Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice

All of the characteristics of the square-ice model are observed in an artificial square-ice system in which the two sublattices of nanomagnets are slightly vertically separated. Geometric frustration in atomic lattices, such as those in water ice and magnetic materials called spin ice, leads to rich physical behaviour. So-called artificial spin ice, consisting of two-dimensional lattices of nanomagnets, was developed as a means of modelling these systems. Magnetic ordering, where neighbouring magnets try to align such that one is spin-up and the other spin-down, cannot be optimized owing to geometric frustration in these systems, which can be directly imaged. Until now it has not been possible to realize a fundamental 'square ice' model in these artificial systems that is of direct relevance to real materials. Nicolas Rougemaille and colleagues have solved this problem by designing a square lattice in which one of the two sublattices of nanomagnets is slightly vertically displaced. This enables them to directly observe the predicted spin-liquid state. Artificial spin-ice systems are lithographically patterned arrangements of interacting magnetic nanostructures that were introduced as way of investigating the effects of geometric frustration in a controlled manner1,2,3,4. This approach has enabled unconventional states of matter to be visualized directly in real space5,6,7,8,9,10,11,12,13,14,15,16,17,18, and has triggered research at the frontier between nanomagnetism, statistical thermodynamics and condensed matter physics. Despite efforts to create an artificial realization of the square-ice model—a two-dimensional geometrically frustrated spin-ice system defined on a square lattice—no simple geometry based on arrays of nanomagnets has successfully captured the macroscopically degenerate ground-state manifold of the model19. Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality1,20,21,22,23,24,25,26. Here we show that all of the characteristics of the square-ice model are observed in an artificial square-ice system that consists of two sublattices of nanomagnets that are vertically separated by a small distance. The spin configurations we image after demagnetizing our arrays reveal unambiguous signatures of a Coulomb phase and algebraic spin-spin correlations, which are characterized by the presence of ‘pinch’ points in the associated magnetic structure factor. Local excitations—the classical analogues of magnetic monopoles27—are free to evolve in an extensively degenerate, divergence-free vacuum. We thus provide a protocol that could be used to investigate collective magnetic phenomena, including Coulomb phases28 and the physics of ice-like materials.