We consider invariant matrix models with log-normal (asymptotic) weight. It is known that their eigenvalue distribution is intermediate between Wigner-Dyson and Poissonian, which candidates these models for describing a system intermediate between th
e extended and localized phase. We show that they have a much richer energy landscape than expected, with their partition functions decomposable in a large number of equilibrium configurations, growing exponentially with the matrix rank. Within each of these saddle points, eigenvalues are uncorrelated and confined by a different potential felt by each eigenvalue. The equilibrium positions induced by the potentials differ in different saddles. Instantons connecting the different equilibrium configurations are responsible for the correlations between the eigenvalues. We argue that these instantons can be linked to the SU(2) components in which the rotational symmetry can be decomposed, paving the way to understand the conjectured critical breaking of U(N) symmetry in these invariant models.
The Aharonov-Bohm effect is the prime example of a zero-field-strength configuration where a non-trivial vector potential acquires physical significance, a typical quantum mechanical effect. We consider an extension of the traditional A-B problem, by
studying a two-dimensional medium filled with many point-like vortices. Systems like this might be present within a Type II superconducting layer in the presence of a strong magnetic field perpendicular to the layer, and have been studied in different limits. We construct an explicit solution for the wave function of a scalar particle moving within one such layer when the vortices occupy the sites of a square lattice and have all the same strength, equal to half of the flux quantum. From this construction we infer some general characteristics of the spectrum, including the conclusion that such a flux array produces a repulsive barrier to an incident low-energy charged particle, so that the penetration probability decays exponentially with distance from the edge.
