Order from structural disorder in $XY$ pyrochlore antiferromagnet $\rm Er_2Ti_2O_7$

Effect of structural disorder is investigated for an $XY$ pyrochlore antiferromagnet with continuous degeneracy of classical ground states. Two types of disorder, vacancies and weakly fluctuating exchange bonds, lift degeneracy selecting the same subset of classical ground states. Analytic and numerical results demonstrate that such an "order by structural disorder" mechanism competes with the effect of thermal and quantum fluctuations. Our theory predicts that a small amount of nonmagnetic impurities in $\rm{Er_2Ti_2O_7}$ will stabilize the coplanar $\psi_3$ ($m_{x^2-y^2}$) magnetic structure as opposed to the $\psi_2$ ($m_{3z^2-r^2}$) state found in pure material.