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Universal Velocity Correlations in Disordered and Chaotic Systems.pdf

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Universal Velocity Correlations in Disordered and Chaotic Systems

Universal Velocity Correlations in Disordered and ChaoticSystemsB D Simons and Boris L AltshulerDepartment of Physics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139AbstractThe response of weakly disordered metallic grains to Aharonov-Bohm uxsuggests a rescaling in which statistical correlators become universal. Wederive an exact expression for a correlator of level \velocities, and providenumerical evidence which suggests that the universality extends to a widerclass of systems and generalizes to arbitrary perturbations providing a newcharacterization of quantum chaos. These results are applied to Fermi veloc-ities of complex lattices. Typeset using REVTEX1 The Wigner-Dyson distribution of level spacings [1] provides an accurate description ofa variety of complex systems ranging from those with many degrees of freedom and stronginteractions (e.g. atomic nuclei) to the quantum mechanical motion of particles in irregularpotentials (e.g. quantum dots, disordered metallic grains). The distribution serves as auniversal classi cation of quantum chaos [2]. With such correlations, the dependence on thesystem enters only through the mean level spacing, . Here we provide exact analyticalresults which suggest that the response of the energy levels, Ei to an external perturbation,controlled by some parameter X [3,5,6] relies on just one additional parameter,C(0) = 12D@Ei(X)@X 2E; (1)where h i denotes a statistical average which for example can performed over a typicalrange of levels, and/or X. We propose that after rescaling,x = qC(0)X; i(x) = Ei(X)=; (2)the statistical properties of the random functions, i(x) are universal independent of thenature of X. The physical interpretation of C(0) as a \generalized conductance can befound through a universal uctuation-dissipation theorem [7,3] which can be derived for thisclass of system. The non-universality of C(0) prevents a more precise general de nition.We will seek to establish th