Complex systems with half-integer spins: symplectic ensembles.

We study the statistical behavior of the Hermitian operators of complex systems with half-integer angular momentum and time-reversal symmetry. The complexity leads to randomization of the operators which can then be modeled, following maximum entropy hypothesis, by multiparametric Gaussian ensembles of real-quaternion matrices. The modeling shows that it is possible to classify the statistical behavior of spin-based complex systems into a continuum of universality classes characterized by a single parameter which is a function of all system parameters.