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Physical Review A

Composite pairs and effective two-body scattering in a many-body medium

R. F. Bishop, H. B. Ghassib, Michael R. Strayer

Published Apr 1, 1976• 45 citations
Publisher

Abstract

The generalized off-shell scattering of pairs of constituent fermions embedded in an infinite system of identical particles having thermodynamic properties characterized by arbitrary values of the temperature and chemical potential, is examined using a temperature-dependent Green's-function formalism, which treats particle-particle and hole-hole excitations symmetrically. The important features which emerge are (i) the possible existence of bound-state pairs (of quasiparticles which are linear combinations of particle and hole states) which correspond to poles in the scattering amplitude at negative total energies depending parametrically on the properties of the medium and the total momentum of the interacting pair; (ii) a generalized unitarity condition from which one obtains a parametrization of the on-shell positive-energy scattering amplitude by a set of real effective phase shifts; (iii) that these phase shifts completely determine the large-separation asymptotic behavior of the scattering (positive energy) pair wave functions in a manner which is form equivalent to two-particle scattering in free space; and (iv) that the small- and large-momentum asymptotic limits of the phase shifts are related to the number of bound-state pairs in the medium via an analogous, but modified, form of Levinson's theorem for free scattering. The effects of these pairing singularities on the single-particle excitation spectrum, and hence on the thermodynamic behavior of the system are outlined.

DOI
10.1103/physreva.13.1570

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The generalized off-shell scattering of pairs of constituent fermions embedded in an infinite system of identical particles having thermodynamic properties characterized by arbitrary values of the temperature and chemical potential, is examined using a temperature-dependent Green's-function formalism, which treats particle-particle and hole-hole excitations symmetrically. The important features which emerge are (i) the possible existence of bound-state pairs (of quasiparticles which are linear combinations of particle and hole states) which correspond to poles in the scattering amplitude at negative total energies depending parametrically on the properties of the medium and the total momentum of the interacting pair; (ii) a generalized unitarity condition from which one obtains a parametrization of the on-shell positive-energy scattering amplitude by a set of real effective phase shifts; (iii) that these phase shifts completely determine the large-separation asymptotic behavior of the scattering (positive energy) pair wave functions in a manner which is form equivalent to two-particle scattering in free space; and (iv) that the small- and large-momentum asymptotic limits of the phase shifts are related to the number of bound-state pairs in the medium via an analogous, but modified, form of Levinson's theorem for free scattering. The effects of these pairing singularities on the single-particle excitation spectrum, and hence on the thermodynamic behavior of the system are outlined.
We are concerned with few-particle correlations in a fermionic system at finite temperature and density. Within the many-body Green functions formalism the description of correlations is provided by the Dyson equation approach that leads to effective few-body equations. They contain the dominant medium effects, which are self energy corrections and the Pauli blocking. Hence the effective two-body interactions between quasiparticles are momentum/energy-dependent and therefore they can be usesed in the medium modified, momentum space, integral AGS equations for three- and four-body systems. To investigate correlations and clusters beyond four-body, we employ, instead, the configuration space two-variable integro-differential equations (IDEA) for AAA-body bound systems which are based on Hyperspherical Harmonics and the Faddeev decomposition of the wave function in two-body amplitudes. This requires the transformation of the energy dependent two-body interactions to equivalent local, energy independent, ones. To achieve this we use inverse scattering techniques the resulting interactions being, on-- and (to all practical purposes) off--shell equivalent to the energy dependent potentials. In this way we obtain binding energy results for the 2--, 3--, 4--, and 16--particle in a medium at a finite temperature and various densities. Several aspects of the problem are discussed and the behavior of the potential surfaces obtained in the extreme adiabatic approximation, below and above the Mott transition, is investigated.
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 NNN 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+1N+1N+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 NNN 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.
We study static properties of attractively interacting Bose-Fermi mixtures of uniform atomic gases at zero temperature. Using the Green's function formalism, we calculate the boson-fermion scattering amplitude and the fermion self-energy in the medium to lowest order of the hole-line expansion. We study the ground-state energy and pressure as a function of the scattering length for a few values of the boson-fermion mass ratio mb∕mf{m}_{b}∕{m}_{f}mb​∕mf​ and the number ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​. We find that the attractive contribution to the energy is greatly enhanced for small values of the mass ratio. We study the role of Bose-Fermi pair correlations in the mixture by calculating the pole of the boson-fermion scattering amplitude in the medium. The pole shows a standard quasiparticle dispersion for a Bose-Fermi pair. In addition, we also study the fermion dispersion relation. We find two dispersion branches with the possibility of avoided crossings. This strongly depends on the number ratio Nb∕Nf{N}_{b}∕{N}_{f}Nb​∕Nf​.
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.
To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.