Deep thermalization in constrained quantum systems

The concept of "deep thermalization" has recently been introduced to characterize moments of an ensemble of pure states, resulting from projective measurements on a subsystem, which lie beyond the purview of conventional eigenstate thermalization hypothesis (ETH). In this paper, we study deep thermalization in systems with kinetic constraints, such as the quantum East and the PXP models, which have been known to weakly break ETH by slow dynamics and high sensitivity to the initial conditions. We demonstrate a sharp contrast in deep thermalization between the first and higher moments in these models by studying quench dynamics from initial product states in the computational basis: While the first moment shows good agreement with ETH, higher moments deviate from the uniform Haar ensemble at infinite temperature. We show that such behavior is caused by an interplay of time-reversal symmetry and an operator that anticommutes with the Hamiltonian. We formulate sufficient conditions for violating deep thermalization, even for systems that are otherwise "thermal" in the ETH sense. By appropriately breaking these properties, we illustrate how the PXP model fully deep thermalizes for all initial product states in the thermodynamic limit. Our results highlight the sensitivity of deep thermalization as a probe of physics beyond ETH in kinetically-constrained systems.