{"id":72,"date":"2010-03-30T12:45:02","date_gmt":"2010-03-30T12:45:02","guid":{"rendered":"http:\/\/lptms.u-psud.fr\/christophe_texier\/?page_id=72"},"modified":"2023-09-04T14:25:27","modified_gmt":"2023-09-04T14:25:27","slug":"a-review-article","status":"publish","type":"page","link":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/a-review-article\/","title":{"rendered":"PhD, Habilitation &amp; Review articles"},"content":{"rendered":"<h2>Th\u00e8se :<\/h2>\n<ul>\n<li><a href=\"\/christophe_texier\/files\/2010\/07\/these_christophe_texier.pdf\">Quelques aspects du transport quantique dans les syst\u00e8mes d\u00e9sordonn\u00e9s de basse dimension<\/a><br \/>\n(th\u00e8se de l&rsquo;Universit\u00e9 Paris 6, soutenue le 20 janvier 1999),<br \/>\ndisponible sur <a href=\"https:\/\/tel.archives-ouvertes.fr\/\">TEL<\/a> (archives ouvertes du CNRS), <a title=\"http:\/\/tel.archives-ouvertes.fr\/tel-01088853\" href=\"https:\/\/tel.archives-ouvertes.fr\/tel-01088853\">http:\/\/tel.archives-ouvertes.fr\/tel-01088853<\/a><\/li>\n<\/ul>\n<h2>Habilitation \u00e0 Diriger des Recherches :<\/h2>\n<ul>\n<li><a href=\"http:\/\/tel.archives-ouvertes.fr\/tel-01091550\">D\u00e9sordre, localisation et interaction. Transport quantique dans les r\u00e9seaux m\u00e9talliques<\/a><br \/>\n(HDR de l&rsquo;Universit\u00e9 Paris-Sud, soutenue le 4 octobre 2010),<br \/>\ndisponible sur <a href=\"https:\/\/tel.archives-ouvertes.fr\/\">TEL<\/a>,\u00a0 <a href=\"http:\/\/tel.archives-ouvertes.fr\/tel-01091550\">http:\/\/tel.archives-ouvertes.fr\/tel-01091550<\/a><\/li>\n<\/ul>\n<h1 style=\"text-align: center;\"><\/h1>\n<h1 style=\"text-align: center;\">Review articles<\/h1>\n<h3>Disordered Supersymmetric quantum mechanics.<\/h3>\n<p>Hamiltonians that may be written as a sum of quadratric terms <em>H<\/em>=\u2211<sub><em>i<\/em>=1<\/sub><sup><em>N<\/em><\/sup>(<em>Q<sub>i<\/sub><\/em>)<sup>2<\/sup> are called supersymmetric Hamiltonians, the <em>N<\/em> conserved quantities <em>Q<sub>i<\/sub><\/em> being the supercharges. A simple case (with <em>N<\/em>=2), introduced by Witten (1981) as a toy model, leads to the two supersymmetric partners :<\/p>\n<p style=\"padding-left: 30px;\"><em>H<sub>\u00b1<\/sub><\/em> = \u2013 d<sup>2<\/sup>\/d<em>x<\/em><sup>2<\/sup> + \u03c6(<em>x<\/em>)<sup>2<\/sup> \u00b1 \u03c6\u2019(<em>x<\/em>)<\/p>\n<p style=\"padding-left: 30px;\">i.e. \u00a0<em> H<\/em><sub>+<\/sub>= <em>Q<\/em><sup>\u2020<\/sup><em>Q <\/em>and<em>\u00a0 <em>H<\/em><sub>&#8211;<\/sub>= Q <em>Q<\/em><\/em><sup>\u2020<\/sup><em><em>\u00a0 <\/em><\/em> where Q=\u2013 d\/d<em>x<\/em> + \u03c6(<em>x<\/em>)<\/p>\n<p>\u00a0Supersymmetric structure is closely related to integrability. For example, the solvable cases of quantum mechanics possess some underlying supersymmetric structure : (i) the harmonic oscillator, \u03c6(<em>x<\/em>)=\u03c9<em>x<\/em>, (ii) the P\u00f6schl-Teller potential \u03c6(<em>x<\/em>)=\u03bc tanh(\u03bb<em>x<\/em>), (iii) Morse&rsquo;s potential, (iv) 2D Landau problem <em>H<\/em>=[\u03c3.(<em>p<\/em>&#8211;<em>eA<\/em>)]<sup>2<\/sup>, (v) Hydrogen atom, etc.<br \/>\nBeyond integrability, interest in supersymmetric quantum mechanics relies on the relation to several physical problems : (i) Dirac equation with random mass\u00a0<em>H<\/em><sub>D<\/sub> = i \u03c3<em><sub>y<\/sub><\/em> \u2202<em><sub>x<\/sub><\/em> + \u03c3<em><sub>x<\/sub><\/em> \u03c6(<em>x<\/em>), 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 Schr\u00f6dinger equation for the supersymmetric Hamiltonian can be mapped onto the Fokker-Planck equation \u2202<em><sub>t<\/sub>P<\/em>(<em>x<\/em>;<em>t<\/em>)=\u2202<em><sub>x<\/sub><\/em>[\u2202<em><sub>x<\/sub><\/em>-2\u03c6(<em>x<\/em>)]<em>P<\/em>(<em>x<\/em>;<em>t<\/em>) describing <em>classical<\/em> diffusion in a random force field \u03c6(x) (Sinai problem).<\/p>\n<p>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 <em>n<\/em>-point correlation function of the extended zero mode.<\/p>\n<ul>\n<li>Alain Comtet and Christophe Texier,<br \/>\n<strong>One-dimensional disordered supersymmetric quantum mechanics : A brief survey,<\/strong><br \/>\nin <em>Supersymmetry and Integrable Models<\/em>, edited by H. Aratyn, T. D. Imbo, W. Y. Keung and U. Sukhatme, Lecture Notes in Physics, Vol. 502, pp. 313\u2013328, Springer (1998).<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/cond-mat\/9707313\">cond-mat\/9707313.<\/a><\/li>\n<\/ul>\n<h3>Functional of the Brownian motion, metric graphs and Anderson localisation.<\/h3>\n<p>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.<\/p>\n<ul>\n<li>Alain Comtet, Jean Desbois and Christophe Texier,<br \/>\n<strong>Functionals of the Brownian motion, localization and metric graphs<\/strong><br \/>\n<a href=\"http:\/\/www.iop.org\/EJ\/abstract\/0305-4470\/38\/37\/R01\">J. Phys. A : Math. Gen. <strong>38<\/strong>, R341-R383 (2005).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/cond-mat\/0504513\">cond-mat\/0504513.<\/a><\/li>\n<\/ul>\n<h3>Ordered spectral statistics in 1D disordered quantum mechanics.<\/h3>\n<p>In my article J.Phys.A<strong>33<\/strong> (2000), I have considered the question of ordered spectral statistics for 1D random Schr\u00f6dinger problems. This first work has later known several developements and applications that are reviewed here.<\/p>\n<ul>\n<li>Christophe Texier,<br \/>\n<strong>Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers,<\/strong><br \/>\nProceedings for the meeting \u00ab\u00a0Fundations and Applications of non-equilibrium statistical mechanics\u00a0\u00bb, Nordita, Stockholm, september-october 2011,<br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1402-4896\/86\/5\/058515\/\">Physica Scripta <strong>86<\/strong>, 058515 (2012),<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1205.0151\">cond-mat arXiv:1205.0151<\/a><\/li>\n<\/ul>\n<h3>Lyapunov exponents, 1D Anderson localisation and products of random 2\u00d72 matrices.<\/h3>\n<p>We discuss the close relation between general 2\u00d72 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).\u00a0 Lyapunov exponent is a unufying concept of these two problems. The article also presents a new solvable case.<\/p>\n<ul>\n<li>Alain Comtet,\u00a0 Christophe Texier and Yves Tourigny,<br \/>\n<strong>Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices<\/strong>,<br \/>\n<a href=\"http:\/\/iopscience.iop.org\/1751-8121\/46\/25\/254003\/\">J. Phys. A : Math. Gen. <strong>46<\/strong>, 254003 (2013).<\/a><br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1207.0725\">cond-mat arXiv:1207.0725<\/a><\/li>\n<\/ul>\n<h3>Review on Wigner time delay and related concepts for a special issue in memoriam of Markus B\u00fcttiker<\/h3>\n<ul>\n<li>Christophe Texier<br \/>\n<strong>Wigner time delay and related concepts &#8212; Application to transport in coherent conductors<br \/>\n<\/strong><a href=\"http:\/\/www.sciencedirect.com\/science\/article\/pii\/S1386947715302228\">Physica E <strong>82<\/strong>, 16-33 (2016)<\/a>, contribution to a special issue \u201c<em>Frontiers in quantum electronic transport \u2013 in memory of Markus B\u00fcttiker<\/em>\u201d<br \/>\n<a href=\"http:\/\/arxiv.org\/abs\/1507.00075\">cond-mat arXiv:1507.00075<\/a> (<span style=\"color: #993366;\">the <strong>arXiv version<\/strong> has been <strong>updated<\/strong> <em>after<\/em> publication in Physica E\u00a0 and more recent results reported<\/span>; furthermore several typos introduced by Physica (!) have been corrected)<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Th\u00e8se : Quelques aspects du transport quantique dans les syst\u00e8mes d\u00e9sordonn\u00e9s de basse dimension (th\u00e8se de l&rsquo;Universit\u00e9 Paris 6, soutenue le 20 janvier 1999), disponible sur TEL (archives ouvertes du CNRS), http:\/\/tel.archives-ouvertes.fr\/tel-01088853 Habilitation \u00e0 Diriger des Recherches : D\u00e9sordre, localisation &hellip; <a href=\"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/recherche\/a-review-article\/\">Continuer la lecture <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":36,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-72","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/72","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/comments?post=72"}],"version-history":[{"count":47,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/72\/revisions"}],"predecessor-version":[{"id":2710,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/72\/revisions\/2710"}],"up":[{"embeddable":true,"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/pages\/36"}],"wp:attachment":[{"href":"http:\/\/www.lptms.universite-paris-saclay.fr\/christophe_texier\/wp-json\/wp\/v2\/media?parent=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}