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Journal of Physics Conference Series

Variational calculations with explicit energy functionals for fermion systems at zero temperature

Masatoshi Takano, Takayuki Suzuki, Naoyuki Sakumichi

Published Mar 1, 2016• 1 citation
Publisher

Abstract

The variational method with explicit energy functionals (EEFs) is applied to infinite neutron matter. Starting from the Argonne v8' two-body potential and the repulsive part of the Urbana IX three-body potential, the energy per neutron is expressed explicitly with the spin-dependent central, tensor, and spin-orbit distribution functions. This EEF is constructed as an extension of the previously proposed one for the Argonne v6' potential, including central and tensor forces. The Euler-Lagrange equations derived from this EEF are solved numerically. The obtained fully minimized energy with this EEF is in good agreement with that obtained from the auxiliary field diffusion Monte Carlo calculation.

DOI
10.1088/1742-6596/702/1/012016

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Progress of Theoretical and Experimental Physics•Sep 24, 2019•8 citations
The variational method with explicit energy functionals (EEFs) is applied to infinite neutron matter. Starting from the Argonne v8' two-body potential and the repulsive part of the Urbana IX three-body potential, the energy per neutron is expressed explicitly with the spin-dependent central, tensor, and spin-orbit distribution functions. This EEF is constructed as an extension of the previously proposed one for the Argonne v6' potential, including central and tensor forces. The Euler-Lagrange equations derived from this EEF are solved numerically. The obtained fully minimized energy with this EEF is in good agreement with that obtained from the auxiliary field diffusion Monte Carlo calculation.
A refinement is made on the approximate energy expression proposed previously by Takano and Yamada for infinite zero-temperature systems of spin-1/2 fermions interacting through two-body central forces which may depend on the two-body spin states. This refinement is accomplished through incorporation of a certain part of the three-body-cluster exchange terms into the energy expression. The Euler-Lagrange equation is derived from the refined energy expression and solved numerically for liquid 3He and for neutron matter. The energy per particle for liquid 3He is in better agreement with experimental data, while the effect of this refinement is relatively small for neutron matter. The three-body cluster terms not included in the refined energy expression are evaluated perturbatively and shown to be negligibly small for liquid 3He as well as for neutron matter with cenral forces.
We present calculations of the energy per particle of pure neutron and symmetric nuclear matter with simplified Argonne nucleon-nucleon potentials for different many-body theories. We compare critically the Brueckner-Hartree-Fock results to other formalisms, such as the Brueckner-Bethe-Goldstone expansion up to third order, self-consistent Green's functions, auxiliary field diffusion Monte Carlo, and Fermi hypernetted chain. We evaluate the importance of spin-orbit and tensor correlations in the equation of state and find these to be important in a wide range of densities.
Uniform neutron matter is approximated by a cubical box containing a finite number of neutrons, with periodic boundary conditions. We report variational and Green's function Monte Carlo calculations of the ground state of fourteen neutrons in a periodic box using the Argonne ${v8}^{\ensuremath{'}}$ two-nucleon interaction at densities up to one and half times the nuclear matter density. The effects of the finite box size are estimated using variational wave functions together with cluster expansion and chain summation techniques. They are small at subnuclear densities. We discuss the expansion of the energy of low-density neutron gas in powers of its Fermi momentum. This expansion is strongly modified by the large nn\mathrm{nn}nn scattering length, and does not begin with the Fermi-gas kinetic energy, as assumed in both Skyrme and relativistic mean field theories. The leading term of neutron gas energy is approximately half the Fermi-gas kinetic energy. The quantum Monte Carlo results are also used to calibrate the accuracy of variational calculations employing Fermi hypernetted and single operator chain summation methods to study nucleon matter over a larger density range, with more realistic Hamiltonians including three-nucleon interactions.
Abstract Pure neutron matter is not likely to be found in any of the Earth’s laboratories, but it has been found to exist as the material of neutron stars. It presents an interesting and difficult manybody problem which may be attacked by QMC methods. In this paper the authors report variational, approximate fixed-node, and released-node QMC calculations for uniform neutron matter in a three-dimensional box with periodic boundary conditions. These allow the determination of the energy E(p) as a function of the neutron density p for a realistic two-neutron interaction expression. The results are used to assess the accuracy of earlier variational approaches. The system treated was that of 14 neutrons in a cube at densities in the range of 1 / 4 to 3/2 the density p0 typical for nuclear matter. Most of the calculations were done for the Argonne v8’ two-body interaction potential. The trial wavefunction was a correlated Slater function, identical for the variational and diffusion calculations, specifying the nodes and used in importance sampling. Corrections were made for the effects of finite box dimensions. The released-node calculations were limited to very short time intervals especially for the higher densities. The energies E(p) found in the diffusion QMC calculations were 5 to 10% lower than those of the variational QMC. The releasednode results, with relatively large statistical errors, were not significantly different from the fixed-node results. The energies determined for the lower densities were estimated to be accurate within 2% and those for the higher densities somewhat less accurate. The results showed earlier predictions to be generally correct. Comparisons of results from variational chain summations (VCS) indicated an overall accuracy of about 10% for that method.
Abstract The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4′^\prime′NNNNNN potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function FFF, where FFF can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian H\mathcal{H}H and FFF. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density ρ≃0.20\rho \simeq 0.20ρ≃0.20 fm−3^{-3}−3 in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.