  Conclusive evidence of order by disorder is scarce in real materials. Perhaps
one of the strongest cases presented has been for the pyrochlore XY
antiferromagnet Er2Ti2O7, with the ground state selection proceeding by order
by disorder induced through the effects of quantum fluctuations. This
identification assumes the smallness of the effect of virtual crystal field
fluctuations that could provide an alternative route to picking the ground
state. Here we show that this order by virtual crystal field fluctuations is
not only significant, but competitive with the effects of quantum fluctuations.
Further, we argue that higher-multipolar interactions that are generically
present in rare-earth magnets can dramatically enhance this effect. From a
simplified bilinear-biquadratic model of these multipolar interactions, we show
how the virtual crystal field fluctuations manifest in Er2Ti2O7 using a
combination of strong coupling perturbation theory and the random phase
approximation. We find that the experimentally observed psi2 state is indeed
selected and the experimentally measured excitation gap can be reproduced when
the bilinear and biquadratic couplings are comparable while maintaining
agreement with the entire experimental spin-wave excitation spectrum. Finally,
we comments on possible tests of this scenario and discuss implications for
other order-by-disorder candidates in rare-earth magnets.
