Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales

Motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter $\lambda$ suitably defined on the unfolded scale. Using results for the spectral two-point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For $\lambda$ << 1 the correlations are given by Dawson's integral while for $\lambda$ >> 1 we derive a novel analytical formula for the two-point function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticable can be identified. We also derive an exact expansion of the local level density for finite level number.Comment: 15 pages, Revtex, no figures, to be published in special issue of J. Math. Phys. (1997

Between Poisson and GUE statistics: Role of the Breit-Wigner width

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation function $X_2(r)$ and the number variance $\Sigma^2(r)$. The graded eigenvalue approach leads to an expression for $X_2(r)$ which is valid for all values of the parameter $\lambda$ governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. For $\lambda \gg 1$ the Breit-Wigner width $\Gamma_1$ measured in units of the mean level spacing $D$ is much larger than unity. In this limit, closed analytical expression for $X_2(r)$ and $\Sigma^2(r)$ can be derived by (i) evaluating the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations of $\Gamma_1$ manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by $\sqrt{\Gamma_1}$. This is rigorously shown and discussed in great detail. The Breit-Wigner $\Gamma_1$ width itself governs the approach to the Poisson limit for $r\to\infty$. Our analytical findings are confirmed by numerical simulations of an ensemble of $500\times 500$ matrices, which demonstrate the universal validity of our results after proper unfolding.Comment: 25 pages, revtex, 5 figures, Postscript file also available at http://germania.ups-tlse.fr/frah

Conductance length autocorrelation in quasi one-dimensional disordered wires

Employing techniques recently developed in the context of the Fokker--Planck approach to electron transport in disordered systems we calculate the conductance length correlation function $$ for quasi 1d wires. Our result is valid for arbitrary lengths L and $\Delta L$. In the metallic limit the correlation function is given by a squared Lorentzian. In the localized regime it decays exponentially in both L and $\Delta L$. The correlation length is proportional to L in the metallic regime and saturates at a value approximately given by the localization length $\xi$ as $L\gg\xi$.Comment: 23 pages, Revtex, two figure

Effective σ\sigma Model Formulation for Two Interacting Electrons in a Disordered Metal

We derive an analytical theory for two interacting electrons in a $d$--dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear $\sigma$ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.Comment: 4 pages, revtex, no figure

Bottleneck effects in turbulence: Scaling phenomena in r- versus p-space

We (analytically) calculate the energy spectrum corresponding to various experimental and numerical turbulence data analyzed by Benzi et al.. We find two bottleneck phenomena: While the local scaling exponent $\zeta_r(r)$ of the structure function decreases monotonically, the local scaling exponent $\zeta_p(p)$ of the corresponding spectrum has a minimum of $\zeta_p(p_{min})\approx 0.45$ at $p_{min}\approx (10 \eta)^{-1}$ and a maximum of $\zeta_p(p_{max})\approx 0.77$ at $p_{max}\approx 8 L^{-1}$. A physical argument starting from the constant energy flux in p--space reveals the general mechanism underlying the energy pileups at both ends of the p--space scaling range. In the case studied here, they are induced by viscous dissipation and the reduced spectral strength on the scale of the system size, respectively.Comment: 9 pages, 3figures on reques

Antikaon-Nukleon-Wechselwirkung im Mesonaustauschbild

Within the meson exchange framework of the Bonn NN-potential a modelfor the $\overline{K}$N interaction near threshold is constructed. Employing a coupled-channelsformalism we calculate total cross sections for six particle channels(pK$^{-}$, n$\overline{K}^{0}$, $\Lambda \pi^{0}$, $\Sigma^{0} \pi^{0}$, $\Sigma^{+} \pi^{-}$ ,$\Sigma^{-} \pi^{+}$) and the $\Lambda$(1405) mass spectrum. The closeconnection between K N and $\overline{K}$N diagrams due to G-parity conservation is exploitedin order to examine the nature of short-range repulsion in the K N system.Our investigation suggests that this repulsion is not entirely due to $\omega$-exchangebut in addition consists of so far unknown processes of mixed G-parity. Furthermore,fourth-order contributions to the $\overline{K}$N potential are shown to provide animportant isospin dependent part of the interaction while Coulomb correctionsturn out to be small, but not totally negligible.Our model definitely predicts $\Lambda$(1405) to be a $\overline{K}$N quasibound state. Theobserved mass spectrum is reproduced without the inclusion of a genuine threequark state. Therefore we consider $\Lambda$(1670) to be the first excited $\frac{1}{2}$-state of theuds-system and include corresponding pole graphs in our model