1244 nm long, 170 nm wide, and 10 nm thick. Four strips of permalloy (NiFe)—only about a fifth the width of a human hair—are arranged at the vertices of a square. When an external magnetic field is applied to align this collection of tiny magnets into an initial state and then removed, the system "relaxes" through magnetic interactions toward its most stable configuration.
This relaxation process resembles rolling a ball down from a mountaintop. The ball rolls toward the lowest point, but depending on the shape of the slope, it may temporarily settle into a dip along the way, from which the direction it rolls next becomes probabilistically branched. In nanomagnet systems, these "dips"—known as intermediate states—determine the certainty of the output. The higher the energy of an intermediate state, the more the system branches there, making the final state probabilistic.
A research team led by Hanu Arava of the Materials Science Division at Argonne National Laboratory has shown that the appearance of these intermediate states can be controlled solely through the geometric arrangement of the magnets. In a paper published in Communications Materials on April 2, 2026 (DOI: 10.1038/s43246-026-01147-4), the team arranged four nanomagnets at the vertices of a square plaquette and, through the simple operation of continuously rotating each magnet from 0° to 90° about its central axis, discovered a critical angle at which the nature of the relaxation pathway changes dramatically.
Twenty years since the concept of "designing frustration" emerged
The stage for this research, artificial spin ice, traces back to a concept first experimentally demonstrated in Nature by R.F. Wang and colleagues at Pennsylvania State University in 2006 (Nature 439, 303-306, 2006). In natural spin ice materials (rare-earth pyrochlore oxides), magnetic moments arranged at the vertices of a tetrahedron create "frustration," where all dipole interactions cannot be satisfied simultaneously. Wang and colleagues artificially reproduced this geometric frustration through lithography. By arranging single-domain permalloy nanoislands in a square lattice, they showed that at each vertex, the four magnetic moments obey a "two-in, two-out" ice rule.
Since this discovery, artificial spin ice has served both as an experimental platform for fundamental physics and as a candidate for computing device applications. The orientation of the magnets carries information, and dipole interactions between magnets realize logical operations. Since no electric current needs to flow, the system can theoretically operate at energies far below those of CMOS. Indeed, in an experiment reported by Bérut and colleagues in 2012 in the predecessor journal to Science Advances, the dissipated energy for single-bit erasure in a nanomagnet was shown to match the Landauer limit (approximately 2.8 zJ at room temperature, i.e., J). Given that even the most advanced CMOS nodes consume on the order of several aJ ($10^{-18}$ J) per operation, nanomagnet systems could in principle approach roughly 1/1000th of that.
However, this path faced a major obstacle: intermediate states appearing during relaxation rendered the output uncertain.
Intermediate states that "distance" alone could not fully control
In earlier work published by Arava himself in 2019 while at ETH Zurich (Physical Review Applied 11, 054086, 2019), a method was demonstrated for designing relaxation pathways by varying the distance between magnets, switching between deterministic and probabilistic outputs. Bringing magnets closer together strengthens the dipole interaction, while moving them apart weakens it. This intuitive parameter enabled certain types of logic gates to function.
The problem was that adjusting distance only changed the "strength" of interactions—it did not provide a principle for controlling the "structure of the pathway" of relaxation itself. When and why do intermediate states appear? What is the boundary separating conditions under which the system behaves deterministically from those under which it behaves probabilistically? Conventional modeling addressing this question assumed a single spin-flip from the initial state to the final low-energy state, and did not sufficiently treat the role of intermediate states.
To fill this gap, Arava's team deliberately chose a minimal system: four nanomagnets. Their strategy was to extract just a single unit of the square lattice—a single vertex—and thoroughly investigate its behavior.
The "pathway switching" brought about by rotation angle Φ
The experimental design is simple and clear. Four nanomagnets are placed at the midpoints of the sides of a square, and each magnet is rotated by an angle Φ about the axis perpendicular to the plane, passing through its center. At Φ = 0°, the magnets align along the sides of the square in an "open" configuration (a Greek cross shape); at Φ = 90°, the magnets point along the diagonals of the square in a "closed" configuration.
The team fabricated two lengths of nanomagnets—λ (1244 nm × 170 nm × 10 nm) and σ (512 nm × 170 nm × 10 nm)—at Argonne National Laboratory's Center for Nanoscale Materials (a DOE Office of Science user facility), and statistically measured the post-relaxation magnetic configurations at each Φ using magnetic force microscopy (MFM). In parallel, they performed theoretical calculations using a dipole Hamiltonian and dumbbell model, and analyzed the energy landscape using multipolar analysis.
The results were clear.
At Φ = 90° (closed configuration), relaxation proceeds monotonically. Because the energy of the intermediate state is lower than that of the initial state, the system heads straight toward the ground state, as if "rolling downhill." In experiments, the fraction reaching the ground state (state 0) at Φ = 90° reached 75% for λ and 93% for σ.
At Φ = 0° (open configuration), relaxation becomes intermittent. A high energy barrier exists at the intermediate state, and whether the system can cross that barrier branches probabilistically. In experiments, the fraction reaching the most common final state (state 2b) at Φ = 0° was only 46% for λ and 52% for σ—yielding different outputs with roughly 50% probability.
The boundary between these two behaviors lay at Φ = 45°. At this angle, the interaction length between magnet 2 and magnet 4 coincides with the interaction length between magnet 2 and magnet 3, producing complete energetic degeneracy. The output probability is distributed evenly across all states.
| Rotation angle Φ | Type of relaxation pathway | Ground state arrival rate (λ / σ) | Physical meaning |
|---|---|---|---|
| 0° (open configuration) | Intermittent | ~0% (state 2b most common at 46–52%) | High intermediate states cause branching |
| 45° (critical angle) | Complete degeneracy | Evenly distributed across all states | Interaction lengths become equal, pathway is indeterminate |
| 90° (closed configuration) | Monotonic | 75% / 93% | Intermediate states are low, relaxation proceeds in one direction |
The team further confirmed that as Φ was lowered to 67.5°, 45°, 22.5°, and 0°, the ground state arrival rate (for λ) decreased continuously to 17%, 7%, 3%, and 0%. The pathway switch is not abrupt but rather the probability is continuously modulated according to angle. This means designers can use angle as a "dial" to arbitrarily tune output probability.
What multipole analysis revealed about the role of "invisible charge"
Why does geometric arrangement alone determine the relaxation pathway? The team answered this question through multipole expansion.
In artificial spin ice, the N and S poles of each magnet are treated as virtual "magnetic charges" (+q, -q). When four magnets converge at a vertex, an effective magnetic charge appears at the vertex—a magnetic dipole, quadrupole, or so-called "emergent monopole"—depending on the configuration. Arava's analysis identified which multipolar textures drive the flipping of which magnets at each stage of relaxation.
In the open configuration (Φ < 45°), only a single charged intermediate state with high energy exists at the vertex, and this is the cause of the probabilistic branching. In the closed configuration (Φ > 45°), the energy of the intermediate state falls below that of the initial state, and the system slides down to the ground state unimpeded by any barrier.
The team further revealed that while the self-energy of the monopole (Q) governs relaxation, the interaction between charge q and the dipole or quadrupole terms makes the energy landscape "far more complex than what is suggested by Coulomb interactions or dipole interactions alone."
What rules from a minimal cluster offer for large-scale device design
In an Argonne National Laboratory press release, Arava stated: "What we identified is a structure-function relationship. Here, structure refers to the geometric arrangement between magnets, and function refers to how energy moves through the system. We showed that geometric arrangement alone determines how energy moves. This insight offers a new way to design computing devices."
The design principle indicated by this statement is structurally different from the earlier approach of "adjusting the distance between magnets." Distance changes the strength of interaction but offers no guarantee of switching the type of pathway (monotonic versus intermittent). Rotation angle, on the other hand, changes the symmetry of the interaction itself, thereby switching the type of pathway. This yields a clear design rule: pathways can be classified as "vertex type" (probabilistic) or "loop type" (deterministic) with 45° as the boundary.
In a practical device, thousands of identical clusters would need to operate in coordination. By adjusting the rotation angle of each cluster as a design parameter, one could choose whether the entire system behaves deterministically or intentionally generates probabilistic outputs. The latter could find applications in random number generation, probabilistic inference, and neuromorphic computing.
Remaining questions and the wall of scale-up
What this research presents is, after all, a proof of principle in a minimal cluster of just four magnets. Several important questions remain unresolved.
First, there is the issue of scale-up. When thousands of clusters are coupled together, there is no guarantee that interactions between adjacent clusters will not disturb local relaxation pathways. The paper itself states that "in larger magnetic devices, identical clusters need to operate in coordination," without claiming this has been demonstrated.
Second, there is the matter of operating speed and practicality. This experiment was based on field-induced relaxation and static measurement via MFM; high-speed operation at the nanosecond scale and integration with electrical read/write have not been verified. Another study published on arXiv in 2026 (arXiv: 2604.09420) reports an attempt to give directionality to artificial spin ice to realize information propagation, but integration into CMOS-compatible processes remains at an early stage.
Third, there is the question of the universality of the 45° critical angle. This value depends on the specific symmetry of the square plaquette. In other artificial spin ice geometries, such as honeycomb or kagome lattices, the corresponding critical condition may differ.
| Item | Earlier work (Arava 2019, Phys. Rev. Applied) | This study (Arava 2026, Commun. Mater.) |
|---|---|---|
| Control parameter | Distance between magnets | Rotation angle Φ of magnets |
| Physical meaning of control | Interaction strength | Interaction symmetry (pathway type) |
| Pathway classification | Deterministic / probabilistic (continuous variation with distance) | Monotonic / intermittent (clear switch at 45°) |
| Treatment of intermediate states | Implicit | Explicitly identified via multipole analysis |
| System scale | Tens of magnets (logic gates) | 4 (minimal cluster) |
From the standpoint of energy consumption, the proximity to the Landauer limit supports the principle that nanomagnet systems are advantageous relative to CMOS. However, extracting that advantage in an actual device still leaves engineering challenges: applying the cluster geometric design at scale, while also enabling high-speed operation. This discovery—that output probability can be continuously tuned by turning the rotation angle—marks the first calibration point showing which direction that dial should be turned.