Quantum order by disorder or virtual crystal-field selection in $\mathrm{Er_2Ti_2O_7}$?
In plain words
$\mathrm{Er_2Ti_2O_7}$ 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 $\mathrm{Er_2Ti_2O_7}$ 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.
Results
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