Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum spin clusters where every pair of spins is connected by an XY antiferromagnetic bond. The dimer with two spin-S spins provides the simplest example -it maps to a quantum particle on a ring (S 1 ). The trimer is more complex, equivalent to a particle that lives on two disjoint rings (S 1 ⊗ Z2). It has an additional subtlety for half-integer S values, wherein both rings must be threaded by π-fluxes to obtain a satisfactory mapping. This is a consequence of the geometric phase incurred by spins. For both the dimer and the trimer, the validity of the effective theory can be seen from a path-integral-based derivation. This approach cannot be extended to the quadrumer which has a non-manifold ground state space, consisting of three tori that touch pairwise along lines. In order to understand the dynamics of a particle in this space, we develop a tight-binding model with this connectivity. Remarkably, this successfully reproduces the low-energy spectrum of the quadrumer. For half-integer spins, a geometric phase emerges which can be mapped to two π-flux tubes that reside in the space between the tori. The non-manifold character of the space 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 formed around singularities. Physically, this manifests as an order-by-disorder-like preference for collinear ground states. However, unlike order-by-disorder, this 'order by singularity' persists even in the classical limit. We discuss consequences for field theoretic studies of magnets.

A fundamental motif in frustrated magnetism is the fully mutually coupled cluster of N spins, with each spin coupled to every other spin. Clusters with N = 2 and 3 have been extensively studied as building blocks of square and triangular lattice antiferromagnets. In both cases, large-S semiclassical descriptions have been fruitfully constructed, providing insights into the physics of macroscopic magnetic systems. Here, we develop a semiclassical theory for the N = 4 cluster. This problem has rich mathematical structure with a ground state space that has non-trivial topology. We show that ground states are appropriately parametrized by a unit vector order parameter and a rotation matrix. Remarkably, in the low energy description, the physics of the cluster reduces to that of an emergent free spin-S spin and a rigid rotor. This successfully explains the spectrum of the quadrumer and its associated degeneracies. However, this mapping does not hold in the vicinity of collinear ground states due to a subtle effect that arises from the non-manifold nature of the ground state space. We demonstrate this by an analysis of soft fluctuations, showing that collinear states have a larger number of soft modes. Nevertheless, as these singularities only occur on a subset of measure zero, the mapping to a spin and a rotor provides a good description of the quadrumer. We interpret thermodynamic properties of the quadrumer that are accessible in molecular magnets, in terms of the rotor and spin degrees of freedom. Our study paves the way for field theoretic descriptions of systems such as pyrochlore magnets.

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Accidental ground state degeneracies -those not a consequence of global symmetries of the Hamiltonian -are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-bydisorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order-by-thermaldisorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, ∆, and show that at low temperatures it scales as the square root of temperature, √ T . We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.