Thermal pseudo-Goldstone gaps in frustrated quantum magnets

In systems with competing interactions, continuous degeneracies can arise at the classical (mean-field) level that are not protected by any symmetry of the parent Hamiltonian. These "accidental" degeneracies are typically lifted by fluctuations, both thermal and quantum in nature, by a mechanism known as order-by-disorder. As a consequence of order-by-disorder, the excitation spectrum acquires a small energy gap, known as a pseudo-Goldstone gap. While the properties of this gap are known to leading order in the spin wave interactions, in both the zero-temperature and classical limits, the pseudo-Goldstone gap in quantum magnets at finite temperature has yet to be characterized. Using non-linear spin wave theory, we calculate the pseudo-Goldstone gap to leading order in the 1/S expansion and at low temperature, for a variety of quantum spin systems. We discuss the temperature dependence of the pseudo-Goldstone gap and how it relates to the spectrum of non-interacting spin waves. For concreteness, we examine the implications of these results for several pyrochlore oxide compounds: Er₂Ti₂O₇, Yb₂Ge₂O₇, Lu₂V₂O₇, and Y₂V₂O₇, for which there is evidence of order-by-disorder giving rise to different types of long-range ordered phases.