Strongly Interacting Electrons in One and Two Dimensions

Abstract: In this thesis I present a theoretical study of three different strongly correlated systems: Luttinger liquids, spin-1 chains and the two-dimensional spin-1/2 Heisenberg model. For every sub-field of our research I summarize the background and provide further details to the work presented in the appended papers. Chapter 1 corresponds to paper I, Chapter 2 to paper II, Chapter 3 and 4 to paper III and IV. In paper I, we calculate with bosonization techniques the local density of states of a finite segment of Luttinger liquid. Besides of being of experimental relevance for Scanning Tunneling Microscopy experiments on carbon nanotubes and cleaved overgrowth wires, the wave functions of a "Luttinger liquid in a box" very nicely visualize the fundamental bosonic nature of the low energy excitations. In paper II, we analyze with field theory methods how the presence of a single-ion anisotropy affects the behavior of a spin-1 chain. We determine in the framework of the Non Linear Sigma Model how the gaps of the anisotropic chain depend on an applied staggered magnetic field. Our theoretical analysis is meant to model compounds like $ m R_2BaNiO_5$ that show coexistence of Haldane phase and long-range order. In paper III, we compute with quantum Monte Carlo simulations the local response to a uniform magnetic field around vacancies in the two-dimensional Heisenberg model. From the full understanding of the numerical results, we can make quantitative predictions of relevance for Nuclear Magnetic Resonance and susceptibility experiments. In particular, we expect, even in the thermodynamic limit, a finite Curie contribution to the total susceptibility arising from the impurities. Finally in paper IV, we consider interaction effects among impurities. We compute the potential between static vacancies in several low-dimensional spin-1/2 antiferromagnets comparing the numerics with exact results from conformal field theory and linear spin wave theory. The potential is directly related to the local valence bond order and its magnitude is a loyal measure of the amount of quantum correlations present into the system.