Order by Disorder Spin Wave Gap in the $XY$ Pyrochlore Magnet ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

The recent determination of a robust spin Hamiltonian for the antiferromagnetic $XY$ pyrochlore ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ reveals a most convincing case of the “Order-by-Quantum-Disorder” mechanism for ground state selection. This mechanism relies on quantum fluctuations to remove an accidental symmetry of the magnetic ground state, and selects a particular ordered spin structure below $T_N=1.2\hspace{0.17em}\mathrm{K}$. The removal of the continuous degeneracy results in an energy gap in the spectrum of spin wave excitations, long wavelength pseudo-Goldstone modes. We have measured the Order-by-Quantum-Disorder spin wave gap at a zone center in ${\mathrm{Er}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, using low incident energy neutrons and the time-of-flight inelastic scattering method. We report a gap of $\mathrm{\Delta }=0.053\pm 0.006\mathrm{meV}$, which is consistent with upper bounds placed on it from heat capacity measurements and roughly consistent with the theoretical estimate of $\sim 0.02\mathrm{meV}$, further validating the spin Hamiltonian that led to that prediction. The gap is observed to vary with the square of the order parameter, and goes to zero for $T\sim T_N$.

Figure 1

(a) through (d): color contour maps of the INS spectrum at the bottom of the spin wave dispersion near the (1,1,1) Bragg peak, along the [1,1,$-1$] direction ($[\mathrm{H},\mathrm{H}-\mathrm{H}]+[\mathrm{K},\mathrm{K},2\mathrm{K}]$, with $0.58\le \mathrm{K}\le 0.74\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$), as shown in a representation of the reciprocal plane (e). Panels (a) and (c) show the measured $S(\mathrm{Q⃗},\omega )$ at $T=60$ and 700 mK, while panels (b) and (d) show fits to these data using Eq. (3), (see text). (f) Energy cuts, binning over $0.30\le \mathrm{H}\le 0.36\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$ Error bars represent 1 standard deviation. Note that the factor $k_f/k_i$, which appears in the INS cross section is removed in all data presented herein.

Figure 2

(a) Intensity vs energy cuts through the measured $S(\mathrm{Q⃗},\omega )$, binning over the region $0.30\le \mathrm{H}\le 0.36$, and $0.58\le \mathrm{K}\le 0.74\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$ (b) the binning region overlaid on a map of the elastic scattering [(111) Bragg peak]. The fitting function allows the gap ($\mathrm{\Delta }$), intrinsic width ($\mathrm{\Gamma }$), and scale factors for both elastic and inelastic intensities [$A$, see Eq. (3)] to vary. The fitted values are shown in (c), where error bars indicate half of the 95% confidence interval.

Figure 3

Temperature dependence of the spin wave gap $\mathrm{\Delta }$, overlaid on that of the Bragg peak intensity [$\propto M{(\mathrm{Q⃗})}^2$], measured in the present work, as well as that reported in Champion et al. [11]. Error bars on the gap data are half of the 95% confidence interval while those on the Bragg intensity (1 standard deviation) are smaller than the markers.