Marginal Fermi Liquid Behaviour in the Hubbard Model with Cut-off

1. We consider the half-filled Hubbard model with a cutoff forbidding momenta close to the angles of the square shaped Fermi surface. By Renormalization Group methods we find a convergent expansion for the Schwinger function up to exponentially small temperatures. We prove that the system is not a Fermi liquid, but on the contrary it behaves like a Marginal Fermi liquid, a behaviour observed in the normal phase of high T c superconductors. 1.1 Motivations. The notion of Fermi liquids, introduced by Landau, refers to a wide class of interacting fermionic systems whose thermodynamic properties (like the specific heat or the resistivity) are qualitatively the same of a gas of non interacting fermions. While there is an enormous number of metals having Fermi liquid behaviour, in recent times new materials has been found whose properties are qualitatively different. In particular the high-temperature superconducting materials (so anisotropic to be considered essentially bidimensional) in their normal phase have a non Fermi liquid behaviour, in striking contrast with previously known superconductors, which are Fermi liquids above the critical temperature. While in Fermi liquids the wave function renormalization Z is Z = 1 + O(λ 2), where λ is the strength of the interaction, in such metals it was found Z ≃ 1 + O(λ 2 log T) for temperatures T above the critical temperature, see [VLSAR] (see also [VNS] for a review); metals behaving in this way were called Marginal Fermi liquids. Such results stimulated an intense theoretical research. It was found by a perturbative analysis, see for instance [AGD] or [Sh], that in a system of weakly interacting fermions in d = 2 Z is essentially temperature independent, at least for circular or " almost " circular Fermi surfaces. Despite doubts appeared about the reliability of results obtained by perturbative expansions [A], such results were indeed confirmed recently by rigorous Renormalization group methods. It was proved in [FMRT] and [DR] that indeed a weakly interacting Fermi system with a circular Fermi surface is a Fermi liquid, up to exponentially small temperatures. Such result was extended in [BGM] to all possible weakly interacting d = 2 fermionic systems with symmetric , smooth and convex Fermi surfaces, up to exponentially small temperatures. These results cannot be obtained by dimensional power counting arguments as such arguments give a bound |Z − 1| ≤ Cλ 2 | log T | from which one cannot distinguish Fermi …