Strongly-coupled quantum critical point in an all-in-all-out antiferromagnet

Dimensionality and symmetry play deterministic roles in the laws of Nature. They are important tools to characterize and understand quantum phase transitions, especially in the limit of strong correlations between spin, orbit, charge, and structural degrees of freedom. Here, using newly-developed, high-pressure resonant X-ray magnetic and charge diffraction techniques, we have discovered a quantum critical point in Cd2Os2O7 as the all-in-all-out antiferromagnetic order is continuously suppressed to zero temperature and, concomitantly, the cubic lattice structure continuously changes from space group Fd-3m to F-43m. Surrounded by three phases of different time reversal and spatial inversion symmetries, the quantum critical region anchors two phase lines of opposite curvature, with striking departures from a mean-field form at high pressure. As spin fluctuations, lattice breathing modes, and quasiparticle excitations interact in the quantum critical region, we argue that they present the necessary components for strongly-coupled quantum criticality in this three-dimensional compound.

F undamental symmetry constraints determine allowable collective states in solids 1 . Of particular recent interest is broken inversion symmetry in non-centrosymmetric materials, which can lead to hidden topological order and odd-parity superconductivity [1][2][3] . With the added consideration of broken time-reversal invariance, exotic magnetic states can emerge. A quantum critical point, where quantum fluctuations tie together competing ground states, is a fertile region to investigate and manipulate intertwined charge, spin, and structural degrees of freedom under different symmetry conditions. However, many experimental systems either manifest first-order quantum phase transitions without critical behavior, as exemplified by itinerant ferromagnets 4 , or simply follow mean-field behavior 5,6 . Stronglycoupled systems with pronounced spin-orbit interactions provide a potential means to move to non-trivial criticality in three dimensions 4,5 . This scenario has been proposed for all-in-all-out (AIAO) antiferromagnetic order on a pyrochlore lattice 4 , but remains to be identified experimentally.
The AIAO arrangement of spins on the pyrochlore lattice is an unusual form of magnetism that preserves the underlying cubic lattice symmetry. With strong spin-orbit coupling and low itinerant electron density, compounds with AIAO spin order are desirable candidates to explore non-trivial quantum critical behavior in three dimensions 5 . AIAO spin order has been observed in FeF 3 , Nd 2 Zr 2 O 7 , A 2 Ir 2 O 7 (A = Sm, Eu, and Nd), and Cd 2 Os 2 O 7 (refs. [7][8][9][10][11][12] and suggested for additional A 2 Ir 2 O 7 systems with A = Y, Lu, Gd, Tb, Dy, Ho, and Yb 13 . For our purposes, Cd 2 Os 2 O 7 is most desirable. With the transition temperature (T N = 227 K) roughly 60% higher than all other A 2 Ir 2 O 7 members, Cd 2 Os 2 O 7 should demonstrate the strongest correlation effects. Moreover, among spin-orbit coupled 5d compounds, only Cd 2 Os 2 O 7 and Nd 2 Ir 2 O 7 consistently manifest both an antiferromagnetic insulating phase and a metallic paramagnetic phase with dρ/dT > 0 (refs. [13][14][15] ). It remains unclear whether iridates such as Eu 2 Ir 2 O 7 and Sm 2 Ir 2 O 7 are metallic or insulating in the paramagnetic phase, presumably due to vacancies and site disorder from the 3+/4+ valence condition 16 . By comparison, Cd 2 Os 2 O 7 exhibits less site disorder from its 2+/5+ valence condition, with no extraneous spin order arising from the A site of the A 2 (Ir,Os) 2 O 7 structure 14 . Both pressure 14,17,18 and chemical tuning 15,18 of the A site drive the insulating transition to lower temperature in the A 2 (Ir,Os) 2 O 7 compounds, but little is known about the behavior of the AIAO magnetic order.
Here, to address this issue directly, we present results on the evolution of the spin, orbit, and lattice degrees of freedom, using polarization analysis of resonantly diffracted X-rays from Cd 2 Os 2 O 7 under diamond anvil cell pressures. This resonant diffraction technique is highly challenging with regard to both maintaining the single-crystal quality at cryogenic temperatures and tens of giga-Pascal pressures, and efficiently detecting weak magnetic diffraction signals from miniature samples of 5 × 10 −5 mm 3 size. Nevertheless, AIAO magnetic order and 5d orbital order can be resolved directly and tracked to the highest pressures under resonant conditions in the polarization switching π−σ channel of the (6, 0, 0) and (4, 2, 0) forbidden lattice orders, respectively 9 . The lattice symmetry and space group assignments were detected via the π-π' scattering channel.

Results
The magnetic quantum phase transition. We summarize in Fig. 1 the symmetry evolution under pressure. At ambient pressure, Cd 2 Os 2 O 7 has the pyrochlore structure of the Fd-3m space group (No. 227). Optical Raman scattering maps out a uniform Fd-3m phase up to P = 29 GPa and between T = 10 and 300 K. The phonon modes characteristic of that space group develop smoothly over the entire range (Fig. 2). The overall cubic symmetry is further verified by X-ray diffraction up to 41 GPa at 4 K ( Fig. 3), as lineshapes of the (1, 1, 1), (2, 2, 2), (0, 2, 2), (0, 4, 4), and (4, 0, 0) diffraction orders remain single peak. The cubic lattice constant decreases smoothly under pressure, without discernable discontinuity at the quantum critical point P c = 35.8 GPa (Fig. 1, discussed below). While the Fd-3m space group is uniquely determined by the unit cell's lattice constant and one free coordinate x for oxygen position on 48f sites that characterizes the trigonal distortion of the OsO 6 cluster 14,19 , diffraction intensities at (1, 1, 1) and (0, 2, 2) show a continuous evolution through the quantum phase transition, with x increasing by a small amount from 0.319 at P = 0 (ref. 14 ) tõ 0.325 at P c (Fig. 3b, c). Neither the AIAO antiferromagnetic order nor the continuous space group evolution within the cubic symmetry ( Fig. 1) would be detectable by macroscopic approaches such as electrical transport. Instead, we address these issues using resonant single crystal diffraction with polarization analysis at the Os L 2 edge.
Resonant diffraction at both (6, 0, 0) and (4, 2, 0) in the π−σ and π−π' polarization channels are displayed in Fig. 4 for pressures that traverse the quantum phase boundary. The magnetic diffraction intensity I (6, 0, 0) in the π−σ channel at E = 12.387 keV, which scales to the ordered staggered moment <m> as I~<m> 2 , decreases continuously with increasing pressure (Fig. 5a). The scatter in the data makes it difficult to identify definitively the quantum critical region, but a phenomenological fit of the intensity data as I~<m> 2~( P c -P) 2β over the whole pressure range gives a critical pressure P c = 35.8 ± 0.7 GPa and an  Fig. 1 Symmetries across the P-T phase diagram. The Cd 2 Os 2 O 7 lattice retains its cubic symmetry throughout the probed P-T phase space, but continuously transitions between Fd-3m and F-43m space groups. The Fd-3m lattice symmetry was verified by optical Raman scattering from 0-29 GPa and 10-300 K (gray crosses), while both phases of magnetism and structure (pink and blue shading) were inferred from X-ray diffraction measurements at T = 4 K. The metallic paramagnetic phase in the lowpressure Fd-3m space group has both spatial inversion (I) and time reversal (T ) symmetries. For the spin degrees of freedom, time reversal symmetry is broken in the low-pressure AIAO phase. On the high-pressure side, the F-43m space group breaks the spatial inversion symmetry, introduces a tetrahedral breathing distortion (inset), and restores the time reversal symmetry with disordered spins. The coincidence of continuous magnetic and structural phase transitions at a single quantum critical point suggests strong coupling between spin, orbit, lattice, and (potentially) charge degrees of freedom in this 5d pyrochlore compound exponent β = 0.40 ± 0.04 for the order parameter <m>. From energetic considerations of localized 3d-5d spins 20,21 , our magnetic diffraction results at the low temperature limit of 4 K also provide a means to estimate the AIAO magnetic phase boundary through T N~Δ L~<m> 2 , where ΔL is the external magnetostriction that develops under <m>. In AIAO-ordered pyrochlores where the cubic lattice symmetry is preserved by the magnetic order and the phase transition is continuous, the external magnetostriction ΔL is difficult to observe over the lattice's thermal expansion. Nevertheless, a non-monotonic evolution of the lattice constant with temperature, a(T), for AIAO order at very low T has been demonstrated in Nd 2 Ir 2 O 7 (T N = 33.5 K and Δa/a~1 10 −4 ) 22 . Following this logic, we identify the magnetic phase boundary T N (P) of Cd 2 Os 2 O 7 in Fig. 1 via the relationship T N~I~< m> 2 . At high pressure, for P > P c , the presence of a charge resonance at (4, 2, 0) verifies that the unoccupied t 2g orbitals remain the same in promoting the resonance behavior (Fig. 4e, f). At the same time, the absence of a resonance at (6, 0, 0) importantly marks the vanishing of the staggered moment <m> and the long-range antiferromagnetic order (Fig. 4c).
The lattice quantum phase transition. With the disappearance of AIAO order at the magnetic critical point P c , the charge diffraction starts to gain intensity in the π-π' channel for both (6, 0, 0) and (4, 2, 0) (Fig. 4), and grows continuously for P > P c in an effectively exponential manner (Fig. 5b). We emphasize that these are forbidden diffraction orders in the Fd-3m space group. The energy dependence manifests no resonance, but is instead consistent with the inverted shape of the Os L 2 absorption edge. At 41 GPa, 15% above P c , there is no diffraction intensity at forbidden orders (3, 0, 0), (5, 0, 0), and (2, 1, 1). With preserved cubic symmetry, F-43m is the only maximal non-isomorphic subgroup of the original Fd-3m space group that is consistent with the measured diffraction results. We note that the pressureinduced AIAO quantum phase transition in Cd 2 Os 2 O 7 differs from the chemical tuning scenario of the A 2 Ir 2 O 7 series (A from Sm to Pr), where the lattice inversion symmetry is preserved throughout 13 .
The space group F-43m breaks lattice inversion symmetry, opening possibilities for a number of exotic states. We did not detect further symmetry breaking such as the small tetragonal distortion in Cd 2 Re 2 O 7 that would remove the three-fold rotational symmetry 23 . The broken inversion symmetry derives from differently sized adjacent tetrahedra of the Os and Cd sublattices, which are no longer centrosymmetric and can be regarded as fully-softened breathing modes 3 with spontaneous symmetry breaking. These breathing modes disappear above T cK |Δ Os | 2 , with the lower F-43m symmetry replaced by the higher Fd-3m symmetry with adjacent tetrahedra of equal size. Here K is the vibrational elastic constant and Δ Os is the amplitude of the Os tetrahedron breathing mode. We expect T c~I , given that the measured X-ray diffraction intensities (Fig. 5b) depend on these lattice distortions as I~|f Os Δ Os + f Cd Δ Cd | 2 , where f Os/Cd are X-ray atomic form factors. Hence T c increases two orders of magnitude in a short pressure interval: from approximately 5 K at 37 GPa tõ 500 K at 41 GPa, leading to the striking concave high-pressure phase boundary in Fig. 1.

Discussion
The reduction of lattice symmetry from Fd-3m to F-43m in Cd 2 Os 2 O 7 has direct implications for phonon coupling 3 to spin fluctuations at the quantum critical point. Although spin-phonon coupling is known to generate unequal bond lengths in spin-Peierls dimers and antiferromagnetic superlattices 20 , the breathing phonons are not fully softened in the AIAO phase and, at least in the static limit, AIAO order only induces an external magnetostriction with a homogenous expansion 22 . In the highpressure phase, a breathing lattice could in principle still permit AIAO spin configurations on different sized tetrahedra, despite a loss of site inversion symmetry in the F-43m space group. Nevertheless, as no long-range commensurate antiferromagnetic order was observed experimentally (Fig. 4c), the magnetic ground state at high pressure is likely spin-disordered, as ferromagnetically interacting Ising moments along local <1,1,1> axes would generate a high level of frustration 7 . Tuning by either chemical doping or pressure drives the ratio of the magnetic interaction strength to the hopping integral smaller in A 2 (Os,Ir) 2 O 7 . With an increasing electron density under 15% volume reduction by pressure, and moving away from the strong interaction strength limit 13 , one would not naturally expect a ferromagnetic ground state. Furthermore, ferromagnetic quantum phase transitions, as well as commensurate-incommensurate antiferromagnetic transitions, are first order, which would contrast with the continuous AIAO quantum phase transition observed in our experiment.
The increased bandwidth under pressure suggests that the electronic properties of Cd 2 Os 2 O 7 in the high-pressure F-43m state become more metallic, potentially even superconducting in analogy to superconducting Cd 2 Re 2 O 7 with its broken inversion symmetry (ref. 23 ). The relationship between the spin and charge transitions is also intriguing. AIAO magnetic order and the metal-insulator transition respond similarly to compression across the P-T phase diagram. Our projected magnetic phase boundary in Fig. 1 gives dT N /dP~−5.0 K GPa −1 at P = 0, which is consistent with dT MIT /dP = −4 K GPa −1 in Cd 2 Os 2 O 7 200 400 600 800 1000 Raman shift (cm -1 ) Intensity (a.u.) The experimentally observed coincidence of magnetic and structural phase transitions at P c = 35.8 GPa, along with the similarities in the pressure evolution of the spin and charge degrees of freedom, point to one critical point within our experimental resolution (~1 GPa in the region around P c ) between the insulating AIAO order and the spin-disordered metallic phase. As Landau's phase transition theory would dictate two quantum phase transitions to be separated or to share a firstorder phase line. If the transitions are separated, then our results point to a narrow intermediate region where the various spin, charge, and structural modes can couple across phases. If the phase boundaries actually meet at one critical point, then we may be in the realm of deconfined quantum criticality 24 . In any case, the concurrence of magnetic and structural phase transitions within our experimental resolution, and the clear deviation from mean-field behavior of the high-pressure phase, stimulate a discussion of strong coupling in the quantum critical region.
From the band structure perspective, the Os 5d t 2g band in Cd 2 Os 2 O 7 is neither degenerate and forming a S = 3/2 state under Hund's rule, as indicated by the reduced staggered moment <m> = 0.59 μ B Os −1 , nor cleanly separated into several narrow bands as demonstrated for a perfect OsO 6 octahedron 14,25,26 . Instead, the t 2g orbitals in Cd 2 Os 2 O 7 extend continuously over a spectral width of order 2 eV from the combined effect of U (~1 eV) 27 , spin-orbit coupling (~0.35 eV) 26 , and trigonal distortion (~0.3 eV) on the OsO 6 octahedron 19 . Through the continuous quantum phase transition, the overall stability of the empty t 2g band is verified by the constant charge resonance profile at (4, 2, 0), with a coarse energy resolution slightly above 1 eV. From a metallic paramagnetic state, the formation of antiferromagnetic order would influence the oscillating dynamic component of the quasiparticle self-energy 28 , and in turn introduce a spin-order dependent repulsion between empty and filled states that could account for the insulating state below P c . This The overall percentage changes of (0, 2, 2) and (1,1,1) give an x increasing from 0.319 at P = 0 (ref. 14 ) to approximately 0.325 at P c effective exchange field due to neighboring opposite spins is a generalized Slater mechanism, even without introducing Brillouin zone folding 28 . If the gap opens through a critical state of singular points at the Fermi surface, instead of a removal of states altogether, its thermodynamics could fit the Lifshitz description 27,29 . The essentially concurrent, continuous quantum phase transitions of antiferromagnetism, structure, and (apparently) charge in Cd 2 Os 2 O 7 provides the necessary ingredients for a generic approach to strongly-coupled, non-mean-field quantum criticality in three-dimensions 5 . With the Fermi surface fully gapped, quasiparticle fluctuations would involve all itinerant states in reducing the screening on Coulomb U, and the increased interaction range would then help stabilize a continuous quantum phase transition 5 . Indeed, at the ambient-pressure metal-insulator transition in Cd 2 Os 2 O 7 , an increase in U from 0.8 to 1.5 eV in the theoretical modeling is consistent with the observed spectral weight shift in infrared conductivity over the broad range of 0-4 eV (ref. 30 ). Above the quantum critical point and in the spindisordered phase space of the Fd-3m space group, soft AIAO spin fluctuations and lattice breathing modes could exist and compete, and further couple to quasiparticle fluctuations. The competition between spin and lattice degrees of freedom might explain the remarkable concave-shaped phase line at high pressure, as T c scales to pressure with a non-trivial exponent much larger than one, a characteristic of strongly-coupled quantum criticality 5 . We note as well that the quantum critical region is asymmetric in P-T phase space, as the magnetic and structural phase lines approach P c with different asymptotic behavior.
The quantum phase transition in Cd 2 Os 2 O 7 , with its interwoven spin, orbit, lattice, and charge degrees of freedom, contrasts sharply with systems that have a partially gapped Fermi surface, exemplified by itinerant spin density waves where persistent carriers screen spin fluctuations and lead to mean-field behavior [4][5][6]31 . The cubic AIAO antiferromagnet also differs from itinerant ferromagnets, where strong spin and charge mode coupling at wave vector q = 0 categorically induces first-order quantum phase transitions 4 . Spin-orbit coupling in 5d systems is regarded as intermediately strong 13 , and pressure drives U/t smaller with increasing kinetic energy t, away from the strongcorrelation limit. Pressure tuning thus likely induces a continuous quantum phase transition while still preserving non-trivial quantum criticality in this 5d antiferromagnet. By Luttinger's π-π' π-σ (4, 2, 0) π-π' × 0.05 π-σ   ) and (d-f) (4, 2, 0) orders from two separate polarization channels (π-σ in red/pink and π-π' in navy/aqua solid circles), expressed in counts s −1 for 100 mA synchrotron storage ring current. The azimuthal angel ψ relative to the (0, 0, 1) vector is specified for each order. At (6, 0, 0), the π-σ channel manifests magnetic resonant spectra whose intensity decreases continuously with pressure. Beyond 36 GPa (c), the sharp resonance disappears, and a background similar to the Os L 2 absorption edge shape remains, which comes from X-ray fluorescence that passed through the polarization analyzer. Its origin is demonstrated by a comparison with an energy scan at (5.98, 0, 0) (gray line, c), where no magnetic diffraction is expected, and also with an Os L 2 absorption spectrum at 39.3 GPa (black line, c). The fluorescence part of the background intensity is dependent on the detector slit size. At (4, 2, 0), the π-σ channel reveals the orbital ordering via the anisotropic tensor susceptibility resonance. For both (6, 0, 0) and (4, 2, 0) orders, the π-π' charge diffraction intensities rise dramatically at pressures above the magnetic phase boundary, with a small leakage into the π-σ channel becoming apparent in f through the polarization analyzer (Methods)

Methods
Resonant X-ray diffraction under high-pressure. Both charge orbital order and magnetic order can induce resonant behavior at an absorption edge in X-ray diffraction 9,12,31 . At the Os L 2 edge, these two types of resonances share the empty part of the t 2g band as the intermediate state, and exhibit the same resonance profiles. The charge resonance originates from the anisotropic tensor susceptibility. In general, it can be observed at many forbidden lattice orders and, specifically, at both (4, 2, 0) and (6, 0, 0) in Cd 2 Os 2 O 7 . To avoid the charge anisotropic tensor susceptibility resonance and to isolate the magnetic resonance of the AIAO order, X-ray diffraction of the (6, 0, 0) order in Cd 2 Os 2 O 7 was performed with a limited azimuthal angle range around 45°relative to the (0, 0, 1) vector, using a horizontal diffraction geometry for both π-σ and π-π' polarization analyses 9 .
For the diamond anvil cell high-pressure environment 33 , we used a transmission (Laue) diffraction geometry in contrast to the typical reflection (Bragg) geometry at ambient pressure 9 . To achieve the diffraction geometry with the specified azimuthal condition at (6, 0, 0), single crystal samples were prepared in thin plate form with a surface normal along (0, −1, 1). This allows access to other diffraction orders such as (1, 1, 1), (4, 0, 0), (0, 2, 2), and (4, 2, 0) within the confined diamond anvil cell geometry, with the forbidden order (4, 2, 0) providing access to the charge-based anisotropic tensor susceptibility resonance. Several plates were polished down to 13 μm thickness, equivalent to one absorption length in Cd 2 Os 2 O 7 for X-rays at the Os L 2 edge (E = 12.387 keV). Unlike the iridates, resonance behavior at both the L 2 and L 3 edges are present in osmates. We chose the resonance at the L 2 edge at high P because the higher energy X-rays had a longer X-ray penetration length through both the pressure environment and the sample.
We used the low-temperature, high-pressure diffraction setup at beamline 4-ID-D of the Advanced Photon Source 33 . To reduce absorption and enhance the signal-to-background ratio, a pair of wide-angle perforated Boehler diamond anvils 33 (SYNTEK Co. LTD., Japan) were used, with culet size varying from 800 to 550 μm. A methanol/ethanol 4:1 mixture was used as the pressure medium inside rhenium gaskets. Pressure was calibrated by a Ag manometer in situ at 4.0 ± 0.5 K using a two-parameter Birch equation of state, with B 0 = 108.85 GPa and B' = dB/ dP = 5.7 over the large pressure range. For X-ray polarization analysis, a highly oriented pyrolytic graphite (HOPG) plate of 5 mm thickness and 0.35°FWHM mosaic was used as the polarization analyzer. The (0, 0, 10) diffraction order of graphite at the Os L 2 edge of 12.387 keV introduces a leakage of approximately 1.3% of the intensity from the π-π' channel to the π-σ channel, and vice versa. Data presented here were collected from a total of 8 samples under pressure for spin (7) and charge (4) resonances. The absence of high-pressure commensurate antiferromagnetic order was verified on two crystals.
Optical Raman scattering. Shards of single crystal Cd 2 Os 2 O 7 with original growth surfaces were individually loaded with a Neon pressure medium in a diamond anvil cell, and subsequently thermally cycled in a liquid helium flow cryostat. Optical Raman scattering was performed using a Horiba LabRam HR Evolution system in the MRSEC facilities at the University of Chicago, equipped with a 633 nm wavelength laser for excitation.
Data availability. The data that support the findings of this study are available from the corresponding authors upon request. a Magnetic diffraction intensity was measured at (6, 0, 0) and in the π−σ channel, with a power law fit (solid line) to model the evolution over the whole pressure range. b Lattice diffraction intensities, measured at both the (6, 0, 0) and (4, 2, 0) orders in the π-π' channel, indicate a continuous switching between the Fd-3m and F-43m space groups with a phase boundary that rises effectively exponentially. All intensities were measured at 4.0 ± 0.5 K with integration of sample mosaic profiles at specified azimuthal angle ψ using 12.387 keV X-rays. Vertical error bars represent 1σ s.d. counting statistics, and horizontal error bars represent the full range of pressure inhomogeneity during a measurement