Regular-triangle trimer and charge order preserving the anderson condition in the pyrochlore structure of csw2o6

ABSTRACT 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 5_d_ electrons are confined in

regular-triangle W3 trimers. This trimer formation represents the self-organization of 5_d_ 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 5_d_ electrons characteristic of the pyrochlore oxides are both likely to play important roles in this

charge-orbital-spin coupled phenomenon. SIMILAR CONTENT BEING VIEWED BY OTHERS IDEAL SPIN-ORBIT-FREE DIRAC SEMIMETAL AND DIVERSE TOPOLOGICAL TRANSITIONS IN Y8COIN3 FAMILY Article Open access

15 November 2024 COBALT-BASED MAGNETIC WEYL SEMIMETALS WITH HIGH-THERMODYNAMIC STABILITIES Article Open access 04 January 2021 STABILIZATION OF THREE-DIMENSIONAL CHARGE ORDER THROUGH

INTERPLANAR ORBITAL HYBRIDIZATION IN PR_X_Y1−_X_BA2CU3O6+_Δ_ Article Open access 19 October 2022 INTRODUCTION Understanding the phase transitions of crystalline solids is a central issue in

materials science. Electronic phase transitions in transition metal compounds with pyrochlore structures, made of three-dimensional networks of corner-sharing tetrahedra, have posed

challenging questions in materials science since their discovery. The classical example is magnetite Fe3O4, which was reported to show a metal−insulator transition accompanied by a charge

order of Fe at 119 K, called the Verwey transition1. Although many studies of this transition have been made, full understanding of its ground state has not yet been reached, and relevant

studies based on new perspectives are continuing2. Recently, metal−insulator transitions accompanied by all-in-all-out-type magnetic order in 5_d_ oxides, such as Cd2Os2O7 and Nd2Ir2O7, have

attracted considerable attention3,4,5, in terms of a ferroic order of extended magnetic octapoles and the formation of Weyl fermions in solids3,6,7,8,9. As described above, rich physics

appears in pyrochlore systems, which might be caused by the high crystal symmetry and a large number of atoms in a unit cell, resulting in the self-organization of _d_ electrons in various

forms. In this study, we report self-organization of 5_d_ electrons at the electronic phase transition of β-pyrochlore oxide CsW2O6, discovered by using high-quality single crystals. CsW2O6

was first synthesized by Cava et al., which was reported to have a cubic lattice with _Fd_\(\bar 3\)_m_ space group at room temperature10. In this structure, W atoms form a pyrochlore

structure and have 5.5+ valence with a 5_d_0.5 electron configuration. Electrical resistivity measurement of polycrystalline samples suggested that a metal−insulator transition occurs at 210

 K11. The crystal structure of the insulating phase was reported to have orthorhombic _Pnma_ space group11; however, this space group was suggested to be incorrect by a theoretical study12.

Electronic structure calculations on the _Fd_\(\bar 3\)_m_ phase pointed out that there is a strong nesting of the Fermi surfaces, which induces a symmetry lowering to the _P_4132 space

group. Recent photoemission experiments of thin films suggested that the valence of W in the insulating phase disproportionates into 5+ and 6+13. RESULTS A PHASE TRANSITION AT 215 K We

prepared single crystals of CsW2O6 (Fig. 1a) and W-deficient CsW1.835O6 in a quartz tube (see “Method” section). As shown in Fig. 1b, the electrical resistivity, _ρ_, of a single crystal of

CsW2O6 strongly increases below _T_t = 215 K with decreasing temperature, as in the cases of a polycrystalline sample and a thin film11,13. This increase is accompanied by a small but

obvious temperature hysteresis, indicating that a first-order phase transition occurs at _T_t. Here, the phases above and below _T_t are named Phase I and II, respectively. The magnetic

susceptibility, _χ_, shown in Fig. 1b strongly decreases below _T_t, which is identical to the polycrystalline case11. However, the line widths of the 133Cs-NMR spectra in Phase II do not

show any significant broadening compared to those in Phase I, as shown in Fig. 1f, indicating that the decrease of _χ_ in Phase II is not caused by antiferromagnetic order. Figure 1c shows

single-crystal X-ray diffraction (XRD) patterns of CsW2O6 measured at 250 (Phase I) and 100 K (Phase II). Each of the diffraction spots at 250 K were indexed on the basis of a cubic cell of

_a_ = 10.321023(7) Å with _Fd_\(\bar 3\)_m_ space group, consistent with previous reports10,11. In the diffraction pattern at 100 K, more diffraction spots appear. All these spots were

indexed on the basis of cubic _P_213 space group with a lattice constant of _a_ = 10.319398(6) Å, which is almost identical to _a_ of Phase I. This change of diffraction spots occurs at

_T_t, as seen in the temperature dependence of the intensity shown in Fig. 1d. Moreover, in Phase II, diffraction spots do not split into multiple spots nor do they change their shapes, even

in the high-angle region, as shown in Fig. 1c. Laue class and crystal system determined by the observed reflections clearly indicated that a structural change that preserves the cubic

symmetry occurs at _T_t and the Phase II has the Laue class of _m_\(\bar 3\) (see Supplementary Note 1 and Supplementary Fig. 3). As seen in the polarization dependence of the Raman spectra

of (111) surface measured at 100 K (Phase II) and room temperature (Phase I) shown in Fig. 1e, the spectra of Phase II are independent of the polarization angle same as in Phase I,

indicating the presence of three-fold rotational symmetry perpendicular to (111), consistent with the inferred cubic symmetry. These results mean that the _Pnma_ structural model proposed

based on the powder diffraction data is incorrect11. In addition, the proposed _Pnma_ structure has a pseudo-tetragonal distortion of approximately 0.03%, which was not observed in the

present study, as shown in Supplementary Figs. 2 and 4. This result is also supported by the 133Cs-NMR data on the single crystals of CsW2O6 discussed below. In a polycrystalline sample of

CsW2O6, W-deficient CsW1.835O6 always exists as an impurity phase. We believe that the fact that the single crystals of CsW2O6 and W-deficient CsW1.835O6 were separately prepared and the

diffraction and physical property measurements were performed by using the high-quality single crystals played a crucial role for elucidating the nature of Phase II. The crystallographic

parameters of CsW2O6 at 250 and 100 K determined by the structural analyses are shown in Supplementary Tables 2 and 4, respectively. Physical properties and crystallographic parameters of

CsW1.835O6 are also shown in Supplementary Fig. 6 and Supplementary Table 6. A point to be noted in the Raman spectra shown in Fig. 1e is a peak appeared at ~55 cm−1, reflecting the rattling

of Cs+ ions14 (see Supplementary Note 6). CRYSTAL STRUCTURE OF PHASE II Here we discuss the solved crystal structure of Phase II. In Phase I with the _Fd_\(\bar 3\)_m_ space group, each of

Cs, W, and O atoms occupies one site, where the Cs and W atoms form diamond and pyrochlore structures, respectively (Fig. 2a). In Phase II with the _P_213 space group, the Cs atoms occupy

two different sites and form a zinc-blende structure, as shown in Fig. 2b. This was further confirmed by the two peaks in 133Cs-NMR spectra correspond to the two Cs sites, which appear as a

small peak split in the 200, 160, and 125 K data shown in Fig. 1f. On the other hand, W atoms occupy two sites with a 1:3 ratio in Phase II, as shown in Fig. 2b, c, which is incompatible

with the W5+–W6+ charge order with a 1:1 ratio of W5+ and W6+ atoms. According to the bond valence sum calculation for the W–O distances determined from single-crystal XRD analyses15, the

valences of the W(1) and W(2) atoms are estimated to be 6.07(3) and 5.79(3) at 100 K (Phase II), respectively. Considering that the reliable bond valence sum parameters of W6+ are available

but those of W5+ are not, it is natural for the W(1) atoms to be W6+ without 5_d_ electrons. In this case, the valence of the W(2) atoms becomes 5.33+ with 5_d_2/3 electron configurations.

The above discussion indicates that a charge order with a noninteger valence occurs at _T_t. In fact, single crystals of W-deficient CsW1.835O6, where all W atoms have 6+ valence without

5_d_ electrons, do not show the transition at _T_t (see Supplementary Fig. 6). In Phase II, the W(2) atoms form a three-dimensional network of small and large regular triangles, which are

alternately connected by sharing their corners, as shown in Fig. 2b. Although the difference of sizes between the large and small triangles are about 2%, arrangements of the occupied 5_d_

orbitals are completely different between them, resulting in a W3 trimer on a small triangle, as discussed later. If there was no alternation of the W3 triangles, the W sublattice would

possess a hyperkagome structure16, as shown in Fig. 2c. The presence of the alternation indicates that ‘breathing hyperkagome’ structure is formed during Phase II17. DISCUSSION The charge

order in Phase II of CsW2O6 is interesting in that the “Anderson condition” is maintained in an unusual way. Anderson pointed out that magnetite has an infinite number of charge ordering

patterns, where all the tetrahedra in a pyrochlore structure have the same total charge, i.e., so-called Anderson condition, and this macroscopic degeneracy strongly suppresses the

transition temperature of the Verwey transition18. This situation can be interpreted as geometrical frustration of electronic charges. However, not only magnetite, but also other

mixed-valent pyrochlore systems, such as CuIr2S4 and AlV2O4, were reported to show a charge order that violated the Anderson condition19,20,21,22,23. In them, the energy gained by _σ_

bonding between _d_ orbitals of adjacent atoms was expected to be large enough to compensate for the loss of Coulomb energy due to violation of the Anderson condition, because spinel-type

compounds comprised the edge-shared octahedra24,25. In contrast, the charge order of CsW2O6 satisfies the Anderson condition, where each tetrahedron consists of three W5.33+ and one W6+

atoms. However, this charge order is different to that proposed by Anderson and Verwey, which has integer valences with a 1:1 ratio18,26. Hyperkagome-type orders often appear in pyrochlore

systems with a 1:3 ratio of two kinds of atoms, such as the uuud spin structure of the half magnetization plateau of Cr spinel oxides and the atomic order in B-site ordered spinel oxides

A2BB_'_3O827,28,29. As far as we are aware, CsW2O6 is the only example to show a hyperkagome-type order where the formation of this order is nontrivial. It is an alternative way to

relieve the geometrical frustration based on the traditional problem in condensed matter physics. Why does such unusual charge order occur in CsW2O6? A key to understand this question is

hidden in Fermi-surface instability of the electronic band structure of Phase I. The left panel of Fig. 3 shows the band structure of Phase I and the right panel shows four overlapping band

structures, which are depicted after the parallel shifts of electronic bands corresponding to a change of the primitive cell from face-centered cubic to primitive cubic. As seen in the right

panel of Fig. 3, band crossing occurs close to all points where electronic bands touch the Fermi energy _E_F, suggesting that the Fermi surfaces are well nested by the parallel shift of the

electronic bands, corresponding to the loss of centering operations. This situation can be called “three-dimensional nesting”, which means that a large electronic energy is gained by the

structural change associated with the above symmetry change. The electronic instability and nesting of the Fermi surfaces in Phase I was also pointed out in the previous study12. It is quite

rare for cubic compounds to have such well-nested Fermi surfaces, except for the filled-skutterudite PrRu4P12. PrRu4P12 shows a metal−insulator transition accompanied by a structural change

from body-centered cubic to primitive cubic30,31 and has a Fermi-surface instability corresponding to this structural change32. The above discussion indicates that three-dimensional nesting

is likely an essential ingredient for the 215 K transition. However, if it is the only driving force, a structural change from _Fd_\(\bar 3\)_m_ to _P_4132 or _P_4332, which are maximal

non-isomorphic subgroups with primitive cubic lattices, must occur, as also discussed in the previous theoretical study12. In this case, the W(2) atoms should form a uniform hyperkagome

structure. This study also suggested that the band gap does not open at the Fermi energy in the _P_4132 and _P_4332 cases12, which is inconsistent with the observed insulating nature of the

Phase II. In reality, the space group of Phase II is _P_213, which is a subgroup of _P_4132 and _P_4332, and W(2) atoms form a breathing hyperkagome structure, where the size of a small

triangle is 2% smaller than that of a large triangle. The _P_213 space group is also different from orthorhombic _P_212121 proposed in ref. 12. In addition, whether the band structure

calculated with the structural parameters of Phase II is gapped or not might be important for understanding of the underlying physics of the phase transition. At present, however, the

calculated results with an overlapped gap have not yet completely converged, due to the large number of atoms in a primitive unit cell. Phenomenologically, orientation of the occupied 5_d_

orbitals is important for the symmetry lowering from _P_4132/_P_4332 (uniform hyperkagome) to _P_213 (breathing hyperkagome). For a W(2)O6 octahedron of Phase II shown in Fig. 2e, the two

apical W(2)-O bonds (gray) are 3–8% shorter than the other four equatorial bonds (blue), indicating that the octahedron is uniaxially compressed. This compression is comparable to the

typical Jahn–Teller distortion in _t_2g electron systems, and the 5_d_ orbitals lying in this equatorial plane should be occupied by electrons. Schematic pictures of the occupied 5_d_

orbitals in small and large triangles are shown in the right and left panels of Fig. 2f, respectively. There is considerable overlap between the occupied 5_d_ orbitals in the small triangle

via an O 2_p_ orbital. In contrast, there is little overlap in a large triangle, indicating that two electrons in three W(2) atoms are confined in a W3 trimer in the small triangle. This

regular-triangle trimer formation might be understood as a three-centered-two-electron (3c2e) bond formation, where two electrons are accommodated in a molecular orbital made of three W 5_d_

orbitals (and the O 2_p_ orbitals hybridize with them). In this case, it is natural to have a nonmagnetic ground state. This regular-triangle trimer is essentially different to those of

famous LiVO2 and LiVS2, where two electrons are shared by two V atoms along each side of a triangle33,34,35. Na3Ir3O8 was pointed out to have regular-triangle Ir3 molecules formed on a

hyperkagome structure36. However, the Ir3 molecules are connected each other, which is essentially different from the fact that the W3 trimers in CsW2O6 are isolated, as seen in Fig. 2b.

Stabilization of the electronic energy by the formation of multiple-centered bonds, where a few electrons are shared by many atoms, often occurs in electron-deficient molecules or cluster

compounds37. To our knowledge, CsW2O6 is the only example where this type of bond formation appears as a phase transition. The formation of regular-triangle trimers itself is also

surprising, because the 3c2e bond usually has a bent shape. According to previous reports, only the H3+ ion has a regular-triangle shape in triatomic molecules formed by the 3c2e bond.

Moreover, H3+ is an interstellar material and it is not stable on Earth, having been observed in astronomical spectra38. This regular-triangle shape of the trimer might be related to its

internal structure, a part of which appears in the atomic displacement parameters (ADPs). As shown in Fig. 2d, the O atoms bridging W(2) atoms in a W3 trimer (O(1) site) in Phase II have

large ADPs perpendicular to the W-W bond. The ADPs of the other O atoms are typical values, suggesting that the ADPs of the O(1) site do not increase by the structural instability of the

β-pyrochlore structure, but rather by the electronic instability of the trimer. The large ADPs perpendicular to the W-W bond indicate that there is a strong fluctuation that changes the

W(2)–O(1)–W(2) angle. In pyrochlore oxides, the change of this angle has a large effect on the orbital overlap39. Therefore, this fluctuation can be interpreted as a strong fluctuation to a

state in which one of the W-W bonds becomes stronger, or in the extreme, to a state in which a W2 dimer is formed. Since ADPs of the W(2) site have typical values, it is unlikely that the

dimers statically and randomly form on the trimers. Instead, the dimer might dynamically fluctuate or resonate. For a complete understanding of the internal structure of the trimers, it

would be desirable to directly observe their dynamical properties in a future study. For the formation of this trimer, electron correlation of the 5_d_ electrons in CsW2O6 might be another

essential factor. The optical conductivity spectra of CsW2O6 measured at room temperature deduced from the reflectivity using the Kramers–Kronig transformation40, shown in the inset of Fig. 

1b, exhibit a broad peak at around 0.6 eV. Extrapolation of the spectra to zero frequency coincides with _ρ_ = 3 mΩ cm at room temperature (the main panel of Fig. 1b). This result indicates

that there is no, or negligibly small, Drude contribution in the spectra, and the conducting carriers are trapped by something with an energy scale of 0.6 eV, resulting in the loss of

coherency, which is supported by _dρ_/_dT_ < 0 in Phase I. Absence of a peak in the far-infrared region indicates that this localization is not due to disorder, but might be reminiscent

of the spectra of lightly-carrier-doped Mott insulators41,42,43. As seen in the band structure shown in Fig. 3a, there are flat parts near _E_F and the energy bands have a narrow width of

0.7 eV, suggesting the presence of a strong electron correlation for a 5_d_ electron system. In 5_d_ or 4_d_ pyrochlore oxides, the 5_d_/4_d_ electrons often have moderately-strong electron

correlation because of the small orbital overlap due to the bent metal–oxygen–metal bonds. In fact, the optical conductivities of Nd2Ir2O7 and Sm2Mo2O7 indicate the presence of incoherent

_d_ electrons9,44, similar to the case of CsW2O6. As a result, 5_d_ pyrochlore oxides often show an electronic phase transition with the order of electronic degrees of freedom. Nd2Ir2O7 with

_J_eff = 1/2 and Cd2Os2O7 with _S_ = 3/2, without charge and orbital degrees of freedom, showed a magnetic order accompanied by a metal−insulator transition3,4,5. Instead, for CsW2O6, the

trimers are formed with the help of the electron correlation as in the case of LiVO245. In the trimer of CsW2O6, the two 5_d_ electrons form a spin-singlet pair, resulting in the nonmagnetic

and insulating ground state. This is an alternative type of self-organization of _d_ electrons realized in a strongly correlated 5_d_ oxide. Finally, we will discuss another structural

transition at 90 K. Phase II looks like a ground state, where most of the degrees of freedom have been lost, but surprisingly another phase transition occurs at 90 K. By indexing the

diffraction spots in the single-crystal XRD data of Phase II, the crystal structure below 90 K, named Phase III, was found to have monoclinic _P_21 space group with a four-times-larger (2 × 

1 × 2) unit cell than that of Phase II, as shown in Supplementary Fig. 1. The procedure performed for the determination of the size of unit cell and space group of Phase III is described in

Supplementary Notes 2 and 3. As seen in the inset of Fig. 1b, the heat capacity divided by temperature, _C_/_T_, shows a small but obvious peak, which corresponds to the entropy change of

~0.4 J K−1 mol−1, at ~90 K, indicating the presence of a bulk phase transition. The _P_21 space group is different from _Pnma_ and _P_212121 space groups proposed in the previous

studies11,12. The atomic positions in Phase III have not yet been determined, because of tiny monoclinic distortion and domain formation, but it is clear that the structural change at 90 K

is small, as seen in Supplementary Fig. 2. In addition, _χ_ does not exhibit an anomaly at 90 K. These results suggest that the 90 K transition is not caused by the spin, charge, and/or

orbital order different to the 215 K transition. What mechanism gives rise to the 90 K transition? The diffuse scattering that appears in the single-crystal XRD patterns might provide a hint

to answering this question. In the single-crystal XRD patterns of Phases I, II, and III shown in Fig. 4, there are diffuse scatterings at the same positions, which follow the extinction

rule of _h_ + _l_ = 4_n_ (for a cubic unit cell) and connect the superlattice spots that emerged in Phase III. This suggests that the structural change from Phase II to III and the diffuse

scatterings have the same origin. The same diffuse scattering pattern also appeared in CsW1.835O6 (Supplementary Fig. 7) and CsTi0.5W1.5O646, which are isostructural to CsW2O6, but only have

W6+ atoms without 5_d_ electrons, suggesting that they are independent of the 215 K transition and might be caused by the structural instability of the β-pyrochlore structure itself. This

discussion also implies that the 215 K transition is irrelevant to this instability and is purely electronic driven. In conclusion, we found that regular-triangle W3 trimers are formed at

the 215 K transition in β-pyrochlore oxide CsW2O6, as determined using structural- and electronic-property measurements of high-quality single crystals. This transition represents the

self-organization of 5_d_ electrons, where geometrical frustration is relieved in a nontrivial way that satisfies the traditional Anderson condition and results in the quite rare cubic−cubic

structural transition. This type of electronic transition is not only unusual, but is only partly understood by the first principles calculations, suggesting that it might be a spin-,

charge-, and orbital-coupled phase transition occurring beyond the existing electronic phase transitions of pyrochlore systems. The above finding shows that the exploration of geometrically

frustrated 5_d_ compounds will lead to the discovery of further interesting electronic phenomena, such as odd-parity multipoles and spin–charge–orbital entangled quantum liquids. METHODS

SAMPLE PREPARATION Single crystals of CsW2O6 were prepared by crystal growth in an evacuated quartz tube under a temperature gradient. A mixture of a 3:1:3 molar ratio of Cs2WO4 (Alfa Aeser,

99.9%), WO3 (Kojundo Chemical Laboratory, 99.99%), and WO2 (Kojundo Chemical Laboratory, 99.99%), with a combined mass of 0.1 g, was sealed in an evacuated quartz tube with 0.1 g of CsCl

(Wako Pure Chemical Corporation, 99.9%). The hot and cold sides of the tube were heated to, and then kept at 973 K and 873 K, respectively, for 96 h, and then the furnace was cooled to room

temperature. The mixture was put on the hot side. The obtained single crystals had an octahedral shape with {111} faces with edges of at most 1 mm. Powder samples of CsW2O6 were prepared by

the solid-state reaction method described in the previous studies10,11. The obtained powder was sintered at 773 K for 10 min using a spark plasma sintering furnace (SPS Syntex). Single

crystals of W-deficient CsW2−_x_O6 were prepared using the flux method. A mixture of a 3:1:3 molar ratio of Cs2WO4 (Alfa Aeser, 99.9%), WO3 (Kojundo Chemical Laboratory, 99.99%), and WO2

(Kojundo Chemical Laboratory, 99.99%), with the combined mass of 0.1 g, and 0.2 g of CsCl (Wako Pure Chemical Corporation, 99.9%) were put in an alumina crucible, which was sealed in an

evacuated quartz tube. The tube was heated to, and then kept at 923 K for 48 h, and then slowly cooled to 873 K at a rate of −0.5 K/h. The obtained single crystals have similar octahedral

shape and mostly have a larger size than those of CsW2O6. The value of the W deficiency, _x_, was estimated to be 0.165 via a structural analysis using the single-crystal XRD data, meaning

that the chemical composition of the single crystal is CsW1.835O6, where the W atoms have no 5_d_ electrons. MEASUREMENTS AND FIRST PRINCIPLES CALCULATIONS The electrical resistivity and

magnetization measurements of the CsW2O6 and CsW1.835O6 single crystals were performed using a Physical Property Measurement System (PPMS, Quantum Design) and Magnetic Property Measurement

System (Quantum Design), respectively. The normal incident reflectivity of (111) surface of a CsW2O6 single crystal was taken at room temperature using a Fourier-type interferometer

(0.005−1.6 eV, DA-8, ABB Bomem; 0.06−1.0 eV, FT-IR6100, Jasco) and a grating spectrometer (0.46−5.8 eV, MSV-5200, Jasco) installed with a microscope40,47. As a reference mirror, we used

either evaporated Au (far- to near-IR region), Ag (near-IR to visible region), or Al (near- to far-UV region) films on a glass plate. The heat capacity of the CsW2O6 sintered sample was

measured using the relaxation method with the PPMS. Single-crystal XRD experiments of the CsW2O6 and CsW1.835O6 samples were performed at BL02B1 in the SPring-8 synchrotron radiation

facility in Japan. The experimental conditions are shown in Supplementary Tables 1, 3, and 5. SORTAV and SHELXL were used for merging the reflection data and the structural

refinement48,49,50. A part of crystal structure views were drawn using VESTA51. Powder XRD experiments of CsW2O6 were performed at BL02B2 in SPring-8. Synchrotron X-rays with energies of

15.5 and 25 keV were used for the measurements below and above 150 K, respectively. Rietveld analyses of the powder XRD data were performed using GSAS. Raman scattering spectra of the CsW2O6

single crystals were measured using a diode-pumped CW solid-state laser with a wavelength of 5614 Å. 133Cs-NMR measurements of a CsW2O6 single crystal were conducted in a magnetic field of

8 T. The NMR spectra were obtained by Fourier transforming the free induction decay signal. The band structure calculations of Phase I of CsW2O6 were performed using the full potential

linear augmented plane wave (FLAPW) method with a local density approximation. The experimentally obtained structural parameters shown in Supplementary Tables 1 and 2 were used for the

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data. _J. Appl. Crystallogr._ 44, 1272 (2011). Article  CAS  Google Scholar  Download references ACKNOWLEDGEMENTS The authors are grateful to Y. Yamakawa, A. Yamakage, A. Koda, R. Kadono, J.

Matsuno, and D. Hirai for helpful discussions and T. Fujii for his help with experiments. Y.O. is also grateful for collaboration with Y. Nagao, J. Yamaura, M. Ichihara, Z. Hiroi, and M.

Yoshida in the early stage of this work using polycrystalline samples. This work was partly carried out under the Visiting Researcher Program of the Institute for Solid State Physics, the

University of Tokyo and supported by JSPS KAKENHI (Grant Number: 18H04314, 19H05823, 16H03848, 15H05886, 15H05882, 20K03829, 17H02918, 18H04310). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS

* Department of Applied Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan Yoshihiko Okamoto, Haruki Amano, Naoyuki Katayama, Hiroshi Sawa, Kenta Niki, Rikuto Mitoka, 

Yasunori Yokoyama, Yuto Nakamura, Hideo Kishida & Koshi Takenaka * Department of Physics, Kobe University, Rokkodai 1-1, Nada-ku, Kobe, 657-8501, Japan Hisatomo Harima * Graduate School

of Integrated Arts and Sciences, Hiroshima University, Kagamiyama 1-7-1, Higashi-Hiroshima, 739-8521, Japan Takumi Hasegawa & Norio Ogita * Institute for Solid State Physics, University

of Tokyo, Kashiwanoha 5-1-5, Kashiwa, 277-8581, Japan Yu Tanaka & Masashi Takigawa * National Institute for Materials Science (NIMS), Sakura 3-13, Tsukuba, 305-0003, Japan Kanji Takehana

 & Yasutaka Imanaka Authors * Yoshihiko Okamoto View author publications You can also search for this author inPubMed Google Scholar * Haruki Amano View author publications You can also

search for this author inPubMed Google Scholar * Naoyuki Katayama View author publications You can also search for this author inPubMed Google Scholar * Hiroshi Sawa View author publications

You can also search for this author inPubMed Google Scholar * Kenta Niki View author publications You can also search for this author inPubMed Google Scholar * Rikuto Mitoka View author

publications You can also search for this author inPubMed Google Scholar * Hisatomo Harima View author publications You can also search for this author inPubMed Google Scholar * Takumi

Hasegawa View author publications You can also search for this author inPubMed Google Scholar * Norio Ogita View author publications You can also search for this author inPubMed Google

Scholar * Yu Tanaka View author publications You can also search for this author inPubMed Google Scholar * Masashi Takigawa View author publications You can also search for this author

inPubMed Google Scholar * Yasunori Yokoyama View author publications You can also search for this author inPubMed Google Scholar * Kanji Takehana View author publications You can also search

for this author inPubMed Google Scholar * Yasutaka Imanaka View author publications You can also search for this author inPubMed Google Scholar * Yuto Nakamura View author publications You

can also search for this author inPubMed Google Scholar * Hideo Kishida View author publications You can also search for this author inPubMed Google Scholar * Koshi Takenaka View author

publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS The samples were prepared by K.N., R.M., and Y.O. Single crystal XRD experiments were performed by

H.A., H.S., and N.K. Electrical resistivity was measured by K.N., Y.O., and K. Takenaka. Reflectivity measurements were done by Y.Y., K. Takenaka, K. Takehana, Y.I., K.N., Y.N., and H.K.

Magnetic susceptibility and heat capacity were measured by Y.O., K.N., and R.M. Raman scattering experiments were performed by T.H., R.M., and N.O. NMR experiments were performed by Y.T. and

M.T. First principles calculations were performed by H.H. The manuscript was drafted by Y.O. and revised by all the authors. CORRESPONDING AUTHOR Correspondence to Yoshihiko Okamoto. ETHICS

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order preserving the Anderson condition in the pyrochlore structure of CsW2O6. _Nat Commun_ 11, 3144 (2020). https://doi.org/10.1038/s41467-020-16873-7 Download citation * Received: 03

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