A continuous metal-insulator transition driven by spin correlations

While Mott insulators induced by Coulomb interactions are a well-recognized class of metal-insulator transitions, insulators purely driven by spin correlations are much less common, as the reduced energy scale often invites competition from other degrees of freedom. Here, we demonstrate a clean example of a spin-correlation-driven metal-insulator transition in the all-in-all-out pyrochlore antiferromagnet Cd2Os2O7, where the lattice symmetry is preserved by the antiferromagnetism. After the antisymmetric linear magnetoresistance from conductive, ferromagnetic domain walls is removed experimentally, the bulk Hall coefficient reveals four Fermi surfaces of both electron and hole types, sequentially departing the Fermi level with decreasing temperature below the Néel temperature, TN = 227 K. In Cd2Os2O7, the charge gap of a continuous metal-insulator transition opens only at T ~ 10 K << TN. The insulating mechanism parallels the Slater picture, but without a folded Brillouin zone, and contrasts sharply with Mott insulators and spin density waves, where the electronic gap opens above and at TN, respectively.

U nlike insulators or semiconductors derived from simple metals such as sodium and lithium 1,2 , metal-insulator transitions in correlated-electron systems reside outside the paradigm of the single-electron band structure. Factors such as reduced dimensionality and randomness 3,4 enrich the description of the critical behavior, with possible separation of spin and charge, and deep connections to exotic states such as high T c superconductivity and quantum spin liquids 5 . In the limit of large electronic correlations, the starting point for discussion is usually the opening of the Hubbard gap 6 , with antiferromagnetic order being a subsidiary effect.
There exist several major experimental challenges in establishing a convincing example of a metal-insulator transition driven by spin correlations 7 . First, the spin-correlation energy is typically much smaller than the direct Coulomb interaction, often at a scale comparable to that of the structural modifications induced by the antiferromagnetic order through magnetostrictive effects and symmetry changes, leading to a chicken-and-egg conundrum between ascribing the insulating transition to the lattice and the magnetism 8,9 . It is thus preferable to search for candidate systems in which the magnetism would preserve the crystalline symmetry. Materials that demonstrate an all-in−allout (AIAO) antiferromagnetic order on a pyrochlore lattice, with all four spins on the corner of a tetrahedron pointing either towards or away from the center (Fig. 1a, b), meet this criterion. The AIAO order induces no symmetry-breaking magnetostriction and causes an isotropic expansion of the cubic unit cell by a minimal Δa/a~10 −4 , only becoming experimentally resolvable when T N drops below 40 K 10,11 . For AIAO order with T N > 100 K, this overall magnetostriction is fully camouflaged by the thermal lattice contraction 10 , which broadens the bandwidths but introduces no band splitting. The family of AIAO antiferromagnets thus provides a highly desirable model system to investigate spin correlations with little lattice interference.
Identifying an AIAO system with a metal-insulator transition represents the next experimental challenge. Accompanying the AIAO order, many 5d oxides, such as R 2 Ir 2 O 7 (R = Eu, Sm, and Nd) and Cd 2 Os 2 O 7 (ref. 12 and references in Refs. 11,13,14 ), also demonstrate a change of temperature dependence in the resistivity at T~T N 9,13,15,16 . However, the resistive behavior of R 2 Ir 2 O 7 (R = Nd, Sm, Eu), especially in the paramagnetic phase, is often inconsistent and raises concerns about their intrinsic, disorder-free behavior 13,17,18 . Cd 2 Os 2 O 7 has presented consistent behavior in both the electron correlation and metal-insulator transition. Samples from several groups 9,15,16 always demonstrate magnetic ordering at T N = 225-227 K, metallic behavior above T N , and a three-to-four-decade rise of the resistivity for T < T N in the best samples. This repeatability from crystal to crystal likely indicates a low level of disorder because of the 2+/5+ valence condition of Cd and Os ions as well as the chemical transport growth procedure at low temperature 9,15,16 .
There exists an additional challenge arising from complications in modeling and understanding the transport data in many 5d AIAO antiferromagnets due to the intrinsically conductive and highly coercive ferromagnetic domain walls 9,19-21 . Ferromagnetic domain walls introduce a Zeeman shift in the metallic paramagnetic band structure, but they are not expected to gap the Fermi surface like the antiferromagnetic bulk, and they are expected to remain metallic down to T = 0. As was pointed out recently 21 , the highly coercive metallic ferromagnetic domain walls generate antisymmetric linear magnetoresistance (MR) of the same functional form as the Hall resistance. Moreover, the antisymmetric linear MR is detectable in Hall channels due to the distorted current paths through the domain walls 21 . This effect is likely the root cause of the widely varying Hall coefficient reported in the literature for Cd 2 Os 2 O 7 9,16 , as the standard procedure for extracting Hall resistance through antisymmetrization with respect to magnetic field direction leads to erroneous results.
Here we present high-fidelity resistivity and Hall coefficient measurements on single-crystal Cd 2 Os 2 O 7 after employing an intricate procedure to eliminate the influence of conductive ferromagnetic domain walls. Unlike the common understanding of a concurrent metal-insulator transition with the AIAO magnetic transition at T N , our results reveal that Cd 2 Os 2 O 7 is metallic for a broad temperature range below T N , despite an increasing resistivity with decreasing T. It only becomes an insulator at T MIT~1 0 K, when four sets of Fermi surfaces have sequentially left the Fermi level to open a true electronic gap. This large separation in temperature for spin order (T N ) and the charge gap (T MIT ), with T N >> T MIT , unambiguously establishes spin ordering as the driving force in Cd 2 Os 2 O 7 's metal-insulator transition. Our methodology in separating the Hall behavior of the bulk from the influences of the domain walls should provide a generic approach to parsing spin and charge effects in correlated antiferromagnetic insulators with metallic domain walls.

Results
Insulating behavior at zero field. Our single-crystal Cd 2 Os 2 O 7 samples demonstrate a monotonic rise of resistivity over three decades (~3000×) from T N to 1.8 K at zero field (Fig. 1c), consistent with the best samples reported in the literature 9,15,16 . Instead of using bar-shaped samples for both resistivity and Hall measurements, our key approach is to utilize the van der Pauw (vdP) configuration of electrical lead placement on plate-shaped samples (ref. 22 , schematics in Figs. 1c, 2c). This choice of putting four leads on an equal footing aims to comprehensively evaluate the effects of the electrical current paths, which constantly redistribute between the conductive domain walls and an increasingly insulating bulk as the temperature decreases. As demonstrated by our samples, the vdP ratio, defined as R vdP2 / R vdP1 at zero field, stays constant only in the paramagnetic phase above T N . Below T N , the vdP ratios of two samples have a strong temperature dependence (Fig. 1c, inset) but still evolve continuously as expected from the lack of lattice symmetry breaking in Cd 2 Os 2 O 7 . The resistivity in Fig. 1c is calculated from R vdP2 and R vdP1 at each T according to the standard vdP procedure 22 , an issue we will revisit below.
Extracting the bulk Hall coefficient. The highly coercive, fieldindependent magnetization M of the conductive ferromagnetic domain walls necessarily introduces antisymmetric linear MR 21 that makes the Hall resistances of the two reciprocal channels, R 12,34 (H, M) and R 43,12 (H, M), have different linear slopes with H ( Fig. 2c). Due to the cubic symmetry of AIAO order, it is not possible to create a single antiferromagnetic domain and remove the domain walls altogether by field cooling. The separation of galvanomagnetic responses of the bulk domains and the domain walls is instead carried out by introducing a variable M(ϕ) through field-cooling along 24 angular directions ϕ within the sample surface plane (Fig. 2a, b, Methods, and ref. 21 ). Here we first examine the ϕ-dependences of the Hall resistivity slopes from two reciprocal channels R Hall1 H ϕ À Á and R Hall2 H ϕ À Á , the vdP ratio, and resistivity ρ(H = 0), at temperatures 195, 30, and 1.8 K (Fig. 2d- follow a second constraint that (2) they are identical at ϕ and ϕ + π respectively ΔR Hall1 , leading to identical ϕaveraged resistivity slopes of both Hall channels. From the previous analysis 21 , constraint (1) reflects the voltage−current reciprocity, while constraint (2) manifests Onsager's reciprocity relation with regard to the ferromagnetic domain wall M. At 1.8 K, the ϕ-dependent Hall resistivity slopes ΔR Hall1 (1) is sustained, while constraint (2) cannot be satisfied by any choice of the ϕ-independent components. The result is that the ϕ-averaged resistivity slopes of individual Hall channels are no longer necessarily equal to each other and to the ϕ-independent component R H ðϕÞ.
The breakdown of constraint (2) seemingly implies a violation of Onsager's reciprocity relation between ϕ and ϕ + π states. However, the established Onsager's reciprocity in our system hinges on the inverse relationship between domain wall magnetizations M(ϕ) and M(ϕ + π) through separate field-cooling processes at ϕ and ϕ + π. Their inverse relationship is generally robust in that both ρ and the vdP ratio at zero field demonstrate a π-periodicity at all temperatures ( Fig. 2df). At 1.8 K, the conduction patterns at ϕ and ϕ + π are similar enough to be reproducibly differentiated from those at neighboring ϕ positions, judging by ρ and the vdP ratio (Fig. 2f). Nevertheless, because of the separate cooldowns, there exist differences beyond an inversion between M(ϕ) and M(ϕ + π). With the increasingly insulating bulk, the electrical current is more concentrated along a fraction of the domain walls, as reflected by the dramatically oscillating vdP ratio from 3:1 to 1:4 ( Fig. 2f). While the entire sample volume brings better averaged galvanomagnetic behavior and demonstrates Onsager reciprocity (Fig. 2d, e), probing only a small number of domain walls enhances the relative difference between M(ϕ) and M(ϕ + π). Even at a fixed ϕ, multiple fieldcoolings can lead to a significant difference in individual Hall channels at 1.8 K (Fig. 2f), despite the consistency at 30 K (Fig. 2e). We note that the assumed uniform medium for the vdP technique 22 is justified a posteriori at 195 K and 30 K, as the calculated ρ(ϕ) varies within ±0.1% and ±3%, respectively. Constraint (2) remains satisfied (Fig. 2d, e). At 1.8 K, ρ(ϕ) calculated from the vdP formula 22 varies by ±50% (Fig. 2f). Although the uniformity assumption no longer holds at 1.8 K to legitimize both the vdP-derived resistivity and constraint (2), our procedure to extract the Hall coefficient R H through the average slope of two Hall resistivity channels is protected by the fundamental principle of voltage−current reciprocity.
With the understanding of how to extract R H through a ϕdependence study at three fixed temperatures, we now explore the temperature evolution of R H by taking the average Hall resistivity slope of two reciprocal channels at a fixed ϕ (Fig. 2g). The R H (T) measured on two different Cd 2 Os 2 O 7 single crystals (Fig. 3a) demonstrates a consistent picture that represents the major finding of this work. R H (T) remains stable above T N = 227 K, and all bands only start to evolve at T N , maintaining a delicate balance between them until the first sharp change at  Fig. 1 Transport signatures of Cd 2 Os 2 O 7 at zero field. a Schematic of the Os sublattice in the cubic pyrochlore oxide Cd 2 Os 2 O 7 , with all Os atoms forming a three-dimensional corner-sharing octahedra network. b Schematic of AIAO spin arrangement on two adjacent octahedral. Note that this spin arrangement does not break the underlying lattice symmetry. c Resistivity ρ(T) of Cd 2 Os 2 O 7 measured over two single crystals (COO-1 in red and COO-2 in blue) using the vdP configuration (Schematics). At each temperature, both vdP channels were measured in order to account for the changing current path between the insulating bulk and conductive domain walls. (Top inset) The resistance ratio R vdP2 /R vdP1 between two vdP channels manifests the changing current path below T N = 227K. (Bottom inset) Details of ρ(T) near T N , showing both metallic behavior above T N , and also an upturn of ρ near but still above T N , which is often attributed to dynamical spin fluctuation effects without long-range order. T N is determined precisely from the magnetic susceptibility χ(T), individual R vdP2 (T) and R vdP1 (T) (not shown), and R H (T) in Fig. 3a.
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-23039-6 ARTICLE 220 K. Despite a rise of nearly three orders of magnitude in ρ (T) from T N down to 10 K, R H (T) remains finite and oscillates between positive and negative values. As ρ(T) reflects a fast drop in total carrier density with decreasing temperature (Fig. 1c), the oscillating R H (T) reflects the changes in the bands, with an alternating dominance by either electrons or holes as the major composition of itinerant carriers. In a multiband model with both electron and hole types of carriers, 2 , with carrier densities N i;j and mobilities μ i;j ¼ eτ i;j =m i;j determined by the carrier relaxation times τ i;j and masses m i;j of individual band i (of holes) or j (of electrons), respectively (Eq. 2.78 of ref. 23 with multiple bands). Here, carrier mobilities are expected to have only a weak temperature dependence. On the other hand, when a band moves out from the Fermi level, the relative contributions of charge carriers (electron or hole) dramatically change, signaled by a rapid turn (over a temperature range~20 K) in the trend followed by R H (T) and similar to other band-gap-opening scenarios exhibited, for example, by charge and spin density waves 24,25 . The first surface moves out from the Fermi level at~220 K (Fig. 3a, top inset), and with R H (T) turning more electron-like, the departed carriers are of hole type. As thermal excitation at finite temperature still populates the departed bands close to the Fermi level, R H (T) remains a smooth function of T. Here Every galvanomagnetic Cd 2 Os 2 O 7 sample (gray plate) is field-cooled in two stages: a field H is aligned parallel to the sample surface at room temperature before cooling down through T N , and b the sample is rotated below T N to have a field perpendicular to its surface for galvanomagnetic measurements. The in-plane magnetizing direction is defined by angle ϕ at 24 discrete positions, set outside the cryostat at room temperature. c Typical raw data of Hall resistances R(H) between two reciprocal channels, measured for sample COO-2 at a specified temperature and magnetization angle ϕ. The difference in slopes indicates the influence of asymmetric linear magnetoresistance from ferromagnetic domain walls 21 . (inset) vdP configuration for the two reciprocal Hall channels. d-f ϕ-dependence of Hall resistivity slopes R H in two reciprocal channels (red and blue) and their average (fresh green), in addition to the resistivity ρ (green) and vdP ratio (orange), measured in sample COO-2 at three temperatures 195, 30, and 1.8 K. (Methods) (g) Hall resistivity slopes R H from two reciprocal channels (red and blue) are plotted alongside their average (fresh green) for two different single-crystal Cd 2 Os 2 O 7 samples COO-1 and COO-2. Although the resistance slopes of an individual Hall channel are very different for each sample, with occasional crossings at various temperatures, the averages are similar in shape between the two samples and determine the bulk Hall coefficient R H (T). This irregular behavior of resistance slopes of individual Hall channels highlights potential experimental deficiencies in previous Hall measurements. All statistical uncertainties are smaller than the symbols.
we count four characteristic temperatures below T N -at 220, 165, 80, and 10 K-that reflect four sets of carriers leaving the Fermi level (arrows and T 1 −T 4 in Fig. 3a insets). Below T~10 K, a sharply divergent R H indicates that a full gap emerges in the bulk, and all itinerant carriers are due to thermal excitation, predominately of electron type from the last departed band.

Discussion
Our results can be compared with existing band structure calculations 26 , adapted here in Fig. 3b, that suggest the Fermi surface is made up of three sets of carriers in the paramagnetic phase of Cd 2 Os 2 O 7 : hole surfaces around the W point, an electron shell at the Γ point, and a family of electron ellipsoids along the Γ−X line, solely of Os 5d t 2g origin. Our observation reveals a fourth set of hole ellipsoids, potentially located along the Γ−X line 26 (Fig. 3b), but which have an energy difference too small to be resolved by the band structure calculation given that they drop below the Fermi level at 220 K, only 7 K below T N . Infrared reflectivity also illuminates the metal-insulator transition in Cd 2 Os 2 O 7 27,28 . While the direct optical gap has been consistently verified, the interpretation varies from either a spin-density-wave gap opening at T N 27 or an indirect gap opening at~210 K because of a Liftshitz type of mechanism 28 . The latter is partially consistent with our observed first set of carriers departing the Fermi level at 220 K.
Our results demonstrate that the transition at T N = 227 K in Cd 2 Os 2 O 7 should be regarded as only the magnetic-ordering transition, and there is no concurrent metal-insulator transition despite the fact that the resistivity ρ(T) changes its temperature dependence around T N (Fig. 1c inset). Given the well-defined changes observed for R H (T) at T N (Fig. 3a), the driving mechanism behind the electronic evolution should be attributed to Os 5d t 2g band renormalization by the AIAO order 7 , with the effect growing with the increasing strength of the magnetic order parameter, the staggered moment <m>, with decreasing T. From a direct X-ray magnetic diffraction study 12 , <m> continues to grow to the zero-temperature limit without saturation. This spindependent shift of the quasiparticles' self-energy happens within the antiferromagnetic phase and the electronic gap is opened by the developed antiferromagnetic order. It is analogous to a Slater mechanism without Brillouin zone-folding 7 .
The entire density of states within ±1.5 eV of the Fermi level is made of the Os 5d t 2g manifold of 12 bands in total (two Cd 2 Os 2 O 7 units in the primitive unit cell, with three t 2g bands from individual Os ions) 26 . Due to the Os 5+ (5d 3 ) valence, these bands are filled between the sixth and seventh bands, with dispersion across the Fermi surface to create the electron-type and hole-type carriers. Spin correlation renormalizes the t 2g manifold to create a true gap at the half level between the sixth and seventh bands.
The Mott−Hubbard picture of the metal-insulator transition splits a single band to open a gap by strong charge correlation energy (U~2−6 eV). In Cd 2 Os 2 O 7 , an AIAO antiferromagnet with no structural instability, the transition relies on a small spincorrelation energy (T N~2 0 meV) to essentially create a band insulator. The spin and charge gaps are widely separated in temperature. However, they remain experimentally obscure until the confounding effects of metallic domain wall conduction can be separated from the intrinsic bulk behavior. The methods introduced here clarify the contributions from the bulk and the role of the spin and charge degrees of freedom. They also promise a means to quantify the conductive properties of coercive ferromagnetic domain walls winding their way through an insulating antiferromagnet.

Methods
Sample origin and preparation. In this work, we used high-quality Cd 2 Os 2 O 7 single crystals from ref. 9 ; the vapor transport growth techniques and catalog of physical characteristics are detailed therein. The single crystals used in this work were of octahedral shape and 0.3−0.5 mm in size. Individual crystals were polished to plates of 17−20 μm thickness with a (1, 1, 0) surface normal 14 . Previous synchrotron X-ray magnetic diffraction studies directly verified the quality of these single-crystal plates, demonstrating typical crystal mosaics of 0.01−0.02 o FWHM  14 . Further, the pyrochlore space group Fd-3m was verified to the highest sensitivity possible by examining the forbidden orders of (4, 2, 0) and (6, 0, 0) using synchrotron-based single-crystal x-ray diffraction 14 . Samples were selected for these transport measurements from unused plates prepared for the X-ray study, which are of better quality than the sample used in a previous galvanomagnetic study 21 . The sample in ref. 21 was cut from a large crystal (3 × 3× 1 mm 3 ) with a significantly broader mosaic (supplementary materials of ref. 21 ), and higher density of domain walls, as indicated by a comparison of Fig. 2d in this work to Fig. 3f of ref. 21 .
Galvanomagnetic measurements. As the current distribution could potentially change under temperature between the domain walls and bulk, a strict Hall barshaped sample geometry is not advantageous, as it could have potentially complex issues in translating from measured resistances to the underlying resistivity. Instead, we employ a van der Pauw (vdP) sample geometry 22 , where the standard conversion depends only on the sample thickness rather than all lateral dimensions. Furthermore, the vdP geometry allows equal access to all combinations of lead configurations in the measurement of both Hall and magnetoresistance (Figs. 1c and 2c insets) 21 . Four 25 μm diameter gold wires were attached to the transport samples of typical 200 μm lateral sizes by conductive silver epoxy.
The transport samples were mounted on the standard sample holder of a horizontal rotator probe in a 14 T Physical Property Measurement System (PPMS, Quantum Design, Inc.). Using the horizontal rotation, we first cool our samples through T N with a 2 T magnetic field applied along an in-plane direction ϕ (Fig. 2a) then the sample is rotated below T N to allow the measurement magnetic field of both the Hall effect and the MR to be applied perpendicular to the sample surface (Fig. 2b). An additional home-built indexing stage on the sample holder provides degree of freedoms to set field-cooling along 24 discrete angular positions of ϕ within the sample surface plane 21 ; the origin of the ϕ-angle has no specific relationship to either the wiring positions of the electrical leads or the crystalline structure. Galvanomagnetic responses of both reciprocal Hall channels and vdP channels were measured at selected temperatures of 195, 30, and 1.8 K for the full ϕ-dependence (Fig. 2d−f). The data set at 195K was measured independently from those at the other two temperatures. At 30 and 1.8 K, measurements were repeated at three and eleven ϕ positions, respectively, for a check of reproducibility through an additional field-cooling process. For measurements at 30 K, all data except two (one each at ϕ = 30 o and 195 o ) have a corresponding measurement at 1.8 K during the same field-cooling process in order to compare the ϕ-dependence at both temperatures. For the full temperature evolution in Figs. 1, 2g, and 3, both samples were field-cooled along one in-plane ϕ position to base temperature; then the galvanomagnetic measurements were performed at each stabilized temperature along the warming trajectory. We observe no degrading or change in our samples after many thermal cycles (>60 in COO-2).
The resistivity was measured using a Lakeshore LS372 AC resistance bridge, working at 9.8 Hz, together with a low-noise 3708 preamp and a home-built vdP switching box based on low-resistance CMOS relay switches. At zero field, the vdP relationship is satisfied to ΔR/R max = (R Hall1 + R vdP1 − R vdP2 )/Max(R vdP1 :R vdP2 ) < ±0.05% at 30 K and <±0.25% at 1.8 K. R(H) curves were measured over a magnetic field loop of ±4 T The coercive field of the ferromagnetic domain walls is known to be >9 T (see, e.g., ref. 20 ), and our measurements over field loops of ±14 T also demonstrate repeatable field dependence. We thus conclude that the domain wall configuration stays constant during each measurement. Since no field hysteresis was observed, R(H) values are averaged at each field, exemplified by Fig. 2c, and fit to a polynomial form to the second order. The linear slope is taken for plots in Figs. 2 and 3. Our samples demonstrate a very low level of positive parabolic MR 0.3% over ±14 T, consistent from 1.65 to 80 K, and is thereby unappreciable at low fields. The low level of MR indicates that the cyclotron frequency ω c and carrier relaxation time τ satisfy ω c τ ( 1, and the measured Hall coefficient R H (T) is in the low-field limit.