Abstract:Année de publication : 2000
The first part is dedicated to quantum spin chains. We first study quantum spin systems that are continuously connected to the Heisenberg spin chain s=1. We build a non-linear sigma model to estimate the spin gap of those systems. Next we study a s=1/2 spin chain doped by non-magnetic impurities but which nevertheless possess a nuclear spin. We use abelian bosonization to calculate the longitudinal relaxation rate of an impurity and its dependence in the temperature. Logarithmic corrections to its behavior are also given. We perform the same analysis on a chiral Luttinger liquid, such as a quantum lead cut in half. The second part deals with disordered systems in low dimension. We shed light upon formal links between disordered systems on a network, random Dirac fermions, non-compact superspin chain and non-linear sigma model. Details are given on the example of the plateau transition in the integer quantum Hall effect. Then we perform exact calculations of the densities of states and typical localization lengths of a Dirac fermion in 1 dimension in various types of disorder. Many condensed matter systems are equivalent to this model, such as the random XX quantum spin chain. Next we study random Dirac fermions in 2 dimensions. Specifically, we are concerned with the problem of Dirac fermions with random mass. This model describes low energy excitations of a disordered $d$-wave superconductor, which impurities are magnetic. A phase diagram is proposed. It is built around the tricritical point of free Dirac fermions and shows an unexpected metallic phase for thermal conduction.