Experimental Realization of the 1D Random Field Ising Model

Biased Ising Model Using Two Coupled Kerr Parametric Oscillators with External Force

Networks of coupled Kerr parametric oscillators (KPOs) are a leading physical platform for analog solving of complex optimization problems. These systems are colloquially known as "Ising machines." We experimentally and theoretically study such a network under the influence of an external force. The force breaks the collective phase-parity symmetry of the system and competes with the intrinsic coupling in ordering the network configuration, similar to how a magnetic field biases an interacting spin ensemble. Specifically, we demonstrate how the force can be used to control the system, and highlight the crucial role of the phase and symmetry of the force. Our Letter thereby provides a method to create Ising machines with arbitrary bias, extending even to exotic cases that are impossible to engineer in real spin systems.

Clocked dynamics in artificial spin ice

Artificial spin ice (ASI) are nanomagnetic metamaterials with a wide range of emergent properties. Through local interactions, the magnetization of the nanomagnets self-organize into extended magnetic domains. However, controlling when, where and how domains change has proven difficult, yet is crucial for technological applications. Here, we introduce astroid clocking, which offers significant control of ASI dynamics in both time and space. Astroid clocking unlocks a discrete, step-wise and gradual dynamical process within the metamaterial. Notably, our method employs global fields to selectively manipulate local features within the ASI. Sequences of these clock fields drive domain dynamics. We demonstrate, experimentally and in simulations, how astroid clocking of pinwheel ASI enables ferromagnetic domains to be gradually grown or reversed at will. Richer dynamics arise when the clock protocol allows both growth and reversal to occur simultaneously. With astroid clocking, complex spatio-temporal behaviors of magnetic metamaterials become easily controllable with high fidelity.

Artificial Magnetic Tripod Ice

We study the collective behavior of interacting arrays of nanomagnetic tripods. These objects have six discrete moment states, in contrast to the usual two states of an Ising-like moment. Our experimental data demonstrate that triangular lattice arrays form a "tripod ice" that exhibits charge ordering among the effective vertex magnetic charges, in direct analogy to artificial kagome spin ice. The results indicate that the interacting tripods have effective moments that act as emergent local variables, with strong connections to the well-studied Potts and clock models. In addition, the tripod moments display a tendency toward a nearest neighbor alignment in our thermalized samples that separates this system from kagome spin ice. Our results open a path toward the study of the collective behavior of nonbinary moments that is unavailable in other physical systems.

Collective Ferromagnetism of Artificial Square Spin Ice

We study the temperature and magnetic field dependence of the total magnetic moment of large-area permalloy artificial square spin ice arrays. The temperature dependence and hysteresis behavior are consistent with the coherent magnetization reversal expected in the Stoner-Wohlfarth model, with clear deviations due to interisland interactions at small lattice spacing. Through micromagnetic simulations, we explore this behavior and demonstrate that the deviations result from increasingly complex magnetization reversal at small lattice spacing, induced by interisland interactions, and depending critically on details of the island shapes. These results establish new means to tune the physical properties of artificial spin ice structures and other interacting nanomagnet systems, such as patterned magnetic media.