Functional Integral approach to quasi-particles in Fermi liquid theory

In this paper we propose a new way of organizing many-body perturbation theory in the Path-integral formulation where a set of quasi-particle wavefunctionals ψ’s are introduced and are identified with quasi-particles in Landau Fermi liquid theory. We show how Fermi liquid theory can be obtained through ψ’s in this new framework of perturbation theory, where the only assumption is adiabaticity between non-interacting and interacting states and the quasi-particle renormalization factor z does not appear explicitly. Consequences of our new formulation are discussed. PACS Numbers: 71.45, 71.10, 67.55 Typeset using REVTEX 1 For four decades Fermi liquid theory has formed the basis for our understanding of interacting fermion systems like electron liquids in metals and normal state of liquid He. The theory was originally proposed phenomenologically by Landau [1,2] and was later put on a firm basis by a formal ’proof’ using diagramatic perturbation theory techniques [3–5]. Although formally exact, the existing perturbation theory formulation of Fermi liquid theory has a drawback. In the original Fermi liquid theory of Landau, the only assumption was adiabaticity or a one-to-one correspondence between non-interacting states (bare particles) and states when interaction is turned on (quasi-particles). However, in the existing diagrammatic technique, perturbation theory was carried out in terms of bare particles and the connection between bare particles and quasi-particles is established by assuming that there is a non-vanishing overlap z ∼ O(1) between the two. It would be much more satisfying if the perturbation theory can be formulated directly in terms of quasi-particles where the introduction of renormalization factor z can be avoided, as is in the original Landau formulation. In this paper we shall show that within the path-integral formulation it is possible to formulate perturbation theory in terms of a set of quasi-particle wave-functionals which can be identified with quasi-particles in Fermi liquid theory. With these wave-functionals Fermi liquid theory can be derived directly without referring to the quasi-particle renormalization factor z. The one-particle Green’s function will be studied where we shall show that the quasi-particle renormalization factor z measures the overlap between bare-particle states and our quasi-particle states on the Fermi surface. For simplicity we shall consider a gas of spinless fermions interacting through a scalar potential v(r). The formulation can be generalized to include spin easily. We shall restrict us to zero temperature where the perturbation theory is formulated in terms of real-time Path Integral and Green’s functions. The Hamiltonian is, in momentum space,