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Low-field behavior of an XY pyrochlore antiferromagnet: emergent clock anisotropies

Using $\rm Er_2Ti_2O_7$ as a motivation, we investigate finite-field properties of $XY$ pyrochlore antiferromagnets. In addition to a fluctuation-induced six-fold anisotropy present in zero field, an external magnetic field induces a combination of two-, three-, and six-fold clock terms as a function of its orientation providing for a rich and controllable magnetothermodynamics. For $\rm Er_2Ti_2O_7$, we predict a new phase transition for ${\bf H}\parallel [001]$. Re-entrant transitions are also found for ${\bf H}\parallel [111]$. We extend these results to the whole family the $XY$ pyrochlore antiferromagnets and show that presence and number of low-field transitions for different orientations can be used for locating a given material in the parameter space of anisotropic pyrochlores. Finite-temperature classical Monte Carlo simulations serve to confirm and illustrate these analytic predictions.