Heisenberg machines with programmable spin-circuits

TL;DR

The work presents a programmable hardware platform that emulates the classical Heisenberg model using arrays of low barrier magnets (LBMs) coupled by spin-circuit channels. It establishes a Lyapunov-based justification, showing that the LBM dynamics minimize the Heisenberg energy $E = - \frac{1}{2} \sum_{i,j} J_{ij} (\hat{m}_i \cdot \hat{m}_j)$ and, under noise, sample Boltzmann-like statistics over spin configurations. Numerical simulations in HSPICE for 2- and 3-magnet networks corroborate Boltzmann sampling and demonstrate tunable couplings $J_{ij}$ via channel length and current, along with readout and biasing strategies. The authors also train a Heisenberg XOR gate and discuss a generalized contrastive-divergence learning paradigm, highlighting potential energy-efficient continuous-variable optimization and probabilistic computing with Heisenberg machines.

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 sub-nanosecond 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 $E = -1/2\sum_{i,j} J_{ij} (\hat{m}_i \cdot \hat{m}_j$). This may seem surprising for a non-equilibrium 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 coefficients $J_{ij}$ 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.

Figures (9)

• Figure 1: Heisenberg machines with programmable spin-circuits. Low barrier magnets (LBMs) and the spin neutral channels form the Heisenberg array, where an external charge current is injected into the LBMs and thereby inducing a spin current inside the channel. Magnets are assumed to have zero energy barrier with no effective anisotropy. The proposed array is analogous to a system of coupled stochastic Landau–Lifshitz–Gilbert (sLLG) equations where each sLLG sends a spin current vector $\vec{I}_{s}$ to its neighbors, where the spin current is defined as a function of the magnet magnetization $\vec{I}_{si} = I_s \hat{m}_i$ . This coupled LBM system minimizes the Heisenberg Hamiltonian with continuous spin states, functioning as a Heisenberg machine, analogous to Ising machines.
• Figure 2: Coupled LBMs with pure spin currents. (a) Two coupled sLLG equations, where each sLLG sends a spin current along its magnetization direction $\vec{I}_{si}=I_s\hat{m}_i$ to the other sLLG. (b) The two coupled sLLG equations are contacted to the Boltzmann numerical solution by the predefined constant $I_0$. (c) Three coupled sLLG equations in a frustrated configuration. (d) The three coupled sLLG equations also match with the Boltzmann law, note that $I_0$ is configuration independent.
• Figure 3: Simulated hardware for the Heisenberg emulator. (a) 2-LBM system that emulates a Heisenberg model of two spins. (b) Simulated configuration in HSPICE using benchmarked circuit models. Transport physics from the LBM to the channel and from the channel to the ground are characterized by an FM$|$NM and NM modules, respectively. Magnetization dynamics of the LBM are described by the sLLG modules. All circuits are described by 4 components: charge, and spins $(z, x, y)$. (c) The Boltzmann solution is related to the proposed hardware through $I_M$ that takes the charge-to-spin current conversions into account. The relation between spin current and interaction strength is given by ($J_{12} = I_s/I_0$). (d) 3-LBM system that emulates a Heisenberg model of three spins in a frustrated configuration. (e) Simulated configuration in HSPICE. (f) Boltzmann solution for the 3-magnet system. We use the same $I_M$ value in the 2-magnet system (even though a new charge-to-spin mapping may need to be measured or calculated, see Appendix \ref{['Mapping_Charge_to_Spin']}). Insets show the probability distribution of the magnets correlation $\cos\theta_{12}$ at a given input $I_c$, which can be analytically derived for two LBMs by the aid of Eq. \ref{['Langevin_equation']}.
• Figure 4: Programmability of the interaction strength $J_{ij}$. At fixed charge current $I_c$, the NM channel between LBMs encodes the magnitude of the coupling strength between magnets. Extending the channel length $L_{ch}$ reduces the correlation between the magnets due to more spins being neutralized. The polarity of the $J_{ij}$ is controlled through the direction of the input current $I_c$. The analytical analysis is provided in Appendix \ref{['Mapping_Charge_to_Spin']}, we assume the channel to be Cu with $400$ nm spin-diffusion length.
• Figure 5: Programming $\pm$ weights. Auxiliary magnets are introduced to choose the polarity of the interaction strength $J_{ij}$, where the input current is fixed such that all magnets have negative correlation ($J_{ij} < 0$). The magnitude of the $J_{ij}$ is tuned by the channel length between any two spins ($\hat{m}_i, \hat{m}_j$).