Pressure-induced collapse of spin-orbital Mott state in the hyperhoneycomb iridate β -Li 2 IrO 3
T. Takayama, A. Krajewska, A. S. Gibbs, A. N. Yaresko, H. Ishii, H. Yamaoka, K. Ishii, N. Hiraoka, N. P. Funnell, C. L. Bull, H. Takagi
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Pressure-induced collapse of spin-orbital Mott state in the hyperhoneycomb iridate β -Li IrO T. Takayama,
1, 2
A. Krajewska,
1, 2
A. S. Gibbs, A. N. Yaresko, H. Ishii, H.Yamaoka, K. Ishii, N. Hiraoka, N. P. Funnell, C. L. Bull, and H. Takagi
1, 2, 7 Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany Institute for Functional Matter and Quantum Technologies,University of Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany ISIS Facility, STFC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan RIKEN SPring-8 Center, Sayo, Hyogo 679-5148, Japan Synchrotron Radiation Research Center, National Institutes for Quantumand Radiological Science and Technology, Sayo, Hyogo 679-5148, Japan Department of Physics, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan (Dated: August 17, 2018)Hyperhoneycomb iridate β -Li IrO is a three-dimensional analogue of two-dimensional honeycombiridates, such as α -Li IrO , which recently appeared as another playground for the physics of Kitaev-type spin liquid. β -Li IrO shows a non-collinear spiral ordering of spin-orbital-entangled J eff = 1/2moments at low temperature, which is known to be suppressed under a pressure of ∼ P S ∼ β -Li IrO above P S was refined, which indicates the formation of Ir dimers on the zig-zag chains,with the Ir-Ir distance even shorter than that of metallic Ir. We argue that the strong dimerizationstabilizes the bonding molecular orbital state comprising the two local d zx -orbitals on the Ir-O -Irbond plane, which conflicts with the equal superposition of d xy -, d yz - and d zx - orbitals in the J eff = 1/2 wave function produced by strong spin-orbit coupling. The results of resonant inelastic x-rayscattering (RIXS) measurements and the electronic structure calculations are fully consistent withthe collapse of the J eff = 1/2 state. A subtle competition of various electronic phases is universalin honeycomb-based Kitaev materials. PACS numbers: 75.10.Kt, 75.25.Dk, 75.70.Tj
I. INTRODUCTION
The Kitaev model, with S = 1/2 spins on a honeycomblattice connected by a bond-dependent Ising coupling,has been attracting considerable interest as it providesan exactly solvable quantum-spin-liquid (QSL) groundstate consisting of the two kinds of Majorana fermions[1]. The materialization of the Kitaev QSL has been pur-sued extensively in honeycomb-based spin-orbital Mottinsulators with heavy transition metal ions with d ( t g )configuration, such as Ir and Ru [2, 3]. In these spin-orbital compounds, the heavy d transition metal ions areoctahedrally coordinated with anions and the octahedraform a honeycomb network by sharing their edges. Thestrong spin-orbit coupling λ SO ∼ and 0.1eV for Ru splits the degenerate t g into the half-filled J eff = 1/2 doublet and the completely filled J eff = 3/2quartet [4]. The magnetism of the candidate compoundstherefore originates from J eff = 1/2 pseudo-spins. Thesuper-exchange coupling between two adjacent J eff = 1/2moments is shown to be a bond-dependent ferromagneticIsing interaction as in the Kitaev model [5].The layered honeycomb iridates Na IrO and α -Li IrO emerged as the first generation of candidate ma-terials for the Kitaev QSL [6, 7]. Their ground state,however, turned out not to be a QSL. They were found to show a magnetic transition to a zigzag-type antiferro-magnetic phase [8, 9] and to a non-coplanar spiral phase[10] respectively at a low temperature. α -RuCl was sub-sequently proposed as the first non-iridium-based candi-date but again was found to show a zigzag-type anti-ferromagnetic ordering as in Na IrO [11–14]. In par-allel with this, three-dimensional (3D) analogues of thetwo-dimensional (2D) honeycomb α -Li IrO , β -Li IrO and γ -Li IrO , were discovered as another platform forKitaev magnetism [15, 16]. These 3D honeycomb com-pounds also show a clear magnetic transition into a com-plex spiral phase [17, 18] similar to that of α -Li IrO ,though a closer proximity to the Kitaev spin liquid thantheir 2D analogues is suggested [19, 20], for example, bythe ferromagnetic Curie-Weiss temperature. The pres-ence of magnetic interactions other than the Kitaev cou-pling, such as a direct Heisenberg exchange and off-diagonal coupling, has been discussed to stabilize thelong-range magnetic ordering instead of a QSL state[2, 3].In the 2D honeycomb iridates, a chemical substitutionof the interlayer Li ions was attempted to tune the mag-netic interactions through a local lattice distortion and tobring the ground state closer to the QSL [21–23]. Withsuch an approach, H LiIr O was very recently found tohost a QSL ground state [24, 25], while the relevanceto Kitaev physics remains yet to be identified. Controlof the ground states using magnetic field was also at-tempted. By applying a magnetic field of µ H c ∼ α -RuCl , a disap-pearance of the magnetic ordering was observed. Theemergence of a QSL-like phase was pointed out and hasbeen a subject of intensive studies [12, 26–30].Another promising approach to control the magneticground state may be the application of pressure. Theemergence of a QSL state under pressure was theoreti-cally proposed in honeycomb-based iridates [31–33]. In-deed, the suppression of long-range magnetic order underhigh pressure was reported in 3D honeycomb iridates β -Li IrO and γ -Li IrO , in x-ray magnetic circular dichro-ism [15, 34], resonant magnetic x-ray scattering [35], andmuon spin rotation measurements [36]. In β -Li IrO ,possibly related to the disappearance of the magnetic or-der at low temperatures, a first order pressure-inducedstructural transition at room temperature and at criticalpressure P S ∼ P S ∼ β -Li IrO is accompanied by theformation of Ir dimers on the one-dimensional zig-zagchains. Through resonant inelastic x-ray scattering andelectronic structure calculations, we show that the dimerconsists of local molecular orbitals derived from Ir t g electrons, indicating a breakdown of the J eff = 1/2 de-scription in the high-pressure phase. The formation ofdimers may give a clue to the origin of the putative QSLbehavior appearing prior to the structural transition, andpoints to a subtle balance of various competing electronicphases in the honeycomb-based iridates. II. EXPERIMENTAL
Neutron diffraction measurements on a powder sam-ple of β -Li IrO were performed under pressure to un-veil the detailed crystal structure of the high pressurephase. The use of neutron diffraction allowed the re-liable and precise refinement of not only the heavy Irpositions but also those of light Li and O to which neu-trons are intrinsically much more sensitive than x-rays.The measurements were conducted at the Pearl beam-line of the ISIS neutron source [37]. Pressure was ap-plied by a Paris-Edinburgh press up to 5.2 GPa [38].The anvils were single-toroidal zirconia-toughened alu-mina (ZTA), and an encapsulated TiZr gasket was used[39]. In order to minimize the neutron absorption by Ir, we prepared an isotope-enriched powder sampleof β - Li IrO [40]. Deuterium substituted methanol-ethanol mixture (4:1 by volume) was used as a hydro-static pressure medium. The applied pressure was cali-brated from the lattice constant of NaCl powder addedas a pressure-marker. All of the measurements were con-ducted at room temperature. The Rietveld refinement of diffraction patterns was performed by assuming the pres-ence of five phases, β -Li IrO , metallic Ir as an impurity,NaCl and anvil materials (ZrO and Al O ), using theGSAS program [41].To investigate the electronic structure under pressure,we performed resonant inelastic x-ray scattering (RIXS)measurements with Ir L -edge on β -Li IrO at BL12XUof SPring-8. A diamond anvil cell (DAC) was used for theapplication of pressure. A small single crystal of 50 µ msize, grown by a flux method, was loaded in a DAC withFluorinert (1:1 mixture of FC-70 and FC-77 by volume)as a pressure medium [42]. Pressure was evaluated by thefluorescence spectra of a ruby ball loaded together withthe sample. A gasket made of beryllium was used so thatthe incident and scattered x-rays go through the gasketwith minimum attenuation. The energy of incident x-rays was tuned to 11.215 keV, which corresponds to Ir2 p / d ( t g ) excitation. The incident x-ray beam wasmonochromated by a Si(111) double-crystal monochro-mator and further by a 4-bounced Si(440) high-resolutionmonochromator, and focused by using a Kirkpatrick-Baez mirror. The scattered x-rays were analyzed by adiced and spherically-bent Si(844) analyzer. The totalenergy resolution, estimated from the full width at half-maximum of the elastic line, was about 100 meV. No q -resolved measurements were performed, and the obtainedspectra are regarded as q -averaged ones. For a reference,we also collected the RIXS spectrum of polycrystalline β -Li IrO at ambient pressure without using a DAC. Alldata were collected at room temperature.The electronic structure calculations were performedwith the crystal structures refined from neutron diffrac-tion data. The calculations were carried out based on thelocal density approximation (LDA) using the fully rela-tivistic linear muffin-tin orbital (LMTO) method imple-mented in the PY LMTO code [46]. Spin-orbit couplingwas taken into account by solving the four-componentDirac equation inside an atomic sphere. This allows toobtain J resolved densities of states. III. RESULTSA. Structural transition under pressure
The result of the structure refinement at ambient pres-sure from neutron diffraction data is shown in Supple-mental Materials Fig. S1(a) and Table S1 [42], whichagrees very well with that obtained by single-crystal x-ray diffraction [15]. β -Li IrO crystallizes in an orderedrock-salt-type structure. Each IrO octahedron sharesits edges with the three neighboring IrO octahedra as in α -Li IrO [15]. The local configuration of bonds aroundan IrO octahedron is illustrated in Fig. 1(a). All of theIrO octahedra are crystallographically equivalent andform a three-dimensional network via the three almost120 ◦ bonds, termed a hyperhoneycomb lattice. The sub-lattice of Ir atoms at room temperature and at ambient TABLE I. Refined structural parameters of β -Li IrO in thehigh-pressure phase at 4.4(1) GPa. The space group is C /c (No. 15) and Z = 8. The lattice parameters are a = 5.8390(4)˚A, b = 8.1297(5) ˚A, c = 9.2240(6) ˚A, and β = 106.658(4) ◦ . g and U iso denote site occupancy and the isotropic displacementparameter, respectively. U iso was constrained to be equalacross sites containing the same element during the refine-ment. The refinement indices are R p = 2.69%, R wp = 2.27%and χ = 2.472. The Rietveld fits to the data are available inSupplemental Materials [42].Atom Site g x y z U iso (˚A )Li1 8 f f f f f f pressure is depicted in Fig. 1 (b). The hyperhoneycomblattice can be viewed as an assembly of Ir zig-zag chains,running along the a + b and the a − b directions alter-nately. The zig-zag chains are bridged by the Ir-Ir bondsalong the c -axis. The three 120 ◦ Ir-Ir bonds can be la-belled as X-, Y-, and Z-bonds; the Z-bond is the bridgingbond along the c -axis and the X and Y-bonds form thezig-zag chains. In the orthorhombic structure at ambi-ent pressure (space group: F ddd ), the X- and Y-bondsare symmetry-equivalent. The X- and the Y-bonds areappreciably longer than the Z-bond by 3% at ambientpressure.With the application of pressure, the orthorhombicunit cell displays an anisotropic contraction as reportedin Ref. [34] (see Supplemental Materials [42]). The b -axis lattice constant shows a stronger pressure depen-dence than those of the a -axis and the c -axis, whichcomes from the rapid contraction of X- and Y- bonds.By comparing the bond lengths at ambient pressure (Fig.1(b)) and at a pressure of 2.6(1) GPa (Fig. 1(d)), the X-and Y-bond lengths decrease by 1.3%, whereas that ofthe Z-bond actually increases by 0.5%. As a result, theX- and Y-bond lengths and Z-bond length become muchcloser at 2.6 GPa than at ambient pressure. The Ir-Obond lengths do not show any appreciable change fromambient pressure to 2.6 GPa, meaning that the changein the Ir-Ir bond lengths are controlled by the Ir-O-Irangle. In accord with this, the Ir-O-Ir angles for X- andY-bonds and Z-bond became much closer. (Fig. 1(c) and(d)).With further increase of pressure, a structural transi-tion from the low-pressure orthorhombic ( F ddd ) to thehigh-pressure monoclinic ( C /c ) structure takes place ataround 3.7 GPa [42]. The result of structural refinementfor the high-pressure phase at 4.4(1) GPa is listed inTable 1. The hyperhoneycomb network made of edge-shared IrO octahedra is maintained (Fig. 1(e)). The av-erage length of X- and Y-bonds along the zig-zag chainsdecreases further to ∼ FIG. 1. Crystal structure of β -Li IrO . (a), (c) local structurearound an IrO octahedron at 0 and 2.6 GPa, respectively.(b), (d) Hyperhoneycomb network of Ir atoms. X, Y and Zdenote the 3 types of Ir-Ir bonds. (e) Crystal structure in thehigh-pressure monoclinic phase at 4.4(1) GPa. (f) Ir networkin the high-pressure phase. The dimerized bond is shownin red. The crystal structures are illustrated using VESTAsoftware [59]. length increases to 3.026(6) ˚A. This appears to be an ex-tension of the anisotropic pressure dependence betweenX(= Y) and Z in the low-pressure phase. A modula-tion of the Ir-Ir bond length in the zig-zag chains, how-ever, makes the high-pressure phase distinct from thelow-pressure phase. As seen in Fig. 1(f), the X-bondand Y-bond are no longer equivalent at 4.4 GPa. Whilethe X-bond length is as long as 3.069(5) ˚A close to oreven longer than the increased Z-bond length, the Y-bond length becomes as short as 2.663(5) ˚A, which givesrise to an alternating arrangement of the short Y-bondsand the long X-bonds along the zig-zag chains. In fact,at a distance of 2.66 ˚A, the Ir-Ir Y-bond is even shorterthan that seen in metallic Ir, indicating the formation ofan Ir dimer molecule in the zig-zag chains [47]. B. Collapse of J eff = 1/2 state in the high-pressuredimerized phase RIXS measurements unveil the drastic reconstructionof the electronic structure associated with this dimeriza-tion. The Ir L -edge RIXS spectrum at ambient pressure,measured on a polycrystalline pellet, is displayed at thebottom of Fig. 2. In addition to the elastic scatteringpeak at 0 eV, there are two pronounced features; a sharppeak at around 0.7 eV and a broad peak centered ataround 3.5 eV. The latter corresponds to the excitationsfrom Ir 5 d t g to e g manifolds. The peak at ∼ J eff =3/2 and the half-filled J eff = 1/2 state, as observed ina number of d iridium oxides [48–51]. The peak energyrepresents the spin-orbit splitting between J eff = 3/2 and J eff = 1/2, namely 3 λ SO /2. This evidences the dominant J eff = 1/2 character of t g holes in β -Li IrO at ambientpressure.RIXS spectra under pressure were collected with a sin-gle crystal loaded in a DAC. At a low pressure of 0.9(1)GPa, the spectrum is almost identical to the polycrys-talline data. This supports the idea that the d - d ex-citations show only a small q -dependence in β -Li IrO .With increasing pressure up to 3.1(1) GPa, the 0.7 eVpeak remains at the same energy but tends to be moder-ately broadened. This likely suggests a pressure-inducedchange of electronic structure, for example, the mixingof J eff = 3/2 and 1/2 state by enhanced hopping. Wenote that a pressure-induced change of electronic struc-ture under pressure, prior to the structural transition,was inferred also by the loss of magnetic field inducedferromagnetic moment and the pronounced change of thebranching ratio of the x-ray absorption spectrum above2 GPa [34].A pronounced change of RIXS spectra was found inthe high-pressure dimerized phase above 4 GPa. The 0.7eV peak is suppressed almost completely. The one at 3.5eV is significantly broadened, more significantly on thehigh-energy side, but remains in the high pressure phase.Instead of the 0.7 eV-peak, a broad continuum spreadsroughly from 0.5 to 2.5 eV and a peak feature around 2.8eV emerges. The drastic change of RIXS spectra, alongwith the suppression of the 0.7 eV peak, clearly pointsto the collapse of the spin-orbit coupling splitting of J eff = 1/2 and J eff = 3/2 states in the high-pressure phase.We note that the spectrum of the low pressure phase wasrecovered after depressurization, excluding the possibilityof irreversible chemical change of the sample due to x-rayirradiation and/or pressure. C. Formation of molecular orbital in the Ir dimers The electronic structure calculations were performedusing the structural parameters in the ambient and thehigh-pressure phases, as shown in the plot of partial den-sities of states from the relevant orbitals in Fig. 3(a) and(b), respectively. Spin-orbit coupling was incorporatedto the calculation. On-site Coulomb U was not explic-itly introduced, which makes the ambient pressure phasemetallic. In Fig. 3(a) for the low pressure phase, thesplitting of t g -derived bands into the J eff = 3/2-derivedbands around -1 eV and J eff = 1/2-derived bands aroundthe Fermi energy by spin-orbit coupling is clearly seen asin the previous calculations [19, 34]. Appreciable mix- ! " $ % " & ’ ( ) * + * , -./012 / ’ ’;’/22’< =7>&?4&$")>2*@’A=)/*1’A=)B*2’A=)C*.’A=)0*B’A=)’(D FIG. 2. RIXS spectra of β -Li IrO under pressure recordedat room temperature. The data from the polycrystalline sam-ple (bottom) were collected at ambient pressure, and the dataat 2.6(1) GPa (top) after the sample was depressurized from7.4(1) GPa. The spectra are shown with arbitrary offsets.The horizontal broken-lines represent the guide base lines, ob-tained by subtracting a constant background from each spec-trum. ing between J eff = 3/2 and 1/2 can be recognized as inother iridium oxides due to the presence of trigonal dis-tortion and/or the J eff = 1/2 ↔ d xy -, the d yz - and the d zx -orbitals, instead of the spin-orbital-entangled J eff = 1/2and 3/2 states, appear to represent the character of thebands in the high-pressure phase in Fig. 3(b). The d zx orbital, directed along the dimer bond (Y-bond) providesthe dominant character of the two sub-bands stemmedout of t g derived bands; the lowest (-1.7 eV) occupiedsub-band and the highest (+ 0.7 eV) empty sub-band.The two sub-bands with predominant d zx character canbe assigned to the bonding and the antibonding states ofIr dimer molecules with a splitting energy of ∼ t g orbitals, d xy - and d yz -orbitals mainly con-tribute to the sub-bands between the d zx - bonding andantibonding sub-bands. Because of the degeneracy of d xy - and d yz -orbitals, strong spin-orbital-entanglement isexpected for these d xy - and d yz -derived sub-bands, whichare denoted as entangled xy - yz orbitals with different col-ors in Fig. 3(b). Since the hybridization of entangled xy - yz orbitals between the nearest neighbor Ir atoms is muchweaker than that of d zx orbitals, it is natural that theyreside in between the bonding and antibonding orbitalsof d zx . As a result, Ir d electrons fill up the bonding d zx !" & ’ ( ) * $ + , - + . / !" !" & ’ ( ) * $ + , - + . / ;6’ " *.2*<2!=!)>?. ) ,@@ )A)$+ ,@@ )A)"+ ,@@ )A)"+ " " %& ),5/.57C,D)&’3’%),5/.57C,D)&’3’%)! " " )%)>?. FIG. 3. Calculated density of states (DOS) for Ir 5 d states.(a) DOS for the ambient-pressure phase ( F ddd ). The Ir t g orbitals are resolved into J eff = 1/2 and 3/2 characters. (b)DOS for the high-pressure phase at 4.4 GPa ( C /c ). The d zx -orbital is directed along the Ir dimer bond. The other t g orbitals, d xy and d yz , are entangled by spin-orbit cou-pling. The total DOS includes the contributions from oxygen2 p states. sub-bands and the four entangled xy - yz sub-bands. Anenergy gap is formed between the entangled xy - yz sub-bands and the empty antibonding d zx sub-band, yield-ing a band insulating state. The bandwidth of occu-pied states increases appreciably as compared with thatat ambient pressure, which is consistent with the broadfeature observed in the RIXS spectra at the low-energyregion up to 2.5 eV. We note that the effect of electroncorrelations may narrow down the bandwidth and makethe RIXS peaks sharper. The e g orbitals are almost de-generate at ambient pressure, but split appreciably at 4.4GPa due to the strong distortion of IrO octahedra. Thisaccounts for the broadening of the RIXS peak at 3.5 eVin the high-pressure phase.The formation of Ir dimers in the hyperhoneycomblattice gives rise to bonding and antibonding molecu-lar orbital states made of d zx orbitals in the bondingplane. The large bonding-antibonding splitting stabilizesa d zx -orbital-dominant antibonding state of t g holes andmakes the system a band insulator, which is consistentwith the negligible XMCD above 4 GPa [15, 34]. Theemergence of the d zx -orbital dominant state results inthe collapse of the J eff = 1/2 state. IV. DISCUSSION
The present result points to a competition betweenthe spin-orbital-entanglement and the dimerization in β -Li IrO . The former mixes up the different orbital statesto create the orbital moment. The latter selects a spe-cific orbital to gain bonding energy. It was theoreti-cally discussed that β -Li IrO shows an intrinsic insta-bility towards the formation of Ir dimers when spin-orbit coupling is neglected [32]. At ambient pressure,the spin-orbit coupling overcomes the dimer instabilityand the J eff = 1/2 state is formed. By increasing theoverlap of orbitals, the instability is enhanced and even-tually the dimer phase shows up. Similar dimerizationunder pressure was recently identified in another honey-comb iridate α -Li IrO [54, 55]. We note that in theprevious theoretical approaches, the dimerization is pre-dicted to take place within the Z-bond [32]. In both α -Li IrO and β -Li IrO , the dimers are formed in thezig-zag chains made of the X- and Y-bonds rather thanthe Z-bond. The anisotropic lattice contraction of thezig-zag chains under pressure seems to be closely relatedto the preferable dimerization in the zig-zag chains, theorigin of which is worthy of further exploration. Thedimerization of transition-metal ions has been frequentlyseen in not only in honeycomb-based 5 d iridates but alsoin a wide variety honeycomb based 3 d and 4 d oxides andhalides, including α -TiCl (3 d ) [56], α -MoCl (4 d ) [57]and Li RuO (4 d ) [58] even at ambient pressure. Theoccurrence of dimerization only under a high pressuremay reflect that the competition with the spin-orbital-entangled phase is much more significant in the 5 d iri-dates with spin-orbit coupling of ∼ P S ∼ β -Li IrO . The disappearance of mag-netic ordering in β -Li IrO in the T = 0 limit was re-ported to occur at ∼ γ -Li IrO shows ananalogous pressure collapse of the magnetic ordering at P c = 1.5 GPa but no signature of structural dimeriza-tion up to 3.3 GPa [35]. It may be interesting to inferthat the instability to dimer formation may be relevantfor the breakdown of magnetic order in such 3D-basedhoneycomb iridates. V. CONCLUSION
We studied the crystal and electronic structures of thehyperhoneycomb iridate β -Li IrO in the high-pressurephase above 4 GPa. The high pressure phase is character-ized by the formation of Ir dimers on the zig-zag chains.The spin-orbital-entangled J eff = 1/2 states break down,associated with the stabilization of the bonding state ofthe neighboring d zx -orbitals in the dimer phase. Suchcompetition of spin-orbital-entanglement and dimer for-mation are indeed widely observed in honeycomb-basediridates, and we argue it is one of the hallmarks of thephysics of these materials. ACKNOWLEDGEMENTS
We are grateful to D. Haskel, L. S. I. Veiga, S. K.Choi, K. Kitagawa and R. Dinnebier for helpful discus- sions. We thank S. K. Choi, H. H. Kim, U. Engelhardtand K. Syassen for their help in the preparation of DACexperiments. We acknowledge the provision of beamtimefor the PEARL experiment (Proposal No. RB1710302)to Science & Technology Facilities Council (STFC),and the allocation of beamtime at BL11XU of SPring-8 (Proposal No. 2017-1-119-1/2017A4253, 2017-1-119-3/2017B4246, and 2017-3-078-3/2018A4260) to the Na-tional Synchrotron Radiation Research Center (NSRRC)and the Japan Synchrotron Radiation Research Institute(JASRI). This work was partly supported by the Alexan-der von Humboldt foundation, Japan Society for the Pro-motion of Science (JSPS) KAKENHI (No. JP15H05852,JP15K21717, 17H01140), and JSPS Core-to-core pro-gram Solid-state chemistry for transition-metal oxides [1] A. Kitaev, Annals of Physics , 2 (2006).[2] J. G. Rau, E. K. H. Lee, and H. Y. Kee, Annu. Rev.Condens. 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