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Progress of Theoretical Physics

Self-Consistent Fermi-Liquid Properties and Density of States for Nuclear Many-Body Approximations

Hiroshi Uechi

Published Jul 1, 1991• 3 citations
Publisher

Abstract

Conserving approximations lead to thermodynamically consistent expressions for Fermi-liquid properties such as compression modulus, symmetry energy and Landau parameters. We will emphasize the need for the precision experiment to determine the density of states at the Fermi surface by discussing nuclear matter approximations in terms of thermodynamic consistency (the Hugenholtz-Van Hove theorem and the virial theorem). The density of states is the primary self-consistent observable to be determined from nuclear many-body approximations. The precise measurement of this observable will establish a rigorous criterion on the validity of nuclear matter approximations.

DOI
10.1143/ptp.86.119

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Conserving approximations lead to thermodynamically consistent expressions for Fermi-liquid properties such as compression modulus, symmetry energy and Landau parameters. We will emphasize the need for the precision experiment to determine the density of states at the Fermi surface by discussing nuclear matter approximations in terms of thermodynamic consistency (the Hugenholtz-Van Hove theorem and the virial theorem). The density of states is the primary self-consistent observable to be determined from nuclear many-body approximations. The precise measurement of this observable will establish a rigorous criterion on the validity of nuclear matter approximations.
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Deuterium atoms form a fascinating spin-one Fermi fluid of moderate density. Several properties of this fluid are evaluated using the self-consistent Green-function method of many-body theory. The aim is both to determine the properties and to see how well Green-function methods can be implemented in a moderately dense Fermi liquid. Approximations begin with the self-energy. This is separated into Brueckner-Hartree-Fock terms and correlation or two-hole--one-particle (HHP) terms. We find ground-state energies E in reasonable agreement with Monte Carlo values. While the HHP terms are not so important for E, they lead to a significant enhancement of the effective mass m∗{\mathit{m}}^{\mathrm{*}}m∗ at the Fermi surface ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{F}}$. The particle-hole interaction is calculated from the self-energy using the conserving Baym-Kadanoff method. This leads to Landau parameters F0s{\mathit{F}}_{0}^{\mathit{s}}F0s​ that are positive and an F1s{\mathit{F}}_{1}^{\mathit{s}}F1s​ that is consistent with m∗{\mathit{m}}^{\mathrm{*}}m∗.
A method for choosing and deriving a suitable single-particle potential for many-body calculations at finite temperature is suggested. With this potential one can readily determine the self-consistent correlated chemical potential and consequently the resulting equations of state rigorously satisfy a number of basic thermodynamic relations. Application of our method to a Brueckner-Hartree-Fock nuclear matter calculation yields results in remarkably good agreement with the Hugenholtz--Van Hove theorem.
We investigate the relation between the binding energy and the Fermi energy and between different expressions for the pressure in cold nuclear matter. For a self-consistent calculation based on a Phi\\PhiPhi derivable T−T-T−matrix approximation with off-shell propagators the thermodynamic relations are well satisfied unlike for a G−G-G−matrix or a T−T-T−matrix approach using quasi-particle propagators in the ladder diagrams.
The second term being the pressure of the system would vanish for a saturating system at ground state i.e. at equilibrium resulting in the Eq. E/A = ǫf . This is a rare theorem in many-body Physics, which has been rigorously shown[1] to be true by its original authors Hugenholtz and Van Hove by taking all orders of perturbation in the frame-work of timeindependent perturbation theory[1–5]. However it should be mentioned here that prior to this rigorous proof Bethe[6] had also visualized the theorem under HF approximation. It is valid for any interacting infinite Fermi system and thereby applicable to liquid He and in particular to nuclear matter. With its
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.