Anisotropic nematic fluctuations above the ferroquadrupolar transition in ${\mathrm{TmVO}}_4$

Ferroquadrupole order of local atomic orbitals provides a specific realization of electronic nematic order. ${\mathrm{TmVO}}_4$ is an insulator and undergoes ferroquadrupolar order associated with the local Tm $4f$ orbitals at $T_Q=2.15$ K. The material is a model system to study nematic order and the roles played by nematic fluctuations. Here we present ${}^{51}\mathrm{V}$ nuclear magnetic resonance data as a function of field orientation in a single crystal. Although the spectra are well understood in terms of direct dipolar hyperfine couplings, the spin-lattice relaxation rate exhibits strong anisotropy that cannot be understood in terms of magnetic fluctuations. We find that the spin-lattice relaxation rate scales with the shear elastic constant associated with the ferroquadrupole phase transition, suggesting that quadrupole (nematic) fluctuations dominate the spin-lattice relaxation for in-plane fields.

Figure 1

(a) ${}^{51}\mathrm{V}$ spectra in ${\mathrm{TmVO}}_4$ at 11.7294 T for ${\mathbf{H}}_0\perp c$ at several different temperatures. The (b) full-width half maximum and (c) quadrupolar splitting versus temperature for several different angles $\theta$. The inset shows the unit-cell structure. Tm is blue, V is green (within the pyramids), and O is red.

Figure 2

Magnetic shift $K$ versus (a) temperature and (b) angle. The dotted lines are fits as described in the text.

Figure 3

$K_{aa}$ and $K_{cc}$ versus (a) temperature and (b) bulk susceptibility. The open points correspond to values reported in Ref. [14]. The solid lines are fits as described in the text. The inset displays the bulk susceptibility versus temperature.

Figure 4

Magnetization recovery curves at 20, 40, and 80 K for $\theta ={88}^{\circ }$. Solid lines are fits as described in the text. The data have been normalized and offset vertically for clarity.

Figure 5

(a) $T_1^{-1}$ vs temperature for multiple angles. (b) Calculated $T_1^{-1}$ versus $\theta$ for magnetic fluctuations (solid lines) and for quadrupole fluctuations (dashed lines). (c) Calculated $T_1^{-1}$ versus temperature and angle using Eq. (5), where the angle is increased in $5^{\circ }$ increments between $0^{\circ }$ and ${90}^{\circ }$.

Figure 6

The shear elastic stiffness coefficient $c_{66}$ (solid line, reproduced from Ref. [26]) and the quantity $1/[1+{\left(a{T}_{1}T\right)}^{-1}]$ as as a function of temperature. Inset shows $1/(c_{66,0}/c_{66}-1)$ versus $T_{1}T$, with temperature implicit. The solid black line is the best linear fit, giving $a=18.5\pm 0.4{\mathrm{s}}^{-1}{\mathrm{K}}^{-1}$.