Title : Late-time ensembles of quantum states in quantum chaotic systems

Quantum states undergoing quantum chaotic dynamics are expected to evolve into featureless states at late times. While this expectation holds true on an average, coarse-grained level, it is unclear if this expectation applies to higher statistical moments, as symmetries typically present in physical systems constrain the exploration of Hilbert space. In this talk, I discuss the universal structure of late-time ensembles obtained from unitary dynamics in quantum chaotic systems with symmetries, such as charge or energy conservation. We identify two limiting universal regimes depending on the initial conditions. When the initial state is typical---all the moments of the symmetry operators are equal to those of pure random states---then the late-time ensemble is indistinguishable from the Haar ensemble in the thermodynamic limit and at the level of higher statistical moments. Otherwise, atypical initial states evolve into non-universal ensembles, which can be distinguished from the Haar ensemble by simple measurements or subsystem properties. Interestingly, such atypical initial conditions can be found in the middle of the spectrum of Hamiltonians known to be `maximally' chaotic even when considering initial product states. In the limiting case of initial states with negligible variance of the symmetry operator (e.g., states with fixed particle number or states with negligible energy variance), the late-time ensemble has universal behavior captured by constrained random matrix theory ensembles. Our work shows that even though midspectrum states do not explore ergodically all of Hilbert space at late times, the late-time ensemble typically---but not always---exhibits the same average and sample-to-sample fluctuations as the Haar ensemble.

[1] SG, Langlett, Hunter-Jones, Rodriguez-Nieva, arxiv: 2409.02187.