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

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. This approach has enabled unconventional states of matter to be visualized directly in real space, 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. Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality. 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 monopoles-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 phases and the physics of ice-like materials.