The Cost of Reliability in Optimal Magnetization Switching: A Longitudinal-Field Front for Van der Waals Magnets

Optimal control theory delivers magnetic-field pulses that reverse a magnetic moment for the least dissipated energy, and recent work shows these pulses reach picosecond, energy-efficient switching in two-dimensional van der Waals magnets. Because the optimal pulse is always perpendicular to the moment, it exerts no restoring torque on the transverse perturbations that thermal fluctuations excite, so the switching becomes unreliable exactly where it matters. A longitudinal field parallel to the moment removes that instability and is invisible to the leading-order dynamics, so it is often described as a cost-free stabilizer. It is not free: it adds the square of its amplitude integrated over the pulse to the switching cost. This preprint computes, for the first time, the cost-reliability front of that longitudinal field. Using an open-source engine (spinoct) that solves optimal control over the Landau-Lifshitz-Gilbert equation, and a stochastic thermostat validated against the Boltzmann distribution, we reproduce the published stabilization physics for monolayer CrSBr, the half-hyperbolic bare reversal path and the counterintuitive success-rate dip near a half anisotropy field, and we add the missing axis: the added switching cost, which grows quadratically with the longitudinal field and reaches about 1.5e-10 T^2 s at 2.5 anisotropy fields for a demanding thermal-stability factor of twenty. The front makes the reliability-versus-energy trade explicit and quantitative, and lets a device designer read off the smallest field that reaches a target reliability. The switching cost is reported in its model-free units of tesla-squared-seconds, never converted to joules without an explicit circuit model. All quantitative results are reproduced from artifacts committed to the repository, and a live instance is open. Engine (MIT): https://github.com/fsantibanezleal/CAOS_SpinOCT . Product and live instance (MIT): https://github.com/fsantibanezleal/CAOS_RES_Espira ; https://espira.fasl-work.com .