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Magnetic systems with frustration often have large classical degeneracy. Weshow that their low-energy physics can be understood as dynamics within thespace of classical ground states. We demonstrate this mapping in a family ofquantum spin clusters where every pair of spins is connected by an $XY$antiferromagnetic bond. The dimer with two spin-$S$ spins provides the simplestexample, it maps to a quantum particle on a ring ($S^1$). The trimer is morecomplex, equivalent to a particle that lives on two disjoint rings ($S^1\otimes\mathbb{Z}_2$). It has an additional subtlety for half-integer $S$ values,wherein both rings must be threaded by $\pi$-fluxes to obtain a satisfactorymapping. This is a consequence of the geometric phase incurred by spins. Forboth the dimer and the trimer, the effective theory can be seen from apath-integral-based derivation. This approach cannot be extended to thequadrumer which has a non-manifold ground state space, consisting of three torithat touch pairwise along lines. In order to understand the dynamics of aparticle in this space, we develop a tight-binding model with thisconnectivity. Remarkably, this successfully reproduces the low-energy spectrumof the quadrumer. For half-integer spins, a geometric phase emerges which canbe mapped to two $\pi$-flux tubes that reside in the space between the tori.The non-manifold character of the space leads to a remarkable effect - thedynamics at low energies is not ergodic as the particle is localized aroundsingular lines of the ground state space. The low-energy spectrum consists ofan extensive number of bound states formed around singularities. Physically,this manifests as an order-by-disorder-like preference for collinear groundstates. However, unlike order-by-disorder, this `order by singularity' persistseven in the classical limit. We discuss consequences for field theoreticstudies of magnets.

Effective theories for quantum spin clusters: Geometric phases and state selection by singularity