Regular-triangle trimer and charge order preserving the Anderson condition in the pyrochlore structure of CsW2O6

Since the discovery of the Verwey transition in magnetite, transition metal compounds with pyrochlore structures have been intensively studied as a platform for realizing remarkable electronic phase transitions. We report on a phase transition that preserves the cubic symmetry of the β-pyrochlore oxide CsW2O6, where each of W 5d electrons are confined in regular-triangle W3 trimers. This trimer formation represents the self-organization of 5d electrons, which can be resolved into a charge order satisfying the Anderson condition in a nontrivial way, orbital order caused by the distortion of WO6 octahedra, and the formation of a spin-singlet pair in a regular-triangle trimer. An electronic instability due to the unusual three-dimensional nesting of Fermi surfaces and the strong correlations of the 5d electrons characteristic of the pyrochlore oxides are both likely to play important roles in this charge-orbital-spin coupled phenomenon.


Supplementary Note 1. Structural analyses on the single crystal and powder XRD data
The experimental conditions during the single-crystal XRD measurements, and the crystallographic parameters obtained by the structural analyses of Phase I (250 K) and Phase II (100 K) of CsW2O6 are shown in Supplementary Tables 1−4, respectively. Those of CsW1.835O6 are shown in Supplementary Tables 5 and 6. The unit cell of Phase III was determined by structural analyses of the single crystal XRD data, as shown in Supplementary Fig. 1. The temperature dependences of the lattice constants determined by the Rietveld analyses of the powder XRD data of CsW2O6 are shown in Supplementary Fig. 2. At the lowest measured temperature of 30 K, the difference between a/2 and b and the deviation of  from 90° are 0.04% and 0.08%, respectively, indicating the monoclinic distortion in Phase III is quite small. In contrast to Phase III, Phase II was found to preserve the cubic symmetry. Supplementary Fig. 3 shows the intensities of symmetrically-equivalent reflections of 1210 in single-crystal XRD of Phase II. Intensities of three reflections of 1120, 0112, and 1201 and those of 1210, 0121, and 1012 are identical within the uncertainties, respectively, indicating that the Phase II has the m3 Laue class. Supplementary Figs. 4A and 4B show a peak profile of 1197 reflection (cubic unit cell) at 30, 100, and 300 K and temperature dependence of full width at half maximum of the 1197 reflection, respectively. In the cubic phase, in addition to symmetrically-equivalent reflections of the 1197 reflection, some reflections such as 1391 and their symmetricallyequivalent reflections overlap at the same diffraction angle, but they split when symmetry lowering from cubic occurs. The increase of the peak width in Phase III below 95 K reflects such symmetry lowering. In contrast, the peak width in Phase II is same as that in Phase I, indicating that the Phase II has the cubic symmetry.    Figure 1. A unit cell of Phase III determined by the structural analyses of the single crystal XRD data. The unit cell of Phase III is 2 × 1 × 2 of that of Phase II indicated as the blue shaded region.

Supplementary Note 2. Twinning and space group of Phase III
In Phase III, superlattice reflections were observed at (h/2, k/2, l/2) in the cubic unit cell of Phase I and II. The size of the unit cell and the space group of Phase III were determined by the following procedure. First, since the lattice distortion appears continuously in Phase III, as shown in Supplementary Figs. 2 and 4, the phase transition from Phase II to III was considered to be second order. In this case, the space group of Phase III should be a subgroup of Phase II.
According to the group-subgroup relation, R3 and P212121 are the maximal subgroup of P213. In the R3 and its subgroup cases, the observed superlattice reflections could not be indexed. In the case of P212121, possible unit cell is a' = 3a, b' = 3b, or c' = 3c, but the superlattice reflections appeared at (h/2, k/2, l/2), indicating that this space group is not applicable.  Fig. 6 shows the temperature dependences of the electrical resistivity, , and magnetic susceptibility, , of the CsW1.835O6 single crystals. The data of the CsW2O6 single crystals are also shown for reference. The  value of the CsW1.835O6 single crystal is two orders of magnitude larger than that of CsW2O6 at room temperature, and it increases with decreasing temperature. The  value of the CsW1.835O6 single crystal above 100 K is comparable to that of CsW2O6 in Phase II. There are no anomalies in the  and  data of CsW1.835O6, which is in contrast to those of CsW2O6. These results indicate that CsW1.835O6 is a nonmagnetic insulator that does not possess d electrons, which is consistent with its chemical composition determined by the structural analyses of the single-crystal XRD data. This yields the W valence of 5.995+, indicating that there are almost no 5d electrons.  of CsW1.835O6 shows Curie-Weiss behavior at low temperatures, suggestive of the presence of impurity spins in the sample. A Curie-Weiss fit of  = C/(T − W) + 0 to the 2−50 K data yielded C = 0.0118(12) cm 3 K mol −1 , W = −9.2(11) K, and 0 = −1.04(2) × 10 −4 cm 3 mol −1 . This C value means that 1.6% of the W atoms have an S = 1/2 spin. Supplementary Figure 6. Temperature dependences of electrical resistivity (upper) and magnetic susceptibility (lower) of the CsW1.835O6 single crystals. The magnetic susceptibility was measured under a magnetic field of 1 T. The data of CsW2O6 are shown for reference. The solid curve in the lower panel shows the result of a Curie-Weiss fit to the CsW1.835O6 data between 2 and 50 K.

Supplementary Note 5. Diffuse scattering in the single-crystal XRD patterns
Diffuse scattering was observed in the single-crystal XRD patterns of CsW2O6 and CsW1.835O6.
The XRD patterns presented in this section emphasize the presence of diffuse scattering. Since the intensity of diffuse scattering is much lower than those of Bragg reflections, the observed diffuse scattering has no effect on the crystal-structure refinement. Supplementary Fig. 7A shows XRD patterns of CsW2O6 and CsW1.835O6 single crystals measured at 250 and 30 K, respectively, which show diffuse scattering with the same pattern. The intensity of the diffuse scattering for CsW2O6 continuously decreases with decreasing temperature, as shown in Supplementary Fig.   7B. No discontinuous change is observed at the phase transitions. On the other hand, that of CsW1.835O6 is almost constant throughout the entire temperature range. Supplementary Fig. 8A is an enlarged view of the XRD pattern of CsW2O6 at 250 K. Diffuse scattering has been formed that connects diffraction spots with the same h and l that satisfy the extinction rule of h + l = 4n.
This extinction rule cannot be explained by the displacement of Cs atoms. The presence of the extinction rule suggests that the diffuse scattering is not caused by imperfections in the crystal.
An atomic displacement pattern that can reproduce this extinction rule is shown in Supplementary

Supplementary Note 6. Raman scattering
Raman spectra of (100) surface of CsW2O6 are shown in Supplementary Fig. 9. Irreducible representation of each peak is also shown. The scattering peaks are appropriately assigned by the Fd3m symmetry, in which the Raman active modes are A1g + Eg + 4T2g. One T2g mode is missing, probably due to its weak intensity. The Cs + ions behave as a rattling ion in the -pyrochlore  (B) Schematic picture of the atomic displacement that can reproduce the observed diffuse scattering.