Probing hidden symmetry via nonlinear transport in an altermagnet candidate Ca_3Ru_2O_7

Domain: physics
Published: 2026-02-23
Editorial outcome: major_revision

Abstract
X-ray and neutron diffraction are foundational tools for structure determination; however, their resolution limits can lead to misassignments in materials with subtle distortions. Here we demonstrate that nonlinear transport provides a powerful complementary approach to uncover hidden crystal symmetries, using Ca3Ru2O7 as a case study. Below the magnetic transition at TS = 48 K, our experiment reveals a previously overlooked lower-symmetry phase. This is evidenced by the emergence of longitudinal nonlinear resistance (NLR), indicating combined translational and time-reversal symmetry breaking, and thus rendering Ca3Ru2O7 an altermagnetic candidate in terms of symmetry classification. DFT calculation suggests that the lower-symmetry phase arises from an extremely subtle lattice distortion (~0.1 pm) below TS, below the detection limit of conventional diffraction. Moreover, NLR is accompanied by nonlinear Hall effect, both enhanced by the large quantum metric associated with Weyl chains. Our findings establish nonlinear transport as a sensitive probe of hidden symmetry breaking and highlight an alternative route to discovering altermagnetic states.

Introduction
Symmetry and symmetry breaking lie at the heart of modern condensed-matter physics. In particular, Ginzburg-Landau theory frames phase transitions in terms of spontaneous symmetry breaking and the emergence of order parameters1. Accurate determination of a material’s crystal symmetry is therefore essential for understanding its physical properties and identifying its ground state. Conventionally, X-ray and neutron diffraction are foundational tools for crystallographic analysis. However, these methods have resolution limits which can obscure subtle structural distortions and lead to incorrect structural assignments, particularly in complex materials with competing phases or correlated electronic states. A notable example is the massive Dirac material BaMnSb2, which was originally assigned a tetragonal space group based on neutron diffraction data2,3. However, recent studies employing scanning transmission electron microscopy (STEM), second-harmonic generation (SHG), and density functional theory (DFT) have shown that BaMnSb2 actually adopts an orthorhombic polar structure with a subtle distortion (~0.3 Å)4, thereby uncovering a bulk spin-valley-locked Dirac state4 and a nonlinear Hall effect (NLHE)5. This case highlights how resolving hidden symmetry breaking can uncover unexpected quantum states of matter. Yet such investigations typically rely on large-scale facilities, specialized instrumentation, and tightly integrated experimental–theoretical efforts. Given that uncovering hidden symmetry breaking often leads to discoveries of novel electronic or topological phenomena, there is a pressing need for alternative, more accessible approaches capable of detecting subtle symmetry changes and their electronic consequences.

In this work, we demonstrate a simple yet powerful alternative: a metrological method for detecting hidden symmetry breaking through nonlinear transport measurements. Recent advances have established nonlinear transport measurements as a sensitive probe of band topology and quantum geometry, particularly in non-centrosymmetric material6–13. We apply this method to Ca3Ru2O7 (CRO), a prototypical strongly correlated oxide known for its rich phase diagram and emergent phenomena14–16, including colossal magnetoresistance15,17, spin-valve effects14, metamagnetic textures18,19, polar semimetal behavior20,21, and massive Dirac fermions22. Upon cooling, the system undergoes two sequential magnetic transitions: a paramagnetic (PM) to A-type antiferromagnetic (AFM-a) phase at TN = 56 K, followed by a spin-reorientation into the AFM-b phase at TS = 48 K (see Fig. 1g). The magnetic transition at TS is accompanied by the opening of a pseudogap and a sharp increase in resistivity, reflecting strong coupling between spin, lattice, and charge degrees of freedom. Recent Raman spectroscopy further reveals a switch in dominant infrared-active phonon modes across this transition, modulating both pseudogap formation and hopping pathways23. At T*=30 K, CRO exhibits a crossover from an insulating to metallic behavior, with most electron-like bands gapped except for small electron and hole pockets near Ma and Mb point21,24. Several mechanisms could account for this crossover, including competition between carrier population24, a Lifshitz transition21, the emergence of magnetic multipoles25, or an incommensurate spin density wave phase—analogous to the sister material Sr3Ru2O726—that shifts the chemical potential. While these scenarios remain under debate, our nonlinear transport measurements indicate that the magnetic transition at TS induces a minute lattice distortion, resulting in a subtle symmetry lowering that likely underpins many of the exotic quantum phenomena observed below TS in CRO.

Despite decades of neutron diffraction refinements assigning the CRO structure to the Bb21m space group at all temperatures, our nonlinear transport measurements, supported by first-principles calculations27–29, reveal the emergence of a lower-symmetry Pn21a phase below TS. This structural transition arises from an exceedingly small lattice distortion (~0.001 Å), induced by the spin-reorientation—below the resolution limit of conventional neutron scattering. Crucially, this symmetry lowering is not merely a structural curiosity but transforms the AFM-b phase into a distinct altermagnetic phase in terms of symmetry classification30–32, characterized by the breaking of combined translational (τ) and time-reversal (T) symmetry, τT. Although the nonrelativistic spin split in the CRO’s altermagnetic phase is expected to be negligibly small due to the minute distortion, and the anomalous Hall effect remains forbidden by the symmetry constraints, our nonlinear transport results demonstrate an alternative route to discovering τT-symmetry-broken altermagnetic states.

In our nonlinear transport measurements on CRO, we observed a pronounced longitudinal nonlinear resistance (NLR) along the polar b-axis in the AFM-b phase. This response serves as direct evidence of τT-symmetry breaking, consistent with the altermagnetic symmetry class. Furthermore, we found that this longitudinal nonlinear response is accompanied by an NLHE. Both longitudinal and transverse nonlinear effects are significantly enhanced by the large quantum metric33–37 associated with Weyl chains28 near the Fermi surface, revealing a complex interplay between crystal symmetry, electronic topology, and transport in CRO (see Fig. 1).

Results
As noted above, prior neutron diffraction structure refinements show that CRO adopts the polar space group Bb21m38 where RuO6 octahedra form bilayer structures with ferromagnetic coupling inside the bilayer and AFM coupling in between bilayers. Below TS, the corresponding magnetic space group of AFM-b is Pn21a (No. 33.154), characterized by a two-fold screw-rotation {C2y|0,b,0}, two glide reflections Gx = {Mx|a,0,c} and Gz = {Mz|a,0,c}, and τT with τ = {a,0,c}. Here, τ connects two spin sublattices and renders the Bb21m phase an ordinary AFM. Recent DFT calculations27,29 suggest a lower-symmetry crystal structure with the space group Pn21a as the ground state of AFM-b. As illustrated in Fig. 1b, the RuO6 octahedra breathe in and out weakly by ~0.001 Å in the Pn21a structure. While this distortion is subtle and undetectable by neutron scattering, it breaks the τT symmetry, thereby lowering the symmetry from an AFM to an altermagnetic symmetry. This symmetry reduction alters the band structure and subsequently induces nonlinear transport effects, as discussed in the following.

Our DFT calculations28 show the existence of Weyl chains near the Fermi energy for both the Bb21m and Pn21a phases. As illustrated in Fig. 1c,d, Weyl bands are symmetric around the M point (the Brillouin zone boundary) for the Bb21m phase because of the τT symmetry while they turn tilted for the Pn21a phase because of τT-breaking. We note that Weyl points extend continuously in glide planes (Gx and Gz) and form interconnected Weyl chains. Weyl bands generate giant quantum metric near Weyl points as shown in Fig. 1c,d where blue and red represent opposite signs of quantum metric distribution. For symmetric Weyl bands in Bb21m, the net quantum metric is strictly zero because of τT-induced cancellation. For the asymmetric Weyl bands of Pn21a, the net quantum metric is significant despite weak tilting, leading to a large NLR along the polar axis b. Additionally, the calculated Fermi pockets in both phases agree with carrier densities determined by our Hall measurement (ne = 7.0 × 1018 cm-3 and nh = 7.3 × 1018 cm-3) (see supplementary Note 1 and supplementary Fig. S1).

The Bb21m and Pn21a phases exhibit qualitatively different symmetry in nonlinear transport, although both are inversion-breaking and present nonlinear transport. Because of τT symmetry, which is equivalent to T to constrain the nonlinear response, Bb21m exhibits only a NLHE39 induced by Berry curvature dipole (BCD) (Fig. 1e). Free from the τT-constraint, Pn21a exhibits both NLHE due to BCD and quantum metric dipole (QMD) and longitudinal NLR caused by QMD and a Drude term37, as illustrated in Fig. 1f. We summarize the symmetry-constrained nonlinear conductivities in Table 1 for both Bb21m and Pn21a. According to our calculations28, QMD is several orders of magnitude larger than BCD and Drude contributions here. Therefore, we expect that NLHE and NLR will increase quickly after entering the AFM-b phase and the existence of NLR is the hallmark to identify the Pn21a phase and the τT-breaking altermagnet. However, conventional methods cannot distinguish between these two phases. For example, optical SHG shows the same nonlinear susceptibilities for both phases (see supplementary Table 1). Likewise, the nonrelativistic spin splitting, which is smaller than 1 meV, and the anomalous Hall effect, which vanishes due to the coexistence of glide mirror symmetries Gx and Gz in CRO, fail to discriminate the altermagnet from an ordinary antiferromagnet. Importantly, the combined τT-symmetry forbids both the altermagnetic phase and the presence of longitudinal NLR. Thus, the detection of NLR provides a clear signature of τT-symmetry breaking, consistent with the emergence of a non-centrosymmetric altermagnetic state in CRO; nevertheless, further experiments will be required for definitive confirmation.

To find if the AFM-b phase exhibits NLR along the b-axis as the theory predicted, we fabricated a microscale Hall bar device with its long axis aligned along the b-axis (denoted as S1) using a combination of photolithography and focused ion beam (FIB) deposition (see Methods) and performed nonlinear transport measurements on the device with a current applied along the b-axis. Microscale devices offer two key advantages: They enhance any existing nonlinear response by enabling the application of larger electric fields without significant heating, while also minimizing the influence of multiple domains. Given the typical domain size in CRO is approximately 20 µm, our devices were fabricated on single-domain regions. The crystallographic axes of the CRO crystals used in the devices were identified via SHG measurements (Supplementary Note 2 and Supplementary Fig. S2) and verified through magnetoresistance (MR) characterization. As previously reported14,17, the b-axis MR exhibits a distinct spin-valve effect near 6 T due to the transition from the AFM-b to a canted AFM (CAFM) state, resulting in a remarkable drop in MR.

Figure 2a displays the image of device S1. Electrodes 1 through 6 were patterned using optical photolithography. Owing to the approximately 1.5 µm thickness of the CRO crystal, the evaporated gold electrodes exhibit a discontinuity near the sample edges. This gap was bridged using platinum (Pt) deposited via FIB, as indicated by the light blue regions in Fig. 1a. The in-plane resistivity along the b-axis, ρbb, measured using contacts 2–3 with current along 1–4, is shown in Fig. 2b (blue curve). The measured ρbb closely matches previously reported results21,40,41, displaying a characteristic downturn at TN, where the PM phase transitions to the AFM-a phase, followed by a sharp upturn at TS, where the transition from AFM-a to AFM-b occurs. The sharp increase in ρbb at TS is attributed to the opening of a pseudogap, as noted above. Unlike previous reports, however, ρbb does not revert to a metallic temperature dependence below 30 K (T*), likely due to a slight contribution from the out-of-plane resistivity component, ρcc. In CRO, the c-axis resistivity is 10³–10⁴ times larger than the in-plane value at low temperatures15,16; our c-axis resistivity measurements show consistent behavior (see below). Consequently, even a slight current misalignment—i.e., the presence of a small c-axis current component—and/or surface terraces can introduce a measurable c-axis contribution to the in-plane resistivity, resulting in an apparent upturn in ρbb(T) below T*, as reported in several previous studies41,42. The surface terraces in our device may arise naturally from the material’s intrinsic layered structure or be introduced during the FIB surface cleaning process.

Through nonlinear transport measurements on device S1, we find compelling evidence that the AFM-b phase is associated with the Pn21a structure. Figure 2c presents the longitudinal second-harmonic voltage response, V2ωb;bb, measured between electrodes 2 and 3 under an AC current (Iω) applied along the b-axis at 3 K. V2ωb;bb is clearly nonzero, with its dependence on Iω well described by a dominant quadratic term and a minor quartic correction, as shown by the dashed curve in Fig. 2c. The quartic term vanishes upon warming to 10 K (see supplementary Note 3). In addition, V2ωb;bb is independent of frequency (Fig. 2d). A similar longitudinal nonlinear response is also observed using electrodes 5 and 6 (see supplementary Note 4 and Fig. S3a & S3c). These features provide strong evidence for a finite NLR along the b-axis, indicating that AFM-b phase adopts the Pn21a structure. This is further supported by the temperature evolution of V2ωb;bb across the magnetic phase transitions. As stated above, CRO exhibits two magnetic transitions upon warming, at TS and TN, enabling us to track the nonlinear transport response across different magnetic phases. Figure 2b (red curve) presents V2ωb;bb as a function of temperature. Notably, as the temperature increases, the magnitude of V2ωb;bb gradually decreases, vanishing between T* and TS, before rising to a small, nearly temperature-independent value for T > TS. These results strongly suggest a correlation between the longitudinal nonlinear signal and magnetic ordering in CRO. The small but finite signal at high temperatures (T > TS) is attributed to extrinsic effects such as the junction artifacts or thermoelectric contributions, as it is absent in other devices (see below).

As discussed earlier, the nonlinear response V2ωb;bb observed below TS should be attributed to the nontrivial band topology associated with a Weyl chain state that emerges in the AFM-b phase. This is further demonstrated by the coupling of V2ωb;bb with magnetism observed in magnetic fields sweep measurements. As seen in the CRO’s magnetic phase diagram in Fig. 1g, the magnetic field applied along the b-axis induces a transition from the AFM-b to the CAFM state at approximately ~6 T for T < 40 K. The CAFM state is characterized by a superposition of AFM and weak FM components, with spins canted 25° from the b-axis14. As noted above, this magnetic transition results in a spin-valve-like effect, i.e., the in-plane resistivity exhibits a steep drop due to significantly suppressed spin scattering. In our experiment, we observed a sharp decrease in ρbb at ~7 T for the field applied along the long axis of device S1 (Fig. 2e), slightly higher than the previously reported spin-valve transition field (HSV ~6 T). This can be ascribed to a slight misalignment of the sample within the holder as HSV increases when the field is rotated away from the b-axis. From the data shown in Fig. 2e, it can be seen that V2ωb;bb shows a sharp change as well as a hysteresis near HSV in response to the AFM-b-to-CAFM transition, with its sign switching from negative to positive. These results indicate that longitudinal second-order current j2ωk switches its sign as the AFM-b phase transitions to the CAFM phase. This information, as well as its temperature dependence revealed in Fig. 2b, f, has been schematically illustrated in the phase diagram in Fig. 1g. Such a coupling of V2ωb;bb with the magnetic transition can be well understood in terms of the variation of band topology across the magnetic transition revealed in our theoretical studies28. As mentioned earlier, in the AFM-b state, CRO exhibits Weyl chains near the Fermi energy, which are protected by the underlying crystal and magnetic symmetries. However, upon transitioning to the CAFM state, spin canting changes the symmetry of the system, which has profound effects on the band structure, leading to the sign change of j2ωk.

In addition to the NLR along the b-axis, the Pn21a structure phase is also predicted to exhibit a strong nonlinear Hall response, driven by the large QMD associated with Weyl chains near the Fermi energy (see Fig. 1f and Table 1), when the current is applied along the a- or c-axis. To probe the nonlinear Hall conductivity with Iω//a (σb;aa), we fabricated a device (denoted S2) with its long axis along the a-axis. Figure 3a shows the image of this device, which was also fabricated on a single-domain crystal. The temperature dependence of the in-plane resistivity along the a-axis (ρaa) and the second-harmonic Hall voltage (V2ωb;aa) measured on this device are presented in Fig. 3b. Like ρbb, ρaa shows a downturn at TN and an upturn at TS. However, near T*(=30 K), ρaa exhibits a peak, followed by an upturn, indicating that the contribution from ρcc, which is smaller in S2 than in S1. To clarify how the admixture of a c-axis resistivity component in ρaa influences the nonlinear effects, we fabricated and measured an additional micrometer-scaled device (device S3). As shown in Fig. 3e, device S3 reproduces the metallic ρaa(T) behavior reported in the literature while exhibiting similar nonlinear transport signatures. This consistency across samples rules out extrinsic contributions from surface terraces and confirms that the observed nonlinear response originates from the bulk crystal.

At temperatures below 30 K, the magnitude of V2ωb;aa measured between voltage leads 3 and 5 increases sharply, reaching ~5 μV at 3 K, but vanishes above TS (see the red curve in Fig. 3b). This nonlinear Hall voltage shows a clear quadratic dependence on Iω and is independent of frequency (Fig. 3d), consistent with expectations for a second-order Hall response. This behavior is also reproduced in a second Hall channel measured between leads 2 and 6 (see Supplementary Note 4 and Fig. S3b, S3d). Meanwhile, the longitudinal second-harmonic voltage V2ωa;aa remains close to zero, consistent with DFT predictions in Table 1. The small residual values of V2ωa;aa are attributed to the slight misalignment of the voltage leads. Similar second-harmonic behavior is reproduced in device S3, whose second-harmonic I–V data are shown in Supplementary Fig. S4. The magnitude of V2ωb;aa in S3 is somewhat larger, but of the same order as that in S2, likely due to differences in sample geometry.

Under an in-plane field along the b-axis, we observe sharp spin-valve transitions and hysteresis in both ρaa and V2ωb;aa (Fig. 3b, e) near 7 T, corresponding to the AFM-b to CAFM transition, for both devices S1 and S3. Notably, V2ωb;aa decreases in magnitude in the CAFM state (see Fig. 3c, f for device S2 and supplementary Fig. S4c for S3). This reduction suggests diminished topological contributions in the CAFM phase, as discussed above. The NLHE driven by an a-axis current is also reproduced in another microscale device (S4) (see Supplementary Fig. S5), as well as in a millimeter-scale device (S5) containing multiple domains (see Supplementary Fig. S6), demonstrating the robustness of this nonlinear Hall response across different device geometries and domain configurations.

To confirm the NLHE induced by the c-axis current (i.e., σb;cc in Table 1), we also performed measurements on single crystals with the current applied along the c-axis. In this experiment, we could not make a device on a single-domain crystal; instead, we used a CRO crystal with the lateral dimensions of ~1 mm and the electrodes being attached using silver epoxy (denoted S6). Although such a crystal has domains, we expect to observe in-plane nonlinear Hall response along the sample edge (i.e., the a/b-axis), as shown in the insert to Fig. 4a. From the measurements on device S6, we indeed observe in-plane second-harmonic Hall response V2ωb;cc (see the red curve in Fig. 4a where we also present the c-axis resistivity ρcc data measured on another bulk sample). Like V2ωb;aa, the magnitude of V2ωb;cc sharply increases below T* and vanishes above TS. V2ωb;cc also exhibits a clear quadratic dependence on Iω and remains frequency independent (Fig. 4b). In-plane magnetic field scans show similar sharp transitions and hysteresis in V2ωb;cc near the spin-valve transition field (Fig. 4c), with a reduction in magnitude in the CAFM state. For comparison, we also conducted transverse nonlinear measurements on device S1 (Fig. 2a). Since the current in this device is applied along the b-axis, NLHE is not theoretically expected for this measurement configuration (see Table 1). In experiments, the measured transverse second-harmonic response V2ωa;bb is indeed negligibly small (Fig. 2c), and its small value is caused by the voltage leads’ misalignment, consistent with the theoretical prediction of the absence of NLHE for this measurement configuration.

One may ask whether the NLR and NLHE observed in the AFM-b phase of CRO could arise from extrinsic contributions such as junction artifacts, thermoelectric effects, or disorder-induced scattering. This possibility can be excluded based on the consistent behavior observed across six independent devices studied in this work. In five of these devices, both the NLR and NLHE emerge only below the magnetic transition temperature TS and are strongly enhanced below T*, but vanish above TS (Figs. 2b, 3b, e and 4a), indicating their intrinsic link to the magnetic phase transition. Furthermore, both signals are suppressed across the field-induced AFM-b to CAFM transition, showing a parallel response to the change of magnetic ordering (Figs. 2e and 3f). Such robust coupling to magnetic transitions is not expected for nonlinear signals of extrinsic origin (e.g., disorder scattering11,43,44), which are insensitive to the spin order or magnetic transition. There is only one exception: device S1 shows a weak, constant nonlinear response above TS (Fig. 2b), likely due to a trivial source, as this feature is absent in the other five devices—S2 and S3 (Fig. 3), S4 (Supplementary Fig. S5), S5 (Supplementary Fig. S6) and S6 (Fig. 4). Therefore, the observed longitudinal NLR along the b-axis (Fig. 2), as well as the NLHE for Iω//a (Fig. 3) and Iω//c (Fig. 4) in the AFM-b phase, are intrinsic in nature and originate from nontrivial band topology, as illustrated in Fig. 1f. The agreement of these observations with the nonlinear conductivity tensor components predicted for the Pn21a structure (Table 1) support the conclusion that the magnetic transition at TS—where the magnetic moments of Ru rotate from the a-axis (TN < T < TS) to the b-axis (T < TS)—induces subtle structural distortions undetectable by neutron scattering. These distortions break τT symmetry, as shown in Fig. 1a, b, and are essential for the emergence of the observed nonlinear responses.

Our experimental data also show that the nonlinear Hall response is significantly stronger than the longitudinal nonlinear response along the b-axis. The second-harmonic Hall voltage measured in device S2 reaches approximately 5.5 μV at 3 K (Fig. 3d) under an applied current of 70 μA, while device S3 shows about 9 μV at 3 K (Fig. 3d) at 50 μA, both significantly larger than the longitudinal second-harmonic voltage observed in device S1 (Fig. 2d). Since the crystals used in S1, S2, and S3 have similar dimensions, this difference is intrinsic and aligns with theoretical prediction28. To enable a quantitative comparison with theory, we estimate the nonlinear conductivities using the relation σ2ωi;jj = (V2ωi;jj l3)/(I2j R3jj w2d), where l, w, and d are the sample length, width and thickness, respectively; Ij is the applied current, and Rjj is the linear resistance34. Using measured values of ρbb = 2.5 mΩ.cm for S1, ρaa = 460 μΩ.cm for S2, and ρaa = 410 μΩ.cm for S3 at 3 K, we obtain σ2ωb;bb = 0.20 A/V2 for device S1, σ2ωb;aa = 150 A/V2 for device S2, and σ2ωb;aa = 157 A/V2 for device S3. The temperature and magnetic dependence of σ2ωb;bb and σ2ωb;aa for S1, S2, and S3 are presented in Supplementary Fig. S9 and S10. Theoretical calculation28 yields σ2ωb;bb ≈ 0.05 A/V2 and σ2ωb;aa ≈ 10 A/V2 near the charge neutral point. Although the experimental values are approximately one order of magnitude larger, the ratio between σ2ωb;bb and σ2ωb;aa is consistent between theory and experiment. This trend is reminiscent of similar behavior observed in the magnetic topological insulator MnBi2Te434. Similarly, using the measured value of ρcc = 0.62 Ω.cm, σ2ωb;cc is estimated to be 0.0029 A/V2 for device S6, one order of magnitude smaller than the theoretical value of σ2ωb;cc ≈ 0.05 A/V2. We attribute this lower experimental value to multidomain structure, since both twin domains and 180° domains contribute oppositely to the net nonlinear signal. Additionally, the vanishing σ2ωa;bb (Fig. 2c) and σ2ωa;aa (Fig. 3d) components further support the agreement between our measured nonlinear conductivity tensor and theoretical expectations. Most importantly, the observation of NLR (σ2ωb;bb) in CRO provides direct evidence of the concurrent inversion and τT symmetry breaking—conditions uniquely satisfied in the low temperature AFM-b phase that adopts Pn21a space group (Fig. 5). This structural assignment resolves the discrepancy between the band structure computed for Bb21m and the experimentally determined band structure29.

Based on our theoretical studies28, both the longitudinal nonlinear response and the NLHE of the AFM-b phase should be dominated by QMD as indicated above. In theory, the sign of QMD is dependent on the Néel vector direction and can be reversed if the Néel vector can be reversed by controlling the direction of magnetic fields sweep. The sign switching of QMD would lead to the sign reversal of the nonlinear signal. This was previously observed in a few-layer magnetic topological insulator MnBi2Te434,35,45–47, where the field-driven Néel vector switching is enabled by reduced interlayer coupling, low anisotropy energy, broken inversion symmetry and a strong coupling between magnetic and electronic degrees of freedom. In our field sweep measurements on CRO, we find the nonlinear signal remains the same sign upon positive and negative fields sweep (see Figs. 2e, 3f, 4c). This is because in bulk CRO, both the exchange energy and magnetic anisotropy energy are large, such that the magnetic field alone cannot reverse the Néel vector.

As shown in Figs. 2b, 3b, e, and 4a, both the longitudinal nonlinear response and the NLHE along the b-axis are significantly enhanced below T*=30 K. This enhancement can be attributed to a pronounced reconstruction of the Fermi surface at low temperatures. Previous ARPES studies on CRO20,22,48 have revealed that, below 30 K, the electronic structure becomes highly anisotropic, with bands crossing the Fermi level only near the Ma and Mb points, and the emergence of a Dirac cone around M22. Earlier first-principles calculations further predicted the presence of Weyl nodes embedded within this Dirac cone at M29. Our recent calculations28 indicate that these Weyl points extend and form Weyl chains confined within glide mirror planes. However, resolving these fine topological features via ARPES remains challenging due to the narrow energy window of the Weyl bands (~10 meV), and the need for a clean cleavage along the ac-plane to visualize the associated surface states. Although the Pn21a structure leads to the presence of an altermagnet in terms of symmetry classification, the associated nonrelativistic spin splitting is too small to be resolved by ARPES and the anomalous Hall effect is symmetry-forbidden due to the presence of two glide mirrors. Consequently, unlike in other known altermagnets49–54, the altermagnetic state in CRO is difficult to verify using conventional probes such as ARPES or the anomalous Hall effect. Therefore, the demonstration of the τT symmetry breaking via nonlinear transport measurements provides the first direct experimental signature consistent with the altermagnetic state of CRO (see Fig. 5). Although CRO does not exhibit pronounced spin splitting or an anomalous Hall effect, identifying τT-symmetry breaking establishes a powerful framework for screening and classifying other antiferromagnetic materials that may host altermagnetic states. In addition, a recent theory55 predicted that CRO exhibits another altermagnetic phase under strain with a spin order different from our work.

The common AFM materials have τT and/or PT symmetry, where P stands for the spatial inversion (Fig. 5a, c), while altermagnets are a unique group of AFMs that lack both τT and PT (Fig. 5b, d). Most known altermagnets are inversion symmetric and thus forbid the second-order nonlinear transport (Fig. 5b), though allowing for third-order response56, while inversion-breaking altermagnets can exhibit second-order effects (Fig. 5d). Therefore, the nonlinear transport can serve as a sensitive probe to differentiate the altermagnet from an ordinary AFM with τT and distinguish two types of altermagnets (inversion-breaking or inversion symmetry), as shown in Fig. 5b, d. Beyond the specifics of CRO, our findings underscore the broader significance of nonlinear transport as a universal and sensitive probe of subtle symmetry breaking. Across a wide class of systems, minute symmetry differences—whether driven by magnetic transitions, non-stoichiometric compositions, or mixed-valence states—often give rise to profound changes in electronic topology. However, distinguishing these nearly degenerate structural or magnetic configurations remains a central challenge, particularly when conventional probes such as diffraction or ARPES are limited by spatial resolution. In this context, the nonlinear response, which directly couples to symmetry-breaking components of the quantum geometric tensor, provides a robust bulk-sensitive signature. Thus, our work highlights nonlinear transport as a complementary diagnostic across a range of quantum materials, enabling direct access to hidden symmetry breaking and associated topological phenomena. Once the device geometry is defined, the measurements can be carried out using conventional lock-in transport setups and readily extended to a broad range of materials, making this a scalable probe of τT-symmetry breaking. This approach can efficiently narrow the search space for altermagnetic candidates, after which characteristic features—such as spin splitting or anomalous Hall effects—may be verified using more specialized techniques like spin-resolved ARPES or Hall measurements. Notably, this method is experimentally accessible within standard condensed-matter transport laboratories and does not require large-scale infrastructure such as synchrotron X-ray, neutron, or spin-resolved ARPES facilities.

In summary, we have demonstrated intrinsic quantum metric-induced nonlinear transport in the strongly correlated oxide Ca3Ru2O7, representing the first observation of such phenomena in an oxide material. Our nonlinear transport measurements unambiguously resolve the longstanding ambiguity regarding CRO’s ground state structure, establishing the lower-symmetry Pn21a phase, which supports an altermagnetic state in terms of symmetry classification. The exceptional sensitivity of nonlinear transport to the minute symmetry change associated with the spin-reorientation transition highlights its power as a diagnostic tool for uncovering hidden symmetries and topological states in quantum materials. These findings establish nonlinear transport as a versatile and accessible approach for probing subtle symmetry breaking and emergent topology in correlated systems.

Methods
Single crystals of Ca3Ru2O7 (CRO) were synthesized using the floating zone method. Microscale devices were fabricated via photolithography and focused ion beam (FIB) technique using an FEI Scios 2 dual-beam scanning electron microscope (SEM). Gold electrodes were patterned using standard photolithography techniques. Platinum electrical contacts were deposited in situ within the SEM chamber. A cleaning step with an ion beam was performed to remove spurious debris and eliminate potential electrical shorts. The FIB-fabricated devices were placed on SiO₂ substrates to avoid spurious nonlinear signals, as previously indicated when Pt is deposited on Si substrates57.

Nonlinear transport measurements were conducted using standard lock-in detection methods. Alternating current (AC) and direct current (DC) were supplied by a Keithley 6221 and a Keithley 6220 precision current source, respectively. Corresponding voltage signals were measured using a Stanford Research Systems SR860 lock-in amplifier for AC and a Keithley 2182A nanovoltmeter for DC measurements.

Out-of-plane contact fabrication on FIB-patterned devices requires additional manipulation steps that can introduce surface damage and affect measurement reliability; hence, out-of-plane measurements were performed on millimeter-scale devices. A small sample dimension is essential for obtaining a measurable second-harmonic signal; therefore, we made two devices: a smaller device, S6, with four contacts for second-harmonic measurements and a larger device for linear out-of-plane transport measurements. The principal in-plane axes of S6 were determined using spin valve measurements; however, the presence of multiple twin domains cannot be ruled out.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Upon transferring their manuscript to Nature Communications, the authors corrected the major deficiencies pointed out in my first report in a satisfactory manner, by performing additional experiments, inserting additional references, and augmenting their discussion. The manuscript is now technically correct, and it merits publication in its current form. The question whether it should be published in Nature Communications does not have an exact answer. While it would not be unreasonable to do so, I recommend publication in a more specialized journal, because the very subtle transport anomalies reported here do not have any major implications and will likely not be of interest beyond a narrow community of specialists.", "reviewer_2": "I thank the authors for their careful responses to my previous questions and for addressing them in the revised manuscript. I appreciate the effort that has gone into the revision.\nI have one remaining request regarding the analysis of the nonlinear Hall response. In my previous report, I asked the authors to present the second-order conductivity rather than the second-order voltage in the temperature dependence, which they have now done and I appreciate this improvement. It may also be informative to include an analogous plot for the magnetic-field dependence, to further demonstrate that the sharp transition and hysteresis in the second-order response are not solely driven by changes in resistivity. I do not intend this as a major additional burden to the authors, but rather as a potentially helpful piece of information for readers.\nAside from this minor suggestion, I support the publication of this paper."}, "author_response": "We thank both the referees for their careful evaluation of the revised manuscript and their helpful comments. Below, we provide our responses to the remaining comments, with our replies shown in blue.\nResponse to Reviewer #1\nReviewer #1 (Remarks to the Author):\nUpon transferring their manuscript to Nature Communications, the authors corrected the\nmajor deficiencies pointed out in my first report in a satisfactory manner, by performing\nadditional experiments, inserting additional references, and augmenting their\ndiscussion. The manuscript is now technically correct, and it merits publication in its\ncurrent form. The question whether it should be published in Nature Communications\ndoes not have an exact answer. While it would not be unreasonable to do so, I\nrecommend publication in a more specialized journal, because the very subtle transport\nanomalies reported here do not have any major implications and will likely not be of\ninterest beyond a narrow community of specialists.\nResponse:\nWe thank the reviewer for their careful re-evaluation and their positive assessment that\nthe manuscript is now technically correct and merits publication.\nWe appreciate the referee’s thoughtful comment regarding the breadth of interest. We\nbelieve that the nonlinear transport properties reported in this manuscript have broader\nrelevance beyond a narrow community of specialists, as they demonstrate nonlinear\ntransport as a sensitive diagnostic to subtle symmetry-breaking and phase transitions,\nwith implications for a wide range of materials where conventional probes fail to\ndetermine correct crystal structure. Overall, our findings establish nonlinear transport as\na sensitive probe of hidden symmetry breaking and highlight an alternative route to\ndiscovering altermagnetic candidates, which is among central topics in contemporary\ncondensed matter physics.\nResponse to Reviewer #2\nReviewer #2 (Remarks to the Author):\nI thank the authors for their careful responses to my previous questions and for\naddressing them in the revised manuscript. I appreciate the effort that has gone into the\nrevision.\nI have one remaining request regarding the analysis of the nonlinear Hall response. In\nmy previous report, I asked the authors to present the second-order conductivity rather\nthan the second-order voltage in the temperature dependence, which they have now\ndone and I appreciate this improvement. It may also be informative to include an\nanalogous plot for the magnetic-field dependence, to further demonstrate that the sharp\ntransition and hysteresis in the second-order response are not solely driven by changes\nin resistivity. I do not intend this as a major additional burden to the authors, but rather\nas a potentially helpful piece of information for readers.\nAside from this minor suggestion, I support the publication of this paper.\nResponse:\nWe thank the reviewer for this helpful suggestion. Following their advice, we have now\nincluded an additional plot showing the magnetic-field dependence of the second-order\nconductivity in Supplementary Fig. 10."}]

Decision letter