Evidence for strong electron correlations in a non-symmorphic Dirac semimetal

Metallic iridium oxides (iridates) provide a fertile playground to explore new phenomena resulting from the interplay between topological protection, spin-orbit and electron-electron interactions. To date, however, few studies of the low energy electronic excitations exist due to the difficulty in synthesising crystals with sufficiently large carrier mean-free-paths. Here, we report the observation of Shubnikov-de Haas quantum oscillations in high-quality single crystals of monoclinic SrIrO$_3$ in magnetic fields up to 35~T. Analysis of the oscillations reveals a Fermi surface comprising multiple small pockets with effective masses up to 4.5 times larger than the calculated band mass. \textit{Ab-initio} calculations reveal robust linear band-crossings at the Brillouin zone boundary, due to its non-symmorphic symmetry, and overall we find good agreement between the angular dependence of the oscillations and the theoretical expectations. Further evidence of strong electron correlations is realized through the observation of signatures of non-Fermi liquid transport as well as a large Kadowaki-Woods ratio. These collective findings, coupled with knowledge of the evolution of the electronic state across the Ruddlesden-Popper iridate series, establishes monoclinic SrIrO$_3$ as a topological semimetal on the boundary of the Mott metal-insulator transition.


INTRODUCTION
Band topology and strong electron correlations represent two of the most active research themes in quantum materials [1], with much attention now focused on the search for physics arising from their coexistence. To date, however, few examples of correlated topological materials are known to exist [2][3][4][5]. Among these, iridium oxides have emerged as one of the most promising material platforms on which to investigate the interplay between spin-orbit and electron-electron interactions [6][7][8], due to the comparable energy scales of spinorbit coupling (Λ), Coulomb repulsion (U ) and electron bandwidth (W ) [9].
The Ruddlesden-Popper series Sr n+1 Ir n O 3n+1 , for example, exhibits a plethora of intriguing properties, such as spinorbit-coupled Mottness [10], pseudogap phenomenology [11][12][13][14], odd-parity hidden order [15], and metal-insulator transitions [16][17][18][19]. The spin-orbit-coupled J eff = 1/2 and J eff = 3/2 bands, resulting from crystal-field splitting of the e g and t 2g orbitals, are often treated as the starting point for their understanding [7]. In single-layer Sr 2 IrO 4 (n = 1), the electron bandwidth of the half-filled J eff = 1/2 band is * yute.hsu@ru.nl † nigel.hussey@ru.nl ‡ robin.perry@ucl.ac.uk such that U/W > 1, resulting in a spin-orbit-coupled Mott insulator. With increasing n, electron hopping (and thereby W ) increases while at the same time U is reduced. Consequently, bilayer Sr 3 Ir 2 O 7 is found to be only weakly insulating [16] while the infinite-layer end member SrIrO 3 is itinerant, though its semimetallic ground state and low carrier concentration appear at odds with theoretical expectations of a metallic state with a half-filled J eff = 1/2 band and a large Fermi surface. Recently, the semimetallicity of SrIrO 3 was proposed to have a topological origin [20][21][22]. SrIrO 3 crystallizes in two polymorphs: an ambientpressure monoclinic phase and a high-pressure orthorhombic phase, which is also stabilized under epitaxial strain [17,18]. Three-dimensional (3D) Dirac points are created at the zone boundary of the non-symmorphic lattice by glide symmetry and are protected against gapping caused by spin-orbit coupling. This feature, coupled with their perceived proximity to a Mott transition, makes them attractive candidates for realizing a correlated topological semimetallic state. Until now, however, studies of their intrinsic electronic structure and properties have been restricted due to the fact that orthorhombic SrIrO 3 (o-SIO 3 ) can only be synthesized in thin-film form at ambient pressure [9,[17][18][19][23][24][25]), whereas monoclinic SrIrO 3 (m-SIO 3 ), which can be synthesized in bulk form, has thus far remained largely unexplored.
Here, we report the determination of the full Fermi surface of m-SIO 3 via the observation of Shubnikov-de Haas oscillations in high-quality single crystals. A number of small pock-ets are detected, in excellent agreement with first-principles density-functional theory (DFT) calculations. According to DFT, the electronic band structure of m-SIO 3 is found to be highly sensitive to atomic displacements, though the presence of linear (Dirac) band-crossings at the A-and M-points in the first Brillouin zone remains robust. Significantly, the experimentally determined effective masses are found to be substantially enhanced compared to the DFT band mass, signifying the presence of strong electron correlations. We find a good overall agreement between the theoretically derived angular dependence of the quantum oscillation frequencies with the experimental observation, indicating that correlation effects play only a minor role in determining the Fermi surface topology of m-SIO 3 . Further evidence for strong correlation effects is provided by the observation of a linear-in-temperature (T ) component in the low-T resistivity as well as an unusually large Kadowaki-Woods ratio. Overall, this study demonstrates that m-SIO 3 is a rare iridate topological semimetal displaying signatures of both a correlated Fermi liquid and a non-Fermi liquid ground state, possibly linked to its proximity to a Mott transition. The availability of high-quality single crystals and the sensitivity of its electronic structure to small perturbations make m-SIO 3 a promising platform on which to explore correlation-driven physics in a topological system. Figure 1a shows field-dependent resistivity ρ xx (B) curves measured at T = 0.36 K and two angles θ = 70 • and 83 • relative to the [001]-axis (see inset to Fig. 1a for experimental alignment). Prominent Shubnikov-de Haas (SdH) oscillations emerge above 20 T (most evident in the θ = 83 • trace). By plotting the field-derivative dρ xx /dB in Fig. 1b, the full oscillatory signals become visible for both angles. The oscillation waveform at θ = 83 • , which consists of four oscillatory components as revealed by the fast Fourier transform (FFT) spectrum, is dominated by the F SdH = 650 T component. At θ = 70 • , all four constituent frequencies, with distinct peaks at F SdH = 75, 422, 653, and 907 T, can be tracked over a range of temperatures, as shown in Fig. 1c, thereby enabling a more reliable determination of the corresponding quasiparticle effective masses m * . Each frequency F SdH is related to an extremal cross-sectional area (A k ) of the Fermi surface perpendicular to the field direction via the Onsager relation F SdH = ( /2πe)A k . By fitting the T -dependence of the oscillation amplitude A FFT (T ) at θ = 70 • using the Lifshitz-Kosevich expression for the thermal damping term [26] R T = X/ sinh X, where X = 14.69 m * T /B, we obtain m * values associated with each frequency ranging from 1.9 to 4.8 m e . Such high masses are exceptional for a Dirac semimetal.

DFT calculations
In order to understand the origin of the observed SdH oscillations, we performed DFT band structure calculations using the generalized gradient approximation with effects of spin-orbit coupling included (GGA+SOC). Figure 2 shows the calculated band structures using the structural parameters reported in [27], refined at room temperature (RT), and those found on our own single crystals, refined at low temperature (LT; 13 K) (see Supplementary Methods). The RT band structure agrees well with a previous report [22], with linear bandcrossings near the Fermi energy F at the M-and A-points. The LT band structure, however, reveals a markedly different picture, though notably the linear band-crossings at M-and A-points are still preserved.
The effects of SOC on each band structure are illustrated in Fig. 2c,d. The inclusion of SOC leads to a separation of the low-lying bands near F by up to ∼ 0.3 eV, while the degeneracy at the M-and A-points remains unaffected. The resultant Fermi surfaces from the two GGA+SOC band structures, shown in Fig. 2e,f, show a number of notable differences which we attribute to the extreme sensitivity of the electronic structure of m-SIO 3 with respect to the precise atomic positions -known to be a challenging issue for iridates [25,28] LT structure (this work) -rather than to a structural phase transition. The different types of samples used for the structural refinements -polycrystalline powder in [27] and crushed single crystals in our study -can cause minute structural differences and, in turn, lead to differences in the corresponding band structures. Nevertheless, the linear band-crossings near F at the M-and Apoints are found to be a robust feature independent of such structural details, confirming their protection by the underlying non-symmorphic lattice symmetry.
In order to identify the appropriate Fermi surface model from the DFT calculations, we compare in Fig. 3 the angular dependence of the observed quantum oscillation frequencies F SdH (θ) with the predictions emerging from each model. As shown in Fig. 3a, the four distinct frequency branches disperse differently with angle θ and range from 50 T to 2000 T. Overall, we find a better agreement between the experiment with the LT-SOC model (referred to as the LT model hereafter), which reproduces the close tracking between F 1 and F 2 , the rapid increase in F SdH as θ → 0, as well as the moderate variation of F 3 with respect to θ. This observation implies all the observed pockets originate from the metallic bulk. While the apparent divergence of frequencies F 1,2 as B approaches [001] suggests a possible surface state originating in the (100) plane, we can exclude such a possibility by examining its precise angle dependence. For a surface orbit, the SdH frequency is expected to follow a 1/ cos φ dependence as B is tilted at an angle φ away from the surface normal in which the orbit is hosted. As shown in Supplementary Fig. 3, by plotting a F SdH (θ) measured in two samples, S1 and S2, cut from different crystals of the same growth batch. Four distinct branches can be identified, labelled as F1 to F4. b, c F SdH (θ) expected from the LT and RT Fermi surface models as shown in Fig. 2, with the electron and hole-pockets colour-coded accordingly. Note that the y-axis for all panels is the same.
the tilt angle between B and [100], we find F 1,2 cos φ deviates strongly from a constant value as φ exceeds 30 • , inconsistent with expectations for a surface origin. We therefore conclude that there is no surface contribution to the observed SdH frequencies. We note that the LT model underestimates the F SdH for the F 1,2 branch as θ → 90 • , indicating the electron pocket at M-point is less elongated in reality than that illustrated in Fig. 2f, possibly due to subtle atomic displacements and/or correlation effects. Nevertheless, despite the technical challenges associated with DFT calculations for this complex oxide, the overall agreement between the experimentally observed SdH frequencies and the theoretical expectations from the LT model suggests that the full Fermi surface of m-SIO 3 has been determined by our study. The Dirac points lie ≈ 15 and 50 meV below the Fermi level at the M-and A-points, respectively, at which the band dispersion remains linear (see Fig. 2d). In order to identify the correct FS model from the DFT calculations, we compare in Fig. 3 the angular dependence of the observed SdH frequencies F SdH (θ) with the predictions emerging from each model. As shown in Fig. 3(a), the four distinct frequency branches disperse differently with angle θ and range from 50 T to 2000 T. While both models predict four branches at most angles and an electron-pocket associated with the highest frequency branch ( Fig. 3(b, c)), we find better overall agreement with the LT-SOC model (referred to as the LT model hereafter). Importantly, the close tracking between F 1 and F 2 , the rapid increase in F SdH as θ → 0, and the moderate variation of F 3 with respect to θ are all wellreproduced in the LT model.
The rapid increase of F 1,2 as B is tilted away from [100] may suggest a surface state located within the (100) plane. For such an orbit, the SdH frequency should follow a 1/ cos φ dependence as B is tilted away from the surface. We examine this possibility by plotting F 1,2 cos φ versus φ, where φ = 90 • − θ denotes the tilt angle between B and [100], as shown in Supplemental Material Fig. S2. We find F 1,2 cos φ deviates strongly from a constant value as φ exceeds 30 • , inconsistent with expectations for a surface state. We therefore conclude that all the observed pockets arise from the metallic bulk. Despite the technical challenges associated with DFT calculations for this complex oxide, the remarkable agreement in the experimentally observed angular dependence of SdH frequencies and the theoretical expectations from the LT model suggests that the full FS of m-SIO 3 has been accurately determined by our study.

Signatures of strong electronic correlations
Having established the appropriate Fermi surface model, we proceed to quantify the extent of electron correlation in m-SIO 3 by comparing the measured effective masses m * with the calculated band masses m b . Here we use the definition m b = 2 (dA k /dE)| F , namely the change of area enclosed by the cyclotron orbit in k-space with respect to the change in energy at the Fermi level, to describe the band mass of quasiparticles with an arbitrary dispersion (k). As shown in Table 1, we find a substantial mass renormalization factor m * /m b for all frequencies, ranging from 1.4 to 4.5, signifying strong electron correlations. We note that m * /m b is larger for the hole pocket (F 3 ) compared to the electron pockets with linear band-crossings, possibly reflecting the different impact of correlation effects on the conventional and Diraclike bands.
The inclusion of U up to 3 eV in our calculations leads to a considerable modification in the Fermi surface topology, as shown in Supplementary Fig. 4. The best agreement with experiment, however, is found when U = 0, indicating that the influence of electron correlations cannot be effectively captured for complex iridates using the GGA+SOC+U approach [23]. A more accurate accounting of correlation effects will likely require the implementation of computationally demanding dynamical mean-field theory calculations, which will be the subject of a future investigation. Nevertheless, the remarkable agreement between the LT model and experiment demonstrates that correlation effects play only a minor role in determining the Fermi surface topology, as found for related systems such as o-SIO 3 [23] and the topological semimetal ZrSiS [29].
The anomalous nature of the electronic state in m-SIO 3 is revealed through its transport and thermodynamic properties at low temperatures. ρ xx (T ) below 15 K follows an unusual T + T 2 -dependence, as revealed by the finite intercept in dρ xx /dT at T = 0 (Fig. 4a, b). The T -linear term A 1 = 0.1 µΩ cm K −1 , though small, persists down to the lowest temperatures, possibly indicating the presence of low-energy critical fluctuations. By contrast, the specific heat and magnetic susceptibility of m-SIO 3 , shown in Fig. 4c, d, are more characteristic of a paramagnetic Fermi-liquid state. Using the coefficient of the T 2 term A 2 = 0.048 µΩ cm K −2 and the electronic specific heat coefficient γ 0 = 4.47 (7) mJ mol −1 K −2 , the corresponding Kadowaki-Woods ratio (R KW ) is found to be A 2 /γ 2 0 ≈ 2400 µΩ cm K 2 mol 2 J −2 or A 2 /γ 2 V ≈ 3.5 µΩ cm K 2 cm 6 J −2 , where γ V is the volume form of γ 0 [30]. These values, though high relative to other correlated metals [30][31][32], are nonetheless consistent with theoretical estimates derived for a 3D Fermi surface [30] (see Supplementary Discussion for details), implying that electron-electron scattering dominates the transport behaviour at low T . Recently, it has been shown that the strength of electron interactions correlates with R KW in the most general case [32], and overall consistency in our analysis indicates that electron-electron scattering is responsible for the large R KW and the highly renormalized m * in m-SIO 3 , possibly due to the SOC-renormalized bandwidth and/or proximity to a quantum critical point [33,34].
The topological character of the electronic bands in m-SIO 3 is examined by extracting the phase factor of the SdH oscillations (see Supplementary Fig. 5). While we find a nontrivial phase factor ≈ π, as expected for a topologically nontrivial band, recent theoretical findings have suggested that the phase factor in quantum oscillations can comprise of multiple contributions and therefore cannot be considered as a conclusive evidence for a non-trivial band topology [35]. Nonetheless, we emphasise that the topological non-trivial character for the electron band in m-SIO 3 is further supported by the good agreement between the DFT calculations -with their topologically-protected Dirac points at M and A -and the observed quantum oscillation angular dependence.
Recently, transport signatures of Dirac quasiparticles have  ). l, m * , and m b represent quasiparticle mean free path, effective mass, and DFT band mass, respectively. All parameters are extracted at θ = 70 • . Note that a reliable extraction of l associated with F4 is precluded by the dominating influence of the higher-frequency oscillations, and the parameters marked by † are an average over the multiple small orbits (< 30 T) predicted by the DFT model that cannot be resolved experimentally.   . χm is largely T -independent down to ≈ 30K, with a constant χ0 as marked by the grey band. Below 30 K, χm shows a small upturn, which we attribute to an impurity contribution that follows a Curie-Weiss behaviour. e Transport coefficient A2 plotted against γ 2 0 , known as the Kadowaki-Woods plot, for selected correlated oxides (diamonds), iron pnictides (hexagons), heavy fermion materials (circles) ( [30][31][32] and references therein), and monoclinic SrIrO3 (elongated diamond). Red line corresponds to A2/γ 2 0 = 10 µΩ cm mol 2 K 2 mJ −6 , known to describe many heavy fermion materials well. been reported in a closely-related iridate CaIrO 3 [36,37]. The resultant Fermi surface areas and effective masses, however, were found to be small (F SdH = 3.2 − 11.2 T and m * = 0.12 − 0.31 m e ), suggesting that the Fermi level in CaIrO 3 is much closer to the Dirac point and that electron correlations are considerably weaker than in m-SIO 3 . A recent photoemission experiment on o-SIO 3 thin films, on the other hand, found an electron-like band with linear dispersion and parabolic holelike bands with heavy quasiparticle masses m * = 2.4 − 6.0 m e [23], largely in agreement with our findings on m-SIO 3 . Collectively, these findings identify SIO 3 as a rare example of a topological semimetal with enhanced electron correlations, possibly induced through proximity to a Mott instability [9]. The extreme sensitivity of its low-energy electronic structure to atomic displacement, highlighted by the DFT calculations, further identifies m-SIO 3 as a viable platform on which to explore the Mott transition in a topological semimetal, by tuning the relative strength of the electronic bandwidth and Coulomb repulsion via doping or via structural tuning parameters such as hydrostatic pressure or uniaxial strain.

Crystal growth and characterization
Single crystals of m-SIO 3 were grown using the flux method described in Ref. [38]. Low-temperature structural refinement using X-ray diffraction (XRD) was performed at 13 K (see Supplementary Methods for details).

Measurements
Electrical resistivity measurements were performed on Hall bars of (600×100×60) µm 3 in size, cut from pristine crystals, with an ac current of 3 mA applied along the bar identified as the crystallographic axis [100] by Laue XRD. Magnetic fields up to 35 T were generated using a Bitter magnet at the High Field Magnet Laboratory (HFML) in Nijmegen, the Netherlands. Zero-field resistivity measurements were performed using a Cryogenic Free Measurement System by Cryogenic Limited; specific heat and magnetic susceptibility measurements were performed using a Physical Properties Measurement System and a Magnetic Properties Measurement System XL, respectively, by Quantum Design Inc.
Band structure calculations DFT band structure calculations were carried out using abinitio VASP code [39] with energy cut-off set to 400 eV and Brillouin zone sampled by the 9 × 9 × 5 Monkhorst-Pack k-point mesh. We used WANNIER90 package [40] to construct the corresponding tight-binding (TB) Hamiltonian and ensured that the resulting VASP and TB electronic band structures are identical near the Fermi level. The visualization of Fermi surfaces was done using XCrysDen code [41] and the oscillation frequency expected from the DFT Fermi surface model was calculated using the SKEAF code [42].