Heisenberg machines with programmable spin circuits

Abstract

We show that we can harness two recent experimental developments to build a compact hardware emulator for the classical Heisenberg model in statistical physics. The first is the demonstration of spin-diffusion lengths in excess of microns in graphene even at room temperature. The second is the demonstration of low-barrier magnets (LBMs) whose magnetization can fluctuate rapidly even at subnanosecond rates. Using experimentally benchmarked circuit models, we show that an array of LBMs driven by an external current source has a steady-state distribution corresponding to a classical system with an energy function of the form $E=\ensuremath{-}(1/2)\ensuremath{\sum}_{i,j}{J}_{ij}({\stackrel{^}{m}}_{i}\ensuremath{\cdot}{\stackrel{^}{m}}_{j}$). This may seem surprising for a nonequilibrium system, but we show that it can be justified by a Lyapunov function corresponding to a system of coupled Landau--Lifshitz--Gilbert (LLG) equations. The Lyapunov function we construct describes LBMs interacting through the spin currents they inject into the spin-neutral substrate. We suggest ways to tune the coupling coefficient ${J}_{ij}$ so that it can be used as a hardware solver for optimization problems involving continuous variables represented by vector magnetizations, similar to the role of the Ising model in solving optimization problems with binary variables. Finally, we train a Heisenberg xor gate based on a network of four coupled stochastic LLG equations, illustrating the concept of probabilistic computing with a programmable Heisenberg model.

Topics & Concepts