Contents

This thesis deals with the general properties of numerical Real-Space Renormalization Group methods (RSRG) and, in particular, the Density-Matrix Renormalization Group (DMRG). We provide an introduction to these methods and discuss the properties of the approximate states obtained from them. We only consider the so called innite lattice algorithms. In the paper we deal with the question of how the DMRG describes gapless systems, i.e. systems with quasi long-ranged correlations. To do this, we focus on a system consisting of free fermions on a lattice, with a staggered on-site potential. We study the convergence properties of the DMRG and nd that the method converges to a xed point, thus producing nitely correlated states. Furthermore, we investigate the DMRG correlation functions as the number of kept states, m, is changed. We nd that the particle-hole correlation length scales as / m1:3. Moreover, we discuss how symmetries restrict the possibility of long range order in the system