On a Matrix Ensemble for Arbitrary Complex Quantum Systems

We present an exhaustive analytical exploration of a variation of the eigenvector ensemble initially proposed by Deutsch for the foundations of the Eigenstate Thermalization Hypothesis (ETH) that drastically (but not fully) increases system-dependent information, and therefore serves as a model for the study of complex quantum systems beyond the universal Random Matrix Theory (RMT) regime. We especially focus on the remnants of system-specific information that modify the late-time behavior of correlation functions like the spectral form factor, or the inclusion of explicit system-dependent Hamiltonian corrections not captured by the RMT regime. The consistency of the predictions in the new ensemble with respect to the universal (Haar) one is made explicit by showing how that for arbitrary systems it defines an unitary 1-design and for strongly chaotic systems it becomes an approximated 2-design. While we are able to provide universal expressions for two- and four-point ensemble-averaged correlation functions and show how system-dependent information is spectrally decoupled, we also show that for small energy windows correlation functions defined by the new ensemble reduce to the predictions of the ETH.