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Physical Review A

Landau theory of a moderately dense Boltzmann gas

J. C. Rainwater, Robert F. Snider

Published Mar 1, 1976• 5 citations
Publisher

Abstract

The moderately dense quantum Boltzmann gas at equilibrium is shown to be equivalent to a gas of "dressed particles," or "quasiparticles," like those of Landau's theory of a Fermi liquid. The Landau quasiparticle interaction energy $F(p)$ is related to the second virial coefficient by means of an inverse Laplace transform. Expressions are obtained for $F(p)$, and consequently the "renormalized" single-quasiparticle energy ${\ensuremath{\epsilon}}_{p}$, in terms of both scattering phase shifts and the M\o{}ller operator. The theory is compared with the Hartree approximation, the results of Allen, Nicolis et al., the work of Baerwinkel and Grossmann, and with expressions from Fermi-fluid theory.

DOI
10.1103/physreva.13.1190

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The moderately dense quantum Boltzmann gas at equilibrium is shown to be equivalent to a gas of "dressed particles," or "quasiparticles," like those of Landau's theory of a Fermi liquid. The Landau quasiparticle interaction energy F(p)F(p)F(p) is related to the second virial coefficient by means of an inverse Laplace transform. Expressions are obtained for F(p)F(p)F(p), and consequently the "renormalized" single-quasiparticle energy \ensuremathϵp{\ensuremath{\epsilon}}_{p}\ensuremathϵp​, in terms of both scattering phase shifts and the M\o{}ller operator. The theory is compared with the Hartree approximation, the results of Allen, Nicolis et al., the work of Baerwinkel and Grossmann, and with expressions from Fermi-fluid theory.
We introduce a new renormalisation scheme to construct the Landau quasiparticles of Fermi fluids. The scheme introduces an energy cutoff ΛΛΛ to remove the resonant couplings, enabling the dressing of the particles into quasiparticles via a unitary transformation. The dynamics of the quasiparticles is then restricted to low-energy transitions and is fully determined by an effective Hamiltonian which unifies the Landau function fff, the pair interaction ggg responsible for superfluidity, and the collision amplitude A\mathcal{A}A responsible for transport and equilibration. Studying the flow equation that results from infinitesimal variations of the cutoff, we recover the Bethe-Salpeter relation between fff and the forward limit of A\mathcal{A}A, and we demonstrate an analogue relation between ggg and the frontal limit of A\mathcal{A}A. We show that our effective theory captures all the low-energy phenomena of Fermi liquids, from the equation of state to the transport properties, both in the normal and in the superfluid phase. We apply it to the calculation of non-Fermi liquid corrections to the quasiparticle lifetime. This publication is continued by arXiv:2607.07041 where we apply the effective theory to a Fermi fluid of ultracold atoms.
Low-lying excitations in a liquid or a solid are considered to have a long lifetime and to obey the Bose or Fermi statistics. They will be called "quasiparticles" in general. The low-lying energy levels of the system are considered to be characterized by a set of occupation numbers of the quasiparticles. The expressions for the grand partition function and the distribution function (the average occupation number) of the quasiparticles have the same structure as that for the classical lattice gas, under a nonuniform external field and with two-, three-,..., body interactions. Therefore the virial expansion formula for the classical lattice gas can be applied to the system of quasiparticles. The result is simplified by assuming that the energy correction due to the simultaneous excitations of two, three,..., nnn quasiparticles are of $O({V}^{\ensuremath{-}1}),O({V}^{\ensuremath{-}2}),\ensuremath{\cdots},O({V}^{\ensuremath{-}n+1})$, respectively, where VVV is the volume of the system. The resulting expression for the distribution function of the quasiparticles is either a Fermi or a Bose distribution function with an effective energy. The expression for the entropy in terms of the distribution function is the same as that for an ideal gas. This shows that Landau's formula for the distribution function of quasiparticles for a Fermi liquid is valid under conditions which are more general and more practical than those given by Landau. Finally the situation in the Husimi-Temperley model is discussed in comparison with that in the Ising model from the point of view of the present formulation.
Abstract This chapter analyses Landau’s Fermi liquid theory. A fundamental postulate of Landau’s Fermi liquid theory is that quasiparticle energies are a function of the momentum, p, and that the momentum, pF, corresponding to the Fermi energy continues to be given by a free particle expression. Thus, the quasiparticle states are in one-to-one correspondence with those of an ideal Fermi gas and each quasiparticle has a spin quantum number, σ ± 1/2. The remainder of the chapter deals with the quasiparticle interaction function; sum rules; and collisionless collective modes, which are defined as a collective mode as a time- (or frequency) and space- (or wave vector) dependent oscillation of the fluid involving a coordinated motion of its constituent particles. Sample problems are also provided at the end of the chapter.
We show that the Bose liquid can be worked out in a way similar to the Landau Fermi liquid theory by decomposing the diagonal elements of the Boltzmann factor for the system into irreducible diagrams. The energy of the system is expressed by the functional of the occupation number np. The excitation energy is defined by the variation of the energy with respect to np. The Landau f-functions (the second variations of the energy) are closely related to the excitation energy and play important roles in the phase transition. The excitation-energy dependence of Landau f-functions are obtained for the dilute gas.
We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2kF{\mathit{k}}_{\mathit{F}}kF​. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2kF{\mathit{k}}_{\mathit{F}}kF​. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.
We discuss how one-dimensional interacting fermion systems, which in the low-energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional charge and statistics. Our approach is to use Landau's phenomenological approach to Fermi liquid theory, where the quasiparticles are interpreted as adiabatically dressed fermions. In an earlier publication, the local charge carried by these excitations has been shown to be fractional. We focus here on the statistics of the quasiparticles and show that by a change of momentum variables the Landau parameters of the generalized Fermi fluid can be transformed to zero. This change in interaction is compensated by a change of the entropy function, which is consistent with the interpretation of the quasiparticles as satisfying generalized exclusion statistics.