Scrambling Without Chaos in Random Free-Fermionic Systems

We study the role of randomness in the scrambling of quantum information within integrable free-fermionic systems. Considering quadratic Hamiltonians with varying degrees of randomness, we analyze entanglement-based measures to characterize the scrambling structure. We show that the memory effect in the entanglement of disjoint subsystems of Gaussian states vanishes when the local couplings are random, indicating information delocalization. The tripartite mutual information exhibits negative saturation values similar to those in chaotic systems, albeit with a smaller magnitude, revealing weaker scrambling under integrable quadratic dynamics. Despite integrability, spectral analyses reveal that local random models display a spectral-form-factor ramp and a partial crossover in the single-particle level-spacing ratio from Poisson-like to Wigner--Dyson-like behavior within a certain range of random couplings. These results demonstrate that randomness can act as a minimal ingredient for inducing information scrambling in integrable quadratic fermionic models.