Journal of Statistical Physics, {\bf 59}, 541--664, 1990
G. Benfatto, G. Gallavotti
Title: Perturbation theory of the Fermi surface in a quantum liquid.
A general quasi particle formalism and one dimensional systems
Abstract: we develop a perturbation theory formalism for the theory of
the Fermi surface in a Fermi liquid of particles interacting via a
bounded short range repulsive pair potential. The formalism is based
on the renormalization group and provides a formal expansion of the
large distance Schwinger functions in terms of a family of running
couplings consisting of a one and two body quasi particle
potentials. The flow of the running couplings is described in terms of
a beta function, which is studied to all orders of perturbation theory
and shown to obey, in the {$\sf n$}--th order, {$\sf n!$} bounds.
The flow equations are written in general dimension $\sf d{\st \ge}1$
for the spinless case (for simplicity).
The picture that emerges is that on a large scale the system looks like a
system of fermions interacting via a {$\sf\d$}-like interaction potential
({\ait i.e.} a potential approaching {$\sf 0$} everywhere except at the
origin where it diverges although keeping the integral bounded); the theory
is not asymptotically free in the usual sense and the freedom mechanism is
thus more delicate than usual: the technical problem of dealing with
unbounded effective potentials is solved by introducing a mathematically
precise notion of {\ait quasi\ particles}, which turn out to be natural
objects with finite interaction even when the physical potential diverges
as a delta-like function. A remarkable kind of gauge symmetry is associated
with the quasi particles.
To substanciate the analogy with the quasi particles theory we discuss
the mean field theory using our notion of quasi particles: the resulting
selfconsistency relations remind closely those of the BCS model. The
formalism seems suited for a joint theory of normal states of Fermi
liquids and of BCS states: the first are associated with the trivial
fixed point of our flow or with nearby non trivial fixed points (or
invariant sets) and the second may naturally correspond to really non
trivial fixed points (which may nevertheless turn out to be accessible
to analysis because the BCS state is a quasi free state, hence quite
simple unlike the non trivial fixed points of field theory).
The $\sf d=1$ case is deeply different from the $\sf d>1$ case,
for our spinless fermions: we can treat it essentially (see
introduction) completely for
small coupling. The system is not asymptotically free and
presents anomalous renormalization group
flow with a vanishing beta function and the discontinuity of the
occupation number at the Fermi surface is smoothed by the interaction
(remaining singular with a coupling dependent singularity of power type
with exponent identified with the anomalous dimension).
Finally we present a heuristic discussion of the theory for the flow of the
running coupling constants in spinless $\sf d>1$ systems: their structure
is simplified further and the relevant part of the running interaction is
precisely the interaction between pairs of quasi particles which we
identify with the Cooper pairs of superconductivity. The formal
perturbation theory seems to have a chance to work only if the interaction
between the Cooper pairs is repulsive: and to second order we show that in
the spin {$\sf 0$} case this happens if the physical potential is
repulsive. Our results indicate the possibility of the existence of a
normal Fermi surface only if the interaction is repulsive.