Low-field behavior of anXYpyrochlore antiferromagnet: Emergent clock anisotropies

Using ${\mathrm{Er}}₂{\mathrm{Ti}}₂{\mathrm{O}}₇$ as 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 rich and controllable magnetothermodynamics. For ${\mathrm{Er}}₂{\mathrm{Ti}}₂{\mathrm{O}}₇$, we predict a phase transition in a weak magnetic field $\mathbf{H}\ensuremath{\parallel}[001]$. Re-entrant transitions are also found for $\mathbf{H}\ensuremath{\parallel}[111]$. 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.