Critical quantum dynamics of observables at eigenstate transitions

It is an outstanding goal to unveil the key features of quantum dynamics at eigenstate transitions. Focusing on quadratic fermionic Hamiltonians that exhibit localization transitions, we identify physical observables that exhibit scale-invariant critical dynamics at the transition when quenched from the initially localized states. The identification is based on two ingredients: (a) A relationship between the time evolution of observables in a many-body state and the transition probabilities of single-particle states, and (b) scale invariance of transition probabilities, which generalizes a corresponding recent result for survival probabilities [Phys. Rev. Lett. 131, 060404 (2023) and arXiv:2309.16005]. These properties suggest that there is also critical behavior in the quantum-quench dynamics of observables, which share the common eigenbasis with the Hamiltonian before the quench. Focusing on experimentally relevant observables such as site occupations and the particle imbalance we numerically demonstrate their critical behavior at the eigenstate transitions in the three-dimensional Anderson model and the one-dimensional Aubry-Andr\'e model.