Title: Eigenstate thermalization in an open bipartite quantum system and the semiclassical method base on correlations of adjacent local states in phase space

Abstract: We investigate the eigenstate thermalization in terms of a Hermitian operator and the complex eigenkets that follows Gaussian ensemble distribution. With the non-Hermitian open bipartite system, there are, however, some global restrictions such that the elements share some of the properties of Gaussian orthogonal ensemble in diagonal and off-diagonal perspective. Such global restrictions enforce that one of the subsystem contains a nullspace with non-defective degeneracies (the primary subsystem), and the another full-ranked subsystem (the secondary subsystem). For the primary subsystem, the mixed densities in Hermitian and non-Hermitian basis exhibits global fluctuation and unidirectional (non-Hermitian skin effect) fluctuation, respectively. The former is due to the global restrictions of the whole system which plays teh role of environmental disorder, while the latter is due to the nonlocal symmetries which is allowed in the restricted Hilbert space.
We also investigate the integrablity-chaos transition with independent perturbations in terms of the Berry autocorrelation in semicalssical limit, where there is a phase space spanned by the momentum-like projection and the range of local wave function.