In the last decade, relativistic semimetals, in which two or more energy bands touch linearly at nodal points of the Brillouin zone, has become a hot topic in modern condensed matter physics. The low-energy excitations of theses electronic phases behave as massless Dirac fermions, similarly than in their two-dimensional notorious counterpart: graphene.
This Ph.D. tackles the issue of the stability of the touching point of relativistic semimetals with respect to perturbations: disorder in three-dimensional semimetals, and interactions in graphene bilayers with a twist. The renormalisation group approach provides a technical guideline for these works.
Weyl semimetals transit to a metallic phase through a continuous phase transition. The first part of this Ph.D. studies the distribution probability of the wave function at the quantum critical point and their multifractal nature. Notably, long-range correlations of disorder play a crucial role on these exponents. The second part of this Ph.D. probes the spatially-resolved density of states in dirty binodal Weyl and Dirac semimetals, and the effect of disorder on the surface states (Fermi rays, Fermi arcs, and Dirac surface states).
Twisted bilayer graphene displays a rich phase diagram at the magic angle, similar to that of high-temperature cuprates. A nematic insulator was found in several experiments at charge neutrality. The third part of this Ph.D. proposes to identify the relevant instabilities there, through a group-theoretic and non-standard renormalisation group analyses, where interaction and interlayer hopping strengths are treated perturbatively. A gapped nematic instability is indeed favoured.