Abstract
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 non-manifold 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 semi-classical large-$S$ limit, we argue for effective low energy descriptions in terms of a single particle moving on these non-manifold 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.
Highlights
Frustrated magnetism is fertile ground for several interesting phenomena
We start with a general principle that holds in the semiclassical large-S limit: the low-energy physics of a cluster of quantum spins maps to that of a single particle moving on the classical ground-state space (CGSS)
We have demonstrated that the CGSS for the Kitaev square consists of four circles, with the circles intersecting at points
Summary
Frustrated magnetism is fertile ground for several interesting phenomena. This is typically best understood in the S → ∞ limit where frustration gives rise to a large classical ground-state degeneracy. In the case of quantum fluctuations, this is typically captured by small O(1/S) corrections They break the classical degeneracy by their zero-point energies to give rise to ordering. We start with a general principle that holds in the semiclassical large-S limit: the low-energy physics of a cluster of quantum spins maps to that of a single particle moving on the classical ground-state space (CGSS). The low-lying energy states of a spin cluster have a one-to-one relation with those of the corresponding single-particle problem This mapping can be seen from the spin-path-integral formulation combined with a large-S semiclassical approach. As we approach the classical S → ∞ limit, state selection due to ObS becomes stronger This is because the mapping between the spin system and the.
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