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The Journal of Chemical Physics

London equation of state for quantum hard disks

M. A. Solís

Published Dec 22, 1996• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.472925

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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.
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.
Quantum Monte Carlo methods are used to calculate various ground-state properties of charged bosons in two dimensions, throughout the whole density range where the fluid phase is stable. Wigner crystallization is predicted at ${r}_{s}\ensuremath{\simeq}60.$ Results for the ground-state energy and the momentum distribution are summarized in analytic interpolation formulas embodying known asymptotic behaviors. Near freezing, the condensate fraction is less than 1%. The static structure factor S(k)S(k)S(k) and susceptibility $\ensuremath{\chi}(k)$ are obtained from the density-density correlation function in imaginary time, $F(\mathbf{k},\ensuremath{\tau}).$ An estimate of the energy of elementary excitations, given in terms of an upper bound involving S(k)S(k)S(k) and $\ensuremath{\chi}(k),$ is compared with the result obtained via analytic continuation from $F(\mathbf{k},\ensuremath{\tau}).$
The energy and structure of a dilute hard-disks Bose gas are studied in the framework of a variational many-body approach based on a Jastrow correlated ground-state wave function. The asymptotic behaviors of the radial distribution function and the one-body density matrix are analyzed after solving the Euler equation obtained by a free minimization of the hypernetted chain energy functional. Our results show important deviations from those of the available low density expansions, already at gas parameter values $x\ensuremath{\sim}0.001$. The condensate fraction in 2D is also computed and found generally lower than the 3D one at the same xxx.
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.
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.
Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at rs=139{r}_{s}=139rs​=139, where rs{r}_{s}rs​ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at rs=2{r}_{s}=2rs​=2, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).