Order by Disorder in the XY Pyrochlore Antiferromagnet Er$_2$Ti$_2$O$_7$

Crystal field effects associated with Er$^{3+}$ magnetic moments in Er$_2$Ti$_2$O$_7$ give rise to local XY anisotropy and effective quantum S=1/2 spins which are antiferromagnetically coupled on this material’s cubic pyrochlore lattice [1]. Er$_2$Ti$_2$O$_7$ orders into a non-collinear antiferromagnetic $\Psi_2$ state below $\sim$ 1.2 K, in zero magnetic field, but the mechanism for its ground state selection has been a puzzle for more than a decade. We have carried out inelastic neutron scattering measurements on single crystal samples of Er$_2$Ti$_2$O$_7$ at low temperatures and in the presence of a strong [110] magnetic field, allowing us to determine the underlying spin Hamiltonian for this quantum antiferromagnet [2, 3]. These results point to ground state selection via an order-by-quantum-disorder mechanism [3], and a concomitant order-by-disorder gap of $\sim$ 0.05 meV has also been observed [4], associated with the pseudo-Goldstone modes in the low field ordered state. In addition, we have explored the sensitivity of the ground state selection to magnetic dilution by preparing and studying single crystals of Er$_{2-x}$Y$_x$Ti$_2$O$_7$ [5]. These studies are particularly topical in light of two theoretical predictions [6,7] that the $\Psi_2$ ordered state may be unstable to formation of the related $\Psi_3$ phase at low temperatures, in the presence of quenched disorder. [1] J.D.M. Champion et al., Phys. Rev. B 68, 020401 (2003). [2] J.P.C. Ruff et al., Phys. Rev. Lett., 101, 147205 (2005). [3] L. Savary et al. Phys. Rev. Lett., 109, 167201 (2012). [4] K.A. Ross et al. Phys. Rev. Lett. 112, 057201 (2014). [5] J.F. Niven, Proc. R. Soc. A, 470: 20140387 (2014). [6] V. S. Maryasin and M. E. Zhitomirsky, Phys. Rev. B 90, 094412 (2014). [7] A. Andreanov and P. A. McClarty, Phys. Rev. B 91, 064401 (2015).