Evolution of the propagation vector of antiferroquadrupolar phases in ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ under magnetic field

Hidden-order phases that occur in a number of correlated $f$-electron systems are among the most elusive states of electronic matter. Their investigations are hindered by the insensitivity of standard physical probes, such as neutron diffraction, to the order parameter that is usually associated with higher-order multipoles of the $f$ orbitals. The heavy-fermion compound ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ exhibits magnetically hidden order at subkelvin temperatures, known as phase II. Additionally, for magnetic field applied along the [001] cubic axis, another phase ${\mathrm{II}}^{\prime }$ was detected, but the nature of the transition from phase II to phase ${\mathrm{II}}^{\prime }$ remained unclear. Here we use inelastic neutron scattering to argue that this transition is most likely associated with a change in the propagation vector of the antiferroquadrupolar order from (111) to (100). Despite the absence of magnetic Bragg scattering in phase ${\mathrm{II}}^{\prime }$, its ordering vector is revealed by the location of an intense magnetic soft mode at the (100) wave vector, that is orthogonal to the applied field. At the II-${\mathrm{II}}^{\prime }$ transition, this mode softens and transforms into quasielastic and nearly $\mathbf{Q}$-independent incoherent scattering, which is likely related to the non-Fermi-liquid behavior recently observed at this transition. Our experiment also reveals sharp collective excitations in the field-polarized paramagnetic phase, after phase ${\mathrm{II}}^{\prime }$ is suppressed in fields above 4 T.

Figure 1

Schematic graphical representation of the experimental setups listed in Table 1. The cube represents the Brillouin zone, the scattering plane is marked with light blue. Field direction is shown with an arrow. Large/small red dots show the location of the allowed/suppressed (111) magnetic peaks. The (100) and (010) wave vectors, where soft modes develop in phase ${\mathrm{II}}^{\prime }$, are marked correspondingly with white dots. Green and blue dots mark (001) and (110) wave vectors that are discussed in the text.

Figure 2

(a) Magnetic-field dependence of the INS signal measured in the vicinity of the (111) wave vector in various fields applied along the $\left[1\overline{1}0\right]$ axis. The inset shows a change in the fitted peak position as a function of field. (b) Momentum dependence of the INS signal at a constant field of 4.25 T. The peak positions resulting from the fits (solid lines) are listed in the legend.

Figure 3

Magnetic-field dependence of the diffuse INS peak at the (110) wave vector, which represents the tail of the two broad (111) and $\left(11\overline{1}\right)$ peaks located above and below the $\left(HK0\right)$ scattering plane. The data are symmetrized with respect to the mirror plane of the cubic Brillouin zone. Solid lines are Gaussian fits, showing the suppression of intensity towards the boundary of phases II and ${\mathrm{II}}^{\prime }$. A schematic field-temperature phase diagram with the positions of the measured data points is shown to the right.

Figure 4

Constant-energy maps, measured at different magnetic field values using setup 3. Each panel was obtained after integration of the TOF data within the energy range from 0.08 to 0.25 meV. In orthogonal momentum directions with respect to each plane, integration was done within $\pm 0.1$ r.l.u. The initial data were symmetrized about the natural mirror planes of the reciprocal space $\left(H0L\right)$, $\left(0KL\right)$, $\left(HK0\right)$, and $\left(HHL\right)$. Then, in order to plot full $\left(HK0\right)$ scattering plane, the available data were mirrored with respect to the $\left(HHL\right)$ plane. The cuts along the $(1+H1-H0)$ direction shown in Fig. 3 were obtained by integrating these data along the diagonal.

Figure 5

(a)–(d) Energy-momentum cuts through the high-symmetry directions in the $\left(HK0\right)$ scattering plane, measured at different magnetic fields as indicated in (g) with crossed circles: (a) within the AFQ phase II, (b) within the AFQ phase ${\mathrm{II}}^{\prime }$ near the phase boundary, (c),(d) in the field-polarized paramagnetic phase I. (e),(f) Constant-energy cuts through the 1.7 and 3.8 T datasets, respectively, showing the appearance of intensity maxima at ${\mathbf{Q}}_{{\mathrm{II}}^{\prime }}=\left(100\right)$ in phase ${\mathrm{II}}^{\prime }$. The corresponding integration windows in energy, given above the panels, are marked with ‘a’ on the vertical axis in panels (a) and (b), respectively. Additional constant-energy cuts from the same data are also shown in Fig. 6. (g) A schematic field-temperature phase diagram, showing the field and temperature values of the presented datasets.

Figure 6

Constant-energy maps, measured at different magnetic field values using setup 4. Each panel was obtained by integrating the TOF data within the energy range as indicated in every panel. The corresponding integration windows of each row are marked with ‘a’, ‘b’, and ‘c’ on the vertical axis of the energy-momentum cuts in Figs. 5–5. In orthogonal momentum directions with respect to each plane integration was done within $\pm 0.08$ r.l.u. The initial data were symmetrized about the natural mirror planes of the reciprocal space $\left(H0L\right)$, $\left(0KL\right)$, $\left(HK0\right)$, and $\left(HHL\right)$, therefore in order to plot full $\left(HK0\right)$ scattering plane the available data were mirrored with respect to the $\left(HHL\right)$ plane.

Figure 7

The difference of elastic scattering intensity (integrated within $\pm 0.05\mathrm{meV}$), obtained by subtracting the CNCS data sets measured at (a) 3.8 and 1.7 T and (b) 5.5 and 1.7 T. The absence of elastic scattering intensity at the (100) and (010) wave vectors in the first data set confirms the absence of field-induced magnetic Bragg peaks within phase ${\mathrm{II}}^{\prime }$. The presence of positive magnetic intensity around (200) and (020) structural reflections in the second data set indicates the presence of ferromagnetic correlations in the field-polarized paramagnetic phase I.

Figure 8

(a) Constant-$\mathbf{Q}$ cuts through the TOF data at $B=3.8\mathrm{T}$ (setup 4), taken at the (100) and (110) points. The (100) dataset is shifted upwards by 1 unit for clarity. (b)–(d) The low-temperature TAS spectra at $B=3.8\mathrm{T}$ (setup 2), reference background spectra ($T=70\mathrm{K}$, $B=0$), and their corresponding subtractions at three different wave vectors: (330), (003), and near (331), respectively. The subtracted data in panel (b) are shifted down by 30 units for clarity. Arrows mark fitted peak positions. Black horizontal bars indicate energy resolution defined as the full width at half maximum of the elastic line.