Real symmetric random matrix ensembles of Hamiltonians with partial symmetry breaking

We investigate random matrix ensembles E({ε}) containing real symmetric matrices H=H⁽⁰⁾+{ε}V, where H⁽⁰⁾ is block diagonal, each block a member of the Gaussian orthogonal ensemble, coupled together by the Gaussian random elements of {ε}V. E({ε}) could model, for example, a chaotic Hamiltonian apart from an approximate integral of the motion. We focus on transitions in eigenvalue and eigenvector projection statistics of E({ε}) upon variation of their respective scaling parameters. Expressions for the probability density of nearest-neighbor level spacings as well as the spectral rigidity are given, and supported by numerical data, and their application in determining a symmetry-breaking perturbation is discussed. We derive an expression for the probability density of projections of eigenvectors of E({ε}) onto those of E(0) valid for sufficiently small {ε} and use it to calculate ensemble averages and distribution functions. Each of these results is compared with numerical data. We furthermore use E({ε}) at small {ε} to calculate average time-dependent transition probabilities of nonstationary states.