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Physical review. B, Solid state

Self-consistent-field approximation for a one-dimensional many-body system: Fermions

Vera Beatriz Campos, Oscar Hipólito, R. Lobo

Published May 1, 1977• 3 citations
Publisher

Abstract

The self-consistent-field approximation is applied for the study of the energy spectrum, sound velocity, and correlation function on a one-dimensional, spin-1/2 Fermi system, interacting as rigid rods of length $l$. Two very distinct regions are observed, which are named as low- and high-density behaviors, corresponding to small and large Landau's ${F}_{0}$, respectively. The results are very similar to those obtained for a boson system with the same type of interaction, as one might expect. For the very high densities the energy spectrum approaches the $E(q)=0$ line, indicating the existence of an instability, a liquid to crystal transition, which is very neatly followed by observing the behavior of the pair-correlation function as density increases.

DOI
10.1103/physrevb.15.4234

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The self-consistent-field approximation is applied for the study of the energy spectrum, sound velocity, and correlation function on a one-dimensional, spin-1/2 Fermi system, interacting as rigid rods of length lll. Two very distinct regions are observed, which are named as low- and high-density behaviors, corresponding to small and large Landau's F0{F}_{0}F0​, respectively. The results are very similar to those obtained for a boson system with the same type of interaction, as one might expect. For the very high densities the energy spectrum approaches the E(q)=0E(q)=0E(q)=0 line, indicating the existence of an instability, a liquid to crystal transition, which is very neatly followed by observing the behavior of the pair-correlation function as density increases.
The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density $\ensuremath{\rho}$, first interacting through finite rigid rods, of length lll, and again via a repulsive $\ensuremath{\delta}$-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for $\ensuremath{\rho}l<0.5$. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive $\ensuremath{\epsilon}$ potential of strength 2c2c2c, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small $\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}$ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.
The zero-temperature dynamical structure factor $S(q,\ensuremath{\omega})$ of one-dimensional hard rods is computed using state-of-the-art quantum Monte Carlo and analytic continuation techniques, complemented by a Bethe ansatz analysis. As the density increases, $S(q,\ensuremath{\omega})$ reveals a crossover from the Tonks-Girardeau gas to a quasisolid regime, along which the low-energy properties are found in agreement with the nonlinear Luttinger liquid theory. Our quantitative estimate of $S(q,\ensuremath{\omega})$ extends beyond the low-energy limit and confirms a theoretical prediction regarding the behavior of $S(q,\ensuremath{\omega})$ at specific wave vectors ${\mathcal{Q}}_{n}=n2\ensuremath{\pi}/a$, where aaa is the core radius, resulting from the interplay of the particle-hole boundaries of suitably rescaled ideal Fermi gases. We observe significant similarities between hard rods and one-dimensional 4He^{4}\mathrm{He}4He at high density, suggesting that the hard-rods model may provide an accurate description of dense one-dimensional liquids of quantum particles interacting through a strongly repulsive, finite-range potential.
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∗.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.
A central problem in the theory of strongly correlated fermion systems, concerns the nature of their low-lying excitations. It is generally accepted that, except for special cases with symmetry breaking, three dimensional (3D) interacting Fermi systems, are well described in terms of weakly correlated quasi-particles with quantum numbers in one-toone correspondence with those identifying the excitations of a non-interacting Fermi gas. This is the basic assumption underlying the Landau theory of Fermi liquids (FL), which was originally proposed as a phenomenological description of strongly interacting fermions by Landau in early 1956[1), an then established microscopically by Nozieres and Luttinger in 1962[2]. Quite a different scenario occurs in lD interacting Fermi systems. Apart from specific models developing gaps in the excitation spectrum, most fermion systems in one dimension with repulsive interactions, do have low-lying gapless excitations, like ordinary Fermi liquids, but totally different from the quasi-particles predicted by Landau theory. The low-energy behavior of most of these lD interacting fermion systems with gapless linear excitations, can be understood in terms of few model dependent constants, which parametrize all the long-wavelength properties...
A density-functional theory is developed for fermions in one dimension, interacting via a $\ensuremath{\delta}$ function. Such systems provide a natural testing ground for questions of principle, as the local-density approximation should be highly accurate since for this interaction type the exchange contribution to the local-density approximation is intrinsically self-interaction-free. The exact-exchange contribution to the total energy is a local functional of the density. A local-density approximation for correlation is obtained using perturbation theory and Bethe ansatz results for the one-dimensional contact-interacting uniform Fermi gas. The ground-state energies are calculated for two finite systems, the analogs of helium and of Hooke's atom. The local-density approximation is shown to be excellent as expected.