G. V. Chester

The Metropolis Monte Carlo method is used to sample the square of an antisymmetric wave function composed of a product of a Jastrow wave function and a number of Slater determinants. We calculate variational energies for $^{3}\mathrm{He}$ and several models of neutron matter. The first-order Wu-Feenberg expansion is shown always to underestimate the energy, sometimes seriously. The phase diagram for ground-state Yukawa matter is determined. There is a class of Yukawa potentials which do not lead to a crystal phase at any density.

It is shown, by almost rigorous arguments, that there exist many-body states of a system of interacting bosons which exhibit both crystalline order and Bose-Einstein condensation into the zero-momentum eigenstate of the single-particle density matrix. The implications of this result are discussed in relation to theories of superfluidity and the nature of quantum crystals.

We have simulated the two-dimensional classical planar spin model using the Metropolis Monte Carlo technique. The loss of long-range order as a function of the size of the lattice was confirmed. The energy and specific heat were calculated for a square lattice of 900, 3 600, and 10 000 spins. A sharp specific-heat peak was found at $\frac{{k}_{B}T}{J}=1.02$ $J$ is the nearest-neighbor coupling), 15% above the transition temperature $\frac{{k}_{B}{T}_{c}}{J}=0.89$. ${T}_{c}$ was determined by fitting the spin-spin correlation function and the susceptibility to the forms of the Kosterlitz-Thouless theory. The density of vortex pairs was computed and found to increase exponentially with inverse temperature. At ${T}_{c}$ vortex pairs begin to unbind and also larger clusters of vortices appear and unbind as the temperature is increased. These larger clusters may be responsible for the specific-heat peak being sharper and closer to ${T}_{c}$ than simple theories predict.

The exact expressions for the transport coefficients of a metal are used to show that the Wiedemann-Franz law is valid provided that (a) the electrons do not interact with each other and form a degenerate Fermi-Dirac assembly, (b) the scattering of the electrons is due to impurities or lattice vibrations and is elastic. The derivation is valid no matter how strong the scattering and it is therefore more general than the usual weak-coupling derivation of the Wiedemann-Franz law.

In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$, the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-$k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.