The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It is found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed.
The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It was found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed. (auth)
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
(kFc)2, where
kF 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
kF. 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.
Marginal Fermi liquid was originally introduced as a phenomenological description of the cuprates in a part of the metallic doping range which appears to be governed by fluctuations due to a quantum-critical point. An essential result due to the form of the assumed fluctuation spectra is that the large inelastic quasiparticle relaxation rate near the Fermi surface is proportional to the energy measured from the chemical potential, ${\ensuremath{\tau}}_{i}^{\ensuremath{-}1}\ensuremath{\propto}\ensuremath{\epsilon}$. We present a microscopic long-wavelength derivation of the hydrodynamic properties in such a situation by an extension of the procedure that Eliashberg used for the derivation of the hydrodynamic properties of a Landau-Fermi liquid. In particular, the density-density and the current-current correlations and the relation between the two are derived, and the connection to microscopic calculations of the frequency dependence of the optical conductivity with an additional Fermi-liquid correction factor shown to follow. The method used here may be necessary, quite generally, for the correct hydrodynamic theory for any problem of quantum-critical fluctuations in fermions.
The Fermi surface topology in the two-dimensional Hubbard model is particularly relevant for the high-temperature superconductors, whereas its theoretical research encounters with the difficulty of the analytical continuation problem. To this end, we proposed the concept of the momentum-dependent compressibility, defined as the variation of the momentum distribution function with respect to the chemical potential. The surface determined by the maximum of the momentum-dependent compressibility is nearly identical to the Fermi surface in the weakly and intermediate coupling regions according to our numerical results. In the correlated region, this surface also exhibits pocket and arc features, just like the Fermi surface in high-temperature superconductors. Therefore, for theoretical studies, this surface can be used as an alternative to determine the underlying Fermi surface. Considering that the momentum-dependent compressibility is closely related to the charge density correlations, our work also shows a connection between the Fermi surface topology and the charge density fluctuations.
We introduce density-functional theory for inhomogeneous Bose-Fermi mixtures, derive the associated Kohn-Sham equations, and determine the exchange-correlation energy in local-density approximation. We solve numerically the Kohn-Sham system, and determine the boson and fermion density distributions and the ground-state energy of a trapped, dilute mixture beyond mean-field approximation. The importance of the corrections due to exchange correlation is discussed by a comparison with current experiments; in particular, we investigate the effect of the repulsive potential-energy contribution due to exchange correlation on the stability of the mixture against collapse.