Function Monte Carlo Approach Research Articles

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.

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.

Generalizing the self-healing diffusion Monte Carlo approach to finite temperature: A path for the optimization of low-energy many-body bases

A statistical method is derived for the calculation of thermodynamic properties of many-body systems at low temperatures. This method is based on the self-healing diffusion Monte Carlo method for complex functions [F. A. Reboredo, J. Chem. Phys. 136, 204101 (2012)] and some ideas of the correlation function Monte Carlo approach [D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988)]. In order to allow the evolution in imaginary time to describe the density matrix, we remove the fixed-node restriction using complex antisymmetric guiding wave functions. In the process we obtain a parallel algorithm that optimizes a small subspace of the many-body Hilbert space to provide maximum overlap with the subspace spanned by the lowest-energy eigenstates of a many-body Hamiltonian. We show in a model system that the partition function is progressively maximized within this subspace. We show that the subspace spanned by the small basis systematically converges towards the subspace spanned by the lowest energy eigenstates. Possible applications of this method for calculating the thermodynamic properties of many-body systems near the ground state are discussed. The resulting basis can also be used to accelerate the calculation of the ground or excited states with quantum Monte Carlo.

A Green's function Monte Carlo algorithm for the Helmholtz equation subject to Neumann and mixed boundary conditions: Validation with an 1D benchmark problem

In this paper, we present the application of our recently developed Green's function Monte Carlo algorithm to the solution of the one-dimensional Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional Green's function Monte Carlo approach for the solution of partial differential equations subjected to Neumann and mixed boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. Our algorithm, motivated 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. In the past, we have applied it to the solution of the one-dimensional Laplace equation and the modified Helmholtz equation. In this work, we apply it to the solution of the Helmholtz equation. In the case of the Helmholtz equation, unlike the Laplace equation and modified Helmholtz equation, the algorithm is constrained to quarter-wavelength length scales, a constraint that is the result of resonance in the Green's function for the Helmholtz equation. This constraint is also present in the case of the Helmholtz equation subjected to Dirichlet conditions and is not specific to Neumann and mixed boundary conditions. However, within this constraint, excellent agreement has been obtained between an analytical solution and numerical results.

Possible realization of the exciton Bose liquid phase in a hard-core boson model with ring-only exchange interactions

We investigate a hard-core boson model with ring-only exchanges on a square lattice, where a ${K}_{1}$ term acts on 1$\ifmmode\times\else\texttimes\fi{}$1 plaquettes and a ${K}_{2}$ term acts on 1$\ifmmode\times\else\texttimes\fi{}$2 and 2$\ifmmode\times\else\texttimes\fi{}$1 plaquettes, with a goal of realizing a novel exciton Bose liquid (EBL) phase first proposed by Paramekanti et al. [Phys. Rev. B 66, 054526 (2002)]. We construct Jastrow-type variational wave functions for the EBL, study their formal properties, and then use them as seeds for a projective quantum Monte Carlo study. Using the Green's function Monte Carlo approach, we obtain an unbiased phase diagram that at half-filling reveals a charge density wave for small ${K}_{2}$, a valence bond solid for intermediate ${K}_{2}$, and possibly for large ${K}_{2}$ the EBL phase. Away from half-filling, the EBL phase is present for intermediate ${K}_{2}$ and remains stable for a range of densities below 1/2 before phase separation occurs at lower densities.

Convergence theorems for quantum annealing

We prove several theorems to give sufficient conditions for convergence of quantum annealing, which is a protocol to solve generic optimization problems by quantum dynamics. In particular, the property of strong ergodicity is proved for the path-integral Monte Carlo implementation of quantum annealing for the transverse Ising model under a power decay of the transverse field. This result is to be compared with the much slower inverse-log decay of temperature in the conventional simulated annealing. Similar results are proved for the Green's function Monte Carlo approach. Optimization problems in continuous space of particle configurations are also discussed.

Optimization using quantum mechanics: quantum annealing through adiabatic evolution

We review here some recent work in the field of quantum annealing, alias adiabatic quantum computation. The idea of quantum annealing is to perform optimization by a quantum adiabatic evolution which tracks the ground state of a suitable time-dependent Hamiltonian, where '' is slowly switched off. We illustrate several applications of quantum annealing strategies, starting from textbook toy-models—double-well potentials and other one-dimensional examples, with and without disorder. These examples display in a clear way the crucial differences between classical and quantum annealing. We then discuss applications of quantum annealing to challenging hard optimization problems, such as the random Ising model, the travelling salesman problem and Boolean satisfiability problems. The techniques used to implement quantum annealing are either deterministic Schrodinger's evolutions, for the toy models, or path-integral Monte Carlo and Green's function Monte Carlo approaches, for the hard optimization problems. The crucial role played by disorder and the associated non-trivial Landau–Zener tunnelling phenomena is discussed and emphasized.

Euclidean responses of 4He at high momentum transfer.

The Euclidean proton and nucleon responses of $^{4}\mathrm{He}$ at q=10 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}1}$, evaluated with the correlated Glauber approximation using a realistic nucleon-nucleon interaction and nonrelativistic kinematics, are compared to those obtained from the Green's function Monte Carlo approach. The results show that final state interactions play a crucial role even at this large value of q, and that their effect can be quantitatively accounted for with the correlated Glauber approximation.

Scattering in the Framework of the Green's Function Monte-Carlo Method

A method based on the Green's Function Monte-Carlo approach is used to solve the integral equation for the scattering wave function. The algorithm employed is explained for two-body scattering and tested for cases where the exact solution is known.