Renormalization-group approach to the theory of the Fermi surface.

we describe an approach to the theory of the Fermi surface based on the renormalization group. A notion of quasi particle appears naturally in the theory. The beta function is finite to all orders of perturbation theory and we compute and discuss it to second order. The normality of the Fermi surface is linked to the repulsivity of the quasi particle potential: the remarkable cancellations that occur in the beta function lead to think that a normal Fermi surface can perhaps occur, but only when the interaction between the quasi particles is repulsive. The approach was first proposed in an earlier paper, [BG1], and we give here an improved version of it, referring to [BG1] only to make use of technical estimates derived there. A natural approach to the theory of the Fermi surface in an interacting Fermi liquid is to slice momentum space into layers I1, I0, I−1, I−2 . . ., with I1 = {set of (k0, ~k) such that k 0 + [( ~k2 − pF )/2m] 2 ≥ p0} and, if n ≤ 0, In = {set of (k0, ~k) such that (2p0) 2 ≤ k 0 +[( ~k2 − pF )/2m] 2 ≤ (2p0) }; here p0 is an arbitrary momentum scale, pF is the Fermi momentum and m is the particle mass. Then the fermion propagator g(t, ~x) appearing in the perturbation theory [LW] can be written: