Quantum order by disorder or virtual crystal-field selection in Er2Ti2O7?
In plain words
Er2Ti2O7 orders into one particular spin pattern although its main interactions treat a whole continuous family equally. Both quantum fluctuations and virtual excitations to higher atomic energy levels can pick that pattern, and their relative share is unsettled.
Precise statement
For Er2Ti2O7 with anisotropic nearest-neighbor exchange $(J_{zz}, J_{pm}, J_{pmpm}, J_{zpm})$ fitted to neutron data, compute the energy splitting between the $\psi_2$ and $\psi_3$ states and the pseudo-Goldstone gap produced by (a) quantum zero-point fluctuations within the ground doublet and (b) virtual crystal-field excitations, and compare with the measured gap. An answer gives the fraction of the measured gap due to each mechanism.
What would settle it
A calculation including both spin-wave zero-point energy and second-order crystal-field corrections, with exchange fitted to the full neutron spectrum, compared with the measured gap.
Status in the literature
A pseudo-Goldstone gap was observed (Ross et al., PRL 2014) as order by disorder predicts; Rau et al. (PRL 2016) showed virtual crystal-field fluctuations give a comparable selection.