Green's Function Monte Carlo Research Articles

A hybrid method integrating Green’s function Monte Carlo and projected entangled pair states

Abstract This paper introduces a hybrid approach combining Green’s function Monte Carlo (GFMC) method with projected entangled pair state (PEPS) ansatz. This hybrid method regards PEPS as a trial state and a guiding wave function in GFMC. By leveraging PEPS’s proficiency in capturing quantum state entanglement and GFMC’s efficient parallel architecture, the hybrid method is well-suited for the accurate and efficient treatment of frustrated quantum spin systems. As a benchmark, we applied this approach to study the frustrated J 1–J 2 Heisenberg model on a square lattice with periodic boundary conditions (PBCs). Compared with other numerical methods, our approach integrating PEPS and GFMC shows competitive accuracy in the performance of ground-state energy. This paper provides systematic and comprehensive discussion of the approach of our previous work [Phys. Rev. B 109 235133 (2024)].

Ab initio estimation of E2 strengths in Li8 and its neighbors by normalization to the measured quadrupole moment

For electric quadrupole ($E2$) observables, which depend on the large-distance tails of the nuclear wave function, ab initio no-core configuration interaction (NCCI) calculations converge slowly, making meaningful predictions challenging to obtain. Nonetheless, the calculated values for different $E2$ matrix elements, particularly those involving levels with closely-related structure (e.g., within the same rotational band) are found to be robustly proportional. This observation suggests that a known value for one observable may be used to determine the overall scale of $E2$ strengths, and thereby provide predictions for others. In particular, we demonstrate that meaningful predictions for $E2$ transitions may be obtained by calibration to the ground-state quadrupole moment. We test this approach for well-measured low-lying $E2$ transitions in $^7$Li and $^9$Be, then provide predictions for transitions in $^8$Li and $^9$Li. In particular, we address the $2^+\rightarrow1^+$ transition in $^8$Li, for which the reported measured strength exceeds ab initio Green's function Monte Carlo (GFMC) predictions by over an order of magnitude.

Green’s function Monte Carlo method combined with restricted Boltzmann machine approach to the frustrated J 1 – J 2 Heisenberg model

Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green’s function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J 1–J 2 Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.

Combination of tensor network states and Green's function Monte Carlo

We propose an approach to study the ground state of quantum many-body systems in which Tensor Network States (TNS), specifically Projected Entangled Pair States (PEPS), and Green's function Monte Carlo (GFMC) are combined. PEPS, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short range interactions but are hindered by the high computational complexity scaling with bond dimension (D). GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of negative sign problem is the introduction of systematic error by guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPS are very accurate variational wave functions, while at the same time, only contractions of single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of $J_1$-$J_2$ Heisenberg model on square lattice are provided.

Muon capture in nuclei: An ab initio approach based on Green's function Monte Carlo methods

An ab initio quantum Monte Carlo method is introduced for calculating total rates of muon weak capture in light nuclei with mass number $A \leq 12$. As a first application of the method, we perform a calculation of the rate in $^4$He in a dynamical framework based on realistic two- and three-nucleon interactions and realistic nuclear charge-changing weak currents. The currents include one- and two-body terms induced by $\pi$- and $\rho$-meson exchange, and $N$-to-$\Delta$ excitation, and are constrained to reproduce the empirical value of the Gamow-Teller matrix element in tritium. We investigate the sensitivity of theoretical predictions to current parametrizations of the nucleon axial and induced pseudoscalar form factors as well as to two-body contributions in the weak currents. The large uncertainties in the measured values obtained from bubble-chamber experiments (carried out over 50 years ago) prevent us from drawing any definite conclusions.

Isobaric multiplet mass equation within nuclear density functional theory

We extend the nuclear density functional theory (DFT) by including proton–neutron mixing and contact isospin-symmetry-breaking (ISB) terms up to next-to-leading order. Within this formalism, we perform systematic study of the nuclear mirror and triplet displacement energies, or equivalently of the isobaric multiplet mass equation (IMME) coefficients. By comparing results with those obtained within the existing Green Function Monte Carlo (GFMC) calculations, we address the fundamental question of the physical origin of the ISB effects. This we achieve by analyzing separate contributions to IMME coefficients coming from the electromagnetic and nuclear ISB terms We show that the ISB DFT and GFMC results agree reasonably well, and that they describe experimental data with a comparable quality. Since the separate electromagnetic and nuclear ISB contributions also agree, we conclude that the beyond-mean-field electromagnetic effects may not play a dominant role in describing the ISB effects in finite nuclei.

Relativistic effects in ab initio electron-nucleus scattering

The electromagnetic responses obtained from Green's function Monte Carlo (GFMC) calculations are based on realistic treatments of nuclear interactions and currents. The main limitations of this method comes from its nonrelativistic nature and its computational cost, the latter hampering the direct evaluation of the inclusive cross sections as measured by experiments. We extend the applicability of GFMC in the quasielastic region to intermediate momentum transfers by performing the calculations in a reference frame that minimizes nucleon momenta. Additional relativistic effects in the kinematics are accounted for employing the two-fragment model. In addition, we developed a novel algorithm, based on the concept of first-kind scaling, to compute the inclusive electromagnetic cross section of $^4$He through an accurate and reliable interpolation of the response functions. A very good agreement is obtained between theoretical and experimental cross sections for a variety of kinematical setups. This offers a promising prospect for the data analysis of neutrino-oscillation experiments that requires an accurate description of nuclear dynamics in which relativistic effects are fully accounted for.

An investigation of ab initio shell-model interactions derived by no-core shell model

The microscopic shell-model effective interactions are mainly based on the many-body perturbation theory (MBPT), the first work of which can be traced to Brown and Kuo’s first attempt in 1966, derived from the Hamada-Johnston nucleon-nucleon potential. However, the convergence of the MBPT is still unclear. On the other hand, ab initio theories, such as Green’s function Monte Carlo (GFMC), no-core shell model (NCSM), and coupled-cluster theory with single and double excitations (CCSD), have made many progress in recent years. However, due to the increasing demanding of computing resources, these ab initio applications are usually limited to nuclei with mass up to A = 16. Recently, people have realized the ab initio construction of valence-space effective interactions, which is obtained through a second-time renormalization, or to be more exactly, projecting the full-manybody Hamiltonian into core, one-body, and two-body cluster parts. In this paper, we present the investigation of such ab initio shell-model interactions, by the recent derived sd-shell effective interactions based on effective J-matrix Inverse Scattering Potential (JISP) and chiral effective-field theory (EFT) through NCSM. In this work, we have seen the similarity between the ab initio shellmodel interactions and the interactions obtained by MBPT or by empirical fitting. Without the inclusion of three-body (3-bd) force, the ab initio shell-model interactions still share similar defects with the microscopic interactions by MBPT, i.e., T = 1 channel is more attractive while T = 0 channel is more repulsive than empirical interactions. The progress to include more many-body correlations and 3-bd force is still badly needed, to see whether such efforts of ab initio shell-model interactions can reach similar precision as the interactions fitted to experimental data.

A new Green's function Monte Carlo algorithm for the estimation of the derivative of the solution of Helmholtz equation subject to Neumann and mixed boundary conditions

The objective of this paper is the extension and application of a newly-developed Green's function Monte Carlo (GFMC) algorithm to the estimation of the derivative of the solution of the one-dimensional (1D) Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional GFMC approach for the solution of partial differential equations subject to these boundary conditions involves resulting in relatively large computational times. My work, inspired by the work of K.K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. My past work has involved the application of this algorithm to the estimation of the solution of the 1D Laplace equation, the Helmholtz equation and the modified Helmholtz equation. In this work, this algorithm has been adapted to the estimation of the derivative of the solution which is a very important development. In the traditional approach involving reflection, to estimate the derivative at a certain number of points, one has to a priori estimate the solution at a larger number of points. In the case of a one-dimensional problem for instance, to obtain the derivative of the solution at a point, one has to obtain the solution at two points, one on each side of the point of interest. These points have to be close enough so that the validity of the first-order approximation for the derivative operator is justified and at the same time, the actual difference between the solutions at these two points has to be at least an order of magnitude higher than the statistical error in the estimation of the solution, thus requiring a significantly larger number of random-walks than that required for the estimation of the solution. In this new approach, identical amount of computational resources is needed irrespective of if we are trying to estimate the solution or the derivative. This becomes very significant in electromagnetic problems where the scalar/vector potential is the unknown in the PDE of interest, but the quantity of interest is the electric/magnetic field or in heat conduction problems where temperature of an object is the unknown variable in a PDE, but the quantity of interest is the spatial/temporal variation of the temperature. In this work, this algorithm is applied to the estimation of the derivative of the solution of the 1D Helmholtz equation which is the frequency domain version of both Maxwell's equations and the heat conduction equation. As a result the algorithm is an important first step in the development of computationally efficient GFMC algorithms for Neumann and mixed boundary condition problems. The numerical results have been validated by an exact, analytical solution and very good agreement has been observed. The long-term goal of this research is the application of this methodology to the numerical solution of the F region ionization problem in space plasma modeling.

Green’s Function Monte Carlo Calculations with Two- and Three-Nucleon Interactions from Chiral Effective Field Theory

I discuss our recent work on Green's function Monte Carlo (GFMC) calculations of light nuclei using local nucleon-nucleon interactions derived from chiral effective field theory (EFT) up to next-to-next-to-leading order (N$^2$LO). I present the natural extension of this work to include the consistent three-nucleon (3N) forces at the same order in the chiral expansion. I discuss our choice of observables to fit the two low-energy constants which enter in the 3N sector at N$^2$LO and present some results for light nuclei.

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.

A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

Precise Electromagnetic Tests ofAb InitioCalculations of Light Nuclei: States inBe10

In order to test ab initio calculations of light nuclei, we have remeasured lifetimes in 10Be using the Doppler shift attenuation method (DSAM) following the 7Li(7Li,alpha)10Be reaction at 8 and 10 MeV. The new experiments significantly reduce systematic uncertainties in the DSAM technique. The J(pi) = 2(1)(+) state at 3.37 MeV has tau = 205 +/- (5)(stat) +/- (7)(sys) fs corresponding to a B(E2 down) of 9.2(3)e(2) fm(4) in broad agreement with many calculations. The J(pi) = 2(2)(+) state at 5.96 MeV was found to have a B(E2 down) of 0.11(2)e(2) fm(4) and provides a more discriminating test of nuclear models. New Green's function Monte Carlo calculations for these states and transitions with a number of Hamiltonians are also reported and compared to experiment.

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.

The Geophysical Inverse Method Based on Quantum Monte Carlo

AbstractQuantum Monte Carlo method (QMC), including variational Monte Carlo (VMC), Green's function Monte Carlo (GFMC), diffusion Monte Carlo (DMC) and path integral Monte Carlo (PIMC), is a stochastic method based on the quantum mechanics. This paper, firstly, gives a brief summarization about the development of quantum Monte Carlo method and expatiates its principles; then does some model tests, which show that the using of quantum Monte Carlo method for geophysical inverse problems is successful, and its algorithm is also acceptable for some other nonlinear optimazation problems; lastly, gives a perspective on the using of quantum Monte Carlo method for geophysical inversion, and points out its shortcomings.

Magnetism and superconductivity in thet−t′−Jmodel

We present a systematic study of the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model by using the Green's function Monte Carlo (GFMC) technique, implemented within the fixed-node (FN) approximation and a wave function that contains both antiferromagnetism and $d$-wave pairing. This enables us to study the interplay between these two kinds of order and compare the GFMC results with the ones obtained by the simple variational approach. By using a generalization of the forward-walking technique, we are able to calculate true FN ground-state expectation values of the pair-pair correlation functions. In the case of ${t}^{\ensuremath{'}}=0$, there is a large region with a coexistence of superconductivity and antiferromagnetism that survives up to ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.10$ for $J∕t=0.2$, and ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.13$ for $J∕t=0.4$. The presence of a finite ${t}^{\ensuremath{'}}∕tl0$ induces a strong suppression of both magnetic (with ${\ensuremath{\delta}}_{c}\ensuremath{\lesssim}0.03$ for $J∕t=0.2$ and ${t}^{\ensuremath{'}}∕t=\ensuremath{-}0.2$) and pairing correlations. In particular, the latter ones are depressed both in the low-doping regime and around $\ensuremath{\delta}\ensuremath{\sim}0.25$, where strong size effects are present.

Quantum Monte Carlo calculations of electroweak transition matrix elements inA=6,7nuclei

Green's function Monte Carlo (GFMC) calculations of magnetic dipole, electric quadrupole, Fermi, and Gamow-Teller transition matrix elements are reported for $A=6,7$ nuclei. The matrix elements are extrapolated from mixed estimates that bracket the relevant electroweak operator between variational Monte Carlo (VMC) and GFMC propagated wave functions. Because they are off-diagonal terms, two mixed estimates are required for each transition, with a VMC initial (final) state paired with a GFMC final (initial) state. The realistic Argonne ${v}_{18}$ two-nucleon and Illinois-2 three-nucleon interactions are used to generate the nuclear states. In most cases we find good agreement with experimental data.

Quantum annealing of an Ising spin-glass by Green’s function Monte Carlo

We present an implementation of quantum annealing (QA) via lattice Green's function Monte Carlo (GFMC), focusing on its application to the Ising spin glass in transverse field. In particular, we study whether or not such a method is more effective than the path-integral Monte Carlo- (PIMC) based QA, as well as classical simulated annealing (CA), previously tested on the same optimization problem. We identify the issue of importance sampling, i.e., the necessity of possessing reasonably good (variational) trial wave functions, as the key point of the algorithm. We performed GFMC-QA runs using such a Boltzmann-type trial wave function, finding results for the residual energies that are qualitatively similar to those of CA (but at a much larger computational cost), and definitely worse than PIMC-QA. We conclude that, at present, without a serious effort in constructing reliable importance sampling variational wave functions for a quantum glass, GFMC-QA is not a true competitor of PIMC-QA.

Quantum Monte Carlo calculations of excited states inA=6–8nuclei

A variational Monte Carlo method is used to generate sets of orthogonal trial functions, ${\ensuremath{\Psi}}_{T}({J}^{\ensuremath{\pi}};T)$, for given quantum numbers in various light $p$-shell nuclei. These ${\ensuremath{\Psi}}_{T}$ are then used as input to Green's function Monte Carlo (GFMC) calculations of first, second, and higher excited $({J}^{\ensuremath{\pi}};T)$ states. Realistic two- and three-nucleon interactions are used. We find that if the physical excited state is reasonably narrow, the GFMC energy converges to a stable result. With the combined Argonne ${\mathrm{v}}_{18}$ two-nucleon and Illinois-2 three-nucleon interactions, the results for many second and higher states in $A=6\char21{}8$ nuclei are close to the experimental values.

Incorporation of density-matrix wave functions in Monte Carlo simulations: Application to the frustrated Heisenberg model

We combine the Density Matrix Technique (DMRG) with Green Function Monte Carlo (GFMC) simulations. The DMRG is most successful in 1-dimensional systems and can only be extended to 2-dimensional systems for strips of limited width. GFMC is not restricted to low dimensions but is limited by the efficiency of the sampling. This limitation is crucial when the system exhibits a so-called sign problem, which on the other hand is not a particular obstacle for the DMRG. We show how to combine the virtues of both methods by using a DMRG wavefunction as guiding wave function for the GFMC. This requires a special representation of the DMRG wavefunction to make the simulations possible within reasonable computational time. As a test case we apply the method to the 2-dimensional frustrated Heisenberg antiferromagnet. By supplementing the branching in GFMC with Stochastic Reconfiguration (SR) we get a stable simulation with a small variance also in the region where the fluctuations due to minus sign problem are maximal. The sensitivity of the results to the choice of the guiding wavefunction is extensively investigated. We analyse the model as a function of the ratio of the next-nearest to nearest neighbor coupling strength. We observe in the frustrated regime a pattern of the spin correlations which is in-between dimerlike and plaquette type ordering, states that have recently been suggested. It is a state with strong dimerization in one direction and weaker dimerization in the perpendicular direction.