Electronic reconstruction forming a C2-symmetric Dirac semimetal in Ca3Ru2O7

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 Ca3Ru2O7, numerous Fermi surface-sensitive probes indicate a low-temperature electronic reconstruction. Yet, the causality and the reconstructed band structure remain unsolved. Here, we show by angle-resolved photoemission spectroscopy, how in Ca3Ru2O7 a C2-symmetric massive Dirac semimetal is realized through a Brillouin-zone preserving electronic reconstruction. This Dirac semimetal emerges in a two-stage 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 translational-symmetry-breaking density waves—serves as an important test for band structure calculations of correlated electron systems.


INTRODUCTION
A Fermi-surface 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 Fermi-surface 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 Fermi-surface reconstructions are translational-symmetry breaking spin-or charge-density waves. Typically, this reduction of symmetry and the resulting folding of the Brillouin zone lead large Fermi-surface 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, charge-density-wave 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 Fermi-surface 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 A-type 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 b-axis direction 11,12 occurs at T s1 = 48 K. While this lattice-space-group 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 in-plane 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 angle-resolved photoemission spectroscopy (ARPES) data have revealed the existence of a boomerang-shaped Fermi surface 17 . Obviously, a single pocket is insufficient to explain the observed ambipolar electronic properties 14 . Although a density-wave state has never been identified 12 , a common assumption is that a translational-symmetry-breaking 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 short-axis orthorhombic zone boundary and a boomerang-like hole pocket in vicinity to the long-axis 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 zone-boundary Fermi surfaces first appear below T s1 = 48 K and eventually the Dirac fermions settle into the low-temperature structure below T s2 = 30 K. Throughout this temperature evolution, no signature of Brillouin-zone folding is identified, excluding translationalsymmetry breaking density-wave/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 test-bed for ab-initio band structure calculations beyond density-functional-theory concepts.

High-temperature state
The Fermi surface and low-energy electronic structure of the Ca 3 Ru 2 O 7 normal state-above the Néel temperature T N = 56 Kare 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 s-polarised light is used. Part of the Fermi surface consists of straight sectors running diagonally through the orthorhombic Brillouin zone. This quasi-one-dimensional 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 , boomerang-like Fermi surface sheets are found around the M b point. Therefore, in contrast to T > T s1 , the low-temperature 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 photoionization-matrix-element 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 are inequivalent not only in terms of curvature but also in terms of temperature dependence. The M a -Γ band dispersion is temperature-dependent 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 single-band situation, we define Δk F as the reciprocal-space distance between the two MDCintensity maxima. Across T s2 = 30 K, Δk F drops by a factor of two. This observation is independent of incident-light 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 low-temperature 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.
Low-temperature 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 hole-like bands-forming an M-shaped structure-are found [ Fig. 2g]. While the hole-like 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 boomerang-like feature thus forms a closed hole-like Fermi surface. Around the M a point, the electron-like Fermi surface pocket is revealed by a high-resolution map in Fig. 4(a). The electron pocket is elliptical with k a F ¼ 0:04π=a and k b F ¼ 0:07π=b along the M a -Γ and M a -X directions, respectively. The Fermi surface area A FS ¼ πk a F k b F corresponds to 0.23% of the orthorhombic Brillouin zone. Inspecting the band dispersion along the M a -Γ direction reveals a Dirac-cone structure with the Dirac point placed about E D = 15 meV below E F [Fig. 4b]. The two-peak 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 a F ¼ 0:62 eVÅ (95 km/s) and v b F ¼ 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 low-temperature resistivity anisotropy in Ca 3 Ru 2 O 7 , is as large as ρ c /ρ ab~1 000 10 . It is, therefore, reasonable to consider the Fermi surface to be two-dimensional. 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 low-temperature electron pocket around M a corresponds to F = 34 T in agreement with observed quantum oscillation frequencies 28-43 T [14][15][16] . The hole-like boomerang structure comprises a Fermi surface area that is too small to be quantified accurately by our ARPES experiments. However, it should produce a low-frequency 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 Ã ¼ _ 2 2π ∂AFS ∂ε 21,22 on the electron pocket. Lifshitz-Kosevich analysis of the~35 T quantum-oscillation frequency yield m Ã e ¼ 0:6m e 16 , where m e is the free electron mass. Assuming a parabolic band dispersion m Ã e ¼ _ 2 k a F k b F =2ε F where ε F = 15 meV is the Fermi energy, the effective mass m Ã e ¼ 0:25m e is significantly lower than that inferred from quantum-oscillation experiments. A linear band dispersion E k = v F k provides a much better agreement 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 M-shaped 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 quantum-oscillation measurements 15 -is also about three times smaller than the electron sheet, we estimate the hole-like carriers to have a comparable effective mass of m Ã h ¼ 0:49m 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 two-dimensional band dispersions without bi-layer splitting and used the twodimensional expression γ ¼ πN A k 2 B ab=3_ 2 P 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 A-type 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 two-stage transformation of the electronic structure has a clear impact on all transport coefficients. A remarkable increase of in-and out-of-plane 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 hole-like carriers are contributing 24 . Furthermore, the low-temperature 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 quantum-oscillation knowledge that n e = 7.8 × 10 18 cm −3 and n h ≈ 2 × n e /3, a two band model 25  k MDCs at E F (integrated within ± 3 meV) for temperatures as indicated. Vertical dashed lines define MDC peak maxima. A clear difference in the peak separation, Δk F , is found across 30 K. At 30 K, a three-peak structure is found and indicated by the black arrow and the two dashed lines. l Δk F plotted as a function of temperature. The error bars represent 3σ of the fitting with σ being the standard deviation. For comparison, the Nernst coefficient ν 14 is plotted as ν/T versus T. Both experiments suggest an electronic transformation across T = 30 K.
(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 hole-like 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 bi-layer. The point group of such a bi-layer is C 2v with a mirror plane between the two layers, as well as a glide plane perpendicular to the mirror and a two-fold screw axis along the crystalline b axis (the longer in-plane 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 A-type AFM phase 11 , the generating point group of the bi-layer is reduced to C 2v (C s ) for the AFM-a phase and C 2v (C 2 ) for the AFM-b phase. Here, the notation G (G 0 ) denotes the generating point group G with G 0 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 single-layer model, for two reasons. Firstly, the "one-dimensional" 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 matrix-element 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 c-axis compression at 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 tight-binding 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 low-temperature dispersions. We thus conclude that the lowtemperature low-energy 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 density-of-state 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. Density-wave orders breaking translation symmetry are, however, excluded since the reconstruction preserves the original Brillouin-zone boundaries. This leads us to speculate alternative scenarios, with electron correlations likely involved in some way. If so, the situation resembles that of the single-layer counterpart Ca 2 RuO 4 where the instability toward a Mott-insulating state triggers a large c-axis lattice contraction [28][29][30] . Indeed, a c-axis 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 Rashba-type spin-orbit coupling that the electronic reconstruction across T = 48 K can be understood from the magnetic-moment reorientation alone without the need for additional hidden order.

Sample characterisation
High-quality 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 thermo-mechanical device 34 and monitored by polarised light microscopy. The resulting monodomain constitutes 99% (or more) of the sample volume according to X-ray diffraction measurements (see Supplementary Fig. S10).

ARPES experiments
ARPES experiments were carried out at the SIS 35 , CASSIOPEE (https:// www.synchrotron-soleil.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 top-post cleaving at T > T s1 (80 K). Incident photons hν = 31−115 eV, providing high in-plane and modest out-of-plane 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.