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arXiv:1005.2349

Particle-Hole Optical Model: Fantasy or Reality?

M. H. Urin

Published May 13, 2010• 0 citations
arXivHTMLPDF

Abstract

An attempt to formulate the optical model of particle-hole-type excitations (including giant resonances) is undertaken. The model is based on the Bethe--Goldstone equation for the particle-hole Green function. This equation involves a specific energy-dependent particle-hole interaction that is due to virtual excitation of many-quasiparticle configurations and responsible for the spreading effect. After energy averaging, this interaction involves an imaginary part. The analogy between the single-quasiparticle and particle-hole optical models is outlined.

DOI
10.48550/arxiv.1005.2349

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An attempt to formulate the optical model of particle-hole-type excitations (including giant resonances) is undertaken. The model is based on the Bethe--Goldstone equation for the particle-hole Green function. This equation involves a specific energy-dependent particle-hole interaction that is due to virtual excitation of many-quasiparticle configurations and responsible for the spreading effect. After energy averaging, this interaction involves an imaginary part. The analogy between the single-quasiparticle and particle-hole optical models is outlined.
In an accompanying paper, we have described a new approach to the computation of ground-state energies and of correlated particle-hole excitations in terms of a phenomenological pseudo-Hamiltonian containing two- and more-body smooth effective potentials which sum the results of short-range correlations and of density fluctuation amplitudes which describe long-range correlations. In this paper we study the problem of relating the pseudopotentials to a (possibly singular) microscopic interaction with the aid of tools developed within the framework of the coupled cluster theory of K\"ummel and his collaborators, the results containing both familiar and unfamiliar elements. For example, the formulas derived, which depend on Bethe-Goldstone and Bethe-Faddeev amplitudes, include new definitions of particle-hole scattering matrix elements. An important consistency check is satisfied, in that the problem of defining the phenomenological potentials in terms of the microscopic ones must yield two separate but equivalent solutions, once within the framework of the theory of the ground-state energy and a second time within the framework of the theory of excitations. The entire package is studied with the aid of a modified version of coupled cluster theory, and shown to form a self-consistent entity. Among the desirable features of the formalism is that the large gap in the single-particle energy spectrum often utilized in existing formalisms is naturally absent from the current one.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
This dissertation deals with aspects of many-body problem in the electronic structure of solids. More specifically, we study how the introduction of Green’s function of many bodies and their equation of motion leads to information about the interactions between particles. Thus, we get to the GW and Bethe-Salpeter formalisms. We show in specific cases how the GW is able to describe energy gaps at a level that goes beyond the Density Functional Theory. Likewise, we investigate the inclusion of excitonic effects through the Bethe-Salpeter equation, showing how the approach can provide optical spectra in good agreement with experimental results. We exemplify the above points with the discussion of specific cases.
Low-energy theory of the Nambu-Goldstone modes of an ultracold 6Li−^6Li-6Li− 40K^{40}K40K mixture in an optical lattice
A low-energy theory of the Nambu-Goldstone excitation spectrum and the corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and Potassium-40 atoms in a two-dimensional optical lattice at finite temperatures with the Fulde-Ferrell order parameter has been formulated. It is assumed that the two-species interacting Fermi gas is described by the one-band Hubbard Hamiltonian with an attractive on-site interaction. The discussion is restricted to the BCS side of the Feshbach resonance where the Fermi atoms exhibit superfluidity. The quartic on-site interaction is decoupled via a Hubbard-Stratonovich transformation by introducing a four-component boson field which mediates the Hubbard interaction. A functional integral technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green's function and the Bethe-Salpeter equation for the two-particle Green's function and the associated collective modes. The numerical solution of the Bethe-Salpeter equation in the generalized random phase approximation shows that there exist two distinct sound velocities in the long-wavelength limit. In addition to the long-wavelength mode (Goldstone mode), the two-species Fermi gas has a superfluid phase revealed by two rotonlike minima in the asymmetric collective-mode energy.
A low-energy theory of the Nambu-Goldstone excitation spectrum and the corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and Potassium-40 atoms in a two-dimensional optical lattice at finite temperatures with the Fulde-Ferrell order parameter has been formulated. It is assumed that the two-species interacting Fermi gas is described by the one-band Hubbard Hamiltonian with an attractive on-site interaction. The discussion is restricted to the BCS side of the Feshbach resonance where the Fermi atoms exhibit superfluidity. The quartic on-site interaction is decoupled via a Hubbard-Stratonovich transformation by introducing a four-component boson field which mediates the Hubbard interaction. A functional integral technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green’s function and the Bethe-Salpeter equation for the two-particle Green’s function and the associated collective modes. The numerical solution of the Bethe-Salpeter equation in the generalized random phase approximation shows that there exist two distinct sound velocities in the long-wavelength limit. In addition to low-energy (Goldstone) mode, the two-species Fermi gas has a superfluid phase revealed by two roton-like minima in the asymmetric collective-mode energy.