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 does nothing to damp the perturbations that thermal fluctuations excite. A longitudinal field parallel to the moment removes that instability and is invisible to the leading-order dynamics, so it is often described as free. It is not free: it costs the square of its amplitude integrated over the pulse. We compute the cost-reliability front of that longitudinal field with an open-source engine that solves optimal control over the Landau-Lifshitz-Gilbert equation and a thermostat validated against the Boltzmann distribution. For monolayer CrSBr we reproduce the published stabilization physics, the half-hyperbolic bare path and the success-rate dip near one anisotropy field, and we add the missing axis: the added switching cost, which grows quadratically with the field. The price is not small: one anisotropy field costs 2.5 times the bare optimal switching cost, and two cost ten times it. Nor is the benefit universal. At a thermal-stability factor of twenty the bare optimal pulse already reverses every one of 400 copies, so the field buys nothing; the reliability gain appears only below a stability factor of about ten, where it lifts the success rate from 0.81 to 0.96 at a factor of two. The switching cost is reported in tesla-squared-seconds and is not converted to joules without an explicit circuit model. Code and data: engine (MIT) https://github.com/fsantibanezleal/CAOS_SpinOCT ; product, committed artifacts and live instance (MIT) https://github.com/fsantibanezleal/CAOS_RES_Espira ; https://espira.fasl-work.com .