Author : Giovanni Antinucci
Title:: Interacting fermions on the half-line: boundary counterterms and boundary corrections
Abstract: Recent years witnessed an extensive development of the theory of the critical point in two-dimensional statistical systems, which allowed to prove existence and conformal invariance of the scaling limit for two-dimensional Ising model and dimers in planar graphs. Unfortunately, we are still far from a full understanding of the subject: so far, exact solutions at the lattice level, in particular de- terminant structure and exact discrete holomorphicity, play a cucial role in the rigorous control of the scaling limit. The few results about not-integrable (interacting) systems at criticality are still unable to deal with finite domains and boundary corrections, which are of course crucial for getting informations about conformal covariance. In this thesis, we address the question of adapt- ing constructive Renormalization Group methods to non-integrable critical systems in d = 1 + 1 dimensions. We study a system of interacting spinless fermions on a one-dimensional semi-infinite lattice, which can be considered as a prototype of the Luttinger universality class with Dirichlet Boundary Conditions. We develop a convergent renormalized expression for the thermodynamic observables in the presence of a quadratic boundary defect counterterm, polynomially localized at the boundary. In particular, we get explicit bounds on the boundary corrections to the specific ground state energy.