State selection in frustrated magnets

Magnets with frustration often show accidental degeneracies, characterized by a large classical ground-state space (CGSS). Quantum fluctuations may select' one of these ground states -- a phenomenon labeled order by (quantum) disorder’ in literature. In this article, we examine the mechanism(s) by which such state selection takes place. We argue that a magnet, at low energies, maps to a particle moving on the CGSS. State selection corresponds to localization of the particle at a certain point on this space. We distinguish two mechanisms that can bring about localization. In the first, quantum fluctuations generate a potential on the CGSS. If the potential has a deep enough minimum, then the particle localizes in its vicinity. We denote this as order by potential' (ObP). In the second scenario, the particle localizes at a self-intersection point due to bound-state formation -- a consequence of geometry and quantum interference. Following recent studies by the present authors, we denote this scenario as order by singularity’ (ObS). We place our discussion within the context of the one-dimensional spin-$S$ Kitaev model. We map out its CGSS which grows systematically with increasing system size. It resembles a network where the number of nodes increases exponentially. In addition, the number of wires that cross at each node also grows exponentially. This self-intersecting structure leads to ObS, with the low-energy physics determined by a small subset of the CGSS, consisting of `Cartesian’ states. A contrasting picture emerges when an additional XY antiferromagnetic coupling is introduced. The CGSS simplifies dramatically, taking the form of a circle. Spin-wave fluctuations generate a potential on this space, giving rise to state selection by ObP under certain conditions. Apart from contrasting ObS and ObP, we discuss the possibility of ObS in macroscopic magnets.