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Abstract

We introduce a quasiclassical Green function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear $\sigma$-model of disordered systems, the $G \Sigma$-formalism for strong correlations, and real time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective `Thouless time', up to the many-body Heisenberg time. We illustrate the approach with two case studies, (i) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries, and (ii) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.