Origin of the 30 T transition in ${\mathrm{CeRhIn}}_5$ in tilted magnetic fields

We present a comprehensive ultrasound study of the prototypical heavy-fermion material ${\mathrm{CeRhIn}}_5$, examining the origin of the enigmatic 30 T transition. For a field applied at $2^{\circ }$ from the $c$ axis, we observed two sharp anomalies in the sound velocity, at $B_m\approx 20\mathrm{T}$ and $B^{*}\approx 30\mathrm{T}$, in all the symmetry-breaking ultrasound modes at low temperatures. The lower-field anomaly corresponds to the well-known first-order metamagnetic incommensurate-to-commensurate transition. The higher-field anomaly takes place at 30 T, where an electronic-nematic transition was previously suggested to occur. Both anomalies, observed only within the antiferromagnetic state, are of similar shape, but the corresponding changes of the ultrasound velocity have opposite signs. Based on our experimental results, we suggest that a field-induced magnetic transition from a commensurate to another incommensurate antiferromagnetic state occurs at $B^{*}$. With further increasing the field angle from the $c$ axis, the anomaly at $B^{*}$ slowly shifts to higher fields, broadens, and becomes smaller in magnitude. Traced up to ${30}^{\circ }$ from the $c$ axis, it is no longer observed at ${40}^{\circ }$ below 36 T.

Figure 1

(a) Crystal structure of ${\mathrm{CeRhIn}}_5$. Magnetic structure of ${\mathrm{CeRhIn}}_5$ (b) in zero magnetic field (IC) and (c) in a magnetic field higher than 2 T applied along the [100] direction (C). Arrows indicate the orientation of the magnetic moments. Only Ce atoms are shown in (b) and (c) for clarity. (d) Schematic illustration of the symmetry strains, ${\varepsilon }_{ij}$, induced by different ultrasound modes for a tetragonal crystal structure. For each mode, the propagation ($k$) and polarization ($u$) directions are shown by arrows. The associated irreducible representations are shown in brackets.

Figure 2

Field dependence of the relative ultrasound-velocity variation, $\mathrm{\Delta }v/v$, for the $C_{11}$ (a), $C_T=(C_{11}-C_{12})/2$ (b), $C_{33}$ (c), $C_{44}$ (d), and $C_{66}$ (e) modes at $T\simeq 1.4\mathrm{K}$. The ultrasound frequencies for all modes are given in Table 1. The field was applied at $2^{\circ }$ from the $c$ axis for all modes, except for the $C_{33}$ mode, where the angle was $4^{\circ }$. Arrows indicate the anomalies at $B_m, B^{*}$, and $B_p$ discussed in the text. For all the modes, the ultrasound propagation $k$ and polarization $u$ directions are indicated.

Figure 3

Field dependence of the relative ultrasound-velocity variation, $\mathrm{\Delta }v/v$, for the $(C_{11}+C_{12})/2$ mode at $T\simeq 1.4$ K, calculated from the $C_{11}$ and $C_T$ data shown in Figs. 2 and 2, respectively, as explained in the text. The field was applied at $2^{\circ }$ from the $c$ axis. Dashed lines indicate $B_m$ and $B^{*}$ observed in the symmetry-breaking modes.

Figure 4

Relative variation of the ultrasound velocity as a function of magnetic field at different temperatures for the (a) $C_{11}$, (b) $C_T=(C_{11}-C_{12})/2$, (c) $C_{44}$, and (c) $C_{66}$ acoustic modes. The magnetic field was applied at $\theta \simeq 2^{\circ }$. Curves are vertically shifted in (a), (c), and (d) for clarity.

Figure 5

$B-T$ phase diagram of ${\mathrm{CeRhIn}}_5$ obtained from the ultrasound-velocity anomalies observed in all modes. Triangles correspond to the AFM to paramagnetic (PM) transition from Ref. [23]. AFM1 and, presumably, AFM4 are IC phases with different propagation vectors. AFM3 is the C phase discussed in the text. PPM stands for the polarized paramagnetic phase. The solid line is a fit of $T_N\left(B\right)=T_N\left(0\right)[1-{(B/B_c)}^2]$ to the Néel temperature, with $B_c\simeq 52$ T. Dashed lines are guides for the eye.

Figure 6

Magnetic torque divided by field, $\tau /B$, vs $B$ applied at $\theta =2^{\circ }$ from [001] towards the [110] direction at $T=30$ mK.

Figure 7

(a) $\mathrm{\Delta }v/v$ for the $C_T$ mode as a function of $B$ applied at different angles, $\theta$, from the $c$ axis at $T=1.25\mathrm{K}$. Curves are vertically shifted for clarity. (b) and (c) show the curves at the two smallest and largest angles of our measurements, respectively. (d) The angle dependence of the transition fields $B_m$ and $B^{*}$ determined from the minima (maxima) of the first derivative, $d(\mathrm{\Delta }v/v)/dB$, an example of which is shown in the inset. The solid line is a $1/cos({90}^{\circ }-\theta )$ fit to the data above ${10}^{\circ }$. The dashed line is a guide for the eye.

Figure 8

Fast Fourier transform (FFT) spectra of the magnetoacoustic quantum oscillations in the $C_T$ mode (shown in the inset) both below and above $B^{*}$. A nonoscillating background was subtracted prior to performing the FFTs. The FFTs were performed over equidistant $1/B$ ranges indicated by rectangles in the inset.

Figure 9

(a) Field dependence of the relative ultrasound-velocity variation for the modes $C_{33}$ and $C_T$ with the field applied at $\theta =4^{\circ }$ and $2^{\circ }$, respectively. (b) FFT spectra of the magnetoacoustic quantum oscillations in the $C_{33}$ mode from (a), both below and above $B^{*}$. A nonoscillating background was subtracted prior to performing the FFTs. The FFTs were performed over equidistant $1/B$ ranges indicated by rectangles in (a).

Figure 10

FFT spectra of the static-field magnetoacoustic quantum oscillations, obtained by subtracting a nonoscillating background from the $\mathrm{\Delta }v/v$ vs $B$ curve in the $C_{33}$ mode shown in the inset, with $B$ tilted by $4.5^{\circ }$ from the $c$ axis. The FFT spectra are obtained over the same $1/B$ range. For the bottom curve, the range is from $B_{\text{min}}$ = 10 T to $B_{\text{max}}$ = 10.71 T ($B_{\text{avg}}$ = 10.34 T). For each successive curve, $B_{\text{min}}$ is increased by 1 T up to 20 T, and by 0.5 T from there on. Curves are vertically shifted for clarity.