Intermediate level statistics with one-parameter random matrix ensembles

We discuss a formulation of the level statistics for quantum systems that lie in an intermediate regime between the integrability subject to Poisson statistics and the full chaos subject to Gaussian statistics of the typical random matrix theory (RMT). It is based on the idea initiated by Yukawa and also by Nakamura and Lakshmanan, namely, to transcribe the eigenvalue statistics into statistical mechanics of the completely integrable Calogero system with internal degrees of rotation. By analyzing the previous works of Gaudin and Forrester from this viewpoint, we answer the question raised by Mehta and Dyson in early RMT concerning the expectation of compressible level gas, which is realized now in such an intermediate regime. It indicates the general absence of a fractional power dependence of the two-level correlation function under the ordinary thermodynamic-limit procedure as the manifestation that a transition between localized and delocalized states does not occur. The predictability of the transition by modifying the theory is remarked. Two applications of the present scheme are shown: the information loss per level and the correlation hole of survival probability for intermediate circular unitary ensembles.