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Magnetic excitations in the XY-pyrochlore antiferromagnet Er$_2$Ti$_2$O$_7$

The XY-pyrochlore antiferromagnet \ETO\ is studied by heat capacity measurements and electron spin resonance spectroscopy performed on single crystal samples. The magnetic phase diagrams are established for two directions of applied field, $H\parallel [100]$ and $H\parallel [111]$. In the magnetically ordered phase observed below $T_N=1.2$ K, the magnetic excitation spectrum consists of a Goldstone mode acquiring an isotropic gap in an applied field, and another mode with a gap softening in the vicinity of a field-induced phase transition. This second-order transition takes place at a critical field $H_c$ above which the magnetization process is accompanied by a canting of the magnetic moments off their local "easy-planes". The specific heat curves for $H\parallel [100]$ ($H\gg H_c$) are well described by a model presuming a single dispersionless excitation mode with the energy gap obtained from the spectroscopic measurements.