Order by disorder or energetic selection of the ground state in the $XY$ pyrochlore antiferromagnet ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$: An inelastic neutron scattering study

Examples of materials where an “order by disorder” mechanism is at play to select a particular ground state are scarce. It has recently been proposed, however, that the antiferromagnetic XY pyrochlore ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ reveals a most convincing case of this mechanism, with the observation of a spin gap at zone centers having recently been interpreted as a corroboration of this physics. Here we argue, however, that the anisotropy generated by the interaction-induced admixing between the crystal-field ground and excited levels provides for an alternative mechanism. It especially predicts the opening of a spin gap of about 15 $\mu \mathrm{eV}$, which is of the same order of magnitude as the one observed experimentally. We report high-resolution inelastic neutron scattering data which can be well understood within this scenario.

Figure 1

(a) Sketch of the ${\psi }_2$ magnetic configuration. (b) Illustration of the energetic selection by molecular field induced magnetocrystalline anisotropy: the ground-state energy of Hamiltonian ${\mathcal{H}}^{\prime }$ computed for $h=1$ T shows minima along the axes $\varphi =0,60,120,180$ degrees, however slightly tilted out of the XY plane $\left({\theta }_o\ne 90\right)$. (c) shows the corresponding tilt angle as a function of $h$. (d) and (e) show respectively the curvature along $\theta$ and $\varphi$ of the potential wells as a function of the magnetic field $h$. (f) shows the evolution of the spin gap as a function of the renormalization $\eta$ of the Stevens coefficients $B_{nm}$ (see text).

Figure 2

IN5 time of flight spectra taken at 50 mK with an incident wavelength of 6 Å  along various directions, in zero field (a) or under a magnetic field of 2.5 T applied along the $(1,-1,0)$ direction (b). $S(Q,\omega )$ is shown for the RPA (model A) and spin half (model B), taking into account equipopulated ${\psi }_2$ domains, as described in the main text. The global agreement is evident but the arrows point out specific Q regions showing the limits of the two models. (c) INS raw data recorded with an incident wavelength of 8.5 Å at $\mathit{Q}=\left(111\right)$ and $T=50$ mK showing the spin gap (blue line) at 43 $\mu \mathrm{eV}$. (d) Evolution of the spin gap and of the magnetic moment in model A as a function of temperature. The saturated moment is 3.7 ${\mu }_B$ while $T_{\mathrm{N}}=2.1$ K (at the mean-field level).