Effective theories for quantum spin clusters : State selection by singularity

We state a general principle -- the low energy effective theory of a quantum spin cluster reduces to that of a quantum particle moving on the space of classical ground states. We demonstrate this mapping for a family of spin clusters where each pair of spins is connected by an XY antiferromagnetic bond. The simplest member of this family is a dimer-- it maps to a particle on a ring. The trimer, a cluster of three spin-S spins, is more complex -- it maps to a particle on two disjoint rings. Unlike the dimer and the trimer, the classical ground state space of the quadrumer, cluster of four spin-S spins, is non-manifold in nature -- consisting of three tori pairwise touching along lines. Particle moving on this space, successfully, captures the low energy spectrum of the quadrumer. The non-manifold structure leads to a remarkable effect -- the dynamics at low energies is not ergodic as the particle is localized around singular lines of the ground-state space. The low-energy spectrum consists of an extensive number of bound states around singularities. Physically, this manifests as an order-by-disorder like preference for collinear states. However, unlike order-by-disorder, this “order by singularity” gets better as we approach the classical limit.

Ref: Phys. Rev. B 100, 134411(2019)