Fluctuating "order parameter" for a quantum chaotic system with partially broken time-reversal symmetry

The functional $\rho =|\int d\vec{r} \psi^2|^2$ of a chaotic wave function $\psi (\vec{r})$ plays the role of an ``order parameter'' for the transition between Hamiltonian ensembles with orthogonal and unitary symmetry. Upon breaking time-reversal symmetry, $\rho$ crosses over from one to zero. We compute the distribution of $\rho$ in the crossover regime and find that it has large fluctuations around the ensemble average. These fluctuations imply long-range spatial correlations in $\psi$ and non-Gaussian perturbations of eigenvalues, in precise agreement with results by Fal'ko and Efetov and by Taniguchi, Hashimoto, Simons, and Altshuler. As a third implication of the order-parameter fluctuations we find correlations in the response of an eigenvalue to independent perturbations of the system.