Giuseppe Benfatto

In this paper, a fermionic hierarchical model is defined, inspired by the Kondo model, which describes a 1-dimensional lattice gas of spin-1/2 electrons interacting with a spin-1/2 impurity. This model is proved to be exactly solvable, and is shown to exhibit a Kondo effect, i.e. that, if the interaction between the impurity and the electrons is antiferromagnetic, then the magnetic susceptibility of the impurity is finite in the 0-temperature limit, whereas it diverges if the interaction is ferromagnetic. Such an effect is therefore inherently non-perturbative. This difficulty is overcome by using the exact solvability of the model, which follows both from its fermionic and hierarchical nature.

We present the first rigorous derivation of a number of universal relations for a class of models with continuously varying indices (among which are interacting planar Ising models, quantum spin chains and 1D Fermi systems), for which an exact solution is not known, except in a few special cases. Most of these formulas were conjectured by Luther and Peschel, Kadanoff, and Haldane, but only checked in the special solvable models; one of them, related to the anisotropic Ashkin-Teller model, is novel.

We prove the exact relations between the critical exponents and the susceptibility, implied by the Haldane Luttinger liquid conjecture, for a generic lattice fermionic model or a quantum spin chain with short range weak interaction. The validity of such relations was only checked in some special solvable models, but there was up to now no proof of their validity in non-solvable models.