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. A 'square ice' model relevant to real materials 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 manner 1 , 2 , 3 , 4 . This approach has enabled unconventional states of matter to be visualized directly in real space 5 , 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 model 19 . Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality 1 , 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-sp