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

The functional {ρ}=|{∫}dr-{→}ψ²{|}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.