactions in the PM phase leads to a transition to an effective one-band state, in which the upper band becomes empty due to the enhanced $\lambda_{\text{eff}}$. We also construct phase diagrams considering magnetic order. The carrier doping effect on $\lambda_{\text{eff}}$ is also investigated. We find that $\lambda_{\text{eff}}$ enhances in a strongly correlated region around the Mott transition point and it is necessary to include the correlation effects beyond the Hartree-Fock approximation to describe the enhanced SOC properly.

Superconductivity in Bilayer t–t′ Hubbard Models

The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-$T_{\rm c}$ superconductors. In particular, the relationship between the number $n$ of CuO$_2$ layers in a unit cell of cuprate superconductors and the optimum superconducting transition temperature $T_{\rm c}^{\rm opt}$ is intriguing. As experimentally observed in layered cuprates, $T_{\rm c}^{\rm opt}$ increases when $n$ is increased, up to $n=3$, and, then, decreases for larger $n$. However, the mechanism behind the $n$ dependence of $T_{\rm c}^{\rm opt}$ remains elusive although there have been many studies on the $n$ dependence. In this paper, we studied a bilayer $t$-$t'$ Hubbard model to clarify the effects of the adjacent CuO$_2$ layers on the stability of the superconductivity by using a many-variable variational Monte Carlo method. We calculate the superconducting correlation at long distance and zero temperature, and the amplitude of the superconducting gap functions estimated from the momentum distribution as the observables correlated with $T_{\rm c}^{\rm opt}$. It is found that the in-plane superconducting correlation is not enhanced in comparison with that in the single-layer $t$-$t'$ Hubbard model. The superconducting correlations of the bilayer Hamiltonian are significantly small in the overdoped region in comparison with those of single-layer Hamiltonian, which is attributed to the van Hove singularity. In addition, we found that the amplitude of the superconducting gap functions is also similar in both the single-layer and bilayer $t$-$t'$ Hubbard model at the optimal doping. Therefore, we conclude that the adjacent Hubbard layers are not relevant to the enhancement of $T_{\rm c}^{\rm opt}$ in the bilayer cuprates. Possible origins of the enhanced $T_{\rm c}^{\rm opt}$ other than the adjacent layers are also discussed.

Exchange-correlation effect in the charge response of a warm dense electron gas

The study of warm dense matter, widely existing in nature and laboratories, is challenging due to the interplay of quantum and classical fluctuations. We develop a variational diagrammatic Monte Carlo method and determine the exchange-correlation kernel ${K}_{\mathrm{XC}}(q;T)$ of a warm dense electron gas for a broad range of temperature $T$ and momentum $q$. We observe several interesting physics, including the $T$-dependent evolution of the hump structure and the large-$q$ tail and the emergence of a scaling relation. Particularly, by deriving an analytical form for $q\ensuremath{\rightarrow}\ensuremath{\infty}$, we obtain large-$q$ tails of ${K}_{\mathrm{XC}}$ with high precision. It is shown that the ${K}_{\mathrm{XC}}$ data can be reliably transformed into real space, which can be directly used in density-functional-theory calculations of real warm dense matter.

Application of Dual-Source Modal Dispersion and Variational Bayesian Monte Carlo Method for Local Geoacoustic Inversion in Weakly Range-Dependent Shallow Water

Application of Dual-Source Modal Dispersion and Variational Bayesian Monte Carlo Method for Local Geoacoustic Inversion in Weakly Range-Dependent Shallow Water