Normal-conducting mesoscopic systems in contact with a superconductor are classified by the symmetry operations of time reversal and rotation of the electron's spin. Four symmetry classes are identified, which correspond to Cartan's symmetric spaces of type C, CI, D, and DIII. A detailed study is made of the systems where the phase shift due to Andreev reflection averages to zero along a typical semiclassical single-electron trajectory. Such systems are particularly interesting because they do not have a genuine excitation gap but support quasiparticle states close to the chemical potential. Disorder or dynamically generated chaos mixes the states and produces novel forms of universal level statistics. For two of the four universality classes, the n-level correlation functions are calculated by the mapping on a free 1D Fermi gas with a boundary. The remaining two classes are related to the Laguerre orthogonal and symplectic random-matrix ensembles. For a quantum dot with an NS-geometry, the weak localization correction to the conductance is calculated as a function of sticking probability and two perturbations breaking time-reversal symmetry and spinrotation invariance. The universal conductance fluctuations are computed from a maximum-entropy S-matrix ensemble. They are larger by a factor of two than what is naively expected from the analogy with normal-conducting systems. This enhancement is explained by the doubling of the number of slow modes: owing to the coupling of particles and holes by the proximity to the superconductor, every cooperon and diffuson mode in the advanced-retarded channel entails a corresponding mode in the advanced-advanced (or retarded-retarded) channel. 74.80.Fp, 05.45.+b, 74.50.+r, 72.10.Bg
Gaussian random-matrix ensembles defined over the tangent spaces of the large families of Cartan's symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single-particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H, Mr) where G/H is a complex-analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and Mr is a maximal Riemannian submanifold of the support of G/H.
The nuclear matrix element for 2v double-beta decay is calculated within the quasiparticle randomphase approximation. It is shown that the decay matrix element passes through zero as a function of the strength g^^ of the particle-particle component of the spin-isospin polarization force, neglected previously. The analysis of electron capture/^"*" decay rates for semimagic neutron-deficient nuclei suggests values for g^^ in the very vicinity of this zero, which gives rise to long lifetimes. The qualitative features of nuclear fiP decay are illustrated with the example of an exactly soluble model.
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