Cascade of magnetic-field-driven quantum phase transitions in ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$

Magnetically hidden order is a hypernym for electronic ordering phenomena that are visible to macroscopic thermodynamic probes but whose microscopic symmetry cannot be revealed with conventional neutron or x-ray diffraction. In a handful of $f$-electron systems, the ordering of odd-rank multipoles leads to order parameters with a vanishing neutron cross section. Among them, ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ is known for its unique phase diagram exhibiting two distinct multipolar-ordered ground states (phases II and ${\mathrm{II}}^{\prime }$), separated by a field-driven quantum phase transition associated with a putative change in the ordered quadrupolar moment from $O_2^0$ to $O_{xy}$. Using torque magnetometry at sub-kelvin temperatures, here we find another phase transition at higher fields above 12 T, which appears only for low-symmetry magnetic field directions $\mathbf{B}\parallel \langle 11L\rangle$ with $1<L\le 2$. While the order parameter of this new phase ${\mathrm{II}}^{\prime \prime }$ remains unknown, the discovery renders ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ a unique material with two field-driven phase transitions between distinct multipolar phases. They are both clearly manifested in the magnetic-field dependence of the field-induced (111) Bragg intensities measured with neutron scattering for $\mathbf{B}\parallel \left[11\overline{2}\right]$. We also find from inelastic neutron scattering that the number of nondegenerate collective excitations induced by the magnetic field correlates with the number of phases in the magnetic phase diagram for the same field direction. Furthermore, the magnetic excitation spectrum suggests that the new phase ${\mathrm{II}}^{\prime \prime }$ may have a different propagation vector, revealed by the minimum in the dispersion that may represent the Goldstone mode of this hidden-order phase.

Figure 1

(a) The crystal structure of ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ viewed along one of the cubic axes. The isosurfaces of magnetization density at the Ce2 (8c) position, predicted at the DFT/PBE/PAW level of theory, are shown in dark blue. The light-yellow spheres denote the Ce1 atoms. (b) The nearest-neighbor cluster ($T_d$ symmetry) around the magnetic Ce2 site, used to create a point-charge model in CASSCF modeling.

Figure 2

(a) Magnetic-field derivative of the torque, $\partial \tau /\partial B$, measured at $T=20$ mK with torque magnetometry at different field angles in the $\left(1\overline{1}0\right)$ plane. The magnetic-field angles shown beside the curves are given with respect to the [001] cubic axis. Because of space reasons, only curves in the sector between the [112] ($35.3{}^{\circ }$) and [111] ($54.7{}^{\circ }$) directions are shown, shifted vertically for clarity. The arrows mark anomalies corresponding to the transitions between phases III, II, ${\mathrm{II}}^{\prime }$, and ${\mathrm{II}}^{\prime \prime }$. The weak anomaly at the upper boundary of phase ${\mathrm{II}}^{\prime \prime }$ is marked by asterisks. (b), (c) Field-angle magnetic phase diagrams of ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ at $T=20$ mK in polar coordinates, reconstructed from torque magnetometry data for fields rotated in the $\left(1\overline{1}0\right)$ and (001) planes, respectively. The field derivative of the magnetic torque, $\partial \tau /\partial B$, composed of scans like those in panel (a), is plotted as a function of the magnetic field and its direction in the underlaid color map. Here red and blue colors correspond to the positive ($+$) and negative ($-$) signs of $\partial \tau /\partial B$, respectively. The stability regions for different phases are shaded, based on solid lines separating them that were fitted to the data points under constraints of the cubic crystal symmetry.

Figure 3

Field-induced magnetic Bragg intensity $I_{\left(111\right)}\left(B\right)$ at $\mathbf{Q}=\left(111\right)$, measured at $T=50$ mK as a function of the magnetic field applied along the $\left[11\overline{2}\right]$ direction (circles). The dashed lines show phase transitions between multipolar phases according to our magnetic torque data.

Figure 4

(a), (b) A selection of unprocessed INS data taken at $T=0.05$ K at $\mathbf{Q}=\left(111\right)$ and $\left(\overline{1}10\right)$, respectively, in magnetic fields of 7, 9.5, 12, and 13.5 T applied along $\left[11\overline{2}\right]$. (c) The data collected at the highest field of 14.5 T at different $\mathbf{Q}$ vectors from (111) and $\left(\overline{1}10\right)$ as indicated beside each data set. The data in panels (a)–(c) are offset vertically for clarity. (d), (e) Magnetic field dependence of the inelastic peak positions extracted from the fits shown in panels (a)–(c). The straight dashed lines are guides to the eyes, extrapolating the peak energies toward zero in smaller fields. Vertical dashed lines mark phase transitions for the $\left[11\overline{2}\right]$ field directions.

Figure 5

Dispersion of field-induced magnetic excitations in ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Si}}_6$ measured in a magnetic field of 14.5 T applied along the $\left[11\overline{2}\right]$ axis. The color map is composed of energy scans such as those in Fig. 4, measured along high-symmetry directions following the polygonal path $\left(0.40.40.4\right)$–(111)–$\left(\overline{1}10\right)$–$\left(\overline{0.5}0.50\right)$. The fitted peak position are shown as data points; the lines are an empirical fit to the data.

Figure 6

(a) Superimposed Zeeman diagrams (top) and magnetization curves (bottom), calculated in the two-ion model as a function of magnetic field $\mathbf{B}$ for different field directions uniformly distributed in the $\left(1\overline{1}0\right)$ plane. (b) Contour map of the calculated magnetization $M(B,\theta )$ (left) and a color map of its angular derivative $\partial M/\partial \theta$ (equivalent to the experimentally measured $\partial \tau /\partial B$), presented in polar coordinates vs magnetic-field strength up to 18 T and rotation angle $\theta$ in the $\left(1\overline{1}0\right)$ plane. (c) Field-polarized magnetic state at $T=50$ mK and $B=14.5$ T for $\mathbf{B}\parallel \left[11\overline{2}\right]$ with an indication of the $4f$ charge density isosurface. (d) Dispersion of the field-induced magnetic excitations at $B=14.5$ T, plotted along the high-symmetry polygonal path in $\mathbf{Q}$ space for $\mathbf{B}\parallel \left[11\overline{2}\right]\perp \mathbf{Q}$. The color shading represents the expected INS intensities, to be compared with the experimental data in Fig. 5.