On the multifractal dimensions and statistical properties of critical ensembles characterized by the three classical Wigner-Dyson symmetry classes

Abstract

We introduce a power-law banded random matrix model for the third of the three classical Wigner-Dyson ensembles, i.e., the symplectic ensemble. A detailed analysis of the statistical properties of its eigenvectors and eigenvalues, at criticality, is presented. This ensemble is relevant for timereversal symmetric systems with strong spin-orbit interaction. For the sake of completeness, we also review the statistical properties of eigenvectors and eigenvalues of the power-law random banded matrix model for the corresponding systems in the presence and absence of time reversal invariance, previously considered in the literature. Our results show a good agreement with heuristic relations for the eigenstate and eigenenergy statistics at criticality, proposed in previous studies. With this, we provide a full picture of the power-law random banded matrix model corresponding to the three classical Wigner-Dyson ensembles.