Real-space determination of orbital states driving successive phase transitions in FeV2O4

"Direct experimental access to orbital states in strongly correlated materials remains a major challenge, despite their central role in driving coupled structural and magnetic phase transitions. In systems where electronic correlations, electron–lattice coupling, and relativistic spin–orbit interactions compete on comparable energy scales, even first-principles calculations often yield multiple metastable solutions, hindering the unambiguous identification of the ground state. Here, we demonstrate that the orbital states of the spinel oxide FeV2O4, which possesses active orbital degrees of freedom on both Fe and V ions, are uniquely resolved by combining valence electron density (VED) analysis based on state-of-the-art synchrotron x-ray diffraction with spin-polarized density-functional-theory calculations. Our results reveal that temperature-dependent rearrangements of orbital occupations drive successive structural transitions that accompany collinear and noncoplanar ferrimagnetic orders, establishing a direct correspondence between orbital anisotropy and spin structure. More broadly, this work shows that experimentally determined VED provides a decisive real-space constraint on competing theoretical solutions, offering a powerful and broadly applicable framework for elucidating the microscopic mechanisms of complex phase transitions in strongly correlated electron systems."

"The anisotropy of valence electrons is fundamentally intertwined with a variety of emergent quantum phenomena, including charge and orbital order(1–5), unconventional superconductivity(6, 7), quantum spin liquid(8), and multipole order(9). In such systems, orbital configuration directly governs the electronic states and physical properties. However, since external fields that couple directly to orbital degrees of freedom are limited, orbital states are considerably less accessible to direct experimental observation than lattice distortions or magnetic structures. Traditionally, orbital states have been inferred indirectly from Jahn–Teller distortions, which manifest as structural deformations of metal–ligand polyhedra through electron-phonon coupling. While such structural information generally provides valuable insight in strongly correlated electron systems, it is rarely sufficient on its own. Electron correlation, relativistic spin–orbit interaction, and finite-temperature effects act on comparable energy scales, obscuring the simple correspondence between lattice distortions and orbital occupations."

"A prototypical example in which orbital degrees of freedom give rise to exotic electronic states is the spinel-type iron vanadium oxide FeV2O4, which crystallizes in the space group Fd3̅ m at high temperatures [Fig. 1A]. In this compound, each Fe2+ ion (3d6) is coordinated by four O2- ions to form a FeO4 regular tetrahedron, while each V3+ ion (3d2) is surrounded by six O2- ions to form a VO6 octahedron with trigonal distortions. In contrast to AV2O4 (A = Mg, Zn, Cd, Mn)(10–15), both the Fe and V ions host orbital degeneracies in the high-temperature cubic phase in FeV2O4."

"Figure 1B shows the changes in lattice deformations across the successive phase transitions in FeV2O4(15–18). Hereafter, a, b, and c denote the lattice parameters in the standard crystallographic setting of the cubic phase, whereas a′, b′, and c′ are used as a unified notation, referenced to the cubic phase, for all temperature-dependent phases. As shown in Fig. 1B, upon cooling, the cubic lattice first transforms to tetragonal with I41/amd symmetry at Ts = 140 K, in which the lattice is contracted along the c′-axis. Below TN1 = 110 K, collinear ferrimagnetic (C-FM) order emerges, with Fe2+ and V3+ moments aligned antiparallel to each other [Fig. 1C], accompanied by a further lowering of lattice symmetry from tetragonal to orthorhombic, with space group Fddd. The ferrimagnetic moment appears along the longest a′-axis [Fig. 1B]."

"Below TN2 = 70 K, the structure reverts from orthorhombic to tetragonal with space group I41/amd, accompanied by a transition to a noncollinear ferrimagnetic (NC-FM) state(15–18). In contrast to the high-temperature tetragonal phase, the low-temperature tetragonal lattice is elongated along the fourfold axis, unlike in other spinel vanadates AV2O4 with orbitally inactive A ions(10–15). Note that Fig. 1B depicts the low-temperature tetragonal phase using a face-centered tetragonal cell with a′ as the unique axis. The Fe spins remain collinear, whereas the V spins are tilted by approximately 55 degrees Fig. 1C. In the two lower-temperature phases, the spin arrangements of Fe and V are strongly coupled to their orbital states(15, 16, 18)."

"In this study, we address this problem by applying external magnetic fields to a smaller crystal across the magnetic phase transitions. In the low-temperature phases, this approach strongly biases the domain populations, reducing the number of populated domains to two or fewer and thereby allowing us to obtain high-quality diffraction data suitable for precise VED analysis. This enables us to follow the temperature evolution of the VED across the successive phase transitions and to establish a comprehensive experimental picture of how the Fe and V orbital states are reconfigured through the successive phases. Comparison between the experimental VED and spin-polarized density-functional-theory (DFT) calculations further allows us to distinguish between nearly degenerate microscopic solutions for the ground-state orbital configuration and to clarify their relation to the spin structure [Figs. 1D and 1E]. More broadly, our results show that experimentally determined VED provides a practical route to identifying orbital states and resolving their coupling to spin and lattice degrees of freedom in strongly correlated electron systems."

"VED in the Cubic phase: Figure 2A shows the VED distribution at 160 K (Cubic phase), obtained from the CDFS analysis. As shown by yellow iso-density surfaces of 3.9e/Å3, the VED distribution appears isotropic around the O2- ions, which is consistent with the 2s22p6 electronic configuration. On the other hand, weak and pronounced anisotropies are observed around the Fe and V sites, respectively, as shown in the zoomed-in views [Figs. 2B and 2C]. Since the FeO4 forms a regular tetrahedron, the Fe 3d orbitals split into a lower-lying e doublet and a higher-lying t2 triplet. In the high-spin configuration for Fe2+ (3d6), the five majority-spin electrons fully occupy the 3d orbitals, while the remaining minority-spin electron resides in degenerate e orbitals (3z2–r2 and x2–y2) with equal probability [inset, dashed square in Fig. 2B]. Since x-ray diffraction probes orbitals as time- and space-averaged states, the observed VED anisotropy around the Fe site can be interpreted as a manifestation of this orbital disorder(35). A similar interpretation can be applied to the VED distribution around the V 3+ (3d2) ion. In a regular VO6 octahedron, the 3d orbitals split into a higher-lying eg doublet and a lower-lying t2g triplet. However, the VO6 octahedron is slightly elongated along the local three-fold axis, satisfying the.3̅ m symmetry at the V site. As a result, the t2g orbitals are further split into a1g and e'g orbitals. If considering only the crystal field from the adjacent six O2- ions forming the octahedron, the e’g orbitals would lie lower in energy than the a1g orbital. The influence of more distant ions using the Ewald method(39), however, reverses this hierarchy, stabilizing the a1g orbital relative to the e’g orbitals(35). The observed VED anisotropy around the V site confirms the latter situation, where one electron occupies the a1g orbital and the other is equally distributed to the e'g doublet [Fig. 2C]. Starting from this Cubic phase orbital state, we discuss how successive phase transitions lift the degeneracy of Fe and V orbitals and lead to the ground state."

"FeO4 deformation and VED around the Fe site associated with the phase transitions: Figure 3 shows the distortions of FeO4 and VO6 polyhedra associated with the phase transition, as determined by synchrotron x-ray diffraction. As shown in Fig. 3A, the FeO4 distortions contain two-dimensional normal modes of Q2,Fe and Q3,Fe, which cause splitting of the e orbitals [Fig. 2B]. First, we examine the successive change in FeO4 distortion. Figures 3B and 3C present the temperature dependence of Q2,Fe and Q3,Fe for the FeO4 tetrahedron. Orange iso-density surfaces in Fig. 3C show the minority-spin electron densities of Fe predicted from the local FeO4 distortions in each phase. In the Cubic phase, the Q2,Fe and Q3,Fe modes are absent. In the HT-tetra phase at 130 K, negative Q3,Fe coupled to the 3z2–r2 orbital emerges, followed by the emergence of Q2,Fe in the Ortho phase at 100 K. Here we consider the domain with Q2,Fe > 0. In the LT-tetra phase, the FeO4 tetrahedron is elongated along the x-axis and couples with the y2–z2 orbital. To directly elucidate Fe 3d orbital states, we examine the VED distribution. Figure 4A shows the temperature evolution of the VED around the Fe site. In the Cubic phase, high-density regions, shown by orange, are located on the x-, y-, and z-axes. In the HT-tetra phase, enhanced VED develops along the z-axis reflecting the splitting of the e orbitals and minority-spin occupation of 3z2–r2 orbital. Upon further cooling into the Ortho phase, the VED along the y-axis increases, described by a linear combination of the 3z2–r2 and y2–z2 orbitals. In the LT-tetra phase, high-density regions extending within the yz-plane are observed, corresponding to the stabilization of the y2–z2 orbital. Figure 4B presents the calculated VED distribution for the minority-spin of Fe, which reproduces the anisotropy observed in the high iso-density surfaces shown in orange [Fig. 4A]. These results are consistent with those expected from the distortion of the FeO4 tetrahedron [Fig. 3C]."

"The successive changes in Fe 3d orbital state can be understood in terms of the cooperative Jahn–Teller effect below TS and the spin–orbit coupling in the magnetic phases, which stabilize the 3z2–r2 and y2–z2 orbitals, respectively(15, 16, 18, 40). The effective Hamiltonians for the Jahn–Teller coupling and the spin-orbit coupling in the FeO4 tetrahedron are given by HJT=A(τxQ2,Fe−τzQ3,Fe), and HSOC = B/6[(3S2z − S2)τz −√3(S2x − S2y)τx] (41), respectively. Here, A (<0) is an electron-phonon coupling constant, τz and τx are pseudospin operators representing 3z2–r2 (τz = 1/2) and x2–y2 (τz = -1/2) orbitals. S, Sx, Sy, and Sz are spin operators, B (>0) represents a second-order perturbation of spin–orbit coupling. Firstly, we examine the HTtetra phase. Focusing on the anharmonic term of the lattice elastic energy, U = A3Q3cos(3θ), the compressed tetragonal distortion is stabilized. Here, A3 (>0) is the third-order elastic constant, Q = (Q2,Fe2 + Q3,Fe2)1/2, and θ is the polar angle of the distortion vector (Q3,Fe, Q2,Fe) in the Q2,Fe- Q3,Fe plane. This induces cooperative Jahn–Teller distortion, stabilizing the Fe 3z2-r2 orbitals. At lower temperatures where magnetic order emerges, the system undergoes a further structural transition. Under second-order spin-orbit coupling, the favored orbital depends on the spin operation: the y2−z2 orbital is stabilized when the spin moment is aligned along the x-axes. When Fe2+ magnetic order emerges at TN1, the orbital state is therefore modulated by spin–orbit coupling. As the spins align along the x-axis below TN1 in the Ortho phase, the y2−z2 orbital is stabilized by spin–orbit coupling, leading to a hybrid of 3z2–r2 and y2–z2 orbitals(15). Consequently, Q2,Fe becomes non-zero through Jahn–Teller coupling, resulting in an orthorhombic distortion. Here, Q2,Fe > 0, as we adopt the orthorhombicity of a > b > c. Indeed, in FeCr2O4, Cr3+ ions without orbital degrees of freedom occupy the octahedral sites, the ground state remains an orthorhombic phase with conical order of Fe and Cr spins(41, 42). This suggests that the cooperative Jahn–Teller effect and spin–orbit coupling at the Fe site account for the tetragonal-to-orthorhombic transition. However, the reentrant tetragonal symmetry in the ground state of FeV 2O4 cannot be explained within this Fe-only framework, indicating an essential role of the orbital degrees of freedom at the V sites."

"VO6 deformation and VED around the V site associated with the phase transitions: The symmetric distortion of a VO6 octahedron can be decomposed into Eg (Q2,V, Q3,V) and T2g (Q4,V, Q5,V, Q6,V) modes [Fig. 3D]. In the Cubic phase, Q4,V = Q5,V = Q6,V > 0 and their linear combination Qt1 v = (Q4 v + Q5 v + Q6 v)/√3 corresponds to elongation or compression of the VO6 octahedron along the [111] axis. This trigonal distortion leads to the splitting of the t2g into a1g and e’g orbitals. Figure 3E shows the temperature dependence of each Q mode of the VO6 octahedron. The Q2,V and Q3,V values, which directly couple to the splitting of the eg orbitals, remain nearly zero, whereas Q4 v, Q5 v, and Q6 v values exhibit pronounced temperature dependence, reflecting their coupling to the splitting of the t2g orbitals. Notably, the large Qt1 V value is nearly temperature-independent, indicating that one electron continues to occupy the a1g orbital down to the lowest temperatures. To further probe the splitting of the e’g orbitals, we examine Q(z)t2 V = (−Q4 v − Q5 v + 2Q6 v)/√6 and Q(z)t3 V = (−Q4 v + Q5 v)/√2 modes, which are orthogonal to Qt1 V [Fig. 3F]. In the Cubic phase at 160 K, both Q(z)t2 V and Q(z)t3 V values are zero, indicating that the e’g orbitals remain degenerate. In the HT-tetra phase at 130 K, a positive Q(z)t2 V mode emerges while Q(z)t3 V remains zero, corresponding to tetragonal distortion. Subsequently, in the Ortho phase at 100 K, a positive Q(z)t3 V mode appears. Upon entering the LT-tetra phase, Q(z)t3 V gradually increases. We emphasize that the value at 50 K lies along the Q(x)t2 V = (−Q5 v −Q6 v + 2Q4 v)/√6 axis when the Q4,V direction is taken as the principal axis, corresponding to Q(x)t3 V = (−Q5 v +Q6 v)/√2 = 0. The continuous evolution from positive Q(z)t2 V to negative Q(x)t2 V may reflect modifications of the 3d wave functions."

"Next, we turn to the VED distribution around the V site [Fig. 5A], which also continuously changes upon cooling. Here, we define t2g wavefunctions as Ψ1 = C1yz⟩+C2zx⟩+C3xy⟩, Ψ2 = D1yz⟩+D2zx⟩+D3xy>, and Ψ3 = E1yz⟩+E2zx⟩+E3xy>, with C1 2+C2 2+C3 2=1, D1 2+D2 2+D3 2=1, and E1 2+E2 2+E3 2=1, where the coefficients are chosen to satisfy the site-symmetry and mutual-orthogonality constraints (see SI Appendix section3). To elucidate the orbital states, the quantum parameters Ci, Di, and Ei (i = 1 to 3) are optimized by fitting the calculated VED distributions to reproduce the observed anisotropy. Assuming that the trigonal distortion is large enough for one electron to occupy Ψ1, the calculated 3d2 VED is expressed asrhocalc (r) = psi1 2 + ηpsi2 2 + (1 − η)psi3 2, where η represents the electron filling of Ψ2. The evaluation function s for the fitting is defined as s=∑rrhoobs (r) − κ 3 ρcalc (κr)/∑rrhoobs (r), where ρobs (r) is the observed VED around the V site, κ is a variable parameter representing contraction of the VED. The quantum parameters are optimized by minimizing the evaluation function in Eq. (2) within a radial range of 0.25 < r < 0.6 Å around the V site (see SI Appendix Fig. S6). The evaluation function is minimized at approximately η = 0.5 in all the phases."

"Spin-polarized DFT+U calculations in the LT-tetra phase: The 3d2 orbital states of V3+ in the low-temperature phases can be interpreted within two possible frameworks, as shown in Fig. 5D. One possible scenario is that the splitting of the e'g orbitals is sufficiently small that the system resides in a thermally accessible pseudo-degenerate state. Alternatively, the VED with η = 0.5 can also arise from wavefunctions stabilized in magnetically ordered states through relativistic spin–orbit interaction(27). The two scenarios are impossible to distinguish based solely on the VED distribution. To address this issue, we perform spin-polarized DFT+U calculations for the LTtetra phase considering the relativistic spin–orbit interaction. The three V t2g wavefunctions are defined as Ψ1=γ yz⟩ + √(1 − γ2)/2zx⟩ + √(1 − γ2)/2xy⟩, Ψ2=−√1 − γ2 yz⟩ + (γ/√2)zx⟩ + (γ/√2)xy〉, and Ψ3=(1/√2)zx (− 1/sqrt(8)) - (1/sqrt(8))xy〉 [as illustrated in Figs. 1E and 6C]. The Hubbard U raises the energy of unoccupied eg states, thereby promoting gap formation between the t2g and eg orbitals. Interestingly, we find two solutions depending on the mechanism of orbital order. In one solution, Fe spin moments are ferromagnetically arranged along the fourfold axis (−x direction) and V spins are aligned nearly in the opposite direction with small canting arising from the spin–orbit interaction [Fig. 6A]. The Hubbard U effectively enhances the crystal field effect of the compression of VO6 along the xaxis, raising the energy of the Ψ3 orbital, composed of zx and xy orbitals, relative to the Ψ2 orbital, which contains the yz component [Figs. 6B and 6C]. Consequently, orbital ordering consisting of Ψ1 and Ψ2 emerges. The predicted VED is shown in Fig. 6D, which is, however, inconsistent with the experimental observation [Fig. 5A]. In an alternative solution, the V spin moments exhibit a “two-in-two-out” configuration within the V4 tetrahedra of the pyrochlore network [Fig. 6E]. In this case, Ψ1 and (Ψ2 + iΨ3)/√2 constitute the orbital order [Figs. 6F and 6G]. The complex orbital (Ψ2 + iΨ3)/√2 gives an orbital angular moment of approximately 1 μB, oriented close to⟨111⟩, corresponding to the direction toward the center of the V4 tetrahedron. This orientation is nearly opposite to the local spin directions in the two-in-two-out configuration. In fact, an x-ray magnetic circular dichroism study(24) has demonstrated that the V site possesses a finite orbital magnetic moment component parallel to the magnetization. It is likely that spin–orbit interaction triggers orbital ordering, assisted by the gap enhancement due to the Hubbard U."

"Importantly, in the wavefunction Ψ1 = γyz⟩ + √(1 − γ2)/2zx⟩ + √(1 − γ2)/2xy⟩, where γ = 0.76 is determined by fitting to the experimental VED, the occupation of the yz orbital is inequivalent to those of the zx and xy orbitals. The latter two carry equal weights, as dictated by the calculated symmetry. As a result, the threefold rotational symmetry is explicitly broken, whereas the exchange symmetry between zx and xy remains preserved. As a consequence, the local crystal field retains a mirror symmetry that leaves the yz orbital invariant and exchanges zx and xy. This mirror plane contains the local ⟨111⟩ axis, and therefore the orbital angular momentum induced by spin–orbit coupling is constrained to lie within this plane. The spin moment is thus constrained to cant within the same mirror plane, fixing its azimuthal direction. Using the orbital wavefunctions in LT-tetra phase, the direction of the local spin moment is obtained as S ∝ (− γ, −√(1 − γ2)/2, −√(1 − γ2)/2). This corresponds to a polar angle of approximately 41 degrees from the −x direction and about 8 degrees from the direction antiparallel to the local ⟨111⟩ axis, indicating a small but finite canting away from the threefold rotational axis. This estimate is in good agreement with the canting angle of 43 degrees obtained in the DFT calculation. Notably, because the canting direction is confined on the local mirror, the spin configuration on the pyrochlore lattice becomes intrinsically noncoplanar, thereby generating a finite scalar spin chirality."

"The origin of the reentrant tetragonal ground state: The reentrant tetragonal symmetry in the ground state can be understood as a consequence of spin–orbital–lattice cooperation and competition involving both Fe and V sites. The canting of the V spins is primarily determined by competing Fe–V and V–V antiferromagnetic exchange interactions. The evolution of the mode Q(x)t2 V toward negative values in the LT-tetra phase reflects a reconfiguration of the t2g orbital components, corresponding to an increase in the occupancy of the yz orbital component [Fig. 3F]. Through spin-orbit coupling, this orbital reconfiguration further stabilizes the two-in-two-out spin configuration of the V sublattice, providing an additional relativistic energy gain. This evolution in the V sector modifies the balance of spin–orbital–lattice coupling at low temperatures. Consequently, the energetic hierarchy between the anharmonic term of the lattice elastic energy for the FeO4 distortion and the relativistic spin–orbit interaction at the Fe site is altered, effectively reversing the preferred distortion direction and stabilizing the elongated FeO4 distortion along the x axis, which couples to the y2–z2 orbital. Notably, the magnitude of the FeO4 distortion exceeds that of VO6 [Figs. 3C and 3F], indicating that the FeO4 units dominate the overall lattice deformation associated with the structural phase transitions and ultimately give rise to the elongated tetragonal ground state along the a’ axis."

"Finally, we consider the orbital states of the V3+ ion in the intermediate HT-tetra and Ortho phases. Previous thermal conductivity measurements(22) and powder neutron diffraction(17) suggest that V orbital order is not established in these intermediate phases. Consequently, the electron is likely to fluctuate within the Bloch sphere defined by Ψ2 and Ψ3 orbitals. If such fluctuations are sufficiently slow on the relevant experimental time scale, they may give rise to a dynamical Jahn–Teller effect. However, we observe neither x-ray diffuse scattering nor anomalies in anisotropic atomic displacement parameters (see SI Appendix section1). To directly probe this pseudo-degenerate state in the intermediate phases, spectroscopic investigations of phonon dynamics, such as Raman or infrared spectroscopy, would therefore be useful."

"Conclusion: We have directly visualized the temperature evolution of the VED distributions in FeV2O4 across its successive structural and magnetic phase transitions. The observed VED anisotropies around the Fe and V sites reveal how the local orbital states are reconstructed from the orbitally degenerate cubic phase to the low-temperature phases, demonstrating that the phase evolution is governed by cooperative spin–orbital–lattice coupling involving both sublattices. By combining these real-space observations with spin-polarized DFT calculations, we further demonstrate that the lowest-temperature V orbital state can be uniquely identified through direct comparison with experiment, thereby discriminating between competing microscopic scenarios that are nearly degenerate in theory. These results establish experimentally determined VED as a decisive probe of orbital physics in strongly correlated materials, going beyond conventional crystal structure-based inference. More broadly, our work provides a general strategy for identifying hidden orbital states and their coupling to magnetism and lattice symmetry in quantum materials, particularly when multiple metastable solutions compete on comparable energy scales."

"Materials and Methods: X-ray diffraction Single-crystal x-ray diffraction experiments were conducted at the BL02B1 beamline(43) of the SPring-8 synchrotron facility in Japan. The single crystals used in this study were the same as those in Ref. 15. Temperature was controlled using an N2/He gas-blowing device. The wavelengths were 0.30945 Å at 160 and 50 K, and 0.31080 Å at 130 and 100 K. Prior to the measurements in the Ortho and LT-tetra phases, the sample was cooled while applying a magnetic field along the a’-axis of the cubic phase using a Neodymium magnet to align structural domains. Diffraction patterns were recorded using a two-dimensional CdTe PILATUS detector(44) with a dynamic range of 106. Data were acquired using the Fine Slice method by oscillating the crystal and dividing reciprocal space into intervals of Δω = 0.1°(45). The intensities of Bragg reflections were collected by CrysAlisPro(46). Intensities of equivalent reflections were averaged using SORTAV(47), and structural parameters were refined by Jana2006(48). Here, by utilizing only high-angle reflections (sinθ/λ > 0.6 Å−1), where the contribution of spatially spread valence electrons to x-ray diffraction is negligible, structural parameters including atomic displacement parameters were obtained with high accuracy."

"CDFS analysis The CDFS method was used to extract the VED distribution. The [Ar]-type configuration (1s2,2s2,2p6,3s2,3p6) of the Fe and V atoms as well as 1s2 electrons of O atoms were regarded as core electrons. The contribution of the thermal vibration was subtracted from the VED using the atomic displacement parameters determined by the high-angle analysis. The voxel size of the three-dimensional VED distribution was 0.05 Å3. We used the STO-COPPENS Slatertype orbital (STO) library implemented in Jana2006 for the CDFS analysis. The crystal structure and VED distributions were visualized using VESTA(49)."

"DFT calculation The DFT electronic structure calculations for FeV2O4 were performed using Quantum Espresso(50) using the experimental structure of the LT-tetra phase obtained in this study. We employed relativistic norm-conserving pseudopotentials with Perdew-Burke-Ernzerhof (PBE)(51) exchange-correlation functional, which were taken from the PseudoDojo(52). We performed spinpolarized calculations considering the spin–orbit coupling with k-mesh of 7×7×7. The effects of Coulomb interactions of Fe 3d and V 3d orbitals were incorporated within the DFT+U formalism(53) with the Hubbard U parameters of 5 eV. The energy cutoff was set to 100 Ry for the wave functions and 400 Ry for the charge density. Based on the DFT electronic structure, we constructed Wannier orbitals(54, 55) using RESPACK(56, 57) and Wannier90(58) for Fe 3d, V 3d, and O 2p manifold. Since a primitive unit cell consists of two Fe sites, four V sites, and eight O sites, the number of Wannier orbitals was (5×2+5×4+3×8)×2=108, where the factor of two accounted for the summation over spin degrees of freedom."

Acknowledgments We thank A. Nakano, T. Ohashi, Y. Yamanaka, and T. Sasaki for supporting the x-ray diffraction experiments, and T. Katsufuji, T. Hasegawa, K. Siratori, and T. Nakai for the fruitful discussions.

This PDF file includes: Supporting text Figures S1 to S9 Tables S1 to S13 SI References

"Supporting Information Text: 1. Single-crystal structural analysis using synchrotron X-ray diffraction. FeV2O4 undergoes three structural phase transitions upon cooling—from the cubic phase to the high-temperature tetragonal (HT-tetra), orthorhombic (Ortho), and finally the low-temperature tetragonal (LT-tetra) phases, as described in the main manuscript. The structural analysis results for FeV2O4 at 160 K (Cubic), 130 K (HT-tetra), 100 K (Ortho), and 50 K (LT-tetra) are summarized in Tables S1-12. Taking a, b, and c as the crystallographic axes of the cubic phase, the axes of HT-tetra, Ortho, and LT-tetra phases are defined as follows: aHT = 1/2a+bHT = bHT = c HT = c; aO = -a, bO = c, and cO = b+c; aLT = -a', bLT = 1/4b', and cLT is the transformation matrix P shown in Table S13. In this setting, x a', y b' and z c'. 2. Q-mode analysis. To consider the V3+ and Fe2+ orbital states, we analyze the distortion of the VO6 octahedron and the FeO4 tetrahedron in terms of normal (Q) modes [1]. In case of octahedral coordination, symmetric distortion can be decomposed into six modes: A1g(Q1) mode, Eg (Q2, Q3) modes and T2g (Q4, Q5, Q6) modes. The six modes can be expressed as: Q1 = (X1 + Y2 + Z3 − X4 − Y5 − Z6)/√6; Q2 = (X1 − Y2 − X4 + Y5)/2; Q3 = (−X1 − Y2 + 2Z3 + X4 + Y5 − 2Z6)/2√3; Q4 = (Z2 + Y3 − Z5 − Y6)/2; Q5 = (Z1 + X3 − Z4 − X6)/2; and Q6 = (Y1 + X2 − Y4 − X5)/2 where Ui (U = X, Y, Z, and i = 1 to 6) represents the displacement of the ith oxygen atom in the U direction. For the FeO4 tetrahedron, distortion modes are evaluated directly from displacements of O1–O4. The tetrahedral distortion can then be decomposed into a breathing mode (Q1) and two Jahn–Teller–active modes (Q2, Q3). The three modes can be expressed as: Q1 = (X1 − Y1 − Z1 + X2 + Y2 + Z2 − X3 + Y3 − Z3 – X4 – Y4 + Z4)/√12; Q2 = (X1 + Y1 + X2 − Y2 − X3 − Y3 – X4 + Y4)/2√2; and Q3 = (− 2Z1 − X1 + Y1 + 2Z2 − X i - Y i - 2Z3 + X i - Y i + 2Z4 + X i + Y i)/6. 3. Definition of wave functions and determination of orbital states. As shown in the main text, the quantum parameters of orbital configuration at the V site are obtained by fitting the experimental valence electron density (VED) distribution using the 3d (t2g) wave functions determined by site symmetry. For example, in the LT-tetra phase, where symmetry is.2/m, t2g wave functions are written as in Eq. (S4): Ψ1 = C1yz⟩ + √ (1 − γ 2)/2zx⟩ + √(1 - γ 2)/4xy⟩; Ψ2 = −√1 - γ squared yz⟩ + √ 2/sqrt(8)zx⟩ + 1/sqrt(8)xy〉; and Ψ3 = √ (1-γ 2)/4yz⟩ − zx〉 + 1/sqrt(8)xy〉. The calculated 3d2 VED is expressed as (S5): ρcalc (r) = psi1 2 + ηpsi2 2 + (1 − η)psi3 2, where η represents the electron filling of Ψ2. The evaluation function s for the fitting is defined as s=∑rrhoobs (r) − κ 3 ρcalc (κr)/∑rrhoobs (r), where κ is a variable parameter representing contraction of the VED. In general, hybridization between neighboring atoms may modify the radial distribution of the observed VED from that of an isolated-atom model. To account for this effect, a radial scaling parameter κ is introduced."

"Figure S6 shows one-dimensional radial profiles of the calculated (blue dots) and experimentally observed (black dots) VEDs around the V atom in the Cubic phase. Here we use VED calculation using Slater-type orbital (STO) of an isolated atom with the resolution limit of d > 0.28 Å. The quantum parameters were optimized to κ = 1.53, γ = 0.76, and η = 0.50, for which the evaluation function in Eq. (S6) is minimized to s = 0.27."

"Figure S7(a) shows the projected minority spin density of states for the Fe e orbitals in the LT-tetra phase, corresponding to the noncoplanar solution from spin-polarized density-functional theory (DFT) calculations considering Coulomb interactions (U) and spin–orbit coupling. This result indicates that the y2-z2 orbital is occupied by the minority spin, exhibiting orbital ordering. The calculated VED distribution shown in Fig. S7(b) is consistent with the anisotropy observed in the experimental VED [Fig. 4A]."

"Figure S8(b) also shows simulated VED distribution using parameters obtained from fitting and DFT calculations. Both of these models accurately reproduce the experimentally obtained VED distribution anisotropy [Fig. S8(a)]. The iso-density surfaces at 2.9e/Å3 and 3.2e/Å3 are shown in yellow and orange, respectively."

"Figure S9 shows results of spin-polarized DFT+U+SOC calculations performed for the LT-tetra phase with U = 0. (a) Virtual band structure and projected density of states of the V t2g orbitals in the LT-tetra phase in the U=0 case. Here, Ψ1=γ yz⟩ + √(1 − γ2)/2zx⟩ + √(1 - γ2)/4xy⟩, Ψ2=−√1 - γ2 yz⟩ + (γ/√2)zx⟩ + (γ/√2)xy〉, and Ψ3=(1/√8)zx − (1/sqrt(8))xy〉. The virtual projected minority spin density of states for the Fe e orbitals in the case of U=0. Due to the tetragonal distortion, a difference arises in the occupancy of the y2z2 orbital and the 3x2-r2 orbital, resulting in weak anisotropy shown in Fig. S9(d). As U increases, the energy difference between y2-z2 and 3x 2-r squared orbitals becomes larger, leading to an insulating orbital ordered state, as shown in Figs. S7(a) and S7(b)."

Figure 6H displays the calculated VED, which shows good agreement with the observed VED [Fig. 5A].

"The expected noncoplanar arrangement of V spins: In the wavefunction Ψ1 = γyz⟩ + √(1 − γ2)/2zx⟩ + √(1 - γ2)/4xy⟩, where γ = 0.76 is determined by fitting to the experimental VED, the occupation of the yz orbital is inequivalent to those of the zx and xy orbitals. The latter two carry equal weights, as dictated by the calculated symmetry. As a result, the threefold rotational symmetry is explicitly broken, whereas the exchange symmetry between zx and xy remains preserved. As a consequence, the local crystal field retains a mirror symmetry that leaves the yz orbital invariant and exchanges zx and xy. This mirror plane contains the local ⟨111⟩ axis, and therefore the orbital angular momentum induced by spin–orbit coupling is constrained to lie within this plane. The spin moment is thus constrained to cant within the same mirror plane, fixing its azimuthal direction. Using the orbital wavefunctions in LT-tetra phase, the direction of the local spin moment is obtained as S ∝ (− γ, −√(1 − γ2)/2, −√(1 - γ2)/2). This corresponds to a polar angle of approximately 41 degrees from the −x direction and about 8 degrees from the direction antiparallel to the local ⟨111⟩ axis, indicating a small but finite canting away from the threefold rotational axis. This estimate is in good agreement with the canting angle of 43 degrees obtained in the DFT calculation. Notably, because the canting direction is confined on the local mirror, the spin configuration on the pyrochlore lattice becomes intrinsically noncoplanar, thereby generating a finite scalar spin chirality. Generally, carriers endowed with an angular momentum may experience an effective magnetic field arising from the scalar spin chirality, which may give rise to unique transport phenomena such as the magnon Hall effect. These results suggest intertwined spin and orbital degrees of freedom."

"Origin of the reentrant tetragonal ground state: The reentrant tetragonal symmetry in the ground state can be understood as a consequence of spin–orbital–lattice cooperation and competition involving both Fe and V sites. The canting of the V spins is primarily determined by competing Fe–V and V–V antiferromagnetic exchange interactions. The evolution of the mode Q(x)t2 V toward negative values in the LT-tetra phase reflects a reconfiguration of the t2g orbital components, corresponding to an increase in the occupancy of the yz orbital component [Fig. 3F]. Through spin-orbit coupling, this orbital reconfiguration further stabilizes the two-in-two-out spin configuration of the V sublattice, providing an additional relativistic energy gain. This evolution in the V sector modifies the balance of spin–orbital–lattice coupling at low temperatures. Consequently, the energetic hierarchy between the anharmonic term of the lattice elastic energy for the FeO4 distortion and the relativistic spin–orbit interaction at the Fe site is altered, effectively reversing the preferred distortion direction and stabilizing the elongated FeO4 distortion along the x axis, which couples to the y2–z2 orbital. Notably, the magnitude of the FeO4 distortion exceeds that of VO6 [Figs. 3C and 3F], indicating that the FeO4 units dominate the overall lattice deformation associated with the structural phase transitions and ultimately give rise to the elongated tetragonal ground state along the a’ axis."

"Supporting Information: 1. Single-crystal structural analysis using synchrotron X-ray diffraction. FeV2O4 undergoes three structural phase transitions upon cooling—from the cubic phase to the high-temperature tetragonal (HT-tetra), orthorhombic (Ortho), and finally the low-temperature tetragonal (LT-tetra) phases, as described in the main manuscript. The structural analysis results for FeV2O4 at 160 K (Cubic), 130 K (HT-tetra), 100 K (Ortho), and 50 K (LT-tetra) are summarized in Tables S1-12. Taking a, b, and c as the crystallographic axes of the cubic phase, the axes of HT-tetra, Ortho, and LT-tetra phases are defined as follows: aHT = 1/2a+bHT = bHT = c; aO = -a, bO = c, and cO = b+c; aLT = -a', bLT = 1/4b', and cLT is the transformation matrix P shown in Table S13. In this setting, x a', y b' and z c'. 2. Q-mode analysis. To consider the V3+ and Fe2+ orbital states, we analyze the distortion of the VO6 octahedron and the FeO4 tetrahedron in terms of normal (Q) modes [1]. In case of octahedral coordination, symmetric distortion can be decomposed into six modes: A1g(Q1) mode, Eg (Q2, Q3) modes and T2g (Q4, Q5, Q6) modes. The six modes can be expressed as: Q1 = (X1 + Y2 + Z3 − X4 − Y5 − Z6)/√6; Q2 = (X1 − Y2 − X4 + Y5)/2; Q3 = (−X1 − Y2 + 2Z3 + X4 + Y5 − 2Z6)/2√3; Q4 = (Z2 + Y3 − Z5 − Y6)/2; Q5 = (Z1 + X3 − Z4 − X6)/2; and Q6 = (Y1 + X2 − Y4 − X5)/2 where Ui (U = X, Y, Z, and i = 1 to 6) represents the displacement of the ith oxygen atom in the U direction. For the FeO4 tetrahedron, distortion modes are evaluated directly from displacements of O1–O4. The tetrahedral distortion can then be decomposed into a breathing mode (Q1) and two Jahn–Teller–active modes (Q2, Q3). The three modes can be expressed as: Q1 = (X1 − Y1 − Z1 + X2 + Y2 + Z2 − X3 + Y3 − Z3 – X4 – Y4 + Z4)/√12; Q2 = (X1 + Y1 + X2 − Y2 − X3-Y3-X4+Y4)/6; and Q3 = (− 2Z1 − X1 + Y1 + 2Z2 − X i - Y i - 2Z3 + X i - Y i + 2Z4 + X i + Y i)/6. 3. Definition of wave functions and determination of orbital states. As shown in the main text, the quantum parameters of orbital configuration at the V site are obtained by fitting the experimental valence electron density (VED) distribution using the 3d (t2g) wave functions determined by site symmetry. For example, in the LT-tetra phase, where symmetry is.2/m, t2g wave functions are written as in Eq. (S4): Ψ1 = C1yz⟩ + √ (1 - γ 2)/2zx⟩ + √(1 - γ 2)/4xy⟩; Ψ2 = −√1 - γ squared yz⟩ + √ 2/sqrt(8)zx⟩ + 1/sqrt(8)xy〉; and Ψ3 = √ (1-γ 2)/4yz⟩ − zx〉 + 1/sqrt(8)xy〉. The calculated 3d2 VED is expressed as (S5): ρcalc (r) = psi1 2 + ηpsi2 2 + (1 − η)psi3 2, where η represents the electron filling of Ψ2. The evaluation function s for the fitting is defined as s=∑rrhoobs (r) − κ 3 ρcalc (κr)/∑rrhoobs (r), where κ is a variable parameter representing contraction of the VED. In general, hybridization between neighboring atoms may modify the radial distribution of the observed VED from that of an isolated-atom model. To account for this effect, a radial scaling parameter κ is introduced."

"Figure S9 shows results of spin-polarized DFT+U+SOC calculations performed for the LT-tetra phase with U = 0. (a) Virtual band structure and projected density of states of the V t2g orbitals in the LT-tetra phase in the U=0 case. Here, Ψ1=γ yz⟩ + √(1 - γ2)/2zx⟩ + √(1 - γ2)/4xy⟩, Ψ2=−√1 - γ2 yz⟩ + (γ/√2)zx⟩ + (γ/√2)xy〉, and Ψ3=(1/√8)zx − (1/sqrt(8))xy〉. The virtual projected minority spin density of states for the Fe e orbitals in the case of U=0. Due to the tetragonal distortion, a difference arises in the occupancy of the y2z2 orbital and the 3x2-r2 orbital, resulting in weak anisotropy shown in Fig. S9(d). As U increases, the energy difference between y 2-z squared and 3x 2-r squared orbitals becomes larger, leading to an insulating orbital ordered state, as shown in Figs. S7(a) and S7(b)."

"Materials and Methods: X-ray diffraction Single-crystal x-ray diffraction experiments were conducted at the BL02B1 beamline(43) of the SPring-8 synchrotron facility in Japan. The single crystals used in this study were the same as those in Ref. 15. Temperature was controlled using an N2/He gas-blowing device. The wavelengths were 0.30945 Å at 160 and 50 K, and 0.31080 Å at 130 and 100 K. Prior to the measurements in the Ortho and LT-tetra phases, the sample was cooled while applying a magnetic field along the a’-axis of the cubic phase using a Neodymium magnet to align structural domains. Diffraction patterns were recorded using a two-dimensional CdTe PILATUS detector(44) with a dynamic range of 106. Data were acquired using the Fine Slice method by oscillating the crystal and dividing reciprocal space into intervals of Δω = 0.1°(45). The intensities of Bragg reflections were collected by CrysAlisPro(46). Intensities of equivalent reflections were averaged using SORTAV(47), and structural parameters were refined by Jana2006(48). Here, by utilizing only high-angle reflections (sinθ/λ > 0.6 Å−1), where the contribution of spatially spread valence electrons to x-ray diffraction is negligible, structural parameters including atomic displacement parameters were obtained with high accuracy. CDFS analysis The CDFS method was used to extract the VED distribution. The [Ar]-type configuration (1s2,2s2,2p6,3s2,3p6) of the Fe and V atoms as well as 1s2 electrons of O atoms were regarded as core electrons. The contribution of the thermal vibration was subtracted from the VED using the atomic displacement parameters determined by the high-angle analysis. The voxel size of the three-dimensional VED distribution was 0.05 Å3. We used the STO-COPPENS Slatertype orbital (STO) library implemented in Jana2006 for the CDFS analysis. The crystal structure and VED distributions were visualized using VESTA(49). DFT calculation The DFT electronic structure calculations for FeV2O4 were performed using Quantum Espresso(50) using the experimental structure of the LT-tetra phase obtained in this study. We employed relativistic norm-conserving pseudopotentials with Perdew-Burke-Ernzerhof (PBE)(51) exchange-correlation functional, which were taken from the PseudoDojo(52). We performed spinpolarized calculations considering the spin–orbit coupling with k-mesh of 7×7×7. The effects of Coulomb interactions of Fe 3d and V 3d orbitals were incorporated within the DFT+U formalism(53) with the Hubbard U parameters of 5 eV. The energy cutoff was set to 100 Ry for the wave functions and 400 Ry for the charge density. Based on the DFT electronic structure, we constructed Wannier orbitals(54, 55) using RESPACK(56, 57) and Wannier90(58) for Fe 3d, V 3d, and O 2p manifold. Since a primitive unit cell consists of two Fe sites, four V sites, and eight O sites, the number of Wannier orbitals was (5×2+5×4+3×8)×2=108, where the factor of two accounted for the summation over spin degrees of freedom."

"References: 1. A. J. Millis, Lattice effects in magnetoresistive manganese perovskites. Nature 392, 147–150 (1998). 2. P. G. Radaelli et al., Formation of isomorphic Ir3+ and Ir4+ octamers and spin dimerization in the spinel CuIr2S4. Nature 416, 155–158 (2002). 3. M. S. Senn, J. P. Wright, J. P. Attfield, Charge order and three-site distortions in the Verwey structure of magnetite. Nature 481, 173–176 (2012). 4. L. Camerano et al., Multiferroic nematic dwave altermagnetism driven by orbital order on the honeycomb lattice. npj 2D Mater. Appl. 9, 75 (2025). 5. M. Neupane et al., Observation of a novel orbital selective Mott transition in Ca1.8Sr0.2RuO4. Phys Rev Lett. 103, 097001 (2009). 6. G. M. Luke et al., Time-reversal symmetry-breaking superconductivity in Sr2RuO4. Nature 394, 558–561 (1998). 7. Á. A. Carrasco Álvarez et al., Orbital order as the driving mechanism for superconductivity in ruthenates. Nat. Commun. 16, 1458 (2025). 8. H. Takagi et al., Concept and realization of Kitaev quantum spin liquids. Nat Rev Phys. 1, 264–280 (2019)."

"D I Khomskii, T Mizokawa, Orbitally induced Peierls state in spinels. Phys Rev Lett. 94, 156402 (2005). T Manjo et al., Do electron distributions with orbital degree of freedom exhibit anisotropy? Mater. Adv. 3, 3192–3198 (2022). S Kitou et al., Successive dimensional transition in (TMTTF)2PF6 revealed by synchrotron x-ray diffraction. Phys Rev Lett. 119, 065701 (2017). S Kitou et al., Collapse of the simple localized 3d1 orbital picture in Mott insulator. Phys Rev Res 2, 033503 (2020). S Kitou et al., Visualization of spin–orbit-entangled 4f electrons in crystalline materials. Proc. Natl. Acad. Sci. U.S.A. 122, e2500251122 (2025)."

"D I Khomskii, T Mizokawa, Orbitally induced Peierls state in spinels: Phys Rev Lett 94, 156402 (2005). T Manjo et al., Do electron distributions with orbital degree of freedom exhibit anisotropy? Mater. Adv. 3, 3192–3198 (2022). S Kitou et al., Successive dimensional transition in (TMTTF)2PF6 revealed by synchrotron x-ray diffraction: Phys Rev Lett 119, 065701 (2017). S Kitou et al., Collapse of the simple localized 3d1 orbital picture in Mott insulator: Phys Rev Res 2, 033503 (2020). S Kitou et al., Visualization of spin–orbit-entangled 4f electrons in crystalline materials: Proc. Natl. Acad. Sci. U.S.A. 122, e2500251122 (2025)."

"P P Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale: Ann Phys 369, 369–387 (1921). S Nakamura et al., Distinct evidence of orbital order in spinel oxide FeV2O4 by 57Fe Mössbauer spectroscopy. J Phys Soc. Jpn. 85, 014702 (2016)."

"S Ohtani et al., Orbital dilution effect in ferrimagnetic Fe1−xMnxCr2O4: Competition between anharmonic lattice potential and spin-orbit coupling. J Phys.: Condens. Matter 22, 176003 (2010). S Bordács et al., Magnetic-order-induced crystal symmetry lowering in ACr2O4 ferrimagnetic spinels. Phys Rev Lett. 103, 077205 (2009). K Sugimoto et al., Extremely high resolution single crystal diffractometry for orbital resolution using high energy synchrotron radiation at SPring-8. AIP Conf. Proc. 1234, 887–890 (2010). L Krause et al., Accurate high-resolution single-crystal diffraction data from a Pilatus 3X CdTe detector. J Appl Crystallogr. 53, 635 (2020). M Mueller et al., Optimal fine φ-slicing for single-photoncounting pixel detectors. Acta Crystallogr. D 68, 42–56 (2012). Agilent Technologies, CrysAlis PRO (Agilent Technologies Ltd., Yarnton, Oxfordshire, England, 2014). R H Blessing, DREADD—data reduction and error analysis for single-crystal diffractometer data. J Appl Crystallogr. 22, 396–397 (1989). V Petříček et al., Crystallographic computing system JANA2006: General features. Z Kristallogr. 229, 345–352 (14)."

"P Giannozzi et al., Advanced capabilities for materials modelling with Quantum ESPRESSO. J Phys.: Condens. Matter 29, 465901 (2017). J P Perdew, K Burke, M Ernzerhof, Generalized gradient approximation made simple. Phys Rev Lett. 77, 3865–3868 (1996). M J van Setten et al., The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table. Comput. Phys. Commun. 226, 39–54 (2018). A I Liechtenstein, V I Anisimov, J Zaanen, Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators. Phys Rev B 52, R5467 (1995). N Marzari, D Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands. Phys Rev B 56, 12847–12865 (1997). I Souza, N Marzari, D Vanderbilt, Maximally localized Wannier functions for entangled energy bands. Phys Rev B 65, 035109 (2001). K Nakamura et al., RESPACK: An ab initio tool for derivation of effective low-energy model of material. Comput. Phys. Commun. 261, 107781 (2021). M Charlebois et al., Ab initio derivation of low-energy Hamiltonians for systems with strong spin-orbit interaction: Application to Ca5Ir3O12. Phys Rev B 104, 075153 (2021)."

"S Sarkar, T Saha-Dasgupta, Orbital ordering in FeV2O4: Spinel with two orbitally active sites. Phys Rev B 84, 235112 (2011). D Dey et al., Orbital ordering in Fe1−xMnxV2O4: A first-principles study. Phys Rev B 93, 195133 (2016)."

J Krishna et al., Complete description of the magnetic ground state in spinel vanadates. Phys Rev B 100, 081102(R) (2019).

K I Kugel, D I Khomskii, The Jahn-Teller effect and magnetism: Transition metal compounds. Sov. Phys. Usp. 25, 231–256 (1982).

T Suzuki et al., Orbital ordering and magnetic field effect in MnV2O4: Phys Rev Lett 98, 127203 (2007). G J MacDougall et al., Magnons and a two-component spin gap in FeV2O4: Phys Rev B 89, 224404 (2014).

"Y Ishitsuka et al., Comparative studies of the thermal conductivity of spinel oxides with orbital degrees of freedom. Phys Rev B 90, 224411 (2014). K Matsuura et al., X-ray magnetic circular dichroism study of an orbital ordered state in the spinel-type vanadium oxide AV2O4 (A = Mn, Fe). Phys Rev B 92, 035133 (2015)."

"J Okabayashi et al., Local electronic and magnetic properties of ferro-orbital-ordered FeV2O4. Jpn. J. Appl. Phys. 57, 0902BD (2018). M Nakano et al., Coupling between orbital and spin degrees of freedom in Jahn-Teller ions for Co1−xFexV2O4: Phys Rev Lett 135, 186701 (2025)."

"H Tsunetsugu et al., Magnetic transition and orbital degrees of freedom in vanadium spinels. Phys Rev B 68, 060405(R) (2003). O Tchernyshyov, Structural, orbital, and magnetic order in vanadium spinels. Phys Rev Lett. 93, 157206 (2004)."

S Di Matteo et al., Orbital order in vanadium spinels. Phys Rev B 72, 020408(R) (2005).

Y Motome et al., Theory of successive transitions in vanadium spinels and order of orbitals and spins. Prog Theor. Phys. Suppl. 160, 203–213 (2005).

S Sarkar et al., Orbital ordering in FeV2O4: Spinel with two orbitally active sites. Phys Rev B 84, 235112 (2011). D Dey et al., Orbital ordering in Fe1−xMnxV2O4: A first-principles study. Phys Rev B 93, 195133 (2016).

M Nakano et al., Coupling between orbital and spin degrees of freedom in Jahn-Teller ions for Co1−xFexV2O4: Phys Rev Lett 135, 186701 (2025).

"S Di Matteo et al., Orbital order in vanadium spinels. Phys Rev B 72, 020408(R) (2005). Y Motome et al., Theory of successive transitions in vanadium spinels and order of orbitals and spins. Prog Theor. Phys. Suppl. 160, 203–213 (2005)."

"M Nakano et al., Coupling between orbital and spin degrees of freedom in Jahn-Teller ions for Co1−xFexV2O4: Phys Rev Lett 135, 186701 (2025). H Tsunetsugu et al., Magnetic transition and orbital degrees of freedom in vanadium spinels. Phys Rev B 68, 060405(R) (2003). O Tchernyshyov, Structural, orbital, and magnetic order in vanadium spinels. Phys Rev Lett. 93, 157206 (2

As a fastidious and diligent researcher, I have analyzed this paper, Real-space determination of orbital states driving successive phase transitions in FeV2O4. The core scientific contribution is establishing a direct link between experimentally determined valence electron density (VED) anisotropy and the microscopic orbital states that drive coupled structural and magnetic phase transitions in strongly correlated systems.

Here are the specific improvements to AI systems derived from this research, focusing on areas where current computational chemistry/materials science models struggle:

The improved AI system can achieve the following capabilities:

1. A more accurate and robust prediction of ground-state configurations for complex transition metal oxides (TMOs) by incorporating real-space experimental constraints.

2. Automated identification of hidden, nearly degenerate metastable states that are currently computationally inaccessible or ambiguous in purely first-principles calculations.

Here are the specific improvements to AI systems:

1. The system can be trained on the relationship between experimentally measured VED anisotropy maps (derived from synchrotron XRD/CDFS) and theoretical Density Functional Theory (DFT) solutions.

2. The system can perform real-space constraint matching on DFT output, effectively pruning thousands of theoretically possible metastable electronic configurations down to a few physically realizable ground states that match experimental orbital signatures.

3. The system can predict the precise sequence and nature (collinear vs. noncoplanar) of coupled spin/orbital/lattice transitions based on the evolution of the VED anisotropy across different temperature regimes, moving beyond simple energy minimization to capture cooperative effects like Kugel-Khomskii interactions and Jahn-Teller coupling.

The improved AI system can do the following specific tasks:

1. Predict the sequence of structural phase transitions (e.g., Cubic → HT-tetra → Ortho → LT-tetra) in FeV2O4 based on predicted changes in orbital occupation patterns derived from VED evolution.

2. Identify which specific orbital wavefunctions (e.g., stabilizing the 3z2−r2 vs y2−z2 orbitals) are responsible for driving a transition from one structural symmetry to another, linking it directly to spin-orbit coupling strengths and electron-phonon coupling constants derived from DFT+U+SOC calculations.

3. Distinguish between nearly degenerate theoretical solutions (e.g., collinear vs. noncoplanar spin arrangements in the LT-tetra phase) by matching their resulting VED distributions against experimental data, thereby resolving fundamental ambiguities in strong correlation physics that standard energy-based methods cannot resolve alone.

4. Generate a predictive framework for identifying hidden orbital orders in other complex Mott insulators by using experimentally determined VED as a decisive real-space filter to select the physically relevant ground state among competing theoretical solutions.