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Physical Review

Sampson-Seitz Procedures and the Static Responses of Normal Fermion Systems

Ronald M. Rockmore

Published Jan 15, 1963• 3 citations
Publisher

Abstract

It is shown that a differential characterization of Sampson-Seitz methods for calculating the static responses of a normal fermion system at zero temperature yields expressions for these quantities which are identical with those deduced by Landau from a semiphenomenological basis and by the author, in several instances, from spherical, time-independent, many-body perturbation theory. This connection is explicitly demonstrated for the Galilean invariance, magnetic susceptibility, and compressibility of a normal fermion system with translation-invariant interactions. The calculation of the magnetic susceptibility and compressibility of a dense electron gas is examined from both points of view. In the case of the magnetic susceptibility, it is shown that the contribution to ${\ensuremath{\alpha}}_{c}$ from graphs with two virtual excitations vanishes to $O({r}_{s})$ in agreement with the Sampson-Seitz calculation of Brueckner and Sawada. Although this demonstration is trivial by Sampson-Seitz methods, it is only a consequence of detailed calculation in Landau's formulation. It is found that $(\frac{1}{K})(\frac{2}{\ensuremath{\rho}R})={\ensuremath{\alpha}}_{c}\ensuremath{-}0.0676\mathrm{ln}{r}_{s}+0.255$, where $K$ is the compressibility, $\ensuremath{\rho}$ is the density, and $R$ is the rydberg.

DOI
10.1103/physrev.129.926

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It is shown that a differential characterization of Sampson-Seitz methods for calculating the static responses of a normal fermion system at zero temperature yields expressions for these quantities which are identical with those deduced by Landau from a semiphenomenological basis and by the author, in several instances, from spherical, time-independent, many-body perturbation theory. This connection is explicitly demonstrated for the Galilean invariance, magnetic susceptibility, and compressibility of a normal fermion system with translation-invariant interactions. The calculation of the magnetic susceptibility and compressibility of a dense electron gas is examined from both points of view. In the case of the magnetic susceptibility, it is shown that the contribution to \ensuremathαc{\ensuremath{\alpha}}_{c}\ensuremathαc​ from graphs with two virtual excitations vanishes to O(rs)O({r}_{s})O(rs​) in agreement with the Sampson-Seitz calculation of Brueckner and Sawada. Although this demonstration is trivial by Sampson-Seitz methods, it is only a consequence of detailed calculation in Landau's formulation. It is found that (1K)(2\ensuremathρR)=\ensuremathαc\ensuremath−0.0676lnrs+0.255(\frac{1}{K})(\frac{2}{\ensuremath{\rho}R})={\ensuremath{\alpha}}_{c}\ensuremath{-}0.0676\mathrm{ln}{r}_{s}+0.255(K1​)(\ensuremathρR2​)=\ensuremathαc​\ensuremath−0.0676lnrs​+0.255, where KKK is the compressibility, \ensuremathρ\ensuremath{\rho}\ensuremathρ is the density, and RRR is the rydberg.
The calculation of the induced spin density of a normal fermion system, such as the electron gas, in the limit of small wave numbers, is carried beyond the random phase approximation (R.P.A.) by the method of canonical transformation so as to include the first non-R.P.A. corrections. An expression for the induced spin density in the same limit, exact to all orders of particle-particle coupling, is then derived by more general methods of many-body perturbation theory. It depends only on the knowledge, to all orders of interparticle coupling, of the effective mass and the forward scattering of quasi-particles at the Fermi surface, and also yields the results of the extended R.P.A. on appropriate expansion. In the limit of zero wave number, the resulting expression for the magnetic susceptibility is found identical with that deduced by Landau from a phenomenological basis and may be regarded as an additional confirmation of the microscopic validity of the theory of the Fermi liquid.
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​.
We use Landau's theory of a normal Fermi liquid to derive expressions for the static response of a system with a general tensor interaction that conserves the total spin and the total angular momentum of the quasiparticle-quasihole pair. The magnetic susceptibility is calculated in detail, with the inclusion of the center of mass tensor and cross vector terms in addition to the exchange tensor one. We also introduce a new parametrization of the tensor Landau parameters which significantly reduces the importance of high angular harmonic contributions. For nuclear matter and neutron matter we find that the two most important effects of the tensor interaction are to give a contribution from multipair states and to renormalize the magnetic moments. Response to a weak probe may be calculated using similar methods, replacing the magnetic moments with the matrix elements of the weak charges.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.
We consider the response of the density of a fermion ensemble to an applied weak static magnetic field. It is known that, for a noninteracting Fermi gas, this response is fully characterized by the Fermi volume and the Berry curvature on the Fermi surface. Here we show the same result holds for interacting fermions, including a Fermi liquid and a non-Fermi liquid, to all orders in perturbation theory. Our result relies only on the assumption of a well-defined Fermi surface and the general analytic properties of quantum field theory, and is completely model independent.
By the method of ``modes'' introduced by Sawada, the excitation spectra of a fermion system with singular interactions have been obtained in three cases; (1) with one additional particle above and and one hole below the Fermi surface, (2) with one additional particle above the Fermi surface, and (3) with two particles above the Fermi surface. The argument holds not only for the system of low density (nuclear matter with a hard-core repulsion), but also for the case of the interacting electron gas in the high-density region. The results have been compared with the Brueckner-Goldstone perturbation-expansion formulas, by using diagrams in the second- and third-order of the expansion in interaction strength. We show that in the approximation of our treatment, the occupation probability function at temperature T has the same form as the Fermi distribution function.