Quantum dynamics in a single excitation subspace: deviations from eigenstate thermalization via long time correlations

In this work we study a scenario where unitary quantum dynamics in a many-body interacting system is restricted to a single excitation subspace. We ask how dynamics within to such a subspace may in general differ from predictions of the eigenstate thermalization hypothesis (ETH). We show that for certain initial states and observables, if thermalization occurs, it will not fulfil other key predictions of the ETH; instead following differing generic behaviours. We show this by analysing long-time fluctuations, two-point correlation functions, and the out-of-time-ordered correlator; analytically detailing deviation from ETH predictions. We derive instead an ETH-like relation, with non-random off-diagonals for matrix elements of observables, with correlations which alter long-time behaviour and constrain dynamics. Further, we analytically compute the time-dependence of the decay to equilibrium, showing it is proportional to the survival probability of the initial state. We finally note the conditions studied are common in many physical scenarios, such as under the rotating-wave approximation. We show numerically our predictions are robust to perturbations which break this approximation.