Aurélien Grabsch | Research

Aurélien Grabsch

Keywords: random matrix theory, coherent quantum transport, topology in condensed matter, Majorana zero modes, disordered systems, spectral linear statistics, large deviations, stochastic processes

RC Quantum Dot I am interested in the applications of random matrix theory (RMT) to quantum transport. The complex dynamics of chaotic cavities, like quantum dots, can be well described by a statistical approach. This consists in taking a random scattering matrix to characterize transport through the system.
From this matrix can be constructed the Wigner-Smith time-delay matrix, which was shown to belong to the (inverse) Laguerre ensemble of RMT. Many relevant physical quantities (like conductance, resistance,...) take the form of linear statistics of the eigenvalues of this matrix (the proper time delays).
Phase diagram Coulomb
		gas The distribution of these linear statistics is obtained by a Coulomb gas method. This consists in interpreting the eigenvalues of a random matrix as the positions of particles in a 1D gas, with logarithmic repulsion. Determining the distribution of the linear statistics reduces to finding the optimal configuration of the gas under a constraint. An interesting feature is the possibility of phase transitions driven by the constraint.

Braiding edge vortices Topological superconductors can support Majorana zero-modes (midgap states bound to a defect). These zero modes have non-Abelian statistics: they are neither bosons nor fermions, and can be used for the realisation of topologically protected quantum computations. Topological superconductors also possess non-Abelian excitations of the chiral edge modes: the edge vortices. Unlike the Majorana zero-modes which are fixed, the edge vortices have the advantage of propagating along the chiral edge modes. I am investigating the possibility to demonstrate the non-Abelian nature of the edge vortices, and their possible use for the realisation of topologically protected quantum computations.

Lyapunov exponents I am also interested in disordered systems. It is well known that wave functions in 1D in a random potential are localized (Anderson localization). The localization length have been computed for diverse models of disorder. However the 2D case is still out of reach.
We studied models of multichannel disordered wires which describe an intermediate situation. We focused on the multichannel Dirac equation with a random mass, and established a link with a random matrix model. In addition, we showed that this system undergoes topological phase transitions driven by the disorder.