Unhappy vertices in artificial spin ice: new degeneracies from vertex frustration

Abstract

In 1935, Pauling estimated the residual entropy of water ice with remarkable\naccuracy by considering the degeneracy of the ice rule {\\it solely at the\nvertex level}. Indeed, his estimate works well for both the three-dimensional\npyrochlore lattice and the two-dimensional six-vertex model, solved by Lieb in\n1967. The case of honeycomb artificial spin ice is similar: its pseudo-ice\nrule, like the ice rule in Pauling and Lieb's systems, simply extends a\ndegeneracy which is already present in the vertices to the global ground state.\nThe anisotropy of the magnetic interaction limits the design of inherently\ndegenerate vertices in artificial spin ice, and the honeycomb is the only\ndegenerate array produced so far. In this paper we show how to engineer\nartificial spin ice in a virtually infinite variety of degenerate geometries\nbuilt out of non-degenerate vertices. In this new class of vertex models, the\nresidual entropy follows not from a freedom of choice at the vertex level, but\nfrom the nontrivial relative arrangement of the vertices themselves. In such\narrays, loops exist along which not all of the vertices can be chosen in their\nlowest energy configuration: these loops are therefore vertex-frustrated since\nthey contain unhappy vertices. Residual entropy emerges in these lattices as\nconfigurational freedom in allocating the unhappy vertices of the ground state.\nThese new geometries will finally allow for the fabrication of many novel\nextensively degenerate artificial spin ice.\n