Invariant-based control of quantum many-body systems across critical points

Quantum many-body systems are emerging as key elements in the quest for quantum-based technologies and in the study of fundamental physics. In this study, we address the challenge of achieving fast and high-fidelity evolutions across quantum phase transitions, a crucial requirement for practical applications. We introduce a control technique based on dynamical invariants tailored to ensure adiabatic-like evolution within the lowest-energy subspace of the many-body systems described by the transverse-field Ising and long-range Kitaev models. By tuning the controllable parameter according to analytical control results, we achieve high-fidelity evolutions operating close to the speed limit. Remarkably, our approach leads to the breakdown of Kibble-Zurek scaling laws, offering tunable and significantly improved time scaling behavior. We provide detailed numerical simulations to illustrate our findings, demonstrating scalability with the system size and robustness against noisy controls and disorder, as well as its applicability to a non-integrable system.