Distributed twofold ordering in ${\mathrm{URu}}_2{\mathrm{Si}}_2$

In previous ${}^{29}\mathrm{Si}$ NMR basal-plane field rotation measurements, a modest breaking of the fourfold symmetry was suggested in the hidden-order state of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. Here, the angular dependence of the ${}^{29}\mathrm{Si}$ NMR linewidth is analyzed in detail, indicating the occurrence of a distribution of twofold local fields at the Si sites even in a high-quality single crystal. The relation between sample quality and the occurrence of effects having twofold symmetry is discussed.

Figure 1

${}^{29}\mathrm{Si}$ NMR spectra for $\theta =0^{\circ }$ $\left(H\parallel \left[110\right]\right)$ in the paramagnetic (25 K) and ordered (7 K) states of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ (Ref. [9]). Solid blue lines are fitted Lorentzian curves. Resonance peak frequencies and linewidths are parameterized as $f_{\mathrm{res}}\left(0\right)$ and $L\left(0\right)$, respectively.

Figure 2

$\theta$ dependence of $L\left(\theta \right)-L_{\mathrm{para},\mathrm{cal}}\left(\theta \right)$ at 5 and 20 K at 5.19 T using the results presented in Ref. [9]. For the two curves the vertical axis is offset for clarity. Calculated values of $L_{\mathrm{para},\mathrm{cal}}\left(\theta \right)=0.5W_{\mathrm{para}}\cos \left(4\theta \right)$ with $W_{\mathrm{para}}=0.163$ kHz have been subtracted from linewidth data at 5 and 20 K to generate the results in this figure. The quantity $0.5W_{\mathrm{para}}\cos \left(4\theta \right)$ is the $\theta$-dependent part of $L_{\mathrm{para}}\left(\theta \right)$, which is estimated using a least-squares fit based on Eq. (4) to data from the paramagnetic state [9]. The solid line represents the $\theta$-dependent part that appears in the ordered state, i.e., $W_{\mathrm{nem}}\left|\cos \left(2\theta \right)\right|$ in Eq. (5), with $W_{\mathrm{nem}}=0.386$ kHz. The dashed line is a guide to the eye.

Figure 3

(a) $H$ dependence of $L_{\mathrm{nem}}\left(0\right),L_{\mathrm{para}}\left(0\right),L_{c1}$, and $L_{c2}$. $L_{c2}$ is independent of $H$ within experimental error (see the supplement for Ref. [9]). The estimated nuclear dipole-dipole contribution $L_{\mathrm{ndip}}$ is also shown. (b) $H$ dependence of $W_{\mathrm{nem}}$ and $W_{\mathrm{para}}$ (Ref. [9]). Solid lines are obtained via linear least-squares fits. The experimental errors of fitting coefficients are large for $W_{\mathrm{nem}}=(0.046\pm 0.006)H+(0.152\pm 0.093)$, compared with $W_{\mathrm{para}}=(0.026\pm 0.0017)H+(0.025\pm 0.0084)$, where the W's in kHz and H in Tesla.

Figure 4

(a) $T$ dependence of $L_{c2}$ in the HO state. $L_{c2}$ is independent of $H$ and $\theta$. (b) $T$ dependence of $W_{\mathrm{nem}}$ at 5.19 T (Ref. [9]).

Figure 5

Illustration of why a fixed value of $W_{\mathrm{nem}}$ is not a suitable interpretation, using experimental values for $W_{\mathrm{nem}},W_{\mathrm{para}},L_{c1}$, and $L_{c2}$ at $T=5$ K and $H=5.19$ T. (a) Calculated spectra $F_{\mathrm{super}}\left(\theta \right)$ [Eq. (20)] for $\theta =0$ and ${45}^{\circ }$ with $w=W_{\mathrm{nem}}$. While $F_{\mathrm{super}}\left(45\right)$ coincides with $F_{\exp }\left(45\right)$, the linewidth of $F_{\mathrm{super}}\left(0\right)$ is almost the same as that of $F_{\mathrm{super}}\left(45\right)$, which is smaller than that of $F_{\exp }\left(0\right)$. (b) In order to reproduce $F_{\exp }\left(0\right)$ using $F_{\mathrm{super}}\left(0\right),w$ can be optimized as $w=3.4W_{\mathrm{nem}}$. The result, however, is that the superimposed $F_{\mathrm{super}}\left(0\right)$ no longer has a Lorentzian shape. Note that $F_{\mathrm{super}}\left(45\right)$ is independent of $w$. For comparison the experimental result at $\theta =0,T=5$ K, and $H=5.19$ T is shown (open circles).