Field-Angle-Resolved Magnetic Excitations as a Probe of Hidden-Order Symmetry in ${\mathrm{CeB}}_6$

In contrast to magnetic order formed by electrons’ dipolar moments, ordering phenomena associated with higher-order multipoles (quadrupoles, octupoles, etc.) are more difficult to characterize because of the limited choice of experimental probes that can distinguish different multipolar moments. The heavy-fermion compound ${\mathrm{CeB}}_6$ and its La-diluted alloys are among the best-studied realizations of the long-range-ordered multipolar phases, often referred to as “hidden order.” Previously, the hidden order in phase II was identified as primary antiferroquadrupolar and field-induced octupolar order. Here, we present a combined experimental and theoretical investigation of collective excitations in phase II of ${\mathrm{CeB}}_6$. Inelastic neutron scattering (INS) in fields up to 16.5 T reveals a new high-energy mode above 14 T in addition to the low-energy magnetic excitations. The experimental dependence of their energy on the magnitude and angle of the applied magnetic field is compared to the results of a multipolar interaction model. The magnetic excitation spectrum in a rotating field is calculated within a localized approach using the pseudospin representation for the ${\mathrm{\Gamma }}_8$ states. We show that the rotating-field technique at fixed momentum can complement conventional INS measurements of the dispersion at a constant field and holds great promise for identifying the symmetry of multipolar order parameters and the details of intermultipolar interactions that stabilize hidden-order phases.

Popular Summary

Magnetic ordering in materials arising from electron dipolar moments is relatively straightforward to characterize. However, that is not the case for more exotic magnetic architectures arising from higher-order multipoles, such as quadrupoles and octupoles. In this case, typical probes of magnetism that rely on scattering of x rays or neutrons are blind to common signatures of these fields. Here, we work around that hurdle by testing a new approach to the analysis of neutron-scattering data that consists of looking at the field-angular dependence of the measured spectrum rather than at the more typical dispersion in momentum space for a constant field.

To demonstrate the power of this method, we investigate the compound ${\mathrm{CeB}}_6$, a well-studied material known to host long-range order of magnetic multipoles. We find that not only the energy and intensity but also the number of multipolar excitations depend on the direction of the magnetic field. To explain these effects, we also develop an effective theory of multipolar excitations.

Our approach opens up a new way of looking at correlated-electron systems with multipolar order parameters that should be applicable to a wide range of compounds.

Figure 1

Unprocessed INS spectra measured near the zone center ${\mathrm{\Gamma }}^{\prime \prime }(110)$ for $\mathbf{B}\parallel [1\overline{1}0]$ using setup 1.

Figure 2

Unprocessed INS spectra measured near the zone center ${\mathrm{\Gamma }}^{\prime \prime }(110)$ for $\mathbf{B}\parallel [1\overline{1}0]$ using setups 3 (left) and 2 (right). The data are offset vertically for clarity. They illustrate the appearance of resonance $C$ (smaller peak) at a higher energy with respect to resonance $A$ (stronger peak) at magnetic fields above 14 T. Both peak positions shift upward in energy with an increasing field. Solid lines are Lorentzian fits.

Figure 3

Summary of the neutron spectra, plotted as a function of the magnetic field: (a)–(d) at the ${\mathrm{\Gamma }}^{\prime \prime }(110)$ point for different magnetic field directions; (e),(f) at the $R(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ and $M(\frac{1}{2}\frac{1}{2}0)$ points for $\mathbf{B}\parallel [1\overline{1}0]$. Darker blue corresponds to higher INS intensity. Data points mark fitted peak positions. The dashed line in (a) marks the expected position of resonance $B$ seen in ESR data (see the text). The resonances $A$ and $C$ observed in INS experiments are also indicated. We do not extend this notation to other field directions, because the correspondence between these resonances and the three resonances ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ for $\mathbf{B}\parallel [001]$ in (d) is not yet clarified. Dotted lines in (c), (e), (f) are guides to the eyes.

Figure 4

Unprocessed INS data, measured at the $R(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ and $M(\frac{1}{2}\frac{1}{2}0)$ points in different magnetic fields using setups 1 and 3. The curves are shifted vertically for clarity by an amount indicated on the vertical axis with horizontal bars of corresponding colors. Solid lines are fits with Lorentzian peak functions on top of a background. The data measured with setup 1 are additionally multiplied by 3 to obtain similar peak intensity in all curves.

Figure 5

INS intensity vs energy and momentum along straight segments connecting the $R(\frac{1}{2}\frac{1}{2}\frac{1}{2})$, ${\mathrm{\Gamma }}^{\prime \prime }(110)$, and $M(\frac{1}{2}\frac{1}{2}0)$ points: (a) in a magnetic field of 14.5 T, applied along the $[1\overline{1}0]$ direction using setups 1 and 3; (b) in a magnetic field of 10 T, applied along the [001] direction using setup 4. The color map shows raw data after background subtraction and subsequent smoothing with a one-dimensional Gaussian filter characterized by a FWHM of 0.1 meV (to reduce statistical noise). Resonances $A$ and $C$ at the $\mathrm{\Gamma }$ point are marked as such. Markers show the fitted peak positions.

Figure 6

Comparison of the raw INS spectra at the ${\mathrm{\Gamma }}^{\prime \prime }(110)$ point (or very close to it, to avoid Bragg contamination), measured in magnetic fields applied along different crystallographic directions. The reciprocal-space vector, measurement temperature, magnetic field, and its direction are indicated in the legends. Note that the data were measured using different spectrometers in different configurations; therefore, the absolute intensity is not directly comparable. The curves are shifted vertically for clarity by an amount indicated on the vertical axis with horizontal bars of corresponding colors. Solid lines are empirical fits that serve as guides to the eyes.

Figure 7

Comparison of the INS data collected in the vicinity of the $\mathrm{\Gamma }$ point at the base temperature and at an elevated temperature, as indicated in the legend. The panels (a) and (b) have been measured at 6 and 10 T using setups 6 and 4, respectively. The bottom curve in (b), which is shifted down by $-1.5$ units for clarity, shows an equivalent spectrum measured near the ${\mathrm{\Gamma }}^{\prime }(100)$ point to confirm the reproducibility of all three peaks. Note a small difference in $\mathbf{Q}$ between the two top spectra in the bottom panel, which explains a difference in the nonmagnetic background, while the position of all three peaks remains unchanged.

Figure 8

The phase diagram of ${\mathrm{CeB}}_6$ for the magnetic field parallel to [001], reproduced from Ref. [71]. The red rectangle shows the part of the phase diagram covered by ESR measurements of the temperature-dependent $g$ factor by Semeno et al. [68, 69]. Blue squares mark the positions of the 60 GHz resonance at different temperatures.

Figure 9

The calculated dipolar structure function of magnetic excitations along the continuous polygonal path in momentum space: $R(\frac{1}{2}\frac{1}{2}\frac{1}{2})-{\mathrm{\Gamma }}^{\prime \prime }(110)-M(\frac{1}{2}\frac{1}{2}0)-X(\frac{1}{2}00)-{\mathrm{\Gamma }}^{\prime }(100)$. (a) For zero magnetic field; (b) for $h^{\prime }=1.45$ ($B\approx 10\mathrm{T}$) applied along the [001] direction; (c),(d) for fields $h^{\prime }=1$ ($B\approx 7\mathrm{T}$) and $h^{\prime }=2$ ($B\approx 14\mathrm{T}$) applied along the $[1\overline{1}0]$ axis. The left part (segment $R-{\mathrm{\Gamma }}^{\prime \prime }-M$) of (d) and the middle part (segment ${\mathrm{\Gamma }}^{\prime \prime }-M$) of (b) are, therefore, approximately equivalent to the experimental Figs. 5 and 5, respectively.

Figure 10

Calculated magnetic-field dependence of multipolar excitations and their intensities as expected in an INS measurement (dipolar structure function). (a)–(d) At the ${\mathrm{\Gamma }}^{\prime \prime }(110)$ point under different field directions: $[1\overline{1}0]$, $[1\overline{1}1]$, $[1\overline{1}2]$, and [001], respectively. (e),(f) At the $R(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ and $M(\frac{1}{2}\frac{1}{2}0)$ points for $\mathbf{B}\parallel [1\overline{1}0]$. These configurations are identical to those in Fig. 3.

Figure 11

Sketch of the quadrupole (left) and octupole (right) order parameter in a rotating field that points along the unit vector $(\alpha \beta \gamma )$ obtained from the real-space equivalent tesseral harmonics of the Stevens operators in Table 2. The distance from the center to the lobes is proportional to the charge or spin density of the even ${\mathrm{\Gamma }}_5^{+}$ quadrupole or odd ${\mathrm{\Gamma }}_2^{-}$ octupole. The total charge or spin moment is zero. Therefore, equal regions with $\pm$ charge or (N/S) spin densities (different colors for each) appear. When the field unit vector $(\alpha \beta \gamma )$ rotates, the quadrupole charge density rotates with it such that it remains in the plane perpendicular to $\mathbf{H}$, which is a consequence of the threefold degeneracy of ${\mathrm{\Gamma }}_5^{+}$. In contrast, the nondegenerate ${\mathrm{\Gamma }}_2^{-}$ octupole spin density remains fixed with respect to the crystal axes when $\mathbf{H}$ rotates. As a function of field strength $H$, the primary quadrupole order parameter is little affected, while the induced octupole shrinks to zero for $H\to 0$.

Figure 12

Field-angular anisotropies of multipolar excitation branches at the high-symmetry points of the cubic BZ, as indicated beside every panel. We use a polar representation, where the angle gives the field direction and the radius represents the mode frequency plotted from zero (center) to $7.5T_0$ (outer circular boundary). For all plots, the magnetic field strength is fixed to $h^{\prime }=2$ ($B\approx 14\mathrm{T}$). Note that in this figure the field rotation is not in a single plane for the whole range of polar angles: The left and right halves of every plot correspond to the field rotation in the planes orthogonal to [100] and [110], respectively.

Figure 13

Multipolar excitation branches for the ${\mathrm{\Gamma }}^{\prime \prime }(110)$ point in polar representation, where the radial coordinate corresponds to the excitation energy, plotted from 0 to $7.5T_0$, and the angular coordinate to the magnetic field direction, continuously rotating from [001] to $[1\overline{1}0]$ in the plane orthogonal to [110]. The four segments of the figure show results for 4, 7, 10, and 14.5 T. The color map shows the results of the calculation, with the experimental data points along high-symmetry field directions overlaid for comparison. The symbol shapes for different modes are consistent with those in Fig. 3. Filled symbols correspond to direct measurements of the peak position, whereas open symbols are the results of interpolation from the field dependences plotted in Fig. 3.