Abstract
Electronic band structures in solids stem from a periodic potential reflecting the structure of either the crystal lattice or electronic order. In the stoichiometric ruthenate Ca_{3}Ru_{2}O_{7}, numerous Fermi surfacesensitive probes indicate a lowtemperature electronic reconstruction. Yet, the causality and the reconstructed band structure remain unsolved. Here, we show by angleresolved photoemission spectroscopy, how in Ca_{3}Ru_{2}O_{7} a C_{2}symmetric massive Dirac semimetal is realized through a Brillouinzone preserving electronic reconstruction. This Dirac semimetal emerges in a twostage transition upon cooling. The Dirac point and band velocities are consistent with constraints set by quantum oscillation, thermodynamic, and transport experiments, suggesting that the complete Fermi surface is resolved. The reconstructed structure—incompatible with translationalsymmetrybreaking density waves—serves as an important test for band structure calculations of correlated electron systems.
Introduction
A Fermisurface reconstruction refers to the sudden change of the electronic band structure as a function of a tuning parameter. As most electronic properties are governed by electrons in the vicinity of the Fermi level, a change of the Fermisurface topology can have profound ramifications. Those reconstructions that are not obviously linked to a symmetry change of the crystal structure are of particular interest. Common triggers of Fermisurface reconstructions are translationalsymmetry breaking spin or chargedensity waves. Typically, this reduction of symmetry and the resulting folding of the Brillouin zone lead large Fermisurface contours to split into smaller pockets. Once reconstructed, it is, however, often difficult to identify the Fermi surface structure. The hightemperature superconductor YBa_{2}Cu_{3}O_{7−x} is a good example of this. In the underdoped regime, chargedensitywave order^{1,2,3} clearly reconstructs the Fermi surface. Although quantum oscillations^{4,5} and transport^{6,7} experiments have revealed the existence of an electron pocket, the reconstructed Fermisurface topology is, despite strong effort, still not clarified^{8}. Another prominent example is the orthorhombic bilayer ruthenate Ca_{3}Ru_{2}O_{7}. An initial Atype antiferromagnetic (AFM) order setting in at T_{N} = 56 K^{9} has no significant impact on the transport properties^{10}. A spin reorientation from the orthorhombic a to the baxis direction^{11,12} occurs at T_{s1} = 48 K. While this latticespacegroup preserving reorientation^{9} naively might appear to be of minor consequence, the transition at T_{s1} = 48 K marks a dramatic electronic transformation.
Across this transformation, the Seebeck coefficient undergoes a sharp sign change taking large negative values below the transition temperature^{13,14}. T_{s1} is also the onset of inplane anisotropic transport properties^{14}. Although transport and thermodynamic experiments provide information about Fermi surface area and electronic masses^{15,16}, the complete reconstructed Fermi surface has so far remained undetermined. Previously reported angleresolved photoemission spectroscopy (ARPES) data have revealed the existence of a boomerangshaped Fermi surface^{17}. Obviously, a single pocket is insufficient to explain the observed ambipolar electronic properties^{14}. Although a densitywave state has never been identified^{12}, a common assumption is that a translationalsymmetrybreaking order reconstructs the Fermi surface into multiple sheets. However, as long as the complete Fermi surface and its orbital composition remain unidentified, so do the reconstructing mechanism.
Here, we provide by direct ARPES experiments the complete Fermi surface structure of Ca_{3}Ru_{2}O_{7} across the electronic reconstruction. Above the reconstructing temperature, the lowenergy electronic structure resembles that of a strongly correlated metal consistent with its orthorhombic crystal structure. The reconstructed Fermi surface by contrast consists of a small electron pocket formed by massive Dirac fermions along the shortaxis orthorhombic zone boundary and a boomeranglike hole pocket in vicinity to the longaxis zone boundary. As the orthorhombic order parameter ∣a − b∣/(a + b) remains essentially unchanged across the reconstruction^{9}, the sudden emergence of such a dramatic Fermi surface anisotropy is unexpected. We furthermore demonstrate that the Fermi surface transformation appears in two steps. The anisotropic zoneboundary Fermi surfaces first appear below T_{s1} = 48 K and eventually the Dirac fermions settle into the lowtemperature structure below T_{s2} = 30 K. Throughout this temperature evolution, no signature of Brillouinzone folding is identified, excluding translationalsymmetry breaking densitywave/orbital orders as the origin of the phase transition at T_{s1} = 48 K. We argue that the reconstructed Fermi surface should be understood from the d_{xz} and d_{yz} orbitals whereas the d_{xy} sector is not crossing the Fermi level in the reconstructed phase. The revelation of the complete Fermi surface reconstruction provides an ideal testbed for abinitio band structure calculations beyond densityfunctionaltheory concepts.
Results
Hightemperature state
The Fermi surface and lowenergy electronic structure of the Ca_{3}Ru_{2}O_{7} normal state—above the Néel temperature T_{N} = 56 K—are presented in Fig. 1a, d, and e. The orthorhombic zone boundary is indicated by black dashed lines in Fig. 1a. All quasiparticle dispersions are broad irrespective of whether linear p or spolarised light is used. Part of the Fermi surface consists of straight sectors running diagonally through the orthorhombic Brillouin zone. This quasionedimensional structure remains essentially unchanged across the Néel transition at T_{N} = 56 K [see Fig. 1b, f, and g]. Furthermore, the orthorhombic zone boundary points M_{a} = (±π/a, 0) and M_{b} = (0, ±π/b) are virtually indistinguishable (See also Supplementary Fig. S1).
Fermi surface anisotropy
Across the structural transition at T_{s1} = 48 K, however, the electronic structure undergoes a dramatic reconstruction. This is evidenced by the emergence of a fast dispersing band and a tiny Fermi surface around M_{a}—see Fig. 1c, h, and i. Remarkably, this small Fermi surface sheet is absent at M_{b}. Instead, as previously reported^{17}, boomeranglike Fermi surface sheets are found around the M_{b} point. Therefore, in contrast to T > T_{s1}, the lowtemperature structure appears highly anisotropic featuring different Fermi surface topology around M_{a} and M_{b}. This Fermi surface reconstruction appears without change of the crystal lattice space group and with minute (~1%) reduction of the orthorhombic order parameter^{9}.
To exclude the possibility that this C_{2} symmetry is an artefact of photoionizationmatrixelement effects, we follow a standard measurement protocol^{18,19}. That is to carry out Fermi surface mappings with azimuthal angles differing from each other by 90^{∘} [see Fig. 2a, b]. In situ azimuthal rotation implies that the Fermi surface maps in Fig. 2a, b are from the same surface. Here, k_{∥} (k_{⊥}) on the horizontal (vertical) axis represents the momentum parallel (perpendicular) to the electronanalyser slit. The electronic structure with a tiny Fermi pocket around the M_{a} point and boomeranglike features near M_{b} tracks the azimuthal rotation—see Fig. 2a–d. The C_{2}symmetric electronic structure is also revealed by the band dispersions. Along the M_{a}–Γ and M_{b}–Γ directions, the band curvature around M_{a} and M_{b} are clearly different [Fig. 2e, g]. An electron pocket is formed around M_{a} whereas two holelike pockets are found on each side of M_{b}. In a similar fashion, dispersions along the M_{a}–X and M_{b}–X directions are inequivalent [Fig. 2f, h]. Electronlike band curvature is found around M_{a} whereas no Fermilevel crossing is observed along M_{b}–X. These results exclude matrixelement effects as the source of the observed anisotropy.
Twostage Fermi surface reconstruction
Tracking the temperature dependence of the band structure reveals two electronic temperature scales. The electronic band structure along M_{a}–Γ and M_{b}–Γ is shown for temperatures going from 16 to 50 K. Above T_{s1} = 48 K [Fig. 3e, j and 1], all bands appear with broad lineshapes. Once cooled below T_{s1}, welldefined bands around the M_{a} and M_{b} points emerge [see Fig. 3d, i, and k]. The appearance of another electronlike band around the Γ point is accompanied by a gap Δ opening below T_{s1} [see Fig. 3g–i and Supplementary Fig. S4]. The band structures around M_{a} and M_{b} are inequivalent not only in terms of curvature but also in terms of temperature dependence. The M_{a}–Γ band dispersion is temperaturedependent whereas the corresponding structure around M_{b} is virtually insensitive to temperature. Examining the M_{a}–Γ direction, two inequivalent bands with different Fermi momenta are observed for 30 < T < 48 K whereas only a single set of bands is resolved for T < 30 K—see Fig. 3. The two bands around M_{a} display asymmetric matrix elements. Momentum distribution curves (MDCs) at E_{F} and 30 < T < 48 K are therefore not symmetric around M_{a}. Upon cooling below 30 K, the single electron pocket around M_{a} displays symmetric Fermi momenta k_{F} despite the asymmetric matrix elements. To illustrate the transition between the two and singleband situation, we define Δk_{F} as the reciprocalspace distance between the two MDCintensity maxima. Across T_{s2} = 30 K, Δk_{F} drops by a factor of two. This observation is independent of incidentlight polarisation (see Supplementary Figs. S5 and S6). On the other hand, the gap Δ evolves smoothly across T_{s2} [see Fig. 3g–i and Supplementary Fig. S4], suggesting that the states around Γ are not involved in this transition. Although T_{s2} = 30 K remains to be identified as a thermodynamic temperature scale, it does coincide with the onset of a strong negative Nernst effect response^{14} [see Fig. 3l]. We also notice that the reduction of Δk_{F} below T_{s2} is consistent with an increasing Nernst effect ν/T ∝ μ/ε_{F} where μ is the electron mobility and ε_{F} the Fermi energy^{20}, as lower Δk_{F} implies a smaller Fermi energy. The lowtemperature Fermi surface thus emerges as a result of two reconstructions. First below T_{s1} = 48 K, a fast dispersing band appears around M_{a} and M_{b} with a gap opening for other bands. Next, the band dispersion along the M_{a}–Γ direction undergoes a second transformation across T_{s2} = 30 K.
Lowtemperature electronic structure
With the exception of the features around the M_{a} and M_{b} points, all other bands are not crossing the Fermi level for T < T_{s2}—see Fig. 2e–h. Around the M_{b} point, two holelike bands—forming an Mshaped structure—are found [Fig. 2g]. While the holelike band touches E_{F} along the M_{b}–Γ direction [Fig. 2g], the band top sinks below E_{F} upon moving away from it by ~ 0.1π/b. (See Supplementary Fig. S7) consistent with a previous report^{17}. The boomeranglike feature thus forms a closed holelike Fermi surface. Around the M_{a} point, the electronlike Fermi surface pocket is revealed by a highresolution map in Fig. 4(a). The electron pocket is elliptical with \({k}_{{\rm{F}}}^{a}=0.04\pi /a\) and \({k}_{{\rm{F}}}^{b}=0.07\pi /b\) along the M_{a}–Γ and M_{a}–X directions, respectively. The Fermi surface area \({A}_{{\rm{FS}}}=\pi {k}_{{\rm{F}}}^{a}{k}_{{\rm{F}}}^{b}\) corresponds to 0.23% of the orthorhombic Brillouin zone. Inspecting the band dispersion along the M_{a}–Γ direction reveals a Diraccone structure with the Dirac point placed about E_{D} = 15 meV below E_{F} [Fig. 4b]. The twopeak MDC profile found at E_{F} merges into a single peak at E_{D} and then splits again below E_{D} [see Fig. 4c]. This MDC analysis estimates a linear Fermi velocity of \({v}_{{\rm{F}}}^{a}=0.62\) eVÅ (95 km/s) and \({v}_{{\rm{F}}}^{b}=0.37\) eVÅ (57 km/s) (see Supplementary Fig. S8). Our results thus suggest that Ca_{3}Ru_{2}O_{7} at low temperatures is a highly anisotropic Dirac semimetal.
Discussion
The lowtemperature resistivity anisotropy in Ca_{3}Ru_{2}O_{7}, is as large as ρ_{c}/ρ_{ab} ~ 1000^{10}. It is, therefore, reasonable to consider the Fermi surface to be twodimensional. This implies that quantum oscillation and ARPES experiments are directly comparable. Onsager’s relation^{5} links directly Fermi surface areas to quantum oscillation frequencies F = Φ_{0}A_{FS}/(2π^{2}) where Φ_{0} is the flux quantum. The lowtemperature electron pocket around M_{a} corresponds to F = 34 T in agreement with observed quantum oscillation frequencies 28–43 T^{14,15,16}. The holelike boomerang structure comprises a Fermi surface area that is too small to be quantified accurately by our ARPES experiments. However, it should produce a lowfrequency quantum oscillation. Indeed, a frequency corresponding to 0.07% of the Brillouin zone or about 1/3 of the electron pocket has been reported^{15}. It is therefore conceivable that the electron and hole Fermi pockets reported here are those responsible for the quantum oscillations. Our ARPES work unveils the band curvature and position of these pockets within the Brillouin zone.
Combining ARPES and quantum oscillation experiments allows direct comparison of the effective electronic mass \({m}^{* }=\frac{{\hslash }^{2}}{2\pi }\frac{\partial {A}_{{\rm{FS}}}}{\partial \varepsilon }\)^{21,22} on the electron pocket. Lifshitz–Kosevich analysis of the ~35 T quantumoscillation frequency yield \({m}_{e}^{* }=0.6{m}_{e}\)^{16}, where m_{e} is the free electron mass. Assuming a parabolic band dispersion \({m}_{e}^{* }={\hslash }^{2}{k}_{{\rm{F}}}^{a}{k}_{{\rm{F}}}^{b}/2{\varepsilon }_{{\rm{F}}}\) where ε_{F} = 15 meV is the Fermi energy, the effective mass \({m}_{e}^{* }=0.25{m}_{e}\) is significantly lower than that inferred from quantumoscillation experiments. A linear band dispersion E_{k} = v_{F}k provides a much better agreement \({m}_{e}^{* }={\varepsilon }_{{\rm{F}}}/{v}_{{\rm{F}}}^{a}{v}_{{\rm{F}}}^{b}={\hslash }^{2}{k}_{{\rm{F}}}^{a}{k}_{{\rm{F}}}^{b}/{\varepsilon }_{{\rm{F}}}=0.49{m}_{e}\). This fact reinforces the interpretation of Dirac fermions around M_{a}. The boomerang band along Γ–M_{b} has comparable Fermi velocity to that of the electron pocket along Γ–M_{a}. Estimation of the Fermi energy from linear extrapolation of the Mshaped band dispersion yields ε_{F} ~ 5 meV, which is about three times smaller than that obtained for the electron pocket. As the hole pocket area—according to quantumoscillation measurements^{15}—is also about three times smaller than the electron sheet, we estimate the holelike carriers to have a comparable effective mass of \({m}_{h}^{* }=0.49{m}_{e}\). With two hole and one electron pocket per Brillouin zone, a Sommerfeld constant of γ ≈ 2.1 mJ mol^{−1} K^{−2} is found, with mol refering to one formula unit. Here, we assumed twodimensional band dispersions without bilayer splitting and used the twodimensional expression \(\gamma =\pi {N}_{{\rm{A}}}{k}_{{\rm{B}}}^{2}ab/3{\hslash }^{2}{\sum }_{i}{m}_{i}\)^{16}, where N_{A} is the Avogadro constant, k_{B} is the Boltzmann constant, ℏ is the reduced Planck constant, and m_{i} is the effective mass. In addition, spin polarisation within the RuO_{2} plane was employed to treat the Atype AFM order. As our estimate is in reasonable agreement with the value γ ≈ 2.8–3.4 mJ mol^{−1} K^{−2}) obtained by specific heat experiments^{10,16}, we conclude that our experiments reveal the entire bulk Fermi surface.
The twostage transformation of the electronic structure has a clear impact on all transport coefficients. A remarkable increase of in and outofplane resistivity appears across T_{s1} = 48 K^{10,16}. Simultaneously, the Seebeck coefficient changes sign going from weak positive to large negative values across T_{s1}^{14}. Although less sharp, the Hall coefficient also changes sign (from positive to negative) across T_{s1}. While the Hall coefficient takes increasingly large negative values^{14,23}, the Seebeck coefficient displays a complicated temperature dependence that in addition is different along the a and b directions^{14}. This temperature dependence is a typical signature of ambipolar transport behaviour, where both electron and holelike carriers are contributing^{24}. Furthermore, the lowtemperature Hall coefficient R_{H} that (in different studies) ranges from −0.5 × 10^{−7} m^{3} C^{−1} to −1.4 × 10^{−7} m^{3} C^{−1 }^{14,16,23} cannot be explained by the electron pocket that alone should generate R_{H} = − 1/(n_{e}e) = −8.0 × 10^{−7} m^{3} C^{−1}. Using the combined ARPES and quantumoscillation knowledge that n_{e} = 7.8 × 10^{18} cm^{−3} and n_{h} ≈ 2 × n_{e}/3, a two band model^{25} yields R_{H} = (2α^{2}/3 − 1)/(n_{e}e)(2α/3 + 1)^{2} where α = μ_{h}/μ_{e} is the mobility ratio between electrons and holes. The exact experimental values of R_{H}(T = 0) imply that μ_{h} ≈ 0.9–1.1 × μ_{e} and μ_{e} ≈ ∣R_{H}∣/ρ_{xx} = 0.1 T^{−1}. We thus infer that in the T → 0 limit electron and holelike carriers have comparable mobility that in turn generate the ambipolar transport properties.
Having established the existence of small electron pockets with linear dispersion around the M_{a} point, a question arises whether these excitations are massless Dirac fermions or whether they possess a finite mass at M_{a}. While the question cannot be definitively answered from the experimental data due to the finite energy resolution, we discuss here implications from the crystal symmetry. Ca_{3}Ru_{2}O_{7} has the space group Bb2_{1}m (No. 36)^{9}. For our purpose, it is sufficient to focus on a single bilayer. The point group of such a bilayer is C_{2v} with a mirror plane between the two layers, as well as a glide plane perpendicular to the mirror and a twofold screw axis along the crystalline b axis (the longer inplane axis). Together with timereversal symmetry (TRS) in the paramagnetic state, this imposes a Kramer’s degeneracy along M_{b}–X in the Brillouin zone. Furthermore, TRS imposes Kramer’s pairs at the M_{a} and Γ point.
When TRS is broken in the Atype AFM phase^{11}, the generating point group of the bilayer is reduced to C_{2v} (C_{s}) for the AFMa phase and C_{2v} (C_{2}) for the AFMb phase. Here, the notation \({\mathcal{G}}\) (\({\mathcal{G}}^{\prime}\)) denotes the generating point group \({\mathcal{G}}\) with \({\mathcal{G}}^{\prime}\) the subgroup of elements that do not have to be combined with TRS. While Kramer’s degeneracy is preserved along with M_{b}–X, the one at the M_{a} and Γ point is lifted. The Dirac fermions at M_{a} thus possess a finite mass, in other words, the bands hybridise as schematically illustrated in Fig. 4b.
Finally, we can reproduce key features of the lowtemperature semimetallic band structure employing a tightbinding model of the Ru t_{2g} orbitals (see Supplementary Note 1, Supplementary Figs. S1 and S9). We restrict our model to the Ru d_{xz} and d_{yz} orbitals in an effective singlelayer model, for two reasons. Firstly, the "onedimensional” nature of the hightemperature (T > 48 K) Fermi surface resembles the d_{xz}, d_{yz} dominated α and β bands of other ruthenates^{26,27}. Secondly, the matrixelement effect of the electron pocket around M_{a} is incompatible with the expectation of selection rules for the d_{xy} orbital character [see Fig. 1h, i]. Our simple model faithfully reproduces the Fermi surface in the normal state [see Fig. 1a and Supplementary Fig. S1]. Importantly, a rigid band shift in the d_{xz}/d_{yz} sector, as expected due to the caxis compression at T_{s1}^{9}, yields elliptical electron pockets with linear dispersion around the M_{a} point and a holelike boomerang structure around the M_{b} point [Fig. 2c, d]. This d_{xz}/d_{yz} band shift implies a change of orbital polarisation to respect the global charge balance. Finally, the Brillouinzone folding due to the screwaxis opens a gap around the M_{a} Dirac point [see green lines Fig. 4b]. Due to the small gap size and Dirac point distance from the Fermi level, this gap is irrelevant for transport and thermodynamic measurements. While our tightbinding model based on the Ru t_{2g} orbitals is too simplistic to capture all the features and does not include the actual electronic instability, it reproduces the most salient features of both the high and lowtemperature dispersions. We thus conclude that the lowtemperature lowenergy band structure stems primarily from the d_{xz} and d_{yz} Ru orbitals.
A fundamental remaining question links to the triggering mechanism that induces the Dirac semimetal. Specific heat suggests that the phase transition at T_{s1} involves a large entropy change^{10} and unlike other layered ruthenates, the ground state is a low densityofstate semimetal. It has been argued that the reorientation of the magnetic moments alone can not account for this large entropy change. Upon cooling, an energy gain of the system is manifested by an electronic reconstruction that opens a gap leaving only small Fermi surface pockets around the zone boundaries. Most likely, this Fermi surface reconstruction is triggered by an electronic mechanism. Densitywave orders breaking translation symmetry are, however, excluded since the reconstruction preserves the original Brillouinzone boundaries. This leads us to speculate alternative scenarios, with electron correlations likely involved in some way. If so, the situation resembles that of the singlelayer counterpart Ca_{2}RuO_{4} where the instability toward a Mottinsulating state triggers a large caxis lattice contraction^{28,29,30}. Indeed, a caxis lattice contraction is found across the first (48 K) transition though this effect is much less pronounced in Ca_{3}Ru_{2}O_{7}^{9}. Alternatively, it has been proposed that Ca_{3}Ru_{2}O_{7} hosts magnetic anapole order^{31,32}. This would connect Ca_{3}Ru_{2}O_{7} with hidden order problems in the sense that it is very difficult to demonstrate experimentally.
Note added after completion of this work: A recent complementary ARPES study^{33} conducted at T ≧ 30 K suggested using DFT calculations including Rashbatype spinorbit coupling that the electronic reconstruction across T = 48 K can be understood from the magneticmoment reorientation alone without the need for additional hidden order.
Methods
Sample characterisation
Highquality single crystals of Ca_{3}Ru_{2}O_{7} were grown by floating zone technique^{10}. The electronic transition at T_{s1} = 48 K was checked by thermopower measurements (see Supplementary Fig. S10) and found in agreement with existing literatures^{13,14}. Detwinning of orthorhombic domains was achieved with a thermomechanical device^{34} and monitored by polarised light microscopy. The resulting monodomain constitutes 99% (or more) of the sample volume according to Xray diffraction measurements (see Supplementary Fig. S10).
ARPES experiments
ARPES experiments were carried out at the SIS^{35}, CASSIOPEE (https://www.synchrotronsoleil.fr/en/beamlines/cassiopee), and I05^{36} beamlines of the Swiss Light Source, SOLEIL synchrotron, and Diamond Light Source, respectively. Pristine surfaces were obtained by toppost cleaving at T > T_{s1} (80 K). Incident photons hν = 31−115 eV, providing high inplane and modest outofplane^{22} momentum resolution, were used for this study. Consistent results were obtained on different crystals and upon cooling and heating through the critical temperature T_{s1} = 48 K below which the electronic structure is reconstructed. ARPES data are presented using orthorhombic notation with lattice parameters a = 5.37 Å and b = 5.54 Å.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank M. Hoesch for fruitful discussions. M.H., Q.W., K.P.K., D.S., Y.X., and J.C. acknowledge support by the Swiss National Science Foundation. Y.S. is funded by the Swedish Research Council (VR) with a Starting Grant (Dnr. 201705078) and thanks Chalmers Areas of AdvanceMaterials Science. ARPES measurements were carried out at the SIS, CASSIOPEE, and I05 beamlines of the Swiss Light Source, SOLEIL synchrotron, and Diamond Light Source, respectively. We acknowledge Diamond Light Source for time at beamline I05 under proposal SI20259.
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V.G., R.Fi., and A.V. grew and prepared single crystals. L.D. and Y.X. performed thermopower measurements. S.J. and M.H. detwinned single crystals. M.H., Q.W., S.J., and R.Fr. carried out xray and Laue diffraction measurements. M.H., Q.W., K.P.K., Y.S., D.S., A.B., and J.C. prepared and carried out the ARPES experiment with the assistance of T.K.K., C.C., J.E.R., P.L.F., F.B., N.C.P., and M.S. M.H. analyzed the ARPES data. M.H., M.H.F., and J.C. developed the tightbinding model. M.H., D.S., and J.C. conceived the project. All authors contributed to the manuscript.
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Horio, M., Wang, Q., Granata, V. et al. Electronic reconstruction forming a C_{2}symmetric Dirac semimetal in Ca_{3}Ru_{2}O_{7}. npj Quantum Mater. 6, 29 (2021). https://doi.org/10.1038/s41535021003283
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DOI: https://doi.org/10.1038/s41535021003283
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