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Physical Review

Derivation of the Landau Theory of Fermi Liquids. II. Equilibrium Properties and Transport Equation

J. M. Luttinger, Ph. Nozières

Published Sep 1, 1962• 102 citations
Publisher

Abstract

Using the results of the preceding paper, it is shown that a large class of the conclusions of the Landau theory of Fermi liquids may be established within the framework of many-body perturbation theory. Both equilibrium and transport phenomena are discussed. The theory is also carried out for long-ranged Coulomb forces. Finally, it is shown that a rather simple general expression for the quasi-particle distribution function of Landau may be given.

DOI
10.1103/physrev.127.1431

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Using the results of the preceding paper, it is shown that a large class of the conclusions of the Landau theory of Fermi liquids may be established within the framework of many-body perturbation theory. Both equilibrium and transport phenomena are discussed. The theory is also carried out for long-ranged Coulomb forces. Finally, it is shown that a rather simple general expression for the quasi-particle distribution function of Landau may be given.
A Landau-type Fermi-liquid theory for interacting electrons moving in a fixed periodic potential is derived. Two methods are used: the Green's-function method, which gives a more physical description of the system but does not give all the desired results, and the many-body perturbation-theory approach, which is complete but is more complicated. The Green's-function formalism of Kadanoff and Baym is adapted to inhomogeneous systems by rewriting it in operator form. A kinetic equation for an energy-dependent, single-particle density matrix is then derived. The quasiparticle concept is introduced and a formal solution of the kinetic equation is given and discussed. These developments are specialized to the periodic potential and a transport equation of the Landau-Boltzmann type is derived for quasiparticles moving within a single energy band. Expressions for the density and current density are then identified from a continuity equation. Although the present work does not treat real transitions between energy bands, the Green's-function method suggests a plausible form for a Landau-like theory which includes such transitions. The perturbation-theory treatment follows the work of Luttinger and Nozi\`eres, who derived Landau's theory for systems without a lattice potential. When the perturbation theory is expressed in terms of Bloch functions, the quantities involved become matrices between different energy bands. However, it is shown that many of the key results of Luttinger and Nozi\`eres can be suitably generalized. The perturbation-theory calculation shows that when the periodic potential is present the change in quasiparticle energy can still be expressed in terms of the variation of the quasiparticle distribution function, and that the results obtained from the Green's-function approach are exact in the case of slowly varying external fields.
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.
In Landau's phenomenological Fermi-liquid theory (FLT), most physical quantities are derived from the knowledge of the energy variation δE[δn] corresponding to a change δn of the quasi-particle (QP) distribution function n≡{nkσ}. We show that the internal energy E[n] (or, more precisely, the thermodynamic potential Φ[n]), expressed as a function of the QP distribution n, can be interpreted as an effective potential (in the sense of field theory), which is obtained from the free energy by a Legendre transformation. This allows to obtain explicitly δΦ (or δE) starting from a microscopic Hamiltonian and to relate the Landau f function to the forward-scattering two-particle vertex without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-of-equilibrium properties are obtained by extending the definition of the effective potential to space- and time-dependent configurations. Φ[n] is then a functional of the Wigner distribution function n≡{nkσ (r,t)}. It contains information about both the static and dynamic properties of the Fermi liquid. In particular, it yields the quantum Boltzmann equation satisfied by nkσ (r, t). Finally, we show how δΦ[δn] can be derived (in the static case) using a finite-temperature renormalization-group approach. In agreement with previous results based on this technique, we find that the Landau f function is defined by the fixed-point value of the Ω-limit of the forward-scattering two-particle vertex.
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.
For real fermion systems, the particle interaction and the exclusion principle act simultaneously; we are thus led to study degenerate Fermi liquids, in which both effects are important. This chapter explores Landau's semiphenomenological point of view and the applications of the theory to neutral systems such as degenerate liquid. It focuses on the Landau theory of Fermi liquids, which shows clearly all the important features brought about by the interaction. Landau's theory corresponds to an expansion in powers of the fraction of excited particles. Within the two restrictions, the theory is rigorous. It lends itself to a large number of problems: equilibrium properties, transport properties, response to long wavelength external perturbations. Landau's theory is not a "universal panacea": its well-defined limitations should always be kept in mind. Moreover, it uses (and provides) far more information than any experiment will ever sample. The chapter concludes with a brief summary of the range of validity and main achievements of Landau's theory.