  Quantum dynamics with local interactions in lattice models display rich
physics, but is notoriously hard to study. Dual-unitary circuits allow for
exact answers to interesting physical questions in clean or disordered one- and
higher-dimensional quantum systems. However, this family of models shows some
non-universal features, like vanishing correlations inside the light-cone and
instantaneous thermalization of local observables. In this work we propose a
generalization of dual-unitary circuits where the exactly calculable
spatial-temporal correlation functions display richer behavior, and have
non-trivial thermalization of local observables. This is achieved by
generalizing the single-gate condition to a hierarchy of multi-gate conditions,
where the first level recovers dual-unitary models, and the second level
exhibits these new interesting features. We also extend the discussion and
provide exact solutions to correlators with few-site observables and discuss
higher-orders, including the ones after a quantum quench. In addition, we
provide exhaustive parametrizations for qubit cases, and propose a new family
of models for local dimensions larger than two, which also provides a new
family of dual-unitary models.
