The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.
The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.
The methods employed in the previous papers of this series for the ground-state energy of a normal Fermion system are extended to finite temperatures. The basic assumption is that the density matrix of the actual system in thermodynamic equilibrium can be derived by adiabatic transformation from the density matrix of system of noninteracting particles at the same temperature, volume, and chemical potential. However, the spectrum of the noninteracting system must be dependent on the intensive parameters. The self-consistency of the scheme is exhibited by demonstrating that the effective potential characterizing this system can be related to the self-energy operator of the actual system. Two equivalent versions of the theory are derived. In one of these the basic equations are precisely those of the Landau theory of Fermi liquids with the difference, however, that all quantities are given explicitly from first principles.
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.
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.
On the basis of a microscopic model of self-consistent field, the thermodynamics of the many-particle Fermi system at finite temperatures with account of three-body interactions is built and the quasiparticle equations of motion are obtained.It is shown that the delta-like three-body interaction gives no contribution into the selfconsistent field, and the description of three-body forces requires their nonlocality to be taken into account.The spatially uniform system is considered in detail, and on the basis of the developed microscopic approach general formulas are derived for the fermion's effective mass and the system's equation of state with account of contribution from three-body forces.The effective mass and pressure are numerically calculated for the potential of "semi-transparent sphere" type at zero temperature.Expansions of the effective mass and pressure in powers of density are obtained.It is shown that, with account of only pair forces, the interaction of repulsive character reduces the quasiparticle effective mass relative to the mass of a free particle, and the attractive interaction raises the effective mass.The question of thermodynamic stability of the Fermi system is considered and the three-body repulsive interaction is shown to extend the region of stability of the system with the interparticle pair attraction.The quasiparticle energy spectrum is calculated with account of three-body forces.
The individual particle theory of many body systems was studied starting from a microscopic point of view. A technique which is equivalent to a generalized thermodynamics was used to show that, under the appropriate conditions of temperature and external field, the quasi-particle approach is correct for three very dissimilar systemsi the normal degenerate Fermi system, the metallic electron-phonon system, and the polaron. The fact that an approach like this works for such dissimilar systems emphasizes the underlying unity of the quasi-particle models. The concept that the real t~me non-equilibrium self-energy and the incoherent part of the density-density correlation function may be generated as functional derivatives of a generalized thermodynamic potential (~), together with local translational and rotational invariance and Galilean relativity, leads to a transformation from the microscopic particle density variables g«p,m,R,T) to those of the quasi-particle theory n (R,T). ~ - p- It was shown that, for zero temperature, such a ~ derivability is exact; at low temperature and for slowly varying fields, this ~ derivability is approximately true. This was shown by a procedure which is eqUivalent to an expansion in terms of the real scattering probability for quasiparticles. At low temperature and for slowly varying fields, such real scattering is very improbable and the expansion parameter is small. Perturbation theory was used only to determine the actual size of terms which, on physical grounds are necessarily small. As such, this approach is non-perturbative, depending only on properties of the exact selfenergies and correlation functions. In addition to demonstrating that the quasi-particle models are correct, information on the regions of validity for ~hese models is obtained. .As by-products of these derivations, microscopic expressions for the parameters of the quasi-particle theory and simple microscopic expressions for the currents in degenerate Fermi systems and the electron-phonon system are obtained.