Universal temperature dependence of magnon excitation gaps in quantum magnets harboring order-by-disorder

In frustrated magnetic systems with competing interactions, classical ground states often exhibit continuous degeneracies that are not protected by any symmetry of the spin Hamiltonian. These accidental degeneracies are typically lifted by thermal or quantum fluctuations through a mechanism known as order-by-disorder. Using non-linear spin wave theory, we argue that, to leading order in 1/S, the presence of order-by-disorder can be identified by a small spectral gap that increases with temperature according to a universal power law. This observation provides a model-independent framework that distinguishes order-by-disorder from conventional energetic selection of a long-range ordered phase at low temperatures. We discuss the implications of our results in the context of several pyrochlore compounds.