Universal temperature-dependent power law excitation gaps in frustrated quantum spin systems harboring order-by-disorder

When magnetic moments are subject to competing or frustrated interactions, continuous degeneracies that are not protected by any symmetry of the parent Hamiltonian can emerge at the classical (mean-field) level. Such "accidental" degeneracies are often lifted by both thermal and quantum fluctuations via a mechanism known as order-by-disorder (ObD). The leading proposal to detect and characterize ObD in real materials, in a way that quantitatively distinguishes it from standard energetic selection, is to measure a small fluctuation-induced pseudo-Goldstone gap in the excitation spectrum. 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 (PG) gap in quantum magnets at finite temperature has yet to be characterized. Using non-linear spin wave theory, we compute the PG gap to leading order in a $1/S$ expansion at low temperature for a variety of frustrated quantum spin systems. We also develop a formalism to calculate the PG gap in a way that solely uses linear spin-wave theory, circumventing the need to carry out tedious quantum many-body calculations. We argue that, at leading order, the PG gap acquires a distinct power-law temperature dependence, proportional to either $T^{d+1}$ or $T^{d/2+1}$ depending on the gapless dispersion of the PG mode predicted at the mean-field level. Finally, we examine the implications of these results for the pyrochlore oxide compound Er$_2$Ti$_2$O$_7$, for which there is compelling evidence of ObD giving rise to the experimentally observed long-range order.