Local pairing of Feynman histories in many-body Floquet models

We study many-body quantum dynamics using Floquet quantum circuits in one space dimension as simple examples of systems with local interactions that support ergodic phases. Physical properties can be expressed in terms of multiple sums over Feynman histories, which for these models are paths or many-body orbits in Fock space. A natural simplification of such sums is the diagonal approximation, where the only terms that are retained are ones in which each path is paired with a partner that carries the complex conjugate weight. We identify the regime in which the diagonal approximation holds, and the nature of the leading corrections to it. We show that properties are dominated at long times by contributions to orbit sums in which each orbit is paired locally with a conjugate, as in the diagonal approximation, but that in large systems these contributions consist of many spatial domains, with distinct local pairings in neighbouring domains. The existence of these domains is reflected in deviations of the SFF from RMT predictions, and of matrix element correlations from ETH predictions; deviations of both kinds diverge with system size. We also show that, within this picture, the transition from an ergodic to a many-body localised phase can be viewed as symmetry breaking.