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

A particle with finite initial velocity in a disordered potential comes back and in average stops at the original location. This phenomenon dubbed 'quantum boomerang effect' (QBE) has been recently observed in an experiment simulating the quantum kicked-rotor model [Phys. Rev. X 12, 011035 (2022)]. We provide analytical arguments that support QBE in a wide class of disordered systems. Sufficient conditions to observe the real-space QBE effect are (a) Anderson localization, (b) the reality of the spectrum for the case of non-Hermitian systems, (c) the ensemble of disorder realizations $\{H\}$ be invariant under the application of $\mathcal{R\, T}$, and (d) the initial state is an eigenvector of $\mathcal{R\, T}$, where $\mathcal{R}$ is a reflection $x \rightarrow -x$ and $\mathcal{T}$ 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 $\mathcal{T}$ instead of $\mathcal{RT}$. These conditions allow the observation of the QBE in time-reversal symmetry broken models, contrarily to what was expected from previous analyses of the effect, and in a large class of non-Hermitian models. We provide examples of QBE in lattice models with magnetic flux breaking time-reversal symmetry and in a model with 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 $RT$ ($T$) symmetry.