Cooperative Interactions Govern the Fermiology of the Polar Metal Ca$_3$Ru$_2$O$_7$

The antiferromagnetic Ruddlesden-Popper ruthenate Ca$_3$Ru$_2$O$_7$ is a model polar metal, combining inversion symmetry breaking with metallic conductivity; however, its low temperature ($T<48$ K) crystal structure and Fermi surface topology remain ambiguous despite numerous measurements and theoretical studies. Here we perform both first principles calculations with static correlations and angle resolved photoelectron spectroscopy experiments to construct a complete model of Ca$_3$Ru$_2$O$_7$, reconciling inconsistencies among interpretations of electrical transport, thermopower measurements, and momentum- and energy-resolved band dispersions. The solution relies on treating the interplay among Coulomb repulsion, magnetic ordering, spin-orbit interactions, and the RuO$_6$ octahedral degrees-of-freedom on equal footing. For temperatures $30<T<48$ K, we propose weak electron-electron interactions produce a symmetry-preserving metal-semimetal transition with Weyl nodes in proximity to the Fermi level, whereas a new orthorhombic $Pn2_1a$ structure emerges for $T<30$ K, exhibiting charge and spin density waves from enhanced Coulombic interactions.


I. INTRODUCTION
Ruddlesden-Popper (RP) Ca 3 Ru 2 O 7 is a unique polar metal exhibiting a rich phase diagram ( Fig. 1), colossal magnetoresistance [1,2], highly anisotropic electrical resistivity [3], and polar domain structures [4]. It displays multiple magnetic and electronic transitions without changes in crystal symmetry, making it also a member of a rare condensed-matter family exhibiting isosymmetric transitions [5]. Ca 3 Ru 2 O 7 exhibits the polar space group Bb2 1 m from room temperature (RT) to 8 K [6] owing to in-phase and out-of-phase RuO 6 octahedral rotations along the [001] and [110] directions, respectively [4]. Below T N = 56 K, it exhibits antiferromagnetic order (AFM-a) characterized by ferromagnetic perovskite bilayers with a [100] easy axis, i.e., the short axis of the crystal. The bi-layers then couple antiferromagnetically along [001]. At T s = 48 K, it undergoes a first-order isosymmetric transition marked by a discontinuous change in lattice parameters but no change in occupied Wyckoff positions or space group. Interestingly below T s , the lattice constants are nearly temperature independent [6], despite an electrical resistivity transition from a three-to-two-dimensional state. Coincident with T s , a second magnetic transition occurs, AFM-a →AFM-b; the easy axis switches from the [100] to [010] direction (Fig. 2a). Understanding how multiple orders harmonize and evolve with broken inversion and time reversal symmetries as in Ca 3 Ru 2 O 7 is critical for control of topological quasi-particle excitations [7].
Angle resolved photoelectron spectroscopy (ARPES) and optical-pump/optical-probe experiments propose an electronic instability underlies the transport anisotropy at T s by removal of well-defined electron-and hole-like * danilo.puggioni@northwestern.edu † jrondinelli@northwestern.edu  pockets on the Fermi surface (FS) upon cooling [8][9][10][11]. The electron-like band about the Γ point is removed and asymmetric FS changes occur at the zone boundary [11]: A temperature-dependent electron pocket at k = M(π/a, 0) shrinks upon cooling through T = 30 K while a temperature independent hole pocket is found at M (0, π/b). These data suggest either a FS reconstruction [11] or the appearance of a charge density wave at T s [12], respectively; however, there remains no direct evidence for its presence in Ca 3 Ru 2 O 7 . Below 30 K, quasi-2D electron and hole pockets (areas ≈0.3 % of the Brillouin zone) survive and form the low-temperature FS as determined from quantum oscillations [8].
In contrast, thermoelectric measurements show strong asymmetric FS changes with temperature based on assignments of the principal carrier-type with the sign of the thermopower coefficient along the a and b crystal axes [13]. This transport-deduced evolution is inconsistent with the aforementioned optics. Above T s , only hole-like bands are reported, whereas only electronic-like bands contribute for intermediate temperatures, 30  dependencies about k = M(π/a, 0). Below T = 30 K, however, negative and positive thermopower responses were measured along the crystallographic a and b axes, respectively, indicating predominately electron-and hole-like bands along these directions on the FS-features more consistent with the ARPES. Interestingly, all previous band-structure calculations based on density functional theory (DFT) [14,15] are also inconsistent with the ARPES [8,9,11] and thermoelectric transport measurements [13]. This is unusual given that DFT methods typically provide accurate descriptions of weakly correlated metals [16][17][18]. This discrepancy may be due, in part, to underlying structural model assumptions deduced on the reported nearly temperatureindependent changes in the atomic structure below T s and difficulties in resolving weak reflections in layered perovskites [19,20].
Which interactions cause the change in the electronic structure at T s = 48 K and T = 30 K and how can one reconcile the aforementioned discrepancies remains unclarified. Here we provide a model for the low temperature phases of Ca 3 Ru 2 O 7 based on a single-particle DFT plus Hubbard U description, which is fully consistent with all available data. For all values of the Coulomb interaction U , we find a van Hove singularity (vHs) within 20 meV of the Fermi level owing to static Coulomb interactions that enhance broken band degeneracies near k = X(π/a, π/b) split by the spin-orbit interaction (SOI). The filling of these states occurs proximate to Weyl points in momentum space that arise from broken inversion and time-reversal symmetries. Thus, the FS comprises both electron-and hole-like pockets; however, the vHs prohibits using thermopower data alone to deduce the evolution of these bands. We use this electronic structure to propose that a Lifshitz-like electronic transition first occurs at T s = 48 K for U < 1.3 eV and a gapped-out FS persists for 30 < T < 48 K within the known Bb2 1 m structure. Next upon increasing the value of U (cooling below 30 K), we stabilize a unique low-temperature phase (P n2 1 a) with charge/spin density waves that support conclusions of earlier optical spectroscopy experiments [12]. Finally, we compare our T ≤ 30 K band structure calculations for P n2 1 a Ca 3 Ru 2 O 7 with direct ARPES data to confirm both the existence of linear band dispersions below the Fermi level (15 meV) at M(π/a, 0) and that enhanced interactions (U = 1.6 eV) with both SOI and RuO 6 distortions are required to describe the fermiology. Thus, we propose the ground state crystal structure is P n2 1 a and that this phase may persist above T = 30 K and below T s = 48 K.

II. MODEL AND METHODS
We formulate a model that explicitly accounts for the interplay among Coulomb repulsion, magnetic ordering, SOI, and the structural degrees-of-freedom-a combination of features that has not been previously pursued. All previous DFT studies used the reported experimental structure [6], neglecting atomic relaxations [9,14,15]. We include static correlations through a local Coulomb interaction U , which is screened through a temperature dependent electronic bandwidth (W ) as U/W (T ), and further stabilizes states with orbital degeneracies broken by the SOI. We justify this approach as follows: First at T s , the c-axis compression increases the RuO 6 octahedral rotational angles, which are known to narrow the bandwidth (reduce W ) of the low-energy t 2g orbitals in oxides comprising octahedral Ru 4+ . These angles exhibit a further deviation from 180 • upon cooling [6]. Second, finite temperature will effectively reduce the local interaction allowing double occupancy of the d manifold, increasing toward a non-interacting value at high temperature [21]. Thus, the effective U at room-temperature will be smaller than that at 0 K. We perform first-principles DFT+U calculations [22], and vary the on-site Coulomb repulsion from 0 to 2 eV within the Ru 4d manifold. This upper limit on U is close to the constrained random-phase approximation and dynamical mean field theory (DMFT) values for the related ruthenate Ca 2 RuO 4 [23][24][25]. We then interpret these dependencies and assign them to transitions at T s = 48 K and T = 30 K. Previous DFT calculations have shown a strong interplay between SOI and Coulomb repulsion U [14,15]. For this reason, unless otherwise noted, we perform all calculations with SOI and Coulomb repulsion following the Dudarev formalism [26]. We also relax the atomic positions (forces <0.1 meVÅ −1 ) at the DFT + U + SOI level using Gaussian smearing (20 meV width), the experimental lattice parameters at 8 K [6], and the AFM-b magnetic order.

A. Crystal Structure
Upon varying U and performing DFT+U +SOI atomic relaxations, we find that Ca 3 Ru 2 O 7 undergoes a structural transition for U > 1.3 eV characterized by a change in translational symmetry. The Bb2 1 m structure with AFM-b order is dynamically unstable and transforms to the primitive orthorhombic P n2 1 a structure (Fig. 2a). The polar P n2 1 a structure is similar to the Bb2 1 m phase; however, the differential Ru-O bond distortions within the octahedra (Table I), which resemble breathing-like distortions, remove the B-centering operation and permit a charge density wave (CDW). The local distortions in P n2 1 a tile in a rock-salt-like arrangement of small (S) and large (L) RuO 6 octahedra (Fig. 2a). For this reason, while the Bb2 1 m phase is described using two formula units, the P n2 1 a structure is described using four formula units. Also, as indicated by the space group symmetry, the b-glide and mirror plane in the Bb2 1 m structure transform to n-and a-glide operations in the P n2 1 a phase, respectively. This subtle change in crystal-lography gives weak Bragg reflections of the type (0, k, l) with k, l odd for P n2 1 a that are absent in Bb2 1 m. Importantly, convergent-beam electron diffraction (CBED) experiments reported in Ref. 6 are performed in the (h, k, 0) scattering plane and not (0, k, l), which prohibits the reported experiment geometry from differentiating between these two space groups. Fig. 2b shows the octahedral volume (V ) expansion/contraction are modulated by the degree of Coulomb repulsion, which we quantify as B(RuO 6 ) in Fig. 2c using the percent difference in volumes of two adjacent RuO 6 . As U increases in the P n2 1 a phase, the average RuO 6 octahedra volume decreases following the trend exhibited by the Bb2 1 m structure. The average in-plane and out-of-plane Ru − O − Ru angles (Table I and Ref. 27) exhibit a small increase with U , which is in remarkable agreement with the measured temperature dependence below T s [6]. The Ca displacements that lift inversion symmetry undergo minimal changes owing to weak coupling of these displacements to the low-energy states on the Fermi surface [28]. Importantly, we are unable to stabilize the P n2 1 a phase with AFM-a magnetic ordering, confirming the cooperation among Coulomb repulsion, magnetic ordering, and structural degrees of freedom in Coincident with the symmetry reduction, we find disproportionation of the local Ru spin moments, suggesting a spin density wave (SDW) occurs along with the CDW. The small (large) RuO 6 octahedra exhibits a larger (smaller) magnetic moment compared to the adjacent larger (smaller) octahedra (Fig. 2d). This difference increases with increasing U , reaching ≈ 0.2 µ B for U = 2 eV. Owing to the spin disproportionation, the magnetic space group changes from P C na2 1 (No. 33.154, crystal class mm21 [29]) to P na2 1 (No. 33.144), crystal class mm2) [30]. Unlike the breathing distortion B(RuO 6 ), the average magnetic moment increases with increasing Coulomb repulsion indicating overall localization of the itinerant holes within the nominal t 2g manifold of Ru. The coordinations of the small (large) RuO 6 octahedra [27] suggest weak G-type orbital order with the d xy orbital lower (higher) in energy for the large (small) RuO 6 octahedra. Moreover, we find a weak orbital moment of Spin-orbit interaction plays a key role in stabilizing the P n2 1 a phase in the presence of the aforementioned degrees-of-freedom. Upon excluding SOI but allowing full atomic relaxations, we find the P n2 1 a phase remains more stable than the Bb2 1 m phase for U > 1.1 eV (Fig. 2e). For U > 1.4 eV, the total energy difference of the P n2 1 a structures with and without SOI is ≈ 0.1 eV per formula unit (f.u.). However, the spontaneous relaxation of Bb2 1 m → P n2 1 a does not occur-both phases are metastable and separated by an energy barrier, despite the one-dimensional Y 2 order parameter for the symmetry breaking displacements permitting the transition to be continuous. This can be understood as follows: The inclusion of SOI in the calculations reduces the magnetic symmetry from C 1h to C 1 (loss of inversion) allowing energy-lowering displacements due to the broken symmetry. To confirm this behavior, we performed a structural relaxation with U = 1.6 eV excluding SOI starting from the Bb2 1 m phase and removing all symmetries from the calculation. We found that the Bb2 1 m phase relaxes into the triclinic P 1 space group, which exhibits the same type of breathing distortion present in the P n2 1 a phase, but is slightly higher in energy than the P n2 1 a ground state [27]. For these reasons, an accurate atomic ground state description of Ca 3 Ru 2 O 7 requires treating both SOI and Coulomb repulsion. Fig. 3 shows the total density of states (DOS) of Ca 3 Ru 2 O 7 as a function of the Coulomb-repulsion strength. The low-energy electronic structure consists of Ru 4d states, nominally with low-spin Ru 4+ in a d 2 (xy)d 1 (yz)d 1 (xz) configuration hybridized with O 2p states [4]. For U = 0 eV, we find that the valence band is primarily d(xy) with nearly full occupancy while the conduction band consists of primarily unoccupied d(xz)derived states, spanning from the Fermi level (E F ) to ≈ 0.2 eV followed by the d(yz) orbitals located higher in energy (≈ 0.2−0.4 eV). The orbital hierarchy is a consequence of the crystal-field-split states and ratio of the lattice constants b/a > 1; furthermore, the layered structure enhances the bandwidth of the d(xy)-derived bands relative to the d(xz)/d(yz)-derived states. As a consequence, the band structure is nearly semimetallic with multiple bands crossing E F and exhibits quasi-2D character (Fig. 4): The fine structure in the DOS below E F arises from the Ru multiplicity for the d(xy)-derived states, which are degenerate at M (0, π/b) but split off from each other at the zone boundary X(π/a, π/b), as shown in Fig. 4, and the extension of the d(xz)-derived bands below E F . These features evolve with U as we discuss below. SOI also lifts spin degeneracies along particular paths in the Brillouin zone due to Rashba and Dresselhaus interactions, e.g., for the aforementioned d(xy)-derived states spanning E F along the X(π/a, π/b)-M(π/a, 0) trajectory.

B. Electronic Structure
Independent of the crystal structure, a metal-semimetal transition occurs as a pseudogap opens at E F for U > 0 eV. The width of the pseudogap depends on U consistent with the temperature dependence of the pseudogap obtained from ab-plane optical conductivity spectra [12]. Raman scattering performed on single crystals of Ca 3 Ru 2 O 7 indicate a pseudo-gap of ≈ 96 meV appears at T s [31], suggesting that any model of the low-temperature phases of Ca 3 Ru 2 O 7 should include Coulomb interactions (and SOI). The pseudogap in the Bb2 1 m phase arises from the U interaction favoring preferential orbital occupancy and a redistribution of charge among the d(xy) and d(xz) orbitals along with small bandwidth narrowing. Indeed, while the d(xy)-derived bands located at X(π/a, π/b) shift to lower energy with U , the d(xz)-derived bands located at Γ (and d(yz)-derived states) shift to higher energy ( Fig. 3 and Ref. 27).
The SOI interaction with U = 0 increases the energy separation among fractionally occupied states relative to those are occupied [32], because the d(xy), d(xz), and d(yz) states are already split by the C 1 crystal field (Table I) and the d(xy)-like band is close to integer filling. As a result, we observe a van Hove singularity (vHs) near E F (Fig. 3), which is sensitive to small changes in U and hence its position and access to it should be temperature and sample purity dependent. This places Ca 3 Ru 2 O 7 proximate to an electronic instability in the Bb2 1 m phase, and susceptible to a Fermi surface reconstruction. For this reason, we believe the temperature-dependent thermopower data in Ref. 13 should be interpreted with caution since the sign of thermopower cannot be linked directly to the type of carrier when a vHs is within ≈10 meV of E F [33].
To understand the origin of the vHs, we next examine in further detail the band dispersions, focusing on the k z = 0 2D slice of the Brillouin zone (Fig. 4) because dispersions along k z are weak [14]. For U = 0 eV, the Ru bands give rise to quasi-2D hole and electron pockets along M (0, π/b) →X(π/a, π/b) with the electron pockets centered on X(π/a, π/b). We find an electron-like knob at the Γ point along with small quasi-2D electron pockets about Γ and M(π/a, 0). These features are clearly discernible in the FS (Fig. 4b), which presents small degenerate electron pockets centered at X(π/a, π/b) and M(π/a, 0), degenerate hole pockets bordering the electron pocket at X(π/a, π/b), and a feature which almost connects a linear trajectory from M(π/a, 0) to M (0, π/b).
The change in crystal structure also affects the vHs and low-energy electronic structure for U > 1.3 eV. Fig. 3c shows that the vHs splits just below E F , because the change in the local Ru crystal field from the breathing distortions lifts the degeneracy of the single-particle d(xy)derived eigen-energies from different RuO 6 octahedra [see Fig. 4a at X(π/a, π/b) orbitals]. Upon increasing U , these states shift to lower energy while the electron pocket derived from the d(xz) orbital is fully de-occupied at Γ. The bandwidth is further reduced with increasing U , which also enhances the SOI spin splittings. Cooperatively, these effects shift the vHs to lower energy (Fig. 3d) and reduce the FS to comprise small electron pockets at M(π/a, 0) proximate to Weyl nodes (Fig. 4).
These findings confirm the importance of SOI and Coulomb repulsion in the treatment of the low temperature phase of Ca 3 Ru 2 O 7 [14]. For 0 < U < 1.6 eV, the area of the Brillouin zone occupied by the FS is reduced. The volume of the electron pockets centered about M(π/a, 0) decreases and the features along the M (0, π/b)-M(π/a, 0) line and at the Γ point vanish. Due to the shift of the vHs, the electron pockets centered on X(π/a, π/b) also disappear while the nearby hole pockets survive. For U = 1.6 eV, only the electron pockets about M(π/a, 0) survive-all bands are gapped with the exception of the Dirac-like structure that shifts to ∼ 15 meV below E F . In addition, we find that a full charge gap (≈ 30 meV) appears for U = 1.8 eV with SOI. This critical value of U for the metal-insulator transition in Ca 3 Ru 2 O 7 is well below U = 3 eV reported in Ref. [14], which did not allow for local changes in the atomic structure at the DFT + U + SOI level. Importantly, no insulating state is reported for Ca 3 Ru 2 O 7 ; hence, U = 1.8 eV is the upper bound on the interaction strength within our model.

C. Phase Transition Assignments
The addition of Coulomb repulsion within our DFT calculations has three major effects: (i) it shifts the position of the vHs to lower energy, (ii) reduces the overlap between the top of the valence band and the bottom of the conduction band, and (iii) shifts the Dirac-like feature at M(π/a, 0) to higher energy. Therefore, it is natural to ask what is the correct value for U and how can it be used to model the temperature-dependent fermiology. To this purpose, we now compare our calculated electronic band structures as a function of U + SOI with ARPES measured dispersions at 16 K [11]. We find best agreement for U = 1.6 eV (Fig. 5). In particular, the Dirac-like feature at (π/a, 0) and the electronic band structure along Γ-M (0, π/b) are well captured. Note that for U > 1.3 eV, we predict a change in symmetry (Bb2 1 m → P n2 1 a) for Ca 3 Ru 2 O 7 . Using this as an approximate bound on the ground state U value for Ca 3 Ru 2 O 7 , we assign T = 30 K to be a CDW/SDW transition. Furthermore, reasonable values of U to model intermediate temperatures T CDW = 30 < T < T s = 48 K then should span values of 0.5 < U < 1.6 eV. This is justified by the increase of the pseudogap with temperature revealed by optical spectroscopy [12]. For this range of Coulomb interactions, we find that weak static correlations with SOI and RuO 6 distortions drive a Liftshitz-like transition that reconstructs the Fermi surface and opens a pseudo-gap at T s due to filling of the d(xy) bands preferentially to the d(xz) band that is emptied, leaving behind a vHs and Weyl nodes just below E F . In this temperature range, it is plausible that the P n2 1 a phase persists; further experiments are required to conclusively resolve this conjecture.

IV. CONCLUSIONS
Using DFT calculations combined with ARPES measurements, we reconciled existing experiments on the low temperature fermiology of Ca 3 Ru 2 O 7 . The cooperative interplay among Coulomb repulsion U , mag-netic ordering, spin-orbit interactions, and structural degrees of freedom govern the temperature-dependent phase transitions. We assign a a Lifshitz-like transition to T s = 48 K and a charge-(spin-) density wave transition to T CDW = 30 K, which were hypothesized from optical spectroscopy [12], along with broken translational symmetry (Bb2 1 m → P n2 1 a) at T CDW . Last, we validated our temperature-dependent Coulomb interaction model with ARPES experiments at 16 K [11], which indicated that the low-energy physics requires moderate Coulomb repulsion (U = 1.6 eV) and the unique P n2 1 a symmetry. We hope new structural assessments are performed at low temperature with advanced CBED or synchrotron scattering techniques to assess the predicted revised phase diagram. Our work points to the importance of treating multiple interactions on equal footing in correlated metals, which exhibit nontrivial fermions near the Fermi level and fractional orbital filling, as SOI and Coulombic interactions may conspire with lattice and spin degrees-offreedom to produce unexpected electronic and structural transitions.