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.
The Jastrow method is extended to the treatment of tensor forces by the incorporation of suitable state dependence into the two-body correlation operator. The one- and two-body parts of the energy expectation value developed in a factor-cluster (Van Kampen) expansion, are evaluated for nuclear matter. Numerical results are presented for three semirealistic hard-core potentials with differing mixtures of central and tensor components. These results point to the necessity for an examination of three-body cluster corrections to the energy expectation value.
A formulation of the energy eigenvalue problem for a many-particle system is presented in several alternative forms and, following Brueckner and his collaborators, the energy of the system is expressed in terms of solutions to the two-body problem. This work bears considerable resemblance to Brueckner's "linked cluster" expansion. The present expansion is derived with the aid of an ordering operator and involves a sequence of "nearest neighbor" interactions. The particles tend to form "chains," the sequence of approximations involving "two-particle chains," "three-particle chains," etc. The rapidity of convergence of the method depends on the particle density and "temperature" and on the nature of the force between particles. The method lends itself readily to use in statistical mechanics. Some applications to the calculation of the equation of state of a gas are included.
Spin-isospin-dependent liquid structure functions are defined in terms of the projection operators onto particular spin-isospin pair states. Then, necessary conditions on these structure functions are derived in the form of inequalities. The systems studied are liquid 3He, neutron matter, symmetric nuclear matter, asymmetric nuclear matter, and partialy spin-polarized asymmetric nuclear matter. More than one inequalities is obtained for each of these systems, and the number of the inequalities increases with increase of the spin-isospin degree of freedom. It is argued that these conditions can be used to find approximate kinetic-energy expressions of fermion systems.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of
N particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (
N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of
N particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.
This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift
δ(k)=π/2 a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off
Λ such that
δ(k)=0 for
k>Λ. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass
m∗=1. Using only the zero relative momentum component of this interaction the total energy is
ξ=4/9 (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to
ξ=0.24. This result is independent of density and of the cut-off
Λ if
Λ>∼3kf. With
m∗=1 there is however a strong dependence on this cut-off. Including hh-ladders estimates give
ξ=0.4↔0.6, but a reliable result would in this case require a Green's function calculation.