Common explanations of equilibrium thermodynamics depend critically on ergodicity, i.e., the ability to explore all available states, but ergodic kinetics can be constrained by a system’s topology. We report an experimental study of a model nanomagnetic array, in which such constraints visibly affect the behavior. In this system, magnetic excitations are connected in thermally active one-dimensional strings whose motion can be imaged in real-time. At high temperatures, the kinetics include the merging, breaking, and reconnecting of strings, resulting in the system transitioning between topologically distinct configurations. Below a crossover temperature, however, the string motion is dominated by simple changes in length and shape. In this low-temperature regime, the system is energetically stable because of the inability to explore the full set of possible topological configurations. This kinetic crossover suggests a generalizable conception of topologically broken ergodicity and limited equilibration.

To prepare the samples, we use electron-beam lithography to write patterns on Si/SiO x substrates with spin-coated bilayer resists. A layer of ~ 2.5 nm thick permalloy (Ni₈₀Fe₂₀) was then deposited into the patterns via ultrahigh vacuum electron beam evaporation, followed by aluminum capping layers of 2 nm thick to prevent oxidation of the underlying permalloy.

The moment configuration in our samples was measured using x-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM), which yields real-space images of the island magnetic moment,  conducted at the PEEM-3 endstation at beamline 11.0.1.1 of the Advanced Light Source, Lawrence Berkeley National Lab. We heated the sample from 250 K to 360 K in 5–10 K steps and took 100 PEEM images at the Fe L 3 absorption edge at each temperature point. The 100 PEEM images consisted of ten exposures with a left-circularly polarized X-ray beam followed by ten exposures with a right-circularly polarized beam, repeated five times. Each image takes 0.5 seconds, and then there is a delay of 0.5 seconds between images (assuming polarization is not switched), so the time period of the imaging is 1 second. The PEEM image field of view was set at 15  µm x 15 µm to 18  µm x 18 µm and there were about 500 islands within each image. We used MATLAB and Python code to extract the moment direction for each island and emergent strings configuration from every PEEM image. By comparing the string configurations between successive PEEM images, we could therefore observe the thermal kinetics of this system.

We have calculated the magnetostatic energy using the micromagnetic simulation program MuMax3. While doing the simulations, the saturation magnetization is set to be 8.6x10⁵ A/m, the exchange constant is 1.3x10^-11 J/m, and the magnetocrystalline anisotropy is negligible for permalloy.

More detailed information about experimental methods can be found in the related publication of this dataset.