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

TL;DR

This paper investigates a specific order parameter in quantum chaotic systems, analyzing its distribution and fluctuations during the transition from time-reversal symmetry to broken symmetry, revealing long-range correlations and non-Gaussian effects.

Contribution

It provides a detailed computation of the order parameter's distribution during the symmetry crossover and links its fluctuations to long-range correlations and eigenvalue perturbations.

Findings

Order parameter crosses from one to zero as symmetry breaks.

Large fluctuations around the ensemble average of the order parameter.

Fluctuations imply long-range spatial correlations and non-Gaussian eigenvalue perturbations.

Abstract

The functional defined as the squared modulus of the spatial average of the wave function squared, plays the role of an ``order parameter'' for the transition between Hamiltonian ensembles with orthogonal and unitary symmetry. Upon breaking time-reversal symmetry, the order parameter crosses over from one to zero. We compute its distribution in the crossover regime and find that it has large fluctuations around the ensemble average. These fluctuations imply long-range spatial correlations in the eigenfunction 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.