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The Journal of Chemical Physics

Effective pair potentials for the noninteracting spin-1/2 Fermi gas

F. A. Stevens, M. A. Pokrant

Published Nov 1, 1974• 7 citations
Publisher

Abstract

The N-particle configurational probability density P(N) of a noninteracting Fermi gas may be approximated as P(N) ≃ Πi<je−u(rij) / ∫d 3NrΠi<je −u(rij), where u(r) is an effective pair potential. Our reason for considering these potentials is that they have been successfully used in computing properties of interacting Fermi systems. We define the best effective potential to be the one whose corresponding two-body distribution function is closest to the exact two-body distribution function. A method for calculating the best u(r) is not known. Lado has calculated u(r) by using the Born-Green-Yvon integrodifferential equation relating u(r) to the two- and three-body distribution functions. Effective potentials may also be calculated using the hypernetted chain and Percus-Yevick integral equations. It is found that potentials computed by the above three methods are in fairly close agreement. The principal purpose of this paper is to present tables of effective potentials approximating both the spin-summed system and parallel spin subsystem probability densities for a system of noninteracting spin-1/2 fermions.

DOI
10.1063/1.1682563

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The N-particle configurational probability density P(N) of a noninteracting Fermi gas may be approximated as P(N) ≃ Πi<je−u(rij) / ∫d 3NrΠi<je −u(rij), where u(r) is an effective pair potential. Our reason for considering these potentials is that they have been successfully used in computing properties of interacting Fermi systems. We define the best effective potential to be the one whose corresponding two-body distribution function is closest to the exact two-body distribution function. A method for calculating the best u(r) is not known. Lado has calculated u(r) by using the Born-Green-Yvon integrodifferential equation relating u(r) to the two- and three-body distribution functions. Effective potentials may also be calculated using the hypernetted chain and Percus-Yevick integral equations. It is found that potentials computed by the above three methods are in fairly close agreement. The principal purpose of this paper is to present tables of effective potentials approximating both the spin-summed system and parallel spin subsystem probability densities for a system of noninteracting spin-1/2 fermions.
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/2δ(k)=π/2δ(k)=π/2 a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off ΛΛΛ such that δ(k)=0δ(k)=0δ(k)=0 for k>Λk>Λk>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∗=1m^{*}=1m∗=1. Using only the zero relative momentum component of this interaction the total energy is ξ=4/9ξ=4/9ξ=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.ξ=0.24.ξ=0.24. This result is independent of density and of the cut-off ΛΛΛ if Λ>∼3kfΛ> \sim 3k_{f}Λ>∼3kf​. With m∗≠1m^{*}\neq 1m∗=1 there is however a strong dependence on this cut-off. Including hh-ladders estimates give ξ=0.4↔0.6ξ=0.4\leftrightarrow 0.6ξ=0.4↔0.6, but a reliable result would in this case require a Green's function calculation.
A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of vvv representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state vvv representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.
One-dimensional spinor gases with strong $\ensuremath{\delta}$ interaction fermionize and form a spin chain. The spatial degrees of freedom of this atom chain can be described by a mapping to spinless noninteracting fermions and the spin degrees of freedom are described by a spin-chain model with nearest-neighbor interactions. Here, we compute momentum and occupation-number distributions of up to 16 strongly interacting spinor fermions and bosons as a function of their spin imbalance, the strength of an externally applied magnetic field gradient, the length of their spin, and for different excited states of the multiplet. We show that the ground-state momentum distributions resemble those of the corresponding noninteracting systems, apart from flat background distributions, which extend to high momenta. Moreover, we show that the spin order of the spin chain---in particular antiferromagnetic spin order---may be deduced from the momentum and occupation-number distributions of the system. Finally, we present efficient numerical methods for the calculation of the single-particle densities and one-body density matrix elements and of the local exchange coefficients of the spin chain for large systems containing more than 20 strongly interacting particles in arbitrary confining potentials.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
Strongly interacting one-dimensional (1D) Bose--Fermi mixtures form a tunable XXZXXZXXZ spin chain. Within the spin-chain model developed here, all properties of these systems can be calculated from states representing the ordering of the bosons and fermions within the atom chain and from the ground-state wave function of spinless noninteracting fermions. We validate the model by means of an exact diagonalization of the full few-body Hamiltonian in the strongly interacting regime. Using the model, we explore the phase diagram of the atom chain as a function of the boson-boson (BB) and boson-fermion (BF) interaction strengths and calculate the densities, momentum distributions, and trap-level occupancies for up to 17 particles. In particular, we find antiferromagnetic (AFM) and ferromagnetic (FM) order and a demixing of the bosons and fermions in certain interaction regimes. We find, however, no demixing for equally strong BB and BF interactions, in agreement with earlier calculations that combined the Bethe ansatz with a local-density approximation.