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Physical Review C

Self-energy, momentum distribution, and effective masses of a dilute Fermi gas

R. Sartor, Claude Mahaux

Published Apr 1, 1980• 68 citations
Publisher

Abstract

The dependence upon momentum $k$ and upon energy $E$ of the self-energy of a dilute Fermi gas is studied up to terms of order ${({k}_{F}c)}^{2}$, where ${k}_{F}$ denotes the Fermi momentum and $c$ is the positive scattering length. Algebraic expressions are derived for the imaginary part $W(k;E)$ of the self-energy in the whole ($k$,$E$) plane. They are compared with a conjecture recently made by Orland and Schaeffer in their analysis of single-particle states in nuclei. The contributions of core polarization and of ground state correlations to the real part $V(k;E)$ of the self-energy are calculated with the help of subtracted dispersion relations which connect them with $W(k;{E}^{\ensuremath{'}})$. Algebraic expressions are derived for the momentum distribution in the correlated ground state. It is shown that the effective mass of a quasiparticle with momentum $k$ is equal to the bare particle mass at $k=0$ and reaches a local maximum for $k$ close to ${k}_{F}$. This maximum is ascribed to the dependence of $V(k;E)$ upon $E$, which is described in terms of an $E$ mass. We compute the contributions of core polarization and of ground state correlations to this $E$ mass. The dependence of $V(k;E)$ upon $k$ reflects the nonlocality of the self-energy. It is characterized by a $k$ mass that we also calculate. These results shed light on some nuclear matter properties and on the meaningfulness and limitation of nuclear matter calculations that have recently been performed in the framework of the Brueckner-Hartree-Fock approximation.NUCLEAR STRUCTURE Effective mass near the Fermi surface in dilute Fermi gas model.

DOI
10.1103/physrevc.21.1546

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The dependence upon momentum kkk and upon energy EEE of the self-energy of a dilute Fermi gas is studied up to terms of order (kFc)2{({k}_{F}c)}^{2}(kF​c)2, where kF{k}_{F}kF​ denotes the Fermi momentum and ccc is the positive scattering length. Algebraic expressions are derived for the imaginary part W(k;E)W(k;E)W(k;E) of the self-energy in the whole (kkk,EEE) plane. They are compared with a conjecture recently made by Orland and Schaeffer in their analysis of single-particle states in nuclei. The contributions of core polarization and of ground state correlations to the real part V(k;E)V(k;E)V(k;E) of the self-energy are calculated with the help of subtracted dispersion relations which connect them with W(k;E\ensuremath′)W(k;{E}^{\ensuremath{'}})W(k;E\ensuremath′). Algebraic expressions are derived for the momentum distribution in the correlated ground state. It is shown that the effective mass of a quasiparticle with momentum kkk is equal to the bare particle mass at k=0k=0k=0 and reaches a local maximum for kkk close to kF{k}_{F}kF​. This maximum is ascribed to the dependence of V(k;E)V(k;E)V(k;E) upon EEE, which is described in terms of an EEE mass. We compute the contributions of core polarization and of ground state correlations to this EEE mass. The dependence of V(k;E)V(k;E)V(k;E) upon kkk reflects the nonlocality of the self-energy. It is characterized by a kkk mass that we also calculate. These results shed light on some nuclear matter properties and on the meaningfulness and limitation of nuclear matter calculations that have recently been performed in the framework of the Brueckner-Hartree-Fock approximation.NUCLEAR STRUCTURE Effective mass near the Fermi surface in dilute Fermi gas model.
The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.
The quasiparticle concept is an important tool for the description of many-body systems. We study the quasiparticle properties for dilute Fermi systems with short-ranged, repulsive interactions using effective field theory. We calculate the proper self-energy contributions at order (k(F)/Lambda)(3), where A is the short-distance scale that sets the size of the effective range parameters and k(F) the Fermi momentum. The quasiparticle energy, width, and effective mass to O((k(F)/Lambda)(3)) are derived from the calculated self-energy. (C) 2002 Elsevier Science B.V. All rights reserved.
The effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 order (where kF{k}_{\mathrm{F}}kF​ is the system's Fermi momentum and a0{a}_{0}a0​ the sss-wave scattering length) and (with more labor) to extend it up to the order (kFa0)4{({k}_{\mathrm{F}}{a}_{0})}^{4}(kF​a0​)4. The analogous expansion of the ground-state energy of the polarized gas of spin 1/21/21/2 fermions is known only up to the kFa0{k}_{\mathrm{F}}{a}_{0}kF​a0​ order (where kF{k}_{\mathrm{F}}kF​ stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 contribution has been computed (analytically) only using a specific (hard-core type) interaction potential. Here we show that the effective field theory method also allows to easily obtain the order (kFa0)2{({k}_{\mathrm{F}}{a}_{0})}^{2}(kF​a0​)2 correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.
We have examined the ground state of the many-fermion system, using perturbation theory, with particular attention to Brueckner's method. A study of the single-particle propagators in the ground state of the system sheds some light on the relation between the single-particle energies used, for example, in the optical model of nuclear reactions, and Brucekner's self-consistent energies. This leads naturally to the use of a renormalization factor to allow for propagation off the energy shell. It is possible to formulate the perturbation theory entirely in terms of graphs with no fermion self-energy parts, and this method is used to examine the approximations used for this problem, and to relate more closely the calculations of the ground-state energy and of other properties of the ground state.
We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.
We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.