Seminar Event Detail


Integrable Systems and Random Matrix Theory

Date:  Monday, April 03, 2017
Location:  1866 East Hall (4:00 PM to 5:00 PM)

Title:  Integrability, random matrices and Painleve in quantum chaos

Abstract:   Properties of a quantum system become universal in the fully chaotic limit, i.e. when the corresponding classical dynamics obey hyperbolicity (the exponential sensitivity to the initial conditions) and ergodicity (typical classical trajectories fill out the available phase space uniformly). At sufficiently long times, the single particle dynamics is governed by global symmetries rather than by system particularities and demonstrate a universal behavior. In this universal regime the quantum observables can be represented by random matrix integrals. The deform-and-study approach is an adopted string theory method for revealing hidden symmetries in completely integrable systems. It allows to establish non-linear differential identities for the observables in terms of internal parameters of the system. The implementation of the above strategy will be demonstrated for the problem of quantum transport in chaotic cavities. According to the Landauer-type theories the transport observables such as conductance are expressed in terms of the scattering S-matrix. In the universal regime the S-matrix is uniformly distributed over the unitary group, the Landauer conductance distribution takes a form of a matrix integral - an object suitable for the analysis by the deform-and-study technique. It was shown that the cumulant generation function of conductance distribution is expressed in terms of a solution to the Painleve V equation, which, in turn, allows to derive a recurrence for cumulants of conductance. This mean that the problem of quantum transport in chaotic cavities is completely integrable in the universal regime. Some very recent results concerning the power spectrum distribution of eigenvalues in a chaotic system, which were derived with the help of integrable system theory, will be highlighted in the talk as well.

Files:


Speaker:  Vladimir Osipov
Institution:  Lund University

Event Organizer:   Thomas Bothner    bothner@umich.edu

 

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