Ubiquity of the quantum boomerang effect in Hermitian Anderson-localized systems

A particle with finite initial velocity in a disordered potential comes back and on average stops at the original location. This phenomenon, dubbed the "quantum boomerang effect" (QBE), has been recently observed in an experiment simulating the quantum kicked-rotor model [Sajjad et al., Phys. Rev. X 12, 011035 (2022)]. We provide analytical arguments that support the presence of the QBE in a wide class of disordered systems. Sufficient conditions to observe the real-space QBE are (a) Anderson localization, (b) the reality of the spectrum for the case of non-Hermitian systems, (c) the ensemble of disorder realizations {H} being invariant under the application of R T, and (d) the initial state being an eigenvector of R T, where R is a reflection x -> -x and T is the time-reversal operator. The QBE can be observed in momentum space in systems with dynamical localization if conditions (c) and (d) are satisfied with respect to the operator T instead of RT. These conditions allow the observation of the QBE in time-reversal-symmetry-broken models, contrary to what was expected from previous analyses of the effect, and in a large class of non-Hermitian models. We provide examples of the QBE in lattice models with magnetic flux breaking time-reversal symmetry and in a model with an electric field. Whereas the QBE straightforwardly applies to noninteracting many-body systems, we argue that a real-space (momentum-space) QBE is absent in weakly interacting bosonic systems due to the breaking of reflection-timereversal (time-reversal) symmetry.