Fluctuating phase rigidity for a quantum chaotic system with partially broken time-reversal symmetry

The functional ρ=|∫dr-→${\mathrm{\psi }}^2$ ${\mathrm{|}}^2$ measures the phase rigidity of a chaotic wave function ψ(r-→) in the transition between Hamiltonian ensembles with orthogonal and unitary symmetry. Upon breaking time-reversal symmetry, ρ crosses over from one to zero. We compute the distribution of ρ in the crossover regime and find that it has large fluctuations around the ensemble average. These fluctuations imply long-range spatial correlations in ψ and non-Gaussian perturbations of eigenvalues, in precise agreement with results by Fal'ko and Efetov [Phys. Rev. Lett. 77, 912 (1996)] and by Taniguchi et al. [Europhys. Lett. 27, 335 (1994)]. As a third implication of the phase-rigidity fluctuations we find correlations in the response of an eigenvalue to independent perturbations of the system.