Spin ice

[5] Pauling's findings were confirmed by specific heat measurements, though pure crystals of water ice are particularly hard to create.

Rather, the temperature at which the maximum occurs, about 1 K in Dy2Ti2O7, signals a rapid change in the number of tetrahedra where the two-in, two-out rule is violated.

Most importantly, it is the long-range magnetostatic dipole–dipole interaction, and not the nearest-neighbor exchange, that causes the frustration and the consequential two-in, two-out rule that leads to the spin ice phenomenology.

In the context of spin ice, such charges characterizing the violation of the two-in, two-out magnetic moment orientation rule are the aforementioned monopoles.

[2][3][4] In Autumn 2009, researchers reported experimental observation of low-energy quasiparticles resembling the predicted monopoles in spin ice.

[14][15] The effective charge of a magnetic monopole, Q (see figure 3) in both the dysprosium and holmium titanate spin ice compounds is approximately Q = 5 μBÅ−1 (Bohr magnetons per angstrom).

Magnetic ions other than dysprosium (Dy) and holmium (Ho) are required to generate a quantum spin ice, with praseodymium (Pr), terbium (Tb) and ytterbium (Yb) being possible candidates.

In addition, artificial spin ices show potential as reprogrammable magnonic crystals and have been studied for their fast dynamics.

A variety of geometries have been explored, including quasicrystalline systems and 3D structures, as well as different magnetic materials to modify anisotropies and blocking temperatures.

Spontaneous emergence of 2D magnetic vortices was observed in such spin ices, which vortex geometries were correlated with the external bulk remanence.

[23] Future work in this field includes further developments in fabrication and characterization methods, exploration of new geometries and material combinations, and potential applications in computation,[24] data storage, and reconfigurable microwave circuits.

[26] In 2022, another studied achieved an artificial kagome spin ice which could potentially be used in the future for novel high-speed computers with low power consumption.

Figure 1. The arrangement of hydrogen atoms (black circles) about oxygen atoms (open circles) in ice. Two hydrogen atoms (bottom ones) are close to the central oxygen atom while two of them (top ones) are far and closer to the two other (top left and top right) oxygen atoms.
Figure 2. Portion of a pyrochlore lattice of corner-linked tetrahedra. The magnetic ions (dark blue spheres) sit on a network of tetrahedra linked at their vertices. The other atoms (e.g. Ti and O) making the pyrochlore crystal structure are not displayed. The magnetic moments (light blue arrows) obey the two-in, two out spin ice rule over the whole lattice. The system is thus in a spin ice state.
Figure 3. The orientation of the magnetic moments (light blue arrows) considering a single tetrahedron within the spin ice state, as in figure 2. Here, the magnetic moments obey the two-in, two-out rule: there is as much "magnetization field" going in the tetrahedron (bottom two arrows) as there is going out (top two arrows). The corresponding magnetization field has zero divergence. There is therefore no sink or source of the magnetization inside the tetrahedron, or no monopole . If a thermal fluctuation caused one of the bottom two magnetic moments to flip from "in" to "out", one would then have a 1-in, 3-out configuration; hence an "outflow' of magnetization, hence a positive divergence, that one could assign to a positively charged monopole of charge + Q . Flipping the two bottom magnetic moments would give a 0-in, 4-out configuration, the maximum possible "outflow" (i.e. divergence) of magnetization and, therefore, an associated monopole of charge +2 Q .