Green Function Monte Research Articles

Quantum-enhanced Green's function Monte Carlo algorithm for excited states of the nuclear shell model

Quantum-enhanced Green's function Monte Carlo algorithm for excited states of the nuclear shell model

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 (PBC). 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 [1].

Relativistic effects in Green's function Monte Carlo calculations of neutrino-nucleus scattering

Relativistic effects in Green's function Monte Carlo calculations of neutrino-nucleus scattering

Extraction of the Coulomb sum rule, transverse enhancement, and longitudinal quenching from an analysis of all available e−C12 and e−O16 cross section data

We report on a phenomenological analysis of all available electron scattering data on $^{12}\mathrm{C}$ (about 6600 differential cross section measurements) and on $^{16}\mathrm{O}$ (about 250 measurements) within the framework of the quasielastic (QE) superscaling model (including Pauli blocking). All QE and inelastic cross section measurements are included down to the lowest momentum transfer $\mathbf{q}$ (including photoproduction data). We find that there is enhancement of the transverse QE response function (${R}_{T}^{QE}$) and quenching of the QE longitudinal response function (${R}_{L}^{QE}$) at low $\mathbf{q}$ (in addition to Pauli blocking). We extract parametrizations of a multiplicative low $\mathbf{q}$ ``longitudinal quenching factor'' and an additive ``transverse enhancement'' contribution. Additionally, we find that the excitation of nuclear states contribute significantly (up to 30%) to the Coulomb sum rule $SL(\mathbf{q})$. We extract the most accurate determination of $SL(\mathbf{q})$ to date and find it to be in disagreement with random phase approximation (RPA) based calculations but in reasonable agreement with recent theoretical calculations, such as ``first principle Green's function Monte Carlo.''

Path-integral quantum Monte Carlo calculations of light nuclei

We describe a path-integral ground-state quantum Monte Carlo method for light nuclei in continuous space. We show how to efficiently update and sample the paths with spin-isospin dependent and spin-orbit interactions. We apply the method to the triton and alpha particle using both local chiral interactions with next-to-next-to-leading-order %(N$^2$LO) and the Argonne interactions. For operators, like the total energy, that commute with the Hamiltonian, our results agree with Green's function Monte Carlo and auxiliary field diffusion Monte Carlo calculations. For operators that do not commute with the Hamiltonian and for Euclidean response functions, the path-integral formulation allows straightforward calculation without forward walking or the increased variance typical of diffusion methods. We demonstrate this by calculating density distributions, root mean square radii, and Euclidean response functions for single-nucleon couplings.

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.

Variational Monte Carlo Calculations of A≤4 Nuclei with an Artificial Neural-Network Correlator Ansatz.

The complexity of many-body quantum wave functions is a central aspect of several fields of physics and chemistry where nonperturbative interactions are prominent. Artificial neural networks (ANNs) have proven to be a flexible tool to approximate quantum many-body states in condensed matter and chemistry problems. In this work we introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei, and approximately solve the nuclear many-body Schrödinger equation. Using efficient stochastic sampling and optimization schemes, our approach extends pioneering applications of ANNs in the field, which present exponentially scaling algorithmic complexity. We compute the binding energies and point-nucleon densities of A≤4 nuclei as emerging from a leading-order pionless effective field theory Hamiltonian. We successfully benchmark the ANN wave function against more conventional parametrizations based on two- and three-body Jastrow functions, and virtually exact Green's function MonteCarlo results.

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.

Nuclear Charge Radii of ^{10,11}B.

The first laser spectroscopic determination of the change in the nuclear charge radius for a five-electron system is reported. This is achieved by combining high-accuracy abinitio mass-shift calculations and a high-accuracy measurement of the isotope shift in the 2s^{2}2p ^{2}P_{1/2}→2s^{2}3s ^{2}S_{1/2} ground state transition in boron atoms. Accuracy is increased by orders of magnitude for the stable isotopes ^{10,11}B and the results are used to extract their difference in the mean-square charge radius ⟨r_{c}^{2}⟩^{11}-⟨r_{c}^{2}⟩^{10}=-0.49(12) fm^{2}. The result is qualitatively explained by a possible cluster structure of the boron nuclei and quantitatively used to benchmark new abinitio nuclear structure calculations using the no-core shell model and Green's function MonteCarlo approaches. These results are the foundation for a laser spectroscopic determination of the charge radius of the proton-halo candidate ^{8}B.

Measurement of the full excitation spectrum of the 7Li(p,γ)αα reaction at 441 keV

A current challenge for ab initio calculations is systems that contain large continuum contributions such as Be8. We report on new measurements of radiative decay widths in this nucleus that test recent Green's function Monte Carlo calculations.Traditionally, γ ray detectors have been utilized to measure the high energy photons from the Li7(p,γ)αα reaction. However, due to the complicated response function of these detectors it has not yet been possible to extract the full γ ray spectrum from this reaction. Here we present an alternative measurement using large area Silicon detectors to detect the two α particles, which provides a practically background free spectrum and retains good energy resolution.The resulting spectrum is analyzed using a many-level multi channel R-matrix parametrization. Improved values for the radiative widths are extracted from the R-matrix fit. We find evidence for significant non-resonant continuum contributions and tentative evidence for a broad 0+ resonance at 12MeV.

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.

Light-Nuclei Spectra from Chiral Dynamics.

In recent years local chiral interactions have been derived and implemented in quantum MonteCarlo methods in order to test to what extent the chiral effective field theory framework impacts our knowledge of few- and many-body systems. In this Letter, we present Green's function MonteCarlo calculations of light nuclei based on the family of local two-body interactions presented by our group in a previous paper in conjunction with chiral three-body interactions fitted to bound- and scattering-state observables in the three-nucleon sector. These interactions include Δ intermediate states in their two-pion-exchange components. We obtain predictions for the energy levels and level ordering of nuclei in the mass range A=4-12, accurate to ≤2% of the binding energy, in very satisfactory agreement with experimental data.

Short-Range Correlations and the EMC Effect in Effective Field Theory.

We show that the empirical linear relation between the magnitude of the EMC effect in deep inelastic scattering on nuclei and the short-range correlation scaling factor a_{2} extracted from high-energy quasielastic scattering at x≥1 is a natural consequence of scale separation and derive the relationship using effective field theory. While the scaling factor a_{2} is a ratio of nuclear matrix elements that individually depend on the calculational scheme, we show that the ratio is independent of this choice. We perform Green's function MonteCarlo calculations with both chiral and Argonne-Urbana potentials to verify this and determine the scaling factors for light nuclei. The resulting values for ^{3}He and ^{4}He are in good agreement with experimental values. We also present results for ^{9}Be and ^{12}C extracted from variational MonteCarlo calculations.

Quantum Monte Carlo calculations of light nuclei with local chiral two- and three-nucleon interactions

Local chiral effective field theory interactions have recently been developed and used in the context of quantum Monte Carlo few- and many-body methods for nuclear physics. In this work, we go over detailed features of local chiral nucleon-nucleon interactions and examine their effect on properties of the deuteron, paying special attention to the perturbativeness of the expansion. We then turn to three-nucleon interactions, focusing on operator ambiguities and their interplay with regulator effects. We then discuss the nuclear Green's function Monte Carlo method, going over both wave-function correlations and approximations for the two- and three-body propagators. Following this, we present a range of results on light nuclei: Binding energies and distribution functions are contrasted and compared, starting from several different microscopic interactions.

α decay of the T=1, 2+ state in B10 and isospin symmetry breaking in the A=10 triplet

Here, the rate of the T=1, 2+ to T=1, 0 + transition in 10B ( T=1, T z=0) is compared to the analog transitions in 10Be ( T=1, T z=–1) and 10C ( T=1, T z=+1) to provide constraints on ab initio calculations using realistic nuclear forces. The relevant state in 10B, at E x=5.164 MeV, is particle unbound. Therefore, a determination of the B( E2) electromagnetic transition rate requires a precise and accurate determination of the width of the state, as well as the α-particle and γ-ray branching ratios. Previous measurements of the α-particle branching ratio are just barely in agreement. We report on a new study of the α-particle branch by studying the 10B(p,p') 10B* reaction in inverse kinematics with the HELIOS spectrometer. The α-particle branching ratio that we observe, 0.144±0.027, is in good agreement with the evaluated value and improves the associated uncertainty. The resulting experimental B( E2) value is 7.0±2.2 e 2fm 4 and is more consistent with a flat trend across the A=10 triplet than previously reported. This is inconsistent with Green's function Monte Carlo predictions using realistic three-nucleon Hamiltonians, which overpredict the B(E2) value in 10C and 10B.

Electromagnetic scaling functions within the Green's function Monte Carlo approach

We have studied the scaling properties of the electromagnetic response functions of $^4$He and $^{12}$C nuclei computed by the Green's Function Monte Carlo approach, retaining only the one-body current contribution. Longitudinal and transverse scaling functions have been obtained in the relativistic and non relativistic cases and compared to experiment for various kinematics. The characteristic asymmetric shape of the scaling function exhibited by data emerges in the calculations in spite of the non relativistic nature of the model. The results are consistent with scaling of zeroth, first and second kinds. Our analysis reveals a direct correspondence between the scaling and the nucleon-density response functions.

Comparison of the electromagnetic responses of C12 obtained from the Green's function Monte Carlo and spectral function approaches

The electromagnetic responses of carbon obtained from the Green's function Monte Carlo and spectral function approaches using the same dynamical input are compared in the kinematical region corresponding to momentum transfer in the range 300-570 MeV. The results of our analysis, aimed at pinning down the limits of applicability of the approximations involved in the two schemes, indicate that the factorization ansatz underlying the spectral function formalism provides remarkably accurate results down to momentum transfer as low as 300 MeV. On the other hand, it appears that at 570 MeV relativistic corrections to the electromagnetic current not included in the Monte Carlo calculations may play a significant role in the transverse channel.

Electromagnetic Response ofC12: A First-Principles Calculation

The longitudinal and transverse electromagnetic response functions of $^{12}$C are computed in a "first-principles" Green's function Monte Carlo calculation, based on realistic two- and three-nucleon interactions and associated one- and two-body currents. We find excellent agreement between theory and experiment and, in particular, no evidence for the quenching of measured versus calculated longitudinal response. This is further corroborated by a re-analysis of the Coulomb sum rule, in which the contributions from the low-lying $J^\pi\,$=$\, 2^+$, $0^+_2$ (Hoyle), and $4^+$ states in $^{12}$C are accounted for explicitly in evaluating the total inelastic strength.

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.