Catégorie : Collaborations

Abstract : We study products of arbitrary random real 2×22×2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,R)SL(2,R), we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss’ hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.

A. Comtet 1, 2 J. M. Luck 3 C. Texier 1, 4 Y. Tourigny 5
1 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
2 IHP – Institut Henri Poincaré
3 IPHT – Institut de Physique Théorique – UMR CNRS 3681
4 LPS – Laboratoire de Physique des Solides
5 Department of Mathematics [Bristol]

Abstract : The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the localisation length. The Lyapunov exponent also features prominently in the theory of products of random matrices pioneered by Furstenberg. After a brief historical survey, we describe some recent work that exploits the close connections between these topics. We review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case. Finally, we point out some limitations of the Lyapunov exponent as a means of studying localisation properties.

Alain Comtet 1, 2 Christophe Texier 1, 3 Yves Tourigny 4
1 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
2 IHP – Institut Henri Poincaré
3 LPS – Laboratoire de Physique des Solides
4 School of Mathematics [Bristol]

Abstract : We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition which however strongly affects the transient regime, resulting in non-monotonous behavior and power-law relaxation with varying exponents.

Shamik Gupta 1 Alberto Rosso 1 Christophe Texier 1, 2
1 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
2 LPS – Laboratoire de Physique des Solides

Abstract : Using the joint distribution for proper time-delays of a chaotic cavity derived by Brouwer, Frahm & Beenakker [Phys. Rev. Lett. {\bf 78}, 4737 (1997)], we obtain, in the limit of large number of channels NN, the large deviation function for the distribution of the Wigner time-delay (the sum of proper times) by a Coulomb gas method. We show that the existence of a power law tail originates from narrow resonance contributions, related to a (second order) freezing transition in the Coulomb gas.

Christophe Texier 1, 2 Satya N. Majumdar 1
1 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
2 LPS – Laboratoire de Physique des Solides

Abstract : We report on the measurement of phase coherence length in a high mobility two-dimensional electron gas patterned in two different geometries, a wire and a ring. The phase coherence length is extracted both from the weak localization correction in long wires and from the amplitude of the Aharonov-Bohm oscillations in a single ring, in a low temperature regime when decoherence is dominated by electronic interactions. We show that these two measurements lead to different phase coherence lengths, namely LwireΦ∝T−1/3LΦwire∝T−1/3 and LringΦ∝T−1/2LΦring∝T−1/2. This difference reflects the fact that the electrons winding around the ring necessarily explore the whole sample (ergodic trajectories), while in a long wire the electrons lose their phase coherence before reaching the edges of the sample (diffusive regime).

Auteurs:

Thibaut Capron 1 Christophe Texier 2, 3 Gilles Montambaux 3 Dominique Mailly 4 Andreas D. Wieck 5 Laurent Saminadayar 6
1 QuantECA – Circuits électroniques quantiques Alpes
2 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
3 LPS – Laboratoire de Physique des Solides
4 LPN – Laboratoire de photonique et de nanostructures
5 Lehrstuhl für Angewandte Festkörperphysik
6 TPS – Thermodynamique et biophysique des petits systèmes
NEEL – Institut Néel

Richard Legras 1 Yohann Benard 2 Norberto Lopez-Gil
1 LuMIn – Laboratoire Lumière, Matière et Interfaces
2 LAC – Laboratoire Aimé Cotton

François Boulogne 1 L. Pauchard 2 F. Giorgiutti-Dauphiné
1 LPS – Laboratoire de Physique des Solides
2 FAST – Fluides, automatique, systèmes thermiques
Type de document : Article dans une revue

François Boulogne 1 L. Pauchard 2 F. Giorgiutti-Dauphiné
1 LPS – Laboratoire de Physique des Solides
2 FAST – Fluides, automatique, systèmes thermiques

Abstract : A derivation of the spectral determinant of the Schrödinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader. To formulate the spectral determinant we first derive the spectral zeta function of the Schrödinger operator using an appropriate secular equation. The result obtained for the spectral determinant is along the lines of the recent conjecture.

J. M. Harrison 1 K. Kirsten 1 C. Texier 2, 3
1 Department of Mathematics
2 LPS – Laboratoire de Physique des Solides
3 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques

Abstract : Some results on the ordered statistics of eigenvalues for one-dimensional random Schrödinger Hamiltonians are reviewed. In the case of supersymmetric quantum mechanics with disorder, the existence of low energy delocalized states induces eigenvalue correlations and makes the ordered statistics problem nontrivial. The resulting distributions are used to analyze the problem of classical diffusion in a random force field (Sinai problem) in the presence of weakly concentrated absorbers. It is shown that the slowly decaying averaged return probability of the Sinai problem, \meanP(x,t|x,0)∼ln−2t\meanP(x,t|x,0)∼ln−2⁡t, is converted into a power law decay, \meanP(x,t|x,0)∼t−2ρ/g√\meanP(x,t|x,0)∼t−2ρ/g, where gg is the strength of the random force field and ρρ the density of absorbers.

Christophe Texier 1, 2
1 LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
2 LPS – Laboratoire de Physique des Solides