Contact

Address :

Office 276,
Institut de Mathématiques de Bordeaux
351, Cours Libération, 33400, Talence, FRANCE

Email :

eric.vacelet@u-bordeaux.fr
VACELET Eric

Postdoc in mathematics, Équipe EDP-Physique Mathématique , Institut de Mathématiques de Bordeaux

Projet ANR SpecDiMa: Spectrum of Dirac Materials


In condensed matter physics, the Dirac equations govern the behavior of new materials with remarkable properties. This is the case for the graphene (periodic honeycomb material) or for topological insulators (insulating material in the bulk but conductive on the edge). An important objective of these models is to reveal “edge states” which propagate either along the edge of the domain or at an interface created by a variable mass or an electric/magnetic potential. Such phenomena comparable to the quantum Hall effect have been intensively studied for magnetic Schrödinger operators but more recently considered for Dirac operators. The latter present specific difficulties and different behaviors: matrix operator, non-semibounded spectrum, delicate boundary conditions… Our team, made up of recognized specialists in analysis of PDE and spectral analysis of operators in the broad sense (spectral asymptotics, scattering theory, resonance theory, semiclassical analysis, microlocal analysis etc.), aims to contribute to the mathematical description of the new phenomena associated with some of these Dirac type systems and to explore the mathematical questions raised. We can distinguish three themes corresponding to the three important steps of the planned studies:
  • Band functions of fibered operators: Periodic operators with conical band functions and operators invariant in one direction (Dirac “waveguide”). Description of their band functions (or dispersion curves) as well as the associated eigenvectors.
  • Edge states: For some reference operators without perturbation, study of edge states and their localizations. Develop a more general interpretation of edge states in terms of coherent states propagation and Wigner measures for bent interfaces.
  • Influence and interaction of perturbations/impurities: effect of perturbations (electromagnetic potential, obstacle) on the ideal operators, study of the eigenvalues, resonances, scattering matrix and spectral shift function. In addition, study of the stability of edge currents.

PhD in Mathematics, LAREMA, University of Angers

Thesis subject : Semiclassical analysis of two-band models


The Dirac operator, introduced in 1928, is currently experiencing renewed interest through the study of topological insulators. These are materials that are insulating in volume but conductive on the surface. In this thesis, we study propagation in a system composed of two topological insulators without a magnetic field, whose interface is a non-compact, connected, smooth, and boundaryless curve. The dynamics of electrons are then governed by an adiabatic modulation of a Dirac operator with a variable and smooth mass. The first chapter describes the evolution of the semiclassical measurement of the solution using a two-scale Wigner measurement method, after reducing the Hamiltonian to a normal form. The second chapter develops the more specific propagation of wave packets encountering singularities at the interface. Finally, the last two chapters present more theoretical results as well as applications of the superadiabatic projector method. Ongoing research on the Schrödinger magnetic operator in a waveguide with Neumann boundary conditions is presented.