Dynamical signatures of Chaos to integrability crossover in 2x2 generalized random matrix ensembles

We introduce a two-parameter ensemble of generalized 2x2 real symmetric random matrices called the beta-Rosenzweig-Porter ensemble (beta-RPE), parameterized by beta, a fictitious inverse temperature of the analogous Coulomb gas model, and gamma, controlling the relative strength of disorder. beta-RPE encompasses RPE from all of the Dyson's threefold symmetry classes: orthogonal, unitary and symplectic for beta=1,2,4 . Firstly, we study the energy correlations by calculating the density and 2nd moment of the nearest neighbor spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of beta-RPE in the gamma-beta plane. Our results are in qualitative agreement with numerically computed fidelity for N >> 2 matrix ensembles. Furthermore, we observe that for large N the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at gamma = 2.