# Chaînes de Spins, Fermions de Dirac, et Systèmes Désordonnés

Abstract : 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.
Mots-clés :
Document type :
Theses

https://pastel.archives-ouvertes.fr/tel-00001560
Contributor : Marc Bocquet <>
Submitted on : Wednesday, August 14, 2002 - 3:30:00 PM
Last modification on : Wednesday, September 12, 2018 - 2:13:57 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:45:20 PM

### Identifiers

• HAL Id : tel-00001560, version 1

### Citation

Marc Bocquet. Chaînes de Spins, Fermions de Dirac, et Systèmes Désordonnés. Physique mathématique [math-ph]. Ecole Polytechnique X, 2000. Français. ⟨tel-00001560⟩

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