Figure 1

Static magnetic properties of Tb${}_2$Ti${}_2$O${}_7$. (a) Temperature dependence of the $\mu$SR frequency shift $\Delta \nu /{\nu }_{\mathrm{ext}}$ for two fields applied along the [110] direction. For the measurements at $T⩽0.9$ K, two experimental procedures were used: either a zero-field cooling (zfc) or a field-cooling (fc) protocol starting from 3 K. The inset shows that a branching occurs at $T<T_{\mathrm{t}}\approx 0.15$ K between the zfc and fc data taken at 60 mT. A kink is observed in the zfc data at $T_{\mathrm{t}}$. At 0.3 K and below $\Delta \nu /{\nu }_{\mathrm{ext}}$ is field dependent. The dashed lines are guides to the eyes. (b) The temperature dependence of the specific heat $C_{\mathrm{p}}$. The sum of the nuclear and phonon contributions to the specific heat is shown as a dashed line. Details about their evaluation are presented in Appendix . The inset displays the low-temperature electronic specific heat $C_{\mathrm{elec}}$ obtained from the difference between the total specific heat and the nuclear and phonon contributions. An upturn is clearly observed below $T_{\mathrm{t}}$.Reuse & Permissions

Figure 2

Spin dynamics in Tb${}_2$Ti${}_2$O${}_7$. (a) Temperature dependence of the spin-lattice relaxation rate ${\lambda }_Z$ and exponent $\beta$ characterizing the $\mu$SR relaxation function measured with a longitudinal field of 13 mT. In panel (b) we display the half-width at half-maximum of the quasielastic neutron scattering linewidth measured for the wave vector (0,0,2.1) in reduced units. We complement our data with published results from Yasui et al.[33] The dashed lines are guides to the eyes. Both muon and neutron data suggest a crossover to a correlated paramagnetic regime for $T<T_{\mathrm{cp}}\simeq 2$ K. Experimental details about these measurements can be found in Appendixes and .Reuse & Permissions

Figure 3

Temperature dependence of the entropy of electronic origin $S_{\mathrm{elec}}$ per mole of terbium in Tb${}_2$Ti${}_2$O${}_7$. The entropy variation has been computed from $C_{\mathrm{elec}}\left(T\right)$ and has been shifted to match the prediction of the model described below in the temperature range 15–20 K. The prediction shown as a dashed line is for a crystal-field scheme with energy levels lying at 0, 2, 19.5, 21.9, 123, and 171 K in units of temperature. The value of the first excited level has been determined for the entropy to be consistent to the experimental data. The splitting between the levels at 19.5 and 21.9 K was computed to be consistent with the splitting between the ground and first excited states. The last two values for the energy levels (123 and 171 K) result from the predictions for recently determined crystal-field parameters.[15, 16] Note that all these energy levels match very well the transitions which have been observed by inelastic neutron scattering.[16] The subsequent levels are above 400 K and do not affect the entropy in the displayed temperature range. The inset shows the low-temperature part of the experimental data.Reuse & Permissions

Figure 4

Schematic view of the ${}^{159}$Tb nuclear levels. In the presence of the Zeeman interaction only, the four levels are equidistant. When the quadrupole interaction is included, the two levels in the middle are shifted upward. For the sake of clarity the shift is enlarged in the figure. The actual normalized values for the four levels are 0, 1/3, 2/3, 1 and 0, 0.346, 0.679, 1 for the pure Zeeman case and when the quadrupole interaction is added, respectively.Reuse & Permissions

Figure 5

Muon-spin relaxation spectrum measured at 0.7 K on a crystal of Tb${}_2$Ti${}_2$O${}_7$ in a longitudinal field of 13 mT. This field is collinear to the incoming muon polarization and to the [111] crystallographic direction. The full line results from a fit of the spectrum according to the function given in the main text. The lower part of the figure displays the difference between the data and the best fits for two different values of the exponent $\beta$. For the sake of clarity a difference is magnified by a factor of 3.Reuse & Permissions

Figure 6

An example of a quasielastic neutron scattering spectrum recorded at 1.6 K for crystals of Tb${}_2$Ti${}_2$O${}_7$. The value of the wave vector is given in reduced units. The full line results from the fit. We explicitly display the different components to the spectrum: the crystal-field contribution with the dotted line, the incoherent and background scattering with the dashed line, and the quasielastic scattering with the dashed-dotted line.Reuse & Permissions