Entropic sampling in frustrated magnets: Role of self-intersecting spaces

Frustrated magnets typically possess a large space of classical ground states. If this degeneracy is not protected by symmetry, thermal fluctuations may ``select'' certain states via order-by-disorder. In this article, we examine a precursor effect where all ground states are sampled, but with different weights. Geometry plays a key role in determining the weight distribution and its behavior. We demonstrate this with two examples---both clusters with four spins coupled by XY interactions. In the first, the classical ground states form a smooth space. In the second, they form a self-intersecting non-manifold space. Ground-state sampling is very different in these two cases. We first consider the microcanonical ensemble picture, where fluctuations conserve energy. Phase space arguments suggest that the first model exhibits energy-independent probabilities. The second shows a dramatic energy dependence with relative probability increasing as ${\ensuremath{\epsilon}}^{\ensuremath{-}1/2}$, where $\ensuremath{\epsilon}$ is the energy of the system. We simulate low-energy dynamics in both models, confirming the expected behavior. We next consider the canonical ensemble, where the first model produces temperature-independent probabilities. In the second, relative probability rises sharply as ${T}^{\ensuremath{-}1/2}$, where $T$ is the temperature. Our results bring out a classical analog of order-by-singularity, a mechanism that has been recently proposed in the context of quantum spin clusters. The sampling of classical orders is qualitatively different in systems with self-intersecting ground-state spaces. It grows at low energies and becomes singular as $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ (microcanonical ensemble) or $T\ensuremath{\rightarrow}0$ (canonical ensemble). We discuss relevance for disordered phases in macroscopic magnets, particularly for spiral liquids.

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