Statistical quasiparticles, effective green's functions, and Landau descriptions of quantum liquids I. Impurity Systems at Equilibrium

Abstract

This paper deals with Landau descriptions of quantum liquids, which are exact, in the sense that their derivation appeals to no approximation, and relies only on the convergence of the expansions involved. An extension of the validity of Landau's results is obtained, together with diagrammatic expansions for Landau's parameters. In this first part, we restrict to noninteracting particles in a random external field describing impurities. The impurities are distributed either homogeneously or at lattice sites. Such a system, though not a Landau-Fermi liquid in the usual sense, retains several features of the general system of interacting fermions. The method used is straightforward, the only delicate point being raised by the need for a regularization procedure whenever integrands with vanishing denominators show up. The final result expresses thermodynamic observables in terms of an effective Green's function G k eff( y), defined on the real axis. G k eff has a single real pole at y = ε k with a real residue Z k , and describes a “statistical quasiparticle”. The energy ε k replaces here the complex pole of the causal Green's function usually associated with elementary excitations and bears a simple relation with the single-particle level density. The Gibbs potential itself turns out to be exactly the same as for noninteracting particles of energies ε k . Alternatively, it may be obtained as the stationary value of a functional of the statistical quasiparticle energies ε k (in a way reminiscent of Brillouin-Wigner expansions), of the wavefunction renormalization constants Z k , and of the quasiparticle occupation numbers F k . The validity of this formulation is tested by a study of the convergence of a typical subseries. Self-contained appendices are devoted to a discussion of the general regularization problem and to the extension of Lagrange expansions to matrices, respectively.