Spin liquid correlations, anisotropic exchange, and symmetry breaking in Tb${}_2$Ti${}_2$O${}_7$

We have studied the low-energy spin dynamics between 4.6 and 0.07 K in a Tb${}_2$Ti${}_2$O${}_7$ single-crystal sample by means of inelastic neutron scattering experiments. The spectra consist in a dual response, with a static and an inelastic contribution, showing striking $Q$ dependencies. We propose an interpretation involving an anisotropic exchange interaction in combination with a breaking of the threefold symmetry at the rare earth site. Simulations of the $Q$-dependent scattering in the random phase approximation account well for the inelastic response.

Figure 1

(a) Neutron intensity (normalized to monitor) as a function of energy transfer at $T=0.07$ K for $Q=\left(002\right)$. The data were recorded with a final wave vector $k_f=1.2$ Å${}^{-1}$, yielding an energy resolution (FWHM) ${\Delta }_0=0.07$ meV. The lines show the elastic and inelastic contributions; the red one shows the total fitted intensity in the case of a damped harmonic oscillator (DHO) for the inelastic contribution (see text). Inset: thermal evolution of the low-energy scattering. (b and c) Elastic magnetic scattering maps taken respectively at 0.07 and 4.6 K. A $Q$-independent background has been subtracted from the raw data in order to remove the incoherent scattering. It was estimated from the temperature dependence of the total elastic scattering. The color scale is identical for the two temperatures. The red box corresponds to the points actually measured; the remaining has been deduced by symmetry. The pinch point at (111) is marked with an arrow.

Figure 2

Inelastic scattering maps at 0.07 K for an energy transfer: (a) 0.08, (b) 0.11, (c) 0.15, and (d) 0.20 meV. These energies are indicated by arrows in Fig. 1a. The pinch point at (111) is also marked with an arrow.

Figure 3

Inelastic scattering maps in Tb${}_2$Ti${}_2$O${}_7$. On the left, for $T=4.6$ K and energy transfer of 0.15 meV: (a) experimental data; (b, c) simulations, using the anisotropic exchange discussed in the text, respectively with and without a tetragonal distortion $D_Q=0.25$ K. The simulation in (c) has been multiplied by 7000 to obtain the same intensity as in (b). On the right, for $T=0.07$ K and energy transfer of 0.08 meV: (d) experimental data; (e, f) simulations in the presence of a tetragonal distortion $D_Q=0.25$ K, with respectively an anisotropic and an isotropic exchange tensor. From these latter maps, it is clear that the $Q$ dependence of the scattering strongly depends on the anisotropy and that it is not reproduced with isotropic exchange.

Figure 4

Top: experimental data from Ref. 36: (a) SF and (b) NSF. Bottom: simulated diffuse (c) SF and (d) NSF scattering maps at 0.07 K in Tb${}_2$Ti${}_2$O${}_7$.