- Quelques aspects du transport quantique dans les systèmes désordonnés de basse dimension
(thèse de l’Université Paris 6, soutenue le 20 janvier 1999),
disponible sur TEL (archives ouvertes du CNRS), http://tel.archives-ouvertes.fr/tel-01088853
Habilitation à Diriger des Recherches :
- Désordre, localisation et interaction. Transport quantique dans les réseaux métalliques
(HDR de l’Université Paris-Sud, soutenue le 4 octobre 2010),
disponible sur TEL, http://tel.archives-ouvertes.fr/tel-01091550
Disordered Supersymmetric quantum mechanics.
Hamiltonians that may be written as a sum of quadratric terms H=∑i=1N(Qi)2 are called supersymmetric Hamiltonians, the N conserved quantities Qi being the supercharges. A simple case (with N=2), introduced by Witten (1981) as a toy model, leads to the two supersymmetric partners :
H± = – d2/dx2 + φ(x)2 ± φ’(x)
i.e. H+= Q†Q and H–= Q Q† where Q=– d/dx + φ(x)
Supersymmetric structure is closely related to integrability. For example, the solvable cases of quantum mechanics possess some underlying supersymmetric structure : (i) the harmonic oscillator, φ(x)=ωx, (ii) the Pöschl-Teller potential φ(x)=μ tanh(λx), (iii) Morse’s potential, (iv) 2D Landau problem H=[σ.(p–eA)]2, (v) Hydrogen atom, etc.
Beyond integrability, interest in supersymmetric quantum mechanics relies on the relation to several physical problems : (i) Dirac equation with random mass HD = i σy ∂x + σx φ(x), a model relevant in condensed matter physics (ii) one-dimensional metal at half filling (this is related to point i), (iii) organic conductors, (iv) random spin chains, (v) the Schrödinger equation for the supersymmetric Hamiltonian can be mapped onto the Fokker-Planck equation ∂tP(x;t)=∂x[∂x-2φ(x)]P(x;t) describing classical diffusion in a random force field φ(x) (Sinai problem).
The introduction of disorder is natural in several of these contexts. We review several aspects of the problem. The review also contains some original work on the study of the n-point correlation function of the extended zero mode.
- Alain Comtet and Christophe Texier,
One-dimensional disordered supersymmetric quantum mechanics : A brief survey,
in Supersymmetry and Integrable Models, edited by H. Aratyn, T. D. Imbo, W. Y. Keung and U. Sukhatme, Lecture Notes in Physics, Vol. 502, pp. 313–328, Springer (1998).
Functional of the Brownian motion, metric graphs and Anderson localisation.
We give a unified (and brief) presentation of several works scattered in the literature. We emphasize the role of functionals of Brownian motion and probabilistic aspects arising in several questions related to weak and strong localization.
- Alain Comtet, Jean Desbois and Christophe Texier,
Functionals of the Brownian motion, localization and metric graphs
J. Phys. A : Math. Gen. 38, R341-R383 (2005).
Ordered spectral statistics in 1D disordered quantum mechanics.
In my article J.Phys.A33 (2000), I have considered the question of ordered spectral statistics for 1D random Schrödinger problems. This first work has later known several developements and applications that are reviewed here.
- Christophe Texier,
Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers,
Proceedings for the meeting « Fundations and Applications of non-equilibrium statistical mechanics », Nordita, Stockholm, september-october 2011,
Physica Scripta 86, 058515 (2012),
Lyapunov exponents, 1D Anderson localisation and products of random 2×2 matrices.
We discuss the close relation between general 2×2 matrix of SL(2,R) and 1D quantum mechanics with generalised point scatterers. This connection is used to find several solvable cases of random matrix products (i.e. of 1D disordered Hamiltonians). Lyapunov exponent is a unufying concept of these two problems. The article also presents a new solvable case.
- Alain Comtet, Christophe Texier and Yves Tourigny,
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices,
J. Phys. A : Math. Gen. 46, 254003 (2013).