By using the general transport theory of a normal many-fermion system recently developed by R\'esibois, the collision term is calculated up to fourth order in a coupling parameter. It is shown that the higher order terms can be arranged into the form of the quasiparticle collision term suggested by Landau in the Fermi-liquid theory, except for the transformation operator introduced by R\'esibois to connect the particle description to the quasiparticle one. We also analyze the three-body collision operator in transport theory from the same quasiparticle point of view to the lowest nontrivial order in the coupling constant. It is shown that the anomalous terms in this operator contain the renormalization effects on the usual two-body collision operator.
By using the general transport theory of a normal many-fermion system recently developed by R'esibois, the collision term is calculated up to fourth order in a coupling parameter. It is shown that the higher order terms can be arranged into the form of the quasiparticle collision term suggested by Landau in the Fermi-liquid theory, except for the transformation operator introduced by R'esibois to connect the particle description to the quasiparticle one. We also analyze the three-body collision operator in transport theory from the same quasiparticle point of view to the lowest nontrivial order in the coupling constant. It is shown that the anomalous terms in this operator contain the renormalization effects on the usual two-body collision operator.
Renormalized expressions for the transport coefficients of a normal quantum fluid are derived from a nonlocal kinetic equation. As was shown in the classical case by Forster and Martin and by R\'esibois, the expressions for the transport coefficients separate naturally into "kinetic" and "direct" parts. The kinetic terms are proportional to matrix elements of the inverse of the kinetic kernel, and have the same general structure as the "bare" transport coefficients obtained from a local Boltzmann equation. The direct terms are proportional to matrix elements of the kernel itself, and have no counterpart in calculations based on a local kinetic equation, as in the kinetic theory of gases. We evaluate the transport coefficients using the weak-coupling approximation to the kernel derived in a previous paper. Results are given, first, for both Bose and Fermi fluids at arbitrary temperature, and then for the Fermi fluid near T=0, where complete solutions are obtained. It is found that the direct parts of the shear viscosity and thermal conductivity are of higher order in T than the kinetic parts, and are therefore negligible at very low temperature. The kinetic parts have the same leading temperature dependence as the predictions of the Landau theory. For the bulk viscosity, however, the direct and kinetic contributions begin at the same order (T2) in the temperature.
The phenomenological theory of a Fermi liquid is generalized to nonequilibrium conditions. Landau's theory is reformulated to include fluctuations of the distribution function and quasiparticle energy. The method of fluctuation in kinetic equations is used to establish the collision integral of the quasiparticle Boltzmann equation. This collision integral is expressed in terms of the parameters of Landau's theory, and includes dynamic screening effects. The results are compared with recent theories of transport in Fermi liquids.
We introduce a new renormalisation scheme to construct the Landau quasiparticles of Fermi fluids. The scheme introduces an energy cutoff Λ 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 f, the pair interaction g responsible for superfluidity, and the collision amplitude 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 f and the forward limit of A, and we demonstrate an analogue relation between g and the frontal limit of 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.
Abstract The low-temperature properties of a wide range of many-fermion systems spanning metals, quantum gases and liquids to nuclear matter are well understood within the framework of Landau’s theory of Fermi liquids. The low-energy physics of these systems is governed by interacting fermionic quasiparticles with momenta and energies near a Fermi surface in momentum space. Nonequilibrium properties are described by a kinetic equation for the distribution function for quasiparticles proposed by Landau. Quasiparticle interactions with other quasiparticles, phonons, or impurities lead to internal forces acting on a distribution of nonequilibrium quasiparticles, as well as collision processes that ultimately limit the transport of mass, heat, charge, and magnetization, as well as limiting the coherence times of quasiparticles. For Fermi liquids that are close to a second-order phase transition, e.g., Fermi liquids that undergo a superfluid transition, incipient Cooper pairs—long-lived fluctuations of the ordered phase—provide a new channel for scattering quasiparticles, as well as corrections to internal forces acting on the distribution of nonequilibrium quasiparticles. We develop the theory of quasiparticle transport for Fermi liquids in the vicinity of a BCS-type superfluid transition starting from Keldysh’s field theory for nonequilibrium, strongly interacting fermions. The leading corrections to Fermi-liquid theory for nonequilibrium quasiparticle transport near a Cooper instability arise from the virtual emission and absorption of incipient Cooper pairs. Our theory is applicable to quasiparticle transport in superconductors, nuclear matter, and the low-temperature phases of liquid 3He. As an implementation of the theory we calculate the pairing-fluctuation corrections to the attenuation of zero sound in liquid 3He near the superfluid transition and demonstrate quantitative agreement with experimental results.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.
Landau (1958) gave a rigorous microscopic derivation of his phenomenological Fermi -liquid theory starting from the first principles Hamiltonian of strongly interacting Fermions. The derivation is based on an asymptotic expansion in small parameters such as q/kF, bc.>/EF, kT/EF, where q and c.> represent the wave numbers and frequencies of real excitations of the system. Landau showed that the dynamics of these low-lying excitations (quasiparticles and collective modes) is governed by classical transport equations with all quantum many-body effects absorbed into effective interaction parameters. Landau's approach has been generalized to superfluid Fermi liquids and superconductors by Leggett (1965) and, going beyond linear response, by Eliashberg (1971). Their ideas merged into a recent review on the quasiclassical approach to superfluid 3 He by Serene and Rainer (1983). I will use the notation of this review throughout this paper. Landau's transport theory together with its offsprings will be called 'quasiclassical theory in the following. In this context it is important to mention that Eilenberger (1968) and Larkin and Ovchinnikov (1968) have shown that the complete theory of superconductors can be formulated as a quasiclassical transport theory. Surprisingly, this theory permits calculating microscopic properties, e.g. the quasiparticle excitation spectrum, which are predominantly governed by quantum effects.