Deep thermalization under charge-conserving quantum dynamics

"Deep thermalization" describes the emergence of universal wavefunction distributions in quantum many-body dynamics, appearing on a local subsystem upon measurement of its environment. In this work, we study in detail the effect of continuous internal symmetries and associated conservation laws on deep thermalization. Concretely, we consider quantum spin systems with a $U(1)$ symmetry associated with the conservation of magnetization (or `charge'), and analyze how the choice of initial states (specifically, their degree of charge fluctuations) and the choice of measurement basis (specifically, whether or not it can reveal information about the local charge density) determine the ensuing universal wavefunction distributions. We put forth a universal ansatz for the limiting form of the projected ensemble, motivated by maximum-entropy principles rooted in statistical physics and quantum information theory. This limiting form depends on a polynomial amount of data on the initial state and measurement basis, a `coarse-graining' that is an essential feature of bona fide thermodynamic ensembles. We support our ansatz with three complementary approaches: (i) a rigorous proof in the simplest case of no charge fluctuations in either the initial state or the measurement basis; (ii) analytical calculations using a `replica limit' approach, applicable when charge fluctuations are allowed in either the input state or the measurement basis but not both; (iii) extensive numerical simulations of finite-sized systems in the most general case. Our findings demonstrate a rich interplay between symmetries and the information extracted by measurements, which allows deep thermalization to exhibit a range of universal behaviors far beyond regular thermalization.