Interacting fermions on the half-line: boundary counterterms and boundary corrections

TL;DR

This paper develops a constructive Renormalization Group approach to analyze boundary effects in a non-integrable 1+1 dimensional fermionic system, providing explicit bounds on boundary corrections to thermodynamic quantities.

Contribution

It introduces a method to handle boundary corrections in non-integrable critical systems using Renormalization Group, specifically addressing boundary defects in fermionic chains.

Findings

Derived a convergent renormalized expression for boundary effects.

Provided explicit bounds on boundary corrections to ground state energy.

Extended RG techniques to systems with boundary defects.

Abstract

Recent years witnessed an extensive development of the theory of the critical point in two-dimensional statistical systems, which allowed to prove {\it existence} and {\it conformal invariance} of the {\it 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 determinant 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 {\it finite domains} and {\it boundary corrections}, which are of course crucial for getting informations about conformal covariance. In this thesis, we address the question of adapting 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 {\it boundary defect} counterterm, polynomially localized at the boundary. In particular, we get explicit bounds on the boundary corrections to the specific ground state energy.