Operator Growth Across a Quantum Interface

We investigate operator growth across an interface separating chaotic and nonchaotic regions. The underlying behavior is revealed through out-of-time-order correlator (OTOC) computed in both an analytically solvable generalized Sachdev–Ye–Kitaev (SYK) model and a numerical spin model via tensor-network simulations, showing consistent OTOC profiles and front scalings. The resulting operator spreading is effectively captured by coupled reaction–diffusion and dispersion equations. It exhibits hydrodynamic behavior that interpolates between chaotic and dispersive regimes, independent of microscopic details. This unified hydrodynamic framework connects chaotic spreading and coherent dispersive dynamics across the interface.