Crossover from Fast Scrambling to Operator Confinement Tuned by an Auxiliary Qubit

We demonstrate a static, disorder-free spin chain Hamiltonian which, by tuning the coupling to an auxiliary qubit, realizes a crossover between super-ballistic, ancilla-accelerated scrambling and sub-ballistic operator confinement. Our minimal model is the mixed-field Ising chain with a spin-1/2 ancilla coupled to its longitudinal magnetization. The ancilla mediates an effective all-to-all interaction which accelerates operator spreading and entanglement growth when weakly coupled, but rapidly saturates its entanglement and projects spin chain operators into effectively frozen subspaces when the ancilla coupling is strong. We locate this crossover independently through both a divergent peak in the mutual-information saturation time near $λ_c N/h \approx 8$ and an exponential suppression of the late-time OTOC growth rate, $\logα\propto 1/λ$. Through a Feshbach-Fano projection and Schrieffer-Wolff transformation, we reveal an effective hidden symmetry on the chain which confines operators on the chain for a time exponential in the coupling strength. This reconciles the fast, $\log(N)$ scrambling reported for random-unitary-circuit realizations of the star geometry with the confinement previously found in its time-independent Hamiltonian analog, showing both emerge from a single Hamiltonian family as a function of one dimensionless parameter.