Brownian-Motion Model of Quantum-Chaotic Spectra: Exact Two-level Correlations for Transitions to CUE

Abstract

Spectral fluctuations in quantum-chaotic systems undergo transition from one universality class to another as a good symmetry of the system is gradually broken. After suitable rescaling and limits, the transitions also have a universal classification which depends only on the initial and final systems and is describable in terms of Dyson's Brownian-motion ensembles of random matrices. In this article we consider Brownian ensembles which undergo transitions to the circular unitary ensemble (CUE). We give a general method of deriving two-level correlation functions exactly. As a first application, we derive results for the transition with initial spectrum being uniformly spaced. For the study of partitioning symmetries in time-reversal noninvariant quantum-chaotic systems, we obtain the two-level correlation functions for transitions starting with initial ensemble as direct sum of two independent CUEs with lower dimensionalities. For the study of time-reversal noninvariant quantum-chaotic systems whose classical analogs display an abrupt transition from integrability to chaos, we derive the two-level correlation function for the Poisson → CUE transition. We also rederive our earlier results for the COE → CUE and CSE* → CUE transitions, relevant for the study of weakly-broken time reversal symmetry, COE and CSE* denoting here respectively the circular orthogonal and symplectic ensembles with doubly-degenerate eigenvalues in the latter. These results, announced earlier without proofs, are also expected to hold for the corresponding Gaussian ensembles with large dimensionalities.