AHARONOV-BOHM FLUX AND STATISTICS OF ENERGY-LEVELS IN METALS

The statistics of levels in a metallic ring presents a transition from the Gaussian orthogonal ensemble to the Gaussian unitary ensemble, when it is pierced by a magnetic flux-phi. This transition is driven by the dimensionless parameter square-root E(c)phi, where E(c) is the Thouless energy expressed in units of the interlevel distance and phi is the reduced flux-phi = 2-pi-phi/phi-0. It is very well described by a E(c) X E(c) random matrix H = H(S) + i-phi-H(A), where H(S) and H(A) are symmetric and antisymmetric matrices whose elements have variance square-root E(c).