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Proceedings of the Physical Society

The Collective Treatment of a Fermi Gas: I

Sam F Edwards

Published Nov 1, 1958• 17 citations
Publisher

Abstract

The collective approach to a Fermi gas is discussed by considering the wave functions of non-interacting Fermi gases in one and three dimensions, and of the interacting Fermi gas. It is shown how to write the one-dimensional case exactly in terms of collective coordinates, and how these have a well defined existence for long wavelengths even without interaction. This is extended as far as possible to three dimensions, and it is shown how the statistics affect the collective motion in the presence of interaction. A brief discussion is given of how calculations may be performed when redundant sets of coordinates are used.

DOI
10.1088/0370-1328/72/5/301

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•Jan 1, 2008•0 citations
The collective approach to a Fermi gas is discussed by considering the wave functions of non-interacting Fermi gases in one and three dimensions, and of the interacting Fermi gas. It is shown how to write the one-dimensional case exactly in terms of collective coordinates, and how these have a well defined existence for long wavelengths even without interaction. This is extended as far as possible to three dimensions, and it is shown how the statistics affect the collective motion in the presence of interaction. A brief discussion is given of how calculations may be performed when redundant sets of coordinates are used.
The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.
The ground state energy of the free electron gas is calculated using the Rayleigh-Schrödinger variational method with the wave function ψ= D Π i<j f ( x ij ) where D is a determinant of plane waves and f ( x ij ) a correlation function. Consideration of the wave function in terms of the collective coordinates p k' the Fourier components of the density, suggests an accurate approximation for the energy integral which is then evaluated over the coordinates of the particles so that the use of subsidiary conditions is avoided. Effects omitted in the random phase approximation are included and the final results extend continuously over plasma and particle modes and should be valid in the range of densities encountered in real metals. The results agree closely with those of Nozières and Pines, and of Hubbard obtained by more elaborate methods.
The dynamics of a one-dimensional many-particle Fermi system containing one particle spin down (the d particle) in a sea of spin-up particles (u particles) is discussed in the context of an exactly solvable model problem. It is assumed that random variables associated with the d particle are relevant, and the marginal distributions of these random variables are calculated. The marginal distribution of the u-d relative coordinate is found to be constant in time and therefore to provide a connection of dynamics to equilibrium. This constant probability distribution affects the temperature-volume dependence of the partition function and, in the case of an attractive interaction, leads to a phase transition from an unpaired to a paired state at a well-defined critical temperature.
We consider the problem of a single particle interacting with NNN identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the NNN fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one-dimensional problem in the general case with respect to the mass of the single particle.
The collective modes of a spin-density-wave structure are investigated in a one-dimensional model of a Fermi gas. The equations of motion in the random-phase approximation are completely solved at zero temperature and for a $\ensuremath{\delta}$-function interaction. The spin-density-wave structure has been found to be stable if its wavelength is equal to its best variational value. The complete dependence of the energy $\ensuremath{\omega}(q)$ of the collective modes on the momentum qqq is exhibited. A new result is found and proved rigorously: The energy $\ensuremath{\omega}(q)$ vanishes when $\frac{2\ensuremath{\pi}}{q}$ is equal to the wavelength of the spin-density wave.
The question of the breakdown of Fermi liquid theory in two dimensions is examined in the context of perturbation theory for a dilute interacting Fermi gas. The quasiparticle interaction function, fic;u;, is calculated for such a system. The interaction function, calculated to second order in terms of the dimensionless coupling constant, shows various singularities. The most divergent terms appear in the cross channel, but cancel out leaving a much weaker singularity in the limit of two moment a approaching each other (0 —f 0 ). As in the case of the three dimensional Fermi gas, the Cooper channel contains a logarithmic singularity in the limit 0 —f 7r. This singularity can be summed and is known to be harmless to the structure of Fermi liquid theory. A different feature in two dimensions is the existence of such a singularity for 0 —p 0. This feature needs further investigation. Calculations have also been extended to a polarized Fermi gas and the result is equivalent to the unpolarized case and does not show any additional features. In conclusion, the results do not indicate the presence of strong divergences which could cause the breakdown of Fermi liquid theory in two dimensions for a dilute interacting Fermi gas.