Deep thermalization in random quantum circuits with U(1) symmetry

In isolated quantum systems, thermalization occurs when local subsystems approach universal equilibrium states where observables acquire thermal expectation values. Recently, it was shown that equilibration can happen not only at the level of expectation values, but also of statistical distributions of wavefunctions, a phenomenon called “deep thermalization”. In particular, the quantum states in a projected ensemble, obtained on a subsystem after measuring its complement, may also approach a universal form. For systems without conservation laws, this was found to be the unitarily-invariant (Haar) ensemble. Here we consider the projected ensemble constructed from random circuit dynamics with U(1) symmetry and investigate the emergence of late-time universal ensembles. We identify different possible ensembles depending on the choice of initial state and measurement basis; notably, these include the Haar measure on the entire Hilbert space of the subsystem as well as a direct sum of Haar measures in charge sectors. Our results may give new insights into how systems with conserved quantities achieve equilibrium and suggest the possibility of forming (approximate) quantum state designs even in systems constrained by a symmetry, with promising applications in quantum information science.