Scrambling transition in free fermion systems induced by a single impurity

In quantum many-body systems, interactions play a crucial role in the emergence of information scrambling. When particles interact throughout the system, the entanglement between them can lead to a rapid and chaotic spreading of quantum information, typically probed by the growth in operator size in the Heisenberg picture. In this study, we explore whether the operator undergoes scrambling when particles interact solely through a single impurity in generic spatial dimensions, focusing on fermion systems with spatial and temporal random hoppings. By connecting the dynamics of the operator to the symmetric exclusion process with a source term, we demonstrate the presence of an escape-to-scrambling transition when tuning the interaction strength for fermions in three dimensions. As a comparison, systems in lower dimensions are proven to scramble at arbitrarily weak interactions unless the hopping becomes sufficiently long-ranged. Our predictions are validated using both a Brownian circuit with a single Majorana fermion per site and a solvable Brownian SYK model with a large local Hilbert space dimension. This suggests the universality of the theoretical picture for free fermion systems with spatial and temporal randomness.