Using 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 Er_2Ti_2O_7, we predict a new phase transition for H∥ [001]. Re-entrant transitions are also found for H∥ [111]. We

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... 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.