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arXiv:0705.1191 · Physical Review E

Improved quantum hard-sphere ground-state equations of state

M. A. Solís, MANUEL DE LLANO, J. W. Clark, George Allen Baker Jr.

Published Sep 20, 2007• 2 citations
arXivHTMLPDF

Abstract

The London ground-state energy formula as a function of number density for a system of identical boson hard spheres, corrected for the reduced mass of a pair of particles in a "sphere-of-influence" picture, and generalized to fermion hard-sphere systems with two and four intrinsic degrees of freedom, has a double-pole at the ultimate regular (or periodic, e.g., face-centered-cubic) close-packing density usually associated with a crystalline branch. Improved fluid branches are constructed based upon exact, field-theoretic perturbation-theory low-density expansions for many-boson and many-fermion systems, extrapolated to intermediate densities via Padé and other approximants, but whose ultimate density is irregular or random closest close-packing as suggested in studies of a classical system of hard spheres. Results show substantially improved agreement with the best available Green-function Monte Carlo and diffusion Monte Carlo simulations for bosons, as well as with ladder, variational Fermi hypernetted chain, and so-called L -expansion data for two-component fermions.

DOI
10.1103/physreve.76.031125

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The London ground-state energy formula as a function of number density for a system of identical boson hard spheres, corrected for the reduced mass of a pair of particles in a "sphere-of-influence" picture, and generalized to fermion hard-sphere systems with two and four intrinsic degrees of freedom, has a double-pole at the ultimate regular (or periodic, e.g., face-centered-cubic) close-packing density usually associated with a crystalline branch. Improved fluid branches are constructed based upon exact, field-theoretic perturbation-theory low-density expansions for many-boson and many-fermion systems, extrapolated to intermediate densities via Padé and other approximants, but whose ultimate density is irregular or random closest close-packing as suggested in studies of a classical system of hard spheres. Results show substantially improved agreement with the best available Green-function Monte Carlo and diffusion Monte Carlo simulations for bosons, as well as with ladder, variational Fermi hypernetted chain, and so-called L -expansion data for two-component fermions.
A family of extrapolants somewhat more general than Pad\'e approximants is presented and applied to the low-density ground-state energy expansions which systematic quantum-field-theoretic infinite-partial-summation techniques have produced for many-boson and many-fermion systems of hard spheres. The existence of a quantum close-packing ultimate system density, as well as monotonic-increasing energy as function of density, is assumed in narrowing down the possible extrapolants appropriate for moderate to high densities. Of those remaining, excellent agreement is found at physical densities, and almost unique selections at higher densities, which are consistent both with available Green-function Monte Carlo data as well as with a feasible overall picture of ultimate-density pole placements.
For a two-dimensional hard-disk boson system we propose an analytical London-type expression for the ground-state energy as a function of density ranging from zero to close packing. Such an equation interpolates between the leading term of the exact low-density series expansion and the behavior expected at close-packing based on uncertainty-principle arguments and determined by the correct value of the residue which is a dimensionless constant whose value depends on the space dimension as well as on the kind of close packing configuration involved. As for any reasonably accurate equation of state, variational Monte Carlo calculations for both fluid and crystalline energy branches turn out to be close upper bounds to the resulting so-called London–Hubbard energy which includes pair-correlation effects.
Based on the well-known low-density expansion for the ground-state energy per particle of a many-fermion system interacting pairwise via a central potential we develop a perturbation scheme in which the zero-order state is that of the purely-repulsive-spheres fluid. For the hard-core plus square-well pair-interaction case we have deduced, for two-species fermion matter, perturbation formulas up to sixth order in the attractive coupling parameter. We explicitly report results up to fourth order here. Pad\'e-approximant analyses in the density are carried out in each order with the intent of providing the basis necessary for extensive equation-of-state calculations of various many-fermion systems. In particular, for zero order (the fermion hard-sphere fluid) we predict an uncertainty-principle divergence in the energy, and hence pressure, corresponding to the random close packing of spheres, at a density of about 0.5 times the recently obtained value for boson hard spheres and about 0.2 times the well-known empirical value for classical hard spheres.
The thermodynamic perturbation scheme for the very successful construction of classical liquid equations of state is applied in the quantum regime to the ground-state energy of a many-boson system interacting pairwise via a Lennard-Jones inter­ molecular potential. The ensuing density and attractive-coupling double expansion for the energy is series-analyzed with Pade techniques, as well as generalizations thereof. Results through sixth order compare with the Green Function Monte Carlo simula­ tions for the same system at the one-percent level of accuracy.
The authors investigate the calculational scheme for an eventual first-principles description of the ground-state energy of the many-boson Schrodinger equation. They employ a model 4 He intermolecular pair potential consisting of a hard core surrounded by an attractive rectangular well. The study is based on the well known low-density expansion for such a system, where three coefficients are known. The first corrections to the fourth coefficient are determined on the basis of the rearranged series being a thermodynamic perturbation expansion, namely one representing perturbation not about the ideal gas but about the hard-sphere fluid. Pade approximants, and variations thereof, are employed to extrapolate to physical regimes in both resulting series, in density as well as in the attractive coupling constant.