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In this work we study a scenario where dynamics is restricted to a singleexcitation manifold, for particular physical observables with support in themanifold, which we label a `correlated quench'. We ask how such dynamics may ingeneral differ from predictions of the eigenstate thermalization hypothesis(ETH). We show that if thermalization occurs, it will not fulfil other keypredictions of the ETH; instead following differing generic behaviours. We showthis by analysing long-time fluctuations, two-point correlation functions, andthe out-of-time-ordered correlator; analytically detailing deviation from ETHpredictions. We derive instead an ETH-like relation, with non-randomoff-diagonals, for matrix elements of observables, with correlations that alterlong-time behaviour and constrain dynamics. Further, we analytically computethe time-dependence of the decay to equilibrium, showing that it isproportional to the survival probability of the initial state. We finally notethat the conditions studied are common in many physical scenarios, such asunder the rotating-wave approximation. We show numerically that predictions arerobust to perturbations that break this approximation.

Chaotic dynamics in a single excitation subspace: deviations from the ETH via long time correlations