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 high-temperature superconductor YBa2Cu3O7−x is a good example of this. In the underdoped regime, charge-density-wave order1,2,3 clearly reconstructs the Fermi surface. Although quantum oscillations4,5 and transport6,7 experiments have revealed the existence of an electron pocket, the reconstructed Fermi-surface topology is, despite strong effort, still not clarified8. Another prominent example is the orthorhombic bilayer ruthenate Ca3Ru2O7. An initial A-type antiferromagnetic (AFM) order setting in at TN = 56 K9 has no significant impact on the transport properties10. A spin reorientation from the orthorhombic a- to the b-axis direction11,12 occurs at Ts1 = 48 K. While this lattice-space-group preserving reorientation9 naively might appear to be of minor consequence, the transition at Ts1 = 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 temperature13,14. Ts1 is also the onset of in-plane anisotropic transport properties14. Although transport and thermodynamic experiments provide information about Fermi surface area and electronic masses15,16, 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 surface17. Obviously, a single pocket is insufficient to explain the observed ambipolar electronic properties14. Although a density-wave state has never been identified12, 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 Ca3Ru2O7 across the electronic reconstruction. Above the reconstructing temperature, the low-energy 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 reconstruction9, 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 Ts1 = 48 K and eventually the Dirac fermions settle into the low-temperature structure below Ts2 = 30 K. Throughout this temperature evolution, no signature of Brillouin-zone folding is identified, excluding translational-symmetry breaking density-wave/orbital orders as the origin of the phase transition at Ts1 = 48 K. We argue that the reconstructed Fermi surface should be understood from the dxz and dyz orbitals whereas the dxy 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.

Results

High-temperature state

The Fermi surface and low-energy electronic structure of the Ca3Ru2O7 normal state—above the Néel temperature TN = 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 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 TN = 56 K [see Fig. 1b, f, and g]. Furthermore, the orthorhombic zone boundary points Ma = (±π/a, 0) and Mb = (0, ±π/b) are virtually indistinguishable (See also Supplementary Fig. S1).

Fig. 1: Fermi-surface reconstruction in Ca3Ru2O7.
figure 1

a–c Fermi surface maps taken at T = 80, 50, and 20 K, respectively. The photoelectron intensities -- displayed using a false color scheme—are integrated within EF ± 20 meV. White lines in a represent a tight-binding model (see Supplementary Note 1 and Supplementary Fig. S1) of the Fermi surface. Vertical and horizontal dashed lines indicate the orthorhombic zone boundaries. The orthorhombic a and b axes are indicated in top right inset. d–i Energy distribution maps along the Γ–Ma direction for incident light polarisations and temperatures as indicated. Red arrows mark the fast dispersing bands that emerge below Ts1 = 48 K. The photoemission mirror plane along with incident light polarisation and the orientation of the electron analyser slit is shown in the bottom inset. With this setting, matrix elements for even-parity orbitals are expected to be suppressed in the s-polarisation channel.

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Fermi surface anisotropy

Across the structural transition at Ts1 = 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 Ma—see Fig. 1c, h, and i. Remarkably, this small Fermi surface sheet is absent at Mb. Instead, as previously reported17, boomerang-like Fermi surface sheets are found around the Mb point. Therefore, in contrast to T > Ts1, the low-temperature structure appears highly anisotropic featuring different Fermi surface topology around Ma and Mb. This Fermi surface reconstruction appears without change of the crystal lattice space group and with minute (~1%) reduction of the orthorhombic order parameter9.

To exclude the possibility that this C2 symmetry is an artefact of photoionization-matrix-element effects, we follow a standard measurement protocol18,19. 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 electron-analyser slit. The electronic structure with a tiny Fermi pocket around the Ma point and boomerang-like features near Mb tracks the azimuthal rotation—see Fig. 2a–d. The C2-symmetric electronic structure is also revealed by the band dispersions. Along the Ma–Γ and Mb–Γ directions, the band curvature around Ma and Mb are clearly different [Fig. 2e, g]. An electron pocket is formed around Ma whereas two hole-like pockets are found on each side of Mb. In a similar fashion, dispersions along the Ma–X and Mb–X directions are inequivalent [Fig. 2f, h]. Electron-like band curvature is found around Ma whereas no Fermi-level crossing is observed along Mb–X. These results exclude matrix-element effects as the source of the observed anisotropy.

Fig. 2: Breaking of fourfold rotational symmetry.
figure 2

a, b Fermi surface maps recorded at T = 16 K (hν = 75 eV) for two sample azimuthal angles that are 90∘ apart as indicated in the schematics. The azimuthal angle rotation was operated in situ and hence the sample surface is identical. k∥ (k⊥) represents momentum parallel (perpendicular) to the electron-analyser slit. The spectral intensity was integrated within EF ± 20 meV, washing-out spectral gaps within the integration window. c, d Fermi surfaces from the tight-binding model (see Supplementary note 1). The sheets not observed in the experiment are indicated in grey. e–h Energy distribution maps along Ma–Γ, X–Mb, Mb–Γ, and X–Ma as shown in c and d. The energy distribution maps were recorded at hν = 37 eV except for g where hν = 31 eV incident light was selected to optimise the photoelectron matrix element. Systematic photon-energy dependence between 30 and 40 eV is presented in Supplementary Figs. S2 and S3. The band structure along with Ma–Γ (X–Ma) and Mb–Γ (X–Mb) is inequivalent.

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Two-stage Fermi surface reconstruction

Tracking the temperature dependence of the band structure reveals two electronic temperature scales. The electronic band structure along Ma–Γ and Mb–Γ is shown for temperatures going from 16 to 50 K. Above Ts1 = 48 K [Fig. 3e, j and 1], all bands appear with broad line-shapes. Once cooled below Ts1, well-defined bands around the Ma and Mb points emerge [see Fig. 3d, i, and k]. The appearance of another electron-like band around the Γ point is accompanied by a gap Δ opening below Ts1 [see Fig. 3g–i and Supplementary Fig. S4]. The band structures around Ma and Mb are inequivalent not only in terms of curvature but also in terms of temperature dependence. The Ma–Γ band dispersion is temperature-dependent whereas the corresponding structure around Mb is virtually insensitive to temperature. Examining the Ma–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 Ma display asymmetric matrix elements. Momentum distribution curves (MDCs) at EF and 30 < T < 48 K are therefore not symmetric around Ma. Upon cooling below 30 K, the single electron pocket around Ma displays symmetric Fermi momenta kF despite the asymmetric matrix elements. To illustrate the transition between the two- and single-band situation, we define ΔkF as the reciprocal-space distance between the two MDC-intensity maxima. Across Ts2 = 30 K, ΔkF 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 Ts2 [see Fig. 3g–i and Supplementary Fig. S4], suggesting that the states around Γ are not involved in this transition. Although Ts2 = 30 K remains to be identified as a thermodynamic temperature scale, it does coincide with the onset of a strong negative Nernst effect response14 [see Fig. 3l]. We also notice that the reduction of ΔkF below Ts2 is consistent with an increasing Nernst effect ν/T ∝ μ/εF where μ is the electron mobility and εF the Fermi energy20, as lower ΔkF implies a smaller Fermi energy. The low-temperature Fermi surface thus emerges as a result of two reconstructions. First below Ts1 = 48 K, a fast dispersing band appears around Ma and Mb with a gap opening for other bands. Next, the band dispersion along the Ma–Γ direction undergoes a second transformation across Ts2 = 30 K.

Fig. 3: Two-stage temperature evolution of the band structure.
figure 3

a–j Energy distribution maps along Ma--Γ (top panels) and Mb--Γ (bottom panels) for temperatures as indicated. The gap Δ of the electron pocket around Γ is indicated in g–i. For detailed analysis, see Supplementary Fig. S4. f Schematics of the Fermi surface and the high-symmetry cuts used in a–e and g–j. k MDCs at EF (integrated within ± 3 meV) for temperatures as indicated. Vertical dashed lines define MDC peak maxima. A clear difference in the peak separation, ΔkF, 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 ΔkF 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 ν14 is plotted as ν/T versus T. Both experiments suggest an electronic transformation across T = 30 K.

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Low-temperature electronic structure

With the exception of the features around the Ma and Mb points, all other bands are not crossing the Fermi level for T < Ts2—see Fig. 2e–h. Around the Mb point, two hole-like bands—forming an M-shaped structure—are found [Fig. 2g]. While the hole-like band touches EF along the Mb–Γ direction [Fig. 2g], the band top sinks below EF upon moving away from it by ~ 0.1π/b. (See Supplementary Fig. S7) consistent with a previous report17. The boomerang-like feature thus forms a closed hole-like Fermi surface. Around the Ma point, the electron-like Fermi surface pocket is revealed by a high-resolution map in Fig. 4(a). The electron pocket is elliptical with \({k}_{{\rm{F}}}^{a}=0.04\pi /a\) and \({k}_{{\rm{F}}}^{b}=0.07\pi /b\) along the Ma–Γ and Ma–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 Ma–Γ direction reveals a Dirac-cone structure with the Dirac point placed about ED = 15 meV below EF [Fig. 4b]. The two-peak MDC profile found at EF merges into a single peak at ED and then splits again below ED [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 Ca3Ru2O7 at low temperatures is a highly anisotropic Dirac semimetal.

Fig. 4: Dirac cone structure in Ca3Ru2O7.
figure 4

a High-resolution Fermi surface map around the Ma point. b Energy distribution map along the Ma–Γ direction evidencing a Dirac cone structure. Crystal/magnetic symmetry imposes a finite mass as illustrated by green dispersions reproduced by the tight-binding model (see Supplementary Note 1). c MDCs, extracted from b at binding energies as indicated.

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Discussion

The low-temperature resistivity anisotropy in Ca3Ru2O7, is as large as ρc/ρab ~ 100010. 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 relation5 links directly Fermi surface areas to quantum oscillation frequencies F = Φ0AFS/(2π2) where Φ0 is the flux quantum. The low-temperature electron pocket around Ma corresponds to F = 34 T in agreement with observed quantum oscillation frequencies 28–43 T14,15,16. 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 reported15. 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 quantum-oscillation frequency yield \({m}_{e}^{* }=0.6{m}_{e}\)16, where me 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 εF = 15 meV is the Fermi energy, the effective mass \({m}_{e}^{* }=0.25{m}_{e}\) is significantly lower than that inferred from quantum-oscillation experiments. A linear band dispersion Ek = vFk 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 Ma. The boomerang band along Γ–Mb has comparable Fermi velocity to that of the electron pocket along Γ–Ma. Estimation of the Fermi energy from linear extrapolation of the M-shaped band dispersion yields εF ~ 5 meV, which is about three times smaller than that obtained for the electron pocket. As the hole pocket area—according to quantum-oscillation measurements15—is also about three times smaller than the electron sheet, we estimate the hole-like 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 two-dimensional band dispersions without bi-layer splitting and used the two-dimensional expression \(\gamma =\pi {N}_{{\rm{A}}}{k}_{{\rm{B}}}^{2}ab/3{\hslash }^{2}{\sum }_{i}{m}_{i}\)16, where NA is the Avogadro constant, kB is the Boltzmann constant, ℏ is the reduced Planck constant, and mi is the effective mass. In addition, spin polarisation within the RuO2 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 experiments10,16, 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 Ts1 = 48 K10,16. Simultaneously, the Seebeck coefficient changes sign going from weak positive to large negative values across Ts114. Although less sharp, the Hall coefficient also changes sign (from positive to negative) across Ts1. While the Hall coefficient takes increasingly large negative values14,23, the Seebeck coefficient displays a complicated temperature dependence that in addition is different along the a and b directions14. This temperature dependence is a typical signature of ambipolar transport behaviour, where both electron- and hole-like carriers are contributing24. Furthermore, the low-temperature Hall coefficient RH that (in different studies) ranges from −0.5 × 10−7 m3 C−1 to −1.4 × 10−7 m3 C−1 14,16,23 cannot be explained by the electron pocket that alone should generate RH = − 1/(nee) = −8.0 × 10−7 m3 C−1. Using the combined ARPES and quantum-oscillation knowledge that ne = 7.8 × 1018 cm−3 and nh ≈ 2 × ne/3, a two band model25 yields RH = (2α2/3 − 1)/(nee)(2α/3 + 1)2 where α = μh/μe is the mobility ratio between electrons and holes. The exact experimental values of RH(T = 0) imply that μh ≈ 0.9–1.1 × μe and μe ≈ ∣RH∣/ρxx = 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 Ma point, a question arises whether these excitations are massless Dirac fermions or whether they possess a finite mass at Ma. 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. Ca3Ru2O7 has the space group Bb21m (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 C2v 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 time-reversal symmetry (TRS) in the paramagnetic state, this imposes a Kramer’s degeneracy along Mb–X in the Brillouin zone. Furthermore, TRS imposes Kramer’s pairs at the Ma and Γ point.

When TRS is broken in the A-type AFM phase11, the generating point group of the bi-layer is reduced to C2v (Cs) for the AFM-a phase and C2v (C2) for the AFM-b 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 Mb–X, the one at the Ma and Γ point is lifted. The Dirac fermions at Ma 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 low-temperature semimetallic band structure employing a tight-binding model of the Ru t2g orbitals (see Supplementary Note 1, Supplementary Figs. S1 and S9). We restrict our model to the Ru dxz and dyz orbitals in an effective single-layer model, for two reasons. Firstly, the "one-dimensional” nature of the high-temperature (T > 48 K) Fermi surface resembles the dxz, dyz dominated α and β bands of other ruthenates26,27. Secondly, the matrix-element effect of the electron pocket around Ma is incompatible with the expectation of selection rules for the dxy 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 dxz/dyz sector, as expected due to the c-axis compression at Ts19, yields elliptical electron pockets with linear dispersion around the Ma point and a hole-like boomerang structure around the Mb point [Fig. 2c, d]. This dxz/dyz band shift implies a change of orbital polarisation to respect the global charge balance. Finally, the Brillouin-zone folding due to the screw-axis opens a gap around the Ma 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 tight-binding model based on the Ru t2g 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 low-temperature low-energy band structure stems primarily from the dxz and dyz Ru orbitals.

A fundamental remaining question links to the triggering mechanism that induces the Dirac semimetal. Specific heat suggests that the phase transition at Ts1 involves a large entropy change10 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 Ca2RuO4 where the instability toward a Mott-insulating state triggers a large c-axis lattice contraction28,29,30. Indeed, a c-axis lattice contraction is found across the first (48 K) transition though this effect is much less pronounced in Ca3Ru2O79. Alternatively, it has been proposed that Ca3Ru2O7 hosts magnetic anapole order31,32. This would connect Ca3Ru2O7 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 study33 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.

Methods

Sample characterisation

High-quality single crystals of Ca3Ru2O7 were grown by floating zone technique10. The electronic transition at Ts1 = 48 K was checked by thermopower measurements (see Supplementary Fig. S10) and found in agreement with existing literatures13,14. Detwinning of orthorhombic domains was achieved with a thermo-mechanical device34 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 SIS35, CASSIOPEE (https://www.synchrotron-soleil.fr/en/beamlines/cassiopee), and I0536 beamlines of the Swiss Light Source, SOLEIL synchrotron, and Diamond Light Source, respectively. Pristine surfaces were obtained by top-post cleaving at T > Ts1 (80 K). Incident photons hν = 31−115 eV, providing high in-plane and modest out-of-plane22 momentum resolution, were used for this study. Consistent results were obtained on different crystals and upon cooling and heating through the critical temperature Ts1 = 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.