Geometric approach to optimal nonequilibrium control: Minimizing dissipation in nanomagnetic spin systems

Optimal control of nanomagnets has become an urgent problem for the field of spintronics as technological tools approach thermodynamically determined limits of efficiency. In complex, fluctuating systems, such as nanomagnetic bits, finding optimal protocols is challenging, requiring detailed information about the dynamical fluctuations of the controlled system. We provide a physically transparent derivation of a metric tensor for which the length of a protocol is proportional to its dissipation. This perspective simplifies nonequilibrium optimization problems by recasting them in a geometric language. We then describe a numerical method, an instance of geometric minimum action methods, that enables computation of geodesics even when the number of control parameters is large. We apply these methods to two models of nanomagnetic bits: a Landau-Lifshitz-Gilbert description of a single magnetic spin controlled by two orthogonal magnetic fields, and a two-dimensional Ising model in which the field is spatially controlled. These calculations reveal nontrivial protocols for bit erasure and reversal, providing important, experimentally testable predictions for ultra-low-power computing.

Figure 1

Optimal control of the magnetic moment of a thin nanomagnetic film using orthogonal fields $h_x$ and $h_y$, as described in Sec. 4. (a) The optimal protocol as determined by the geometric minimum action method. Inset: The potential energy of the system at the beginning and end of the protocol. (b) A schematic of the control problem: a thin magnetic film is controlled by external fields $h_x$ and $h_y$. (c) The $x$ and $y$ fields as a function of protocol time. Note the significant deviation from the linear interpolations commonly used in experiments.

Figure 2

Optimal control of the 2D Ising model with spatial control of the magnetic field. (a) A schematic of the setup of the system. Boundary conditions are fixed so that there are two distinct metastable states. The external field is tuned independently for the different spin blocks. (b) An optimized protocol is shown (bottom) driving the transition. Representative structures from the configuration space are shown (top).