Magnetoelastic Excitations in the Pyrochlore Spin Liquid ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathbf{O}}_7$

At low temperatures, ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ enters a spin liquid state, despite expectations of magnetic order and/or a structural distortion. Using neutron scattering, we have discovered that in this spin liquid state an excited crystal field level is coupled to a transverse acoustic phonon, forming a hybrid excitation. Magnetic and phononlike branches with identical dispersion relations can be identified, and the hybridization vanishes in the paramagnetic state. We suggest that ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is aptly named a “magnetoelastic spin liquid” and that the hybridization of the excitations suppresses both magnetic ordering and the structural distortion. The spin liquid phase of ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ can now be regarded as a Coulomb phase with propagating bosonic spin excitations.

Figure 1

Overview of the MEM. The sharp mode between the two intense crystal field excitations at 1.5 and 10 meV is the MEM (a). It is weak, but can be clearly identified in constant-energy cuts (b) (background levels offset by 1) and constant-$\mathbf{Q}$ cuts (c) [cut positions are indicated in (a) by dashed lines]. Constant-energy maps show the MEM at (2, 2, 0) (d) and a similar mode at (1, 1, 1) (e). At these small wave vectors, the modes have their intensity parallel to the scattering vector (the arrow and ellipsoids show the scattering vector and highlight the intensity distribution, respectively). Excitations with similar dispersions are visible at large wave vectors [e.g., (0, 0, 8)] (f), but with a transverse intensity distribution.

Figure 2

Interaction of the MEM with the first crystal field excitation. At (2, 2, 0), the MEM intersects with the first CFE (a) [this is the same view as Fig. 1, but with better energy resolution]. The MEM can be seen above the CFE. The MEM is faintly visible in the perpendicular direction (b), where some intensity concentrated around $(h,h,0)$ is integrated in the cut. No sharp feature can be found extending below the CFE, as can also be seen in an intensity map at $E=0.75\mathrm{meV}$ [dashed line in (b)], i.e., in the gap (c). Constant-$\mathbf{Q}$ cuts (d) [at positions indicated by dashed lines in (a)] show a two-component line shape for the CFE far away from (2, 2, 0) and a single, broad, asymmetric, peak where the MEM meets the crystal field level.

Figure 3

Characterization of the excitations in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ at 0.05 K. Polarization analysis (a) shows that the scattering observed in the MEM at (2, 2, 0) is magnetic. The integrated intensities (b) show that the modes at large wave vector have intensities which decrease with $\hslash \omega$, similar to phonons, while those at (2, 2, 0) and (1, 1, 1) have a different character. Fitted peak positions from constant-energy cuts (squares) and constant-$\mathbf{Q}$ cuts (triangles) show that the dispersion of the magnetic mode is exactly the same as the phononlike mode (c). $A_2$ is the second CFE, $E$ and $E^{*}$ are the two components of the first CFE, where they can be distinguished. This level has been fitted with asymmetric Gaussian functions, and the bar indicates the half maximum height of the asymmetric line shapes. Elsewhere, the two bars indicate the error of the fitted peak position and the width of the integral used for the cut. Integrated intensities were extracted from energy slices such as Fig. 1 by summing all of the intensity above background within a ring fixed by the dispersion.

Figure 4

Temperature dependence of the MEM. The MEM intensity collapses above 10 K (a). As the peak intensity falls above 10 K (b), a rising intensity which follows the magnetic form factor of ${\mathrm{Tb}}^{3+}$ (a) (100 K) also appears. This contribution originates from the thermal broadening of the nearby CFEs (it follows the Bose factor). The dashed line in (b) is $n_0-n_1$ (scaled), where $n_0$ and $n_1$ are the thermal population factors of the ground and first excited states, respectively, of a two-level system with $\mathrm{\Delta }=1.4\mathrm{meV}$. TOF peak areas scaled to TAS peak amplitudes at 5 K.