Order by singularity in Kitaev clusters

The Kitaev model is a beautiful example of frustrated interactions giving rise to deep and unexpected phenomena. In particular, its classical version has remarkable properties stemming from exponentially large ground-state degeneracy. Here, we present a study of magnetic clusters with spin-S moments coupled by Kitaev interactions. We focus on two cluster geometries, the Kitaev square and the Kitaev tetrahedron, that allow us to explicitly enumerate all classical ground states. In both cases, the classical ground-state space (CGSS) is large and self-intersecting, with nonmanifold character. The Kitaev square has a CGSS of four intersecting circles that can be embedded in four dimensions. The tetrahedron CGSS consists of eight spheres embedded in six dimensions. In the semiclassical large-S limit, we argue for effective low-energy descriptions in terms of a single particle moving on these nonmanifold spaces. Remarkably, at low energies, the particle is tied down in bound states formed around singularities at self-intersection points. In the language of spins, the low-energy physics is determined by a distinct set of states that lies well below other eigenstates. These correspond to "Cartesian" states, a special class of classical ground states that are constructed from dimer covers of the underlying lattice. They completely determine the low-energy physics despite being a small subset of the classical ground-state space. This provides an example of order by singularity, where state selection becomes stronger upon approaching the classical limit.