Information scrambling in free fermion systems with a sole interaction

It is well established that the presence of single impurity can have a substantial impact on the transport properties of quantum many-body systems at low temperature. In this work, we investigate a close analog of this problem from the perspective of quantum information dynamics. We construct Brownian circuits and Clifford circuits consisting of a free fermion hopping term and a sole interaction. In both circuits, our findings reveal the emergence of operator scrambling. Notably, the growth of the operator can be mapped to the symmetric exclusion process in the presence of a source term localized at a single point. We demonstrate that in the one-dimensional system, both the operator and entanglement exhibit diffusive scaling. Conversely, in scenarios characterized by all-to-all hopping, the operator's size undergoes exponential growth, while the entanglement exhibits a linear increase over time.