Dyson's Brownian Motion and Universal Dynamics of Quantum Systems

We establish a correspondence between the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random Gaussian perturbing matrix, and a Fokker-Planck equation postulated by Dyson. Within this model, we prove the equivalence conjectured by Altshuler et al between the space-time correlations of the Sutherland-Calogero-Moser system in the thermodynamic limit and a set of two-variable correlations for disordered quantum systems calculated by them. Multiple variable correlation functions are, however, shown to be inequivalent for the two cases.Comment: 10 pages, revte

Universal Level dynamics of Complex Systems

. We study the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random perturbation drawn from (i) a generalized Gaussian ensemble (ii) a non-Gaussian ensemble with a measure variable under the change of basis. It turns out that, in the case (i), a redefinition of the parameter governing the evolution leads to a Fokker-Planck equation similar to the one obtained when the perturbation is taken from a standard Gaussian ensemble (with invariant measure). This equivalence can therefore help us to obtain the correlations for various physically-significant cases modeled by generalized Gaussian ensembles by using the already known correlations for standard Gaussian ensembles. For large $N$-values, our results for both cases (i) and (ii) are similar to those obtained for Wigner-Dyson gas as well as for the perturbation taken from a standard Gaussian ensemble. This seems to suggest the independence of evolution, in thermodynamic limit, from the nature of perturbation involved as well as the initial conditions and therefore universality of dynamics of the eigenvalues of complex systems.Comment: 11 Pages, Latex Fil

Vortex dissipation and level dynamics for the layered superconductors with impurities

We study parametric level statistics of the discretized excitation spectra inside a moving vortex core in layered superconductors with impurities. The universal conductivity is evaluated numerically for the various values of rescaled vortex velocities $\kappa$ from the clean case to the dirty limit case. The random matrix theoretical prediction is verified numerically in the large $\kappa$ regime. On the contrary in the low velocity regime, we observe $\sigma_{xx} \propto \kappa^{2/3}$ which is consistent with the theoretical result for the super-clean case, where the energy dissipation is due to the Landau-Zener transition which takes place at the points called ``avoided crossing''.Comment: 10 pages, 4 figures, REVTeX3.

Magnetotunneling spectroscopy of mesoscopic correlations in two-dimensional electron systems

An approach to experimentally exploring electronic correlation functions in mesoscopic regimes is proposed. The idea is to monitor the mesoscopic fluctuations of a tunneling current flowing between the two layers of a semiconductor double-quantum-well structure. From the dependence of these fluctuations on external parameters, such as in-plane or perpendicular magnetic fields, external bias voltages, etc., the temporal and spatial dependence of various prominent correlation functions of mesoscopic physics can be determined. Due to the absence of spatially localized external probes, the method provides a way to explore the interplay of interaction and localization effects in two-dimensional systems within a relatively unperturbed environment. We describe the theoretical background of the approach and quantitatively discuss the behavior of the current fluctuations in diffusive and ergodic regimes. The influence of both various interaction mechanisms and localization effects on the current is discussed. Finally a proposal is made on how, at least in principle, the method may be used to experimentally determine the relevant critical exponents of localization-delocalization transitions.Comment: 15 pages, 3 figures include

Condensation and Lasing of Microcavity Polaritons: Comparison between two Models

Condensation of microcavity polaritons and the substantial influence of pair-breaking disorder and decoherence leading to a laser regime has been recently considered using two different models: a model for direct two band excitons in a disordered quantum well coupled to light and a model where the cavity mode couples instead to a medium of localised excitons, represented by two-level oscillators in the presence of dephasing processes. Even if complementary from the point of view of assumptions, the models share most of the main conclusions and show similar phase diagrams. The issue whether excitons are propagating or localised seems secondary for the polariton condensation and the way in which pair-breaking disorder and decoherence processes influence the condensation and drive the microcavity into a lasing regime is, within the approximations used in each model, generic. The reasons for the similarities between the two physical situations are analysed and explained.Comment: Proceeding of the First International Conference on Spontaneous Coherence in Excitonic Systems (ICSCE'04); 7 pages, 2 eps figure

Thermodynamics of an one-dimensional ideal gas with fractional exclusion statistics

We show that the particles in the Calogero-Sutherland Model obey fractional exclusion statistics as defined by Haldane. We construct anyon number densities and derive the energy distribution function. We show that the partition function factorizes in the form characteristic of an ideal gas. The virial expansion is exactly computable and interestingly it is only the second virial coefficient that encodes the statistics information.Comment: 10pp, REVTE

Universal Correlations of Coulomb Blockade Conductance Peaks and the Rotation Scaling in Quantum Dots

We show that the parametric correlations of the conductance peak amplitudes of a chaotic or weakly disordered quantum dot in the Coulomb blockade regime become universal upon an appropriate scaling of the parameter. We compute the universal forms of this correlator for both cases of conserved and broken time reversal symmetry. For a symmetric dot the correlator is independent of the details in each lead such as the number of channels and their correlation. We derive a new scaling, which we call the rotation scaling, that can be computed directly from the dot's eigenfunction rotation rate or alternatively from the conductance peak heights, and therefore does not require knowledge of the spectrum of the dot. The relation of the rotation scaling to the level velocity scaling is discussed. The exact analytic form of the conductance peak correlator is derived at short distances. We also calculate the universal distributions of the average level width velocity for various values of the scaled parameter. The universality is illustrated in an Anderson model of a disordered dot.Comment: 35 pages, RevTex, 6 Postscript figure

A lecture on the Calogero-Sutherland models

In these lectures, I review some recent results on the Calogero-Sutherland model and the Haldane Shastry-chain. The list of topics I cover are the following: 1) The Calogero-Sutherland Hamiltonian and fractional statistics. The form factor of the density operator. 2) The Dunkl operators and their relations with monodromy matrices, Yangians and affine-Hecke algebras. 3) The Haldane-Shastry chain in connection with the Calogero-Sutherland Hamiltonian at a specific coupling constant.Comment: (2 references added, small modifications

Universality of Parametric Spectral Correlations: Local versus Extended Perturbing Potentials

We explore the influence of an arbitrary external potential perturbation V on the spectral properties of a weakly disordered conductor. In the framework of a statistical field theory of a nonlinear sigma-model type we find, depending on the range and the profile of the external perturbation, two qualitatively different universal regimes of parametric spectral statistics (i.e. cross-correlations between the spectra of Hamiltonians H and H+V). We identify the translational invariance of the correlations in the space of Hamiltonians as the key indicator of universality, and find the connection between the coordinate system in this space which makes the translational invariance manifest, and the physically measurable properties of the system. In particular, in the case of localized perturbations, the latter turn out to be the eigenphases of the scattering matrix for scattering off the perturbing potential V. They also have a purely statistical interpretation in terms of the moments of the level velocity distribution. Finally, on the basis of this analysis, a set of results obtained recently by the authors using random matrix theory methods is shown to be applicable to a much wider class of disordered and chaotic structures.Comment: 16 pages, 7 eps figures (minor changes and reference [17] added

Even-odd correlations in capacitance fluctuations of quantum dots

We investigate effects of short range interactions on the addition spectra of quantum dots using a disordered Hubbard model. A correlation function \cS(q) is defined on the inverse compressibility versus filling data, and computed numerically for small lattices. Two regimes of interaction strength are identified: the even/odd fluctuations regime typical of Fermi liquid ground states, and a regime of structureless \cS(q) at strong interactions. We propose to understand the latter regime in terms of magnetically correlated localized spins.Comment: 3 pages, Revtex, Without figure