High-Resolution Photoemission on Sr2ruo4 Reveals Correlation-Enhanced Eﬀective Spin-Orbit Coupling and Dominantly Local Self-Energies

High-resolution photoemission on Sr2RuO4 reveals correlation-enhanced eﬀective spin-orbit coupling and dominantly local self-energies

A. Tamai,1 M. Zingl,2 E. Rozbicki,3 E. Cappelli,1 S. Riccò,1 A. de la Torre,1 S. McKeown-Walker,1 F. Y. Bruno,1 P.D.C. King,3 W. Meevasana,4 M. Shi,5 M. Radović,5 N.C. Plumb,5 A.S. Gibbs,3, ∗ A.P. Mackenzie,6, 3 C. Berthod,1 H. Strand,2 M. Kim,7, 8 A. Georges,9, 2, 8, 1 and F. Baumberger1, 5 1Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland 2Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA 3SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom 4School of Physics, Suranaree University of Technology and Synchrotron Light Research Institute, Nakhon Ratchasima, 30000, Thailand and Thailand Center of Excellence in Physics, CHE, Bangkok, 10400, Thailand 5Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 6Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany 7Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA 8Centre de Physique ThéoriqueEcole Polytechnique, CNRS, Universite Paris-Saclay, 91128 Palaiseau, France 9Collègede France, 11 place Marcelin Berthelot, 75005 Paris, France (Dated: May 8, 2019) We explore the interplay of electron-electron correlations and spin-orbit coupling in the model Fermi liquid Sr2RuO4 using laser-based angle-resolved photoemission spectroscopy. Our precise measurement of the Fermi surface confirms the importance of spin-orbit coupling in this material and reveals that its effective value is enhanced by a factor of about two, due to electronic correlations. The self-energies for the β and γ sheets are found to display significant angular dependence. By taking into account the multi-orbital composition of quasiparticle states, we determine self-energies associated with each orbital component directly from the experimental data. This analysis demonstrates that the perceived angular dependence does not imply momentum-dependent many-body effects, but arises from a substantial orbital mixing induced by spin-orbit coupling. A comparison to single-site dynamical mean-field theory further supports the notion of dominantly local orbital self-energies, and provides strong evidence for an electronic origin of the observed non-linear frequency dependence of the self-energies, leading to ‘kinks’ in the quasiparticle dispersion of Sr2RuO4.

I. INTRODUCTION Quantum oscillation and angle-resolved photoemission spectroscopy (ARPES) measurements [24–34] further reported a strong enhancement of the quasiparticle effec- The layered perovskite Sr2RuO4 is an important model system for correlated electron physics. Its intriguing su- tive mass over the bare band mass. Theoretical progress perconducting ground state, sharing similarities with su- has been made recently in revealing the important role perfluid 3He [1–3], has attracted much interest and con- of the intra-atomic Hund’s coupling as a key source tinues to stimulate advances in unconventional supercon- of correlation effects in Sr2RuO4 [18, 20, 35]. In this ductivity [4]. Experimental evidence suggest odd-parity context, much attention was devoted to the intrigu- spin-triplet pairing, yet questions regarding the proxim- ing properties of the unusual state above TFL, which ity of other order parameters, the nature of the pairing displays metallic transport with no signs of resistivity mechanism and the apparent absence of the predicted saturation at the Mott-Ioffe-Regel limit [36]. Dynam- edge currents remain open [3–9]. Meanwhile, the normal ical mean-field theory (DMFT) [37] calculations have proven successful in explaining several properties of this state of Sr2RuO4 attracts interest as one of the clean- est oxide Fermi liquids [10–13]. Its precise experimen- intriguing metallic state, as well as in elucidating the tal characterization is equally important for understand- crossover from this unusual metallic state into the Fermi- arXiv:1812.06531v2 [cond-mat.str-el] 7 May 2019 ing the unconventional superconducting ground state of liquid regime [13, 18, 20–23, 38]. Within DMFT, the self-energies associated with each orbital component are Sr2RuO4 [1–9, 14–17], as it is for benchmarking quantitative many-body calculations [18–23]. assumed to be local. On the other hand, the low- Transport, thermodynamic and optical data of temperature Fermi-liquid state is known to display strong magnetic fluctuations at specific wave-vectors, as re- Sr2RuO4 display textbook Fermi-liquid behavior be- vealed, e.g., by neutron scattering [39, 40] and nuclear low a crossover temperature of TFL ≈ 25 K [10–13]. magnetic resonance spectroscopy (NMR) [41, 42]. These magnetic fluctuations were proposed early on to be an important source of correlations [2, 43, 44]. In this pic- ∗ present address: ISIS Facility, Rutherford Appleton Laboratory, ture, it is natural to expect strong momentum depen- Chilton, Didcot OX11 OQX, United Kingdom 2 dence of the self-energy associated with these spin fluctu- experiments. Finally, our results are critically discussed ations. Interestingly, a similar debate was raised long ago and put in perspective in Secs.VII,VIII. in the context of liquid 3He, with ‘paramagnon’ theories emphasizing ferromagnetic spin-fluctuations and ‘quasi- localized’ approaches àla Anderson-Brinkman emphasiz- II. EXPERIMENTAL RESULTS ing local correlations associated with the strong repulsive hard-core, leading to increasing Mott-like localization as the liquid is brought closer to solidification (for a review, A. Experimental methods see Ref. [45]). In this work, we report on new insights into the na- The single crystals of Sr2RuO4 used in our experiments were grown by the floating zone technique and ture of the Fermi-liquid state of Sr2RuO4. Analyzing a comprehensive set of laser-based ARPES data with im- showed a superconducting transition temperature of proved resolution and cleanliness, we reveal a strong an- Tc = 1.45 K. ARPES measurements were performed with gular (i.e., momentum) dependence of the self-energies an MBS electron spectrometer and a narrow bandwidth associated with the quasiparticle bands. We demonstrate 11 eV(113 nm) laser source from Lumeras that was op- that this angular dependence originates in the variation erated at a repetition rate of 50 MHz with 30 ps pulse of the orbital content of quasiparticle states as a func- length of the 1024 nm pump [52]. All experiments were tion of angle, and can be understood quantitatively. In- performed at T ≈ 5 K using a cryogenic 6-axes sample troducing a new framework for the analysis of ARPES goniometer, as described in Ref. [53]. A combined en- data for multi-orbital systems, we extract the electronic ergy resolution of 3 meV was determined from the width of the Fermi-Dirac distribution measured on a polycrys- self-energies associated with the three Ruthenium t2g orbitals with minimal theoretical input. We find that these talline Au sample held at 4.2 K. The angular resolution ◦ orbital self-energies have strong frequency dependence, was ≈ 0.2 . In order to suppress the intensity of the sur- but surprisingly weak angular (i.e., momentum) depen- face layer states on pristine Sr2RuO4 [54], we exposed the dence, and can thus be considered local to a very good cleaved surfaces to ≈ 0.5 Langmuir CO at a temperature approximation. Our results provide a direct experimental of ≈ 120 K. Under these conditions, CO preferentially demonstration that the dominant effects of correlations fills surface defects and subsequently replaces apical oxygen ions to form a Ru–COO carboxylate in which the C in Sr2RuO4 are weakly momentum-dependent and can be understood from a local perspective, provided they end of a bent CO2 binds to Ru ions of the reconstructed are considered in relation to orbital degrees of freedom. surface layer [55]. One of the novel aspect of our work is to directly put the locality ansatz underlying DMFT to the experimental test. We also perform a direct comparison between B. Experimental Fermi surface and quasiparticle DMFT calculations and our ARPES data, and find good dispersions agreement with the measured quasiparticle dispersions and angular dependence of the effective masses. In Fig.1 we show the Fermi surface and selected con- The experimentally determined real part of the self- stant energy surfaces in the occupied states of Sr2RuO4. energy displays strong deviations from the low-energy The rapid broadening of the excitations away from the Fermi-liquid behavior Σ0(ω) ∼ ω(1 − 1/Z) + ··· for Fermi level seen in the latter is typical for ruthenates binding energies |ω| larger than ∼ 20 meV. These de- and implies strong correlation effects on the quasipar- viations are reproduced by our DMFT calculations sug- ticle properties. At the Fermi surface, one can read- gesting that the cause of these non-linearities are local ily identify the α, β and γ sheets that were reported electronic correlations. Our results thus call for a revi- earlier [25, 27, 28]. However, compared with previous sion of earlier reports of strong electron-lattice coupling ARPES studies we achieve a reduced line width and im- in Sr2RuO4 [29–31, 46–50]. We finally quantify the ef- proved suppression of the surface layer states giving clean fective spin-orbit coupling (SOC) strength and confirm access to the bulk electronic structure. This is partic- its enhancement due to correlations predicted theoreti- ularly evident along the Brillouin zone diagonal (ΓX) cally [21, 23, 51]. where we can clearly resolve all band splittings. This article is organized as follows. In Sec.II, we In the following, we will exploit this advance to quan- briefly present the experimental method and report our tify the effects of SOC in Sr2RuO4 and to provide new main ARPES results for the Fermi surface and quasipar- insight into the renormalization of the quasiparticle ex- ticle dispersions. In Sec.III, we introduce the theoret- citations using minimal theoretical input only. To this ical framework on which our data analysis is based. In end we acquired a set of 18 high resolution dispersion Sec.IV, we use our precise determination of the Fermi plots along radial k-space lines (as parameterized by the surface to reveal the correlation-induced enhancement of angle θ measured from the ΓM direction). The subset the effective SOC. In Sec.V we proceed with a direct of data shown in Fig.2 (a) immediately reveals a rich determination of the self-energies from the ARPES data. behavior with a marked dependence of the low-energy Sec.VI presents the DMFT calculations in comparison to dispersion on the Fermi surface angle θ. Along the ΓM 3

Γ X (a) (b) M EF

-10 meV

Γ M -20 meV

-30 meV

β α max γ -40 meV min

FIG. 1. (a) Fermi surface of Sr2RuO4. The data were acquired at 5 K on a CO passivated surface with a photon energy of 11 eV and p-polarization for measurements along the ΓX symmetry line. The sample tilt around the ΓX axis used to measure the full Fermi surface results in a mixed polarization for data away from this symmetry axis. The Brillouin zone of the reconstructed surface layer is indicated by diagonal dashed lines. Surface states and final state umklapp processes are suppressed to near the detection limit. A comparison with ARPES data from a pristine cleave is shown in appendixA. The data in (a) have been mirror-symmetrized for clarity. Original measured data span slightly more than a quadrant of the Brillouin zone. (b) Constant energy surfaces illustrating the progressive broadening of the quasiparticle states away from the Fermi level EF . high-symmetry line our data reproduce the large differ- quasiparticle Fermi velocities of Fig.2 (c) and the cor- β,γ ence in Fermi velocity vF for the β and γ sheet, which responding bare velocities vb of a reference Hamiltonian is expected from the different cyclotron masses deduced Hˆ 0 defined in Sec.IV. As shown in Fig.2 (d), this con- from quantum oscillations [12, 25, 26] and was reported firms a substantial many-body effect on the anisotropy of in earlier ARPES studies [33, 56]. Our systematic data, the quasiparticle dispersion. Along ΓM, we find a strong however, reveal that this difference gradually disappears differentiation with mass enhancements of ≈ 5 for the γ ◦ towards the Brillouin zone diagonal (θ = 45 ), where all sheet and ≈ 3.2 for β, whereas vb/vF approaches a com- three bands disperse nearly parallel to one another. In mon value of ≈ 4.4 for both sheets along the Brillouin Sec.IV we will show that this equilibration of the Fermi zone diagonal. velocity can be attributed to the strong effects of SOC Before introducing the theoretical framework used to around the zone diagonal. quantify the anisotropy of the self-energy and the effects of SOC, we compare our data quantitatively to bulk sen- β,γ sitive quantum oscillation measurements. Using the ex- To quantify the angle dependence of vF from exper- ν perimental Fermi wave vectors k and velocities deter- iment, we determine the maxima kmax(ω) of the mo- F mentum distribution curves (MDCs) over the range of mined from our data on a dense grid along the entire 2-6 meV below the Fermi level EF and fit these k-space Fermi surface, we can compute the cyclotron masses mea- loci with a second-order polynomial. We then define the sured by dHvA experiments, without relying on the ap- Fermi velocity as the derivative of this polynomial at EF . proximation of circular Fermi surfaces and/or isotropic This procedure minimizes artifacts due to the finite en- Fermi velocities used in earlier studies [33, 49, 56, 58]. ergy resolution of the experiment. As shown in Fig.2 (c), Expressing the cyclotron mass m∗ as β,γ the Fermi velocities v obtained in this way show an 2 2 Z 2π F ∗ ~ ∂AFS ~ kF (θ) opposite trend with azimuthal angle for the two Fermi m = = dθ , (1) 2π ∂ 2π 0 ∂/∂k(θ) sheets. For the β band we observe a gentle decrease of vF as we approach the ΓX direction, whereas for γ the veloc- where AFS is the Fermi surface volume, and using the ∗ ity increases by more than a factor of two over the same data shown in Fig.2 (c), we obtain mγ = 17.3(2.0) me ∗ range [57]. This provides a first indication for a strong and mβ = 6.1(1.0) me, in quantitative agreement with 0 ∗ ∗ momentum dependence of the self-energies Σβ,γ , which the values of mγ = 16 me and mβ = 7 me found in we will analyze quantitatively in Sec.V. Here, we limit dHvA experiments [12, 25, 26]. We thus conclude that the discussion to the angle dependence of the mass en- the quasiparticle states probed by our experiments are hancement vb/vF , which we calculate from the measured representative of the bulk of Sr2RuO4[59]. 4

(a) β γ (c) 0 0.8

) β Å max

(meV) -40 (eV F F 0.4 γ v E-E -80 ΓM, θ = 0º θ = 6º θ = 12º θ = 18º θ = 24º min 0.0 β γ α (b) (d) γ 0 X 6 γ α

F 5 β (meV)

-40 / v F b

v 4 E-E -80 3 β θ = 30º θ = 36º θ = 42º θ = 45º θ 0.6 0.8 0.6 0.8 0.6 0.8 0.6 0.8 Γ M 0 45 90 -1 FS angle θ (deg) k// (Å )

FIG. 2. (a) Quasiparticle dispersions measured with p-polarized light, for different azimuthal angles θ as defined in panel (b). (c) Angular dependence of the quasiparticle velocity vF along the β and γ Fermi surface sheets. (d) Angular dependence of the quasiparticle mass enhancement vb/vF . Here, vb is the bare velocity obtained from the single-particle Hamiltonian ˆ 0 ˆ DFT ˆ SOC H =H +HλDFT+∆λ defined in Sec.IV and vF is the quasiparticle Fermi velocity measured by ARPES. Error bars are obtained by propagating the experimental uncertainty on vF . Thin lines are guides to the eye.

III. THEORETICAL FRAMEWORK quantum oscillations in the bulk provides direct evidence for well-defined quasiparticles in Sr2RuO4 down to the In order to define the electronic self-energy and as- lowest energies [25, 27]. sess the effect of electronic correlations in the spectral It is important to note that the Green’s function G, the function measured by ARPES, we need to specify a one- self-energy Σ and the spectral function A are in general particle Hamiltonian Hˆ 0 as a reference point. At this non-diagonal matrices. This has been overlooked thus far stage, we keep the presentation general. The particular in self-energy analyses of ARPES data but is essential choice of Hˆ 0 will be a focus of Sec.IV. The eigenstates to determine the nature of many-body interactions in 0 |ψν (k)i of Hˆ (k) at a given quasi-momentum k and the Sr2RuO4, as we will show in Sec.V. corresponding eigenvalues εν (k) define the ‘bare’ band structure of the system, with respect to which the self- energy Σνν0 (ω, k) is defined in the standard way from the A. Localized orbitals and electronic structure interacting Green’s function −1 Let us recall some of the important aspects of the elec- Gνν0 (ω, k) = [ω + µ − εν (k)] δνν0 − Σνν0 (ω, k) . (2) tronic structure of Sr2RuO4. As shown in Sec.IIB, three In this expression ν and ν0 label the bands and ω denotes bands, commonly denoted ν = {α, β, γ}, cross the Fermi the energy counted from EF . The interacting value of level. These bands correspond to states with t2g symme- the chemical potential µ sets the total electron number. try deriving from the hybridization between localized Ru- ˆ 0 Since µ can be conventionally included in H , we shall 4d (dxy, dyz, dxz) orbitals and O-2p states. Hence, we in- omit it in the following. troduce a localized basis set of t2g-like orbitals |χmi, with The Fermi surfaces and dispersion relations of the basis functions conveniently labeled as m = {xy, yz, xz}. quasiparticles are obtained as the solutions ω = 0 and In practice, we use maximally localized Wannier func- qp ω = ων (k) of tions [60, 61] constructed from the Kohn-Sham eigenbasis 0 of a non-SOC density functional theory (DFT) calcula- det [(ω − ε (k)) δ 0 − Σ 0 (ω, k)] = 0 . (3) ν νν νν tion (see appendixB1 for details). We term the cor- In the above equation Σ0 denotes the real part of the responding Hamiltonian Hˆ DFT. It is important to note self-energy. Its imaginary part Σ00 has been neglected, that the choice of a localized basis set is not unique and i.e., we assume that quasiparticles are coherent with a other ways of defining these orbitals are possible (see, lifetime longer than 1/ωqp. Our data indicate that this e.g., Ref. [62]). is indeed the case up to the highest energies analyzed In the following this set of orbitals plays two important here. At very low frequency, the lifetime of quasiparticles roles. First, they are atom-centered and provide a set cannot be reliably tested by ARPES, since the intrinsic of states localized in real-space |χm (R)i. Secondly, the quasiparticle width is masked by contributions of the ex- unitary transformation matrix to the band basis |ψν (k)i perimental resolution, impurity scattering and inhomo- geneous broadening. However, the observation of strong Umν (k) = hχm (k) |ψν (k)i , (4) 5 allows us to define an ‘orbital’ character of each band ν as is also consistent with general perturbation-theory con- 2 |Umν (k) | . In the localized-orbital basis the one-particle siderations [51], which show a Coulomb-enhancement of Hamiltonian is a non-diagonal matrix, which reads the level splitting in the J basis, similar to a Coulomb- enhanced crystal-field splitting [69]. ˆ 0 X ∗ Hmm0 (k) = Umν (k) εν (k) Um0ν (k) . (5) In the absence of SOC, DFT yields a quasi-crossing ν between the β and γ Fermi surface sheets a few degrees away from the zone diagonal, as displayed on Fig.3 (a). The self-energy in the orbital basis is expressed as Near such a point we expect the degeneracy to be lifted eff X ∗ by SOC, leading to a momentum splitting ∆k = λ /v Σmm0 (ω, k) = Umν (k)Σνν0 (ω, k) Um0ν0 (k) , (6) and to an energy splitting of ∆E = Zλeff [23], as de- νν0 picted schematically in Fig.3 (e). In these expressions, √ and conversely in the band basis as v ≡ vβvγ , with vβ and vγ the bare band velocities in the p absence of SOC and correlations, and Z ≡ ZβZγ in- X ∗ volves the quasiparticle residues Z associated with each Σνν0 (ω, k) = Umν (k)Σmm0 (ω, k) Um0ν0 (k) . (7) ν mm0 band (also in the absence of SOC). It is clear from these expressions that a quantitative determination of λeff is not possible from experiment B. Spin-orbit coupling alone. Earlier studies on Sr2RuO4 [68] and iron-based superconductors [70], have interpreted the energy split- We treat SOC as an additional term to Hˆ DFT, which ting ∆E at avoided crossings as a direct measure of the is independent of k in the localized-orbital basis, but SOC strength λeff . However, in interacting systems ∆E leads to a mixing of the individual orbitals. The single- is not a robust measure of SOC since correlations can particle SOC term for atomic d-orbitals projected to the both enhance ∆E by enhancing λeff and reduce it via t2g-subspace reads [63] the renormalization factor Z. We thus quantify the enhancement of SOC from the momentum splitting ∆k, SOC λ X X † which is not renormalized by the quasiparticle residue Hˆ = c (l 0 · σ 0 ) c 0 0 , (8) λ mσ mm σσ m σ Z. The experimental splitting at the avoided crossing 2 0 0 mm σσ between the β and γ Fermi surface sheets indicated in QP ˚−1 where l are the t2g-projected angular momentum matri- Fig.3 (a) is ∆ k = 0.094(9) A whereas DFT pre- DFT+λ −1 ces, σ are Pauli matrices and λ will be referred to in the dicts ∆k DFT = 0.046 A˚ . We thus obtain an ef- eff QP DFT+λDFT following as the SOC coupling constant. As documented fective SOC strength λ = λDFT∆k /∆k = in appendixB1, the eigenenergies of a DFT+SOC calcu- 205(20) meV, in quantitative agreement with the predic- ˆ DFT ˆ SOC tions in Refs. [21, 23]. We note that despite this large lation are well reproduced by H +Hλ with λDFT = 100 meV. enhancement of the effective SOC, the energy splitting remains smaller than λDFT as shown in Fig.3 (f). When deviations from linearity in band dispersions are small, IV. ENHANCED EFFECTIVE SPIN-ORBIT the splitting ∆E is symmetric around the EF and can COUPLING AND SINGLE-PARTICLE thus be determined from the occupied states probed in HAMILTONIAN experiment. Direct inspection of the data in Fig.3 (f) yields ∆E ≈ 70 meV, which is about 2/3 of λDFT and The importance of SOC for the low-energy physics of thus clearly not a good measure of SOC. Sr2RuO4 has been pointed out by several authors [8, 17, The experimental splitting is slightly larger than that 21, 23, 31, 64–68]. SOC lifts degeneracies found in its expected from the expression ∆E = Zλeff and our the- absence and causes a momentum dependent mixing of oretical determination of Zβ and Zγ at the Fermi sur- the orbital composition of quasiparticle states, which has face (which will be described in Sec.VI). This can be non-trivial implications for superconductivity [8, 17, 68]. attributed to the energy dependence of Z, which, in Signatures of SOC have been detected experimentally on Sr2RuO4, is not negligible over the energy scale of SOC. the Fermi surface of Sr2RuO4 in the form of a small pro- Note that the magnitude of the SOC-induced splitting of trusion of the γ sheet along the zone diagonal [31, 66] the bands at the Γ point reported in Ref. [68] can also and as a degeneracy lifting at the band bottom of the be explained by the competing effects of enhancement by β sheet [68]. These studies reported an overall good correlations and reduction by the quasiparticle weight as agreement between the experimental data and the ef- shown in Ref. [23]. We also point out that the equili- fects of SOC calculated within DFT [31, 66, 68]. This bration of quasiparticle velocities close to the diagonal, is in apparent contrast to more recent DMFT studies apparent from Figs.2 (a,c) and3 (f) is indeed the be- of Sr2RuO4, which predict large but frequency indepen- havior expected close to an avoided crossing [23]. dent off-diagonal contributions to the local self-energy Including the enhanced SOC determined from this that can be interpreted as a contribution ∆λ to the ef- non-crossing gap leads to a much improved theoreti- eff fective coupling strength λ = λDFT + ∆λ [21, 23], This cal description of the entire Fermi surface [71]. As 6

FIG. 3. Correlation enhanced eﬀective SOC. (a) Quadrant of the experimental Fermi surface with a DFT calculation without DFT SOC (Hˆ ) at the experimental kz ≈ 0.4 π/c (grey lines). (b,c) Same as (a) with calculations including SOC (DFT+λDFT) and enhanced SOC (λDFT + ∆λ), respectively. For details, see main text. (d) Comparison of the experimental MDC along the k-space cut indicated in (a) with the diﬀerent calculations shown in (a-c). (e) Schematic illustration of the renormalization of √ 0 √ a SOC induced degeneracy lifting. Here, Z = Zν Zν0 , where ν, ν labels the two bands and v = vν vν0 where vν , vν0 are bare velocities in the absence of SOC [23] (see text). (f) Experimental quasiparticle dispersion along the k-space cut indicated in (a). (g) Orbital character of the DFT+λDFT + ∆λ eigenstates along the Fermi surface.

shown in Fig.3 (b), our high-resolution experimental experimental value of kz ≈ 0.4 π/c the variation is weak Fermi surface deviates systematically from a DFT cal- [72]. The analysis presented here and in Secs.IIIA and culation with SOC. Most notably, Hˆ DFT+Hˆ SOC under- VI is thus robust with respect to a typical uncertainty in λDFT estimates the size of the γ sheet and overestimates the kz. These findings suggest that a natural reference single- β sheet. Intriguingly, this is almost completely corrected particle Hamiltonian is Hˆ 0=Hˆ DFT+Hˆ SOC . This λDFT+∆λ in Hˆ DFT+Hˆ SOC , with λ + ∆λ = 200 meV, as ˆ 0 λDFT+∆λ DFT choice ensures that the Fermi surface of H is very close demonstrated in Fig.3 (c). Indeed, a close inspection to that of the interacting system. From Eq.3, this im- shows that the remaining discrepancies between experi- plies that the self-energy matrix approximately vanishes ment and Hˆ DFT+Hˆ SOC break the crystal symmetry, 0 λDFT+∆λ at zero binding energy: Σνν0 (ω = 0, k) ' 0. We choose suggesting that they are dominated by experimental ar- Hˆ 0 in this manner in all the following. Hence, from now tifacts. A likely source for these image distortions is im- on |ψ (k)i and ε (k) refer to the eigenstates and band perfections in the electron optics arising from variations ν ν structure of Hˆ 0=Hˆ DFT+Hˆ SOC . We point out that of the work function around the electron emission spot on λDFT+∆λ ˆ 0 the sample. Such distortions can presently not be fully although H is a single-particle Hamiltonian, the effec- ˆ 0 eliminated in low-energy photoemission from cleaved sin- tive enhancement ∆λ of SOC included in H is a corre- gle crystals. lation effect beyond DFT. Importantly, the change in Fermi surface sheet volume with the inclusion of ∆λ is not driven by a change in the crystal-field splitting between the xy and xz, yz orbitals V. EXPERIMENTAL DETERMINATION OF (see appendixB). The volume change occurs solely be- SELF-ENERGIES cause of a further increase in the orbital mixing induced by the enhanced SOC. As shown in Fig.3 (g), this mix- A. Self-energies in the quasiparticle/band basis ing is not limited to the vicinity of the avoided crossing but extends along the entire Fermi surface (see Fig.9 Working in the band basis, i.e., with the eigenstates in appendixB1 for the orbital character without SOC). of Hˆ 0, the maximum of the ARPES intensity for a For λDFT + ∆λ we find a minimal dxy and dxz,yz mixing given binding energy ω (maximum of the MDCs) corre- for the γ and β bands of 20/80% along the ΓM direction sponds to the momenta k which satisfy (following Eq.3): 0 with a monotonic increase to ≈ 50% along the Brillouin ω − εν (k) − Σνν (ω, k) = 0. Hence, for each binding en- zone diagonal ΓX. We note that this mixing varies with ergy, each azimuthal cut, and each sheet of the quasi- the perpendicular momentum kz. However, around the particle dispersions, we fit the MDCs and determine the 7

0 FIG. 4. Self-energies extracted in the band and orbital bases. (a) Real part of the self-energy Σνν in the band basis (solid ◦ 0 ˆ 0 symbols) in 9 steps of the Fermi surface angle θ. (b) Illustration of the relation between Σνν , the bare bands given by H (thin 0 lines) and the quasiparticle peak positions (solid symbols). (c) Compilation of Σνν from panel (a). (d) Real part of the 0 self-energy Σmm in the orbital basis.

ν momentum kmax(ω) at their maximum. Using the value zone diagonal (ΓX). This change evolves as a function ν 0 of εν (kmax(ω)) at this momentum yields the following of θ and occurs via a simultaneous increase in Σβ and 0 quantity a decrease in Σγ for all energies as θ is increased from ◦ ◦ ν 0 ν 0 0 (ΓM) to 45 (ΓX). In order to better visualize this ω − εν (kmax(ω)) = Σνν (ω, kmax(ω)) ≡ Σν (ω, θ) . (9) 0 angular dependence, a compilation of Σν (ω) for different This equation corresponds to the simple construction values of θ is displayed in Fig.4 (c). illustrated graphically in Fig.4 (b), and it is a standard way of extracting a self-energy from ARPES, as used in B. Accounting for the angular dependence: local previous works on several materials [32, 48, 73–75]. We self-energies in the orbital basis note that this procedure assumes that the off-diagonal 0 components Σ 0 (ω, k) can be neglected for states close ν6=ν In this section, we introduce a different procedure for to the Fermi surface (i.e., for small ω and k close to a extracting self-energies from ARPES, by working in the Fermi crossing). This assumption can be validated, as orbital basis |χ (k)i. We do this by making two key shown in appendixC. When performing this analysis, we m assumptions: only include the α sheet for θ = 45◦. Whenever the con- 0 straint Σν (ω → 0) → 0 on the self-energy is not precisely 1. We assume that the off-diagonal components are 0 obeyed, a small shift is applied to set it to zero. We negligible, i.e., Σm6=m0 ' 0. Let us note that in chose this procedure to correct for the minor differences Sr2RuO4 even a k-independent self-energy has non- between the experimental and the reference Hˆ 0 Fermi zero off-diagonal elements if Hˆ DFT+Hˆ SOC is con- λDFT wave-vectors because we attribute these differences pre- sidered. Using DMFT, these off-diagonal elements dominantly to experimental artifacts. have been shown to be very weakly dependent on The determined self-energies for each band ν = α, β, γ frequency in this material [23], leading to the no- and the different values of θ are depicted in Fig.4 (a,c). tion of a static correlation enhancement of the ef- For the β and γ sheets they show a substantial de- fective SOC (∆λ). In the present work, these off- pendence on the azimuthal angle. Around ΓM we find diagonal frequency-independent components are al- 0 0 ˆ 0 that Σγ exceeds Σβ by almost a factor of two (at ω = ready incorporated into H (see Sec.IV), and thus −50 meV), whereas they essentially coincide along the the frequency-dependent part of the self-energy is 8

(approximately) orbital diagonal by virtue of the content of quasiparticles in Fig.3 (g). In appendixD 0 ◦ tetragonal crystal structure. we show the back-transform of Σm (ω, θk = 18 ) into 0 ◦ 0 Σν (ω, θk = 0, 18, 45 ). The good agreement with Σνν 2. We assume that the diagonal components of the directly extracted from experiment further justifies the self-energy in the orbital basis depend on the mo- above expression and also confirms the validity of the mentum k only through the azimuthal angle θk: approximations made throughout this section. 0 0 Σmm(ω, k) ' Σm(ω, θk). We neglect the depen- Finally, we stress that expression (12) precisely co- dence on the momentum which is parallel to the incides with the ansatz made by DMFT: within this angular cut. theory, the self-energy is approximated as a local (k-independent) object when expressed in a basis of lo- Under these assumptions, the equation determining calized orbitals, while it acquires momentum dependence the quasiparticle dispersions reads when transformed to the band basis. h 0 ˆ 0 i det (ω − Σm(ω, θk))δmm0 − Hmm0 (k) = 0 . (10) VI. COMPARISON TO DYNAMICAL In this equation, we have neglected the lifetime effects MEAN-FIELD THEORY 00 associated with the imaginary part Σm. In order to extract the functions Σ0 (ω, θ ) directly from the ARPES m k In this section we perform an explicit comparison of data, we first determine the peak positions kν (ω, θ) max the measured quasiparticle dispersions and self-energies for MDCs at a given angle θ and binding energy ω. We to DMFT results. The latter are based on the Hamilto- then compute (for the same ω and θ) the matrix A 0 ≡ mm nian Hˆ DFT, to which the Hubbard-Kanamori interaction ˆ 0 α ωδmm0 − Hmm0 (kmax(ω, θk)) and similarly Bmm0 , Gmm0 β γ with on-site interaction U = 2.3 eV and Hund’s coupling for the β and γ band MDCs, kmax(ω, θ) and kmax(ω, θ), J = 0.4 eV [20] is added. For the details of the DMFT respectively. In terms of these matrices, the quasiparticle calculation and especially the treatment of SOC in this equations (10) read framework, we refer the reader to appendixB. There, 0 0 we also comment on some of the limitations and short- det[A 0 − Σ δ 0 ] = det[B 0 − Σ δ 0 ] = mm m mm mm m mm comings of the current state of the art for DFT+DMFT 0 = det[Gmm0 − Σmδmm0 ] = 0 . (11) calculations in this context. Fig.5 (a) shows the experimental quasiparticle dispersion extracted from our However, when taking symmetry into account, the self- 0 ARPES data (circles) on top of the DMFT spectral func- energy has only two independent components: Σxy and 0 0 tion A (ω, k) displayed as a color-intensity map. Clearly, Σxz = Σyz. Hence, we only need two of the above equa- the theoretical results are in near quantitative agreement tions to solve for the two unknown components of the self- with the data: both the strong renormalization of the energy. This means that we can also extract a self-energy Fermi velocity and the angular-dependent curvature of in the directions where only two bands (β and γ) are the quasiparticle bands are very well reproduced by the present in the considered energy range of ω . 100 meV, 0 purely local, and thus momentum-independent DMFT e.g., along ΓM. The resulting functions Σm(ω, θk) deter- self-energies. This validates the assumption of no mo- mined at several angles θ are displayed in Fig.4 (d). 0 mentum dependence along the radial k-space direction It is immediately apparent that, in contrast to Σνν , the of the self-energy made in Sec.VB for the k/ω range self-energies in the orbital basis do not show a strong an- studied here. The small deviations in Fermi wave vec- gular (momentum) dependence, but rather collapse into tors discernible in Fig.5 are consistent with Fig.3(c) two sets of points, one for the xy orbital and one for the and the overall experimental precision of the Fermi sur- xz/yz orbitals. Thus, we reach the remarkable conclu- face determination. sion that the angular dependence of the self-energy in In Fig.6 (a), we compare the experimental self-energies the orbital basis is negligible, within the range of bind- 0 0 for each orbital with the DMFT results. The overall ing energies investigated here: Σm(ω, θk) ' Σm(ω). This agreement is notable. At low-energy, the self-energies implies that a good approximation of the full momentum are linear in frequency and the agreement is excellent. and energy dependence of the self-energy in the band The slope of the self-energies in this regime controls the (quasiparticle) basis is given by angular-dependence of the effective mass renormalisa-

X ∗ tion. Using the local ansatz (12) into the quasiparticle Σνν0 (ω, k) = Umν (k)Σm(ω) Umν0 (k) . (12) dispersion equation, and performing an expansion around m EF , we obtain

The physical content of this expression is that the an- 0 vb X 1 2 1 ∂Σm gular (momentum) dependence of the quasiparticle self- = |Umν (θ)| , ≡ 1 − . vν (θ) Z Z ∂ω energies emphasized above is actually due to the matrix F m m m ω=0 elements Umν (k) deﬁned in Eq.4. In Sr 2RuO4 the an- (13) ν gular dependence of these matrix elements is mainly due In Fig.6 (b), we show vb/vF (θ) for the β and γ bands to the SOC, as seen from the variation of the orbital using the DMFT values Zxy = 0.18 ± 0.01 and Zxz/yz = 9

β γ β γ α 0

max

(meV) -40 F E-E -80 ΓM, θ = 0º θ = 9º θ = 18º θ = 27º θ = 36º ΓX, θ = 45º min 0.5 0.7 0.5 0.7 0.5 0.7 0.5 0.7 0.7 0.9 0.7 0.9 -1 k// (Å )

FIG. 5. Comparison of experimental quasiparticle dispersions (markers) with DMFT spectral functions (color plots) calculated for diﬀerent Fermi surface angles θ.

0.30 ± 0.01 obtained at 29 K (appendixB2). The over- ditional momentum-dependence of the self-energy. We all angular dependence and the absolute value of the γ further discuss possible contributions of spin fluctuations band mass enhancement is very well captured by DMFT, in Sec.VIII. while the β band is a bit overestimated. Close to the zone-diagonal (θ = 45◦), the two mass enhancements are approximately equal, due to the strong orbital mixing VII. KINKS induced by the SOC. Turning to larger binding energies, we see that the the- The self-energies Σ0 (ω) shown in Figs.4 and6 dis- 0 oretical Σxy is in good agreement with the experimental play a fairly smooth curvature, rather than pronounced data over the full energy range of 2-80 meV covered in our 0 ‘kinks’. Over a larger range, however, Σxy from DMFT experiments. Both the theoretical and experimental self- does show an energy scale marking the crossover from energies deviate significantly from the linear regime down the strongly renormalized low-energy regime to weakly to low energies (∼ 20 meV), causing curved quasiparticle renormalized excitations. This is illustrated in the inset bands with progressively steeper dispersion as the en- to Fig.6 (a). Such purely electronic kinks were reported ergy increases (Fig.5). In contrast, the agreement be- in DMFT calculations of a generic system with Mott- tween theory and experiment for the xz/yz self-energy is Hubbard sidebands [76] and have been abundantly docu- somewhat less impressive at binding energies larger than mented in the theoretical literature since then [13, 20, 77– ∼ 30 meV. Our DMFT self-energy Σ0 overestimates xz/yz 80]. In Sr2RuO4 they are associated with the crossover the strength of correlations in this regime (by 20 − 25%), from the Fermi-liquid behavior into a more incoherent with a theoretical slope larger than the experimental one. regime [18, 20]. The near quantitative agreement of the Correspondingly, the quasiparticle dispersion is slightly frequency dependence of the experimental self-energies 0 steeper in this regime than the theoretical result, as can Σm (ω) and our single-site DMFT calculation provides be also seen in Fig.5. strong evidence for the existence of such electronic kinks There may be several reasons for this discrepancy. in Sr2RuO4. In addition, it implies that the local DMFT Even while staying in the framework of a local self-energy, treatment of electronic correlations is capturing the dom- we note that the present DMFT calculation is performed inant effects. with an on-site value of U which is the same for all or- Focusing on the low-energy regime of our experi- bitals. Earlier cRPA calculations have suggested that mental data, we find deviations from the linear form this on-site interaction is slightly larger for the xy orbital Σ0 (ω) = ω(1 − 1/Z) characteristic of a Fermi liquid for (Uxy = 2.5 eV and Uxz/yz = 2.2 eV) [20] and recent work |ω| > 20 meV, irrespective of the basis. However, this is has advocated the relevance of this for DFT+DMFT cal- only an upper limit for the Fermi-liquid energy scale in culations of Sr2RuO4 [21]. Another possible explanation Sr2RuO4. Despite the improved resolution of our exper- is that this discrepancy is actually a hint of some mo- iments, we cannot exclude an even lower crossover en- mentum dependent contribution to the self-energy, es- ergy to non-Fermi-liquid like excitations. We note that pecially dependence on momentum perpendicular to the the crossover temperature of TFL ≈ 25 K reported from Fermi surface. We note in this respect that the discrep- transport and thermodynamic experiments [10–12] in- ancy is larger for the α, β sheets which have dominant deed suggests a cross-over energy scale that is signifi- xz/yz character. These orbitals have, in the absence of cantly below 20 meV. SOC, a strong one-dimensional character, for which mo- The overall behavior of Σ0 (ω) including the energy mentum dependence is definitely expected and DMFT range where we find strong changes in the slope agrees is less appropriate. Furthermore, these FS sheets are with previous photoemission experiments, which were in- also the ones associated with nesting and spin-density terpreted as evidence for electron-phonon coupling [29– wave correlations, which are expected to lead to an ad- 31, 46, 48]. Such an interpretation, however, relies on 10

(a) VIII. DISCUSSION AND PERSPECTIVES

dxy 200 In this article, we have reported on high-resolution ARPES measurements which allow for a determination of the Fermi surface and quasiparticle dispersions of Sr2RuO4 with unprecedented accuracy. Our data reveal dxz,yz an enhancement (by a factor of about two) of the splitting 100 between Fermi surface sheets along the zone diagonal, in (meV)

m 600 '

Σ comparison to the DFT value. This can be interpreted as a correlation-induced enhancement of the effective SOC, 200 an effect predicted theoretically [21, 23, 51] and demon- 0 strated experimentally here for this material, for the first time. -200 Thanks to the improved cleanliness of our data, we -400 0 400 have been able to determine the electronic self-energies -90 -60 -30 0 directly from experiment, using both a standard proce- ω (meV) dure applied in the band (quasiparticle) basis as well as a novel procedure, introduced in the present article, in (b) ΓX 6 γ band the orbital basis. Combining these two approaches, we have demonstrated that the large angular (momentum) dependence of the quasiparticle self-energies and disper- 5 sions can be mostly attributed to the fact that quasipar- F ticle states have an orbital content which is strongly an- / v b

v gular dependent, due to the SOC. Hence, assuming self- 4 energies which are frequency-dependent but essentially independent of angle (momentum), when considered in the orbital basis, is a very good approximation. This 3 β band provides a direct experimental validation of the DMFT ansatz. 0 45 90 FS angle θ (deg) The key importance of atomic-like orbitals in correlated insulators is well established [82, 83]. The present 0 0 work demonstrates that orbitals retain a considerable FIG. 6. (a) Average of the self-energies Σxz/yz,Σxy shown in Fig.4 (d) compared with DMFT self-energies calculated at physical relevance even in a metal in the low-temperature 0 Fermi-liquid regime. Although the band and orbital 29 K. The self-energies are shifted such that Σm(ω = 0) = 0. The inset shows the DMFT self-energies over a larger en- basis used here are equivalent, our analysis shows that 0 ergy range. Linear fits at low and high energy of Σxy from the underlying simplicity in the nature of correlations DMFT are shown as solid and dashed black lines, respectively. emerges only when working in the latter and taking (b) Angle dependence of the mass enhancement vb/vF from into account the orbital origin of quasiparticles. Beyond ARPES (markers) and DMFT (solid line). The range indi- Sr2RuO4, this is an observation of general relevance to cated by the shaded areas corresponds to mass enhancements multiband metals with strong correlations such as the calculated from the numerical data by using different meth- iron-based superconductors [84, 85]. ods (see appendixB). Error bars on the experimental data are obtained from propagating the estimated uncertainty of Notwithstanding its success, the excellent agreement of the Fermi velocities shown in Fig.2 (c). the DMFT results with ARPES data does raise puzzling questions. Sr2RuO4 is known to be host to strong magnetic fluctuations [39–42], with a strong peak in its spin response χ(Q) close to the spin-density wave (SDW) vec- a linear Fermi-liquid regime of electronic correlations tor Q ∼ (2π/3, 2π/3), as well as quasi-ferromagnetic fluc- over the entire phonon bandwidth of ≈ 90 meV [81], tuations which are broader in momentum around Q = 0. which is inconsistent with our DMFT calculations. More- Indeed, tiny amounts of substitutional impurities induce 0 over, attributing the entire curvature of Σm in our data long-range magnetic order in this material, of either SDW to electron-phonon coupling would result in unrealistic of ferromagnetic type [86, 87]. Hence, it is a prominent coupling constants far into the polaronic regime, which open question to understand how these long-wavelength is hard to reconcile with the transport properties of fluctuations affect the physics of quasiparticles in the Sr2RuO4 [10–12]. We also note that a recent scanning Fermi-liquid state. Single-site DMFT does not capture tunneling microscope (STM) study reported very strong this feedback, and the excellent agreement with the over- kinks in the β and γ sheets of Sr2RuO4 [49], which is in- all quasiparticle physics must imply that these effects consistent with our data. We discuss the reason for this have a comparatively smaller magnitude than the domi- discrepancy in appendixA. nant local effect of correlations (on-site U and especially 11

Hund’s J) captured by DMFT. A closely related ques- (a) tion is how much momentum dependence is present in β γ the low-energy (Landau) interactions between quasipar- 0 ticles. These eﬀects are expected to be fundamental for (meV) subsequent instabilities of the Fermi liquid, into either F -40 the superconducting state in pristine samples or magnetic surface ordering in samples with impurities. Making progress on E - -80 bulk this issue is also key to the understanding of the super- ΓM QPI conducting state of Sr2RuO4, for which the precise na- 0.4 0.6 0.8 ture of the pairing mechanism as well as symmetry of the (b) order parameter are still outstanding open questions [4]. β γ α β γ 0 (meV)

F -40 ACKNOWLEDGMENTS

E - -80 The experimental work has been supported by the Eu- ΓX QPI ΓM QPI ropean Research Council (ERC), the Scottish Funding 0.6 0.8 0.4 0.6 0.8 -1 -1 Council, the UK EPSRC and the Swiss National Science k (Å ) k (Å ) Foundation (SNSF). Theoretical work was supported by // // the ERC grant ERC-319286-QMAC and by the SNSF (NCCR MARVEL). The Flatiron Institute is a division FIG. 8. Comparison with the STM data from Ref. [49]. (a) of the Simons Foundation. AG and MZ gratefully ac- ΓM high-symmetry cut on a pristine cleave. The bulk β and γ and the surface β bands are labeled. (b) Laser-ARPES data knowledge useful discussions with Gabriel Kotliar, An- from CO passivated surface showing the bulk band dispersion drew J. Millis and Jernej Mravlje. along ΓX and ΓM. The dispersion obtained from quasiparticle interference in Ref. [49] is overlaid with red markers.

Appendix A: Bulk and surface electronic structure of Sr2RuO4 In Fig.7 we compare the data presented in the main text with data from a pristine cleave taken with hν = 21 eV at the SIS beamline of the Swiss Light Source. This comparison confirms the identification of bulk and surface bands by Shen et al. [54]. In particular, we find that the larger β sheet has bulk character. This band assignment is used by the vast majority of subsequent ARPES publications [29–31, 33, 46, 56, 88, 89], except for Ref. [47], which reports a dispersion with much lower Fermi velocity and a strong kink at 15 meV for the smaller β sheet that we identify as a surface band. Wang et al. [49] have recently probed the low-energy electronic structure of Sr2RuO4 by STM. Analyzing quasiparticle interference patterns along the ΓX and ΓM high-symmetry directions, they obtained band dispersions with low Fermi velocities and strong kinks at 10 meV and 37 meV. In Fig.8 we compare the band dispersion reported by Wang et al. with our ARPES data. Along both high-symmetry directions, we find a clear discrepancy with our data, which are in quantitative agreement with bulk de Haas van Alphen measurements, as demonstrated in the main text. On the other hand, we FIG. 7. Bulk and surface electronic structure of Sr2RuO4. find a striking similarity between the STM data along Left half: Fermi surface map of a CO passivated surface shown ΓM and the band commonly identified as the surface β in the main text. Right half: Fermi surface acquired on a band [30, 54]. We thus conclude that the experiments re- pristine cleave at 21 eV photon energy. Intense surface states ported in Ref. [49] probed the surface states of Sr2RuO4. are evident in addition to the bulk bands observed in the left This is fully consistent with the strong low-energy kinks panel. and overall enhanced low-energy renormalization seen by ARPES in the surface bands [50, 90]. 12

-1 (eV) k ε

-2 DFT (GGA-PBE) DFT (Wannier90) -3 1

-1 FIG. 10. Orbital character of the DFT FS without SOC at (eV) k

ε kz = 0.4 π/c (left). The orbital character of the DFT+λDFT + ∆λ eigenstates at the same kz is reproduced on the right from -2 Fig.3. DFT + SOC (GGA-PBE) DFT + λDFT (Wannier90) -3 Γ M X Γ Z M X

FIG. 9. DFT band structure along the high-symmetry path ΓMXΓZ compared to the eigenstates of our maximally- DFT localized Wannier Hamiltonian Hˆ for the three t2g bands. λ = 0 meV λ = 100 meV Top: DFT (GGA-PBE) and eigenstates of Hˆ DFT. Bottom: ˆ DFT ˆ SOC DFT+SOC (GGA-PBE) and eigenstates of H +HλDFT ΔεCF = - 80 meV with a local SOC term (Eq.8) and a coupling strength of ΔεCF = - 40 meV λDFT = 100 meV. ΔεCF = 0 meV ΔεCF = 40 meV ΔεCF = 80 meV ARPES Appendix B: Computational details λ = 300 meV λ = 200 meV 1. DFT and model Hamiltonian

These orbitals are centered on the Ru atoms and have t2g symmetry, but are indeed linear combinations of Ru-d and O-p states. We do not add Wannier functions centered on the oxygen atoms, because the resulting three FIG. 11. Fermi surface of Sr2RuO4 for λ = 0 meV (top left), orbital Wannier model already accurately reproduces the λ = 100 meV (top right), λ = 200 meV (bottom right) and three bands crossing the Fermi energy, as demonstrated λ = 300 meV (bottom left) compared to ARPES (dashed in Fig. 9. Also note that the Wannier function construc- black line). λ = 100 meV corresponds to a DFT+SOC cal- tion allows to disentangle the γ band from the bands with culation and λ = 200 meV to an effective SOC enhanced by dominantly O-p character below −2 eV. electronic correlations (see main text). The different shades We generate our theoretical model Hamiltonian Hˆ DFT of red indicate additional crystal-field splittings ∆cf added ˆ DFT with a maximally-localized Wannier function [60, 61] con- to cf = 85 meV of H . struction of t2g-like orbitals for the three bands crossing the Fermi surface. These Wannier orbitals are obtained on a 10 × 10 × 10 k grid based on a non-SOC the eigenstates retain pure orbital character, as shown DFT calculation using WIEN2k [91] with the GGA-PBE in Fig. 10. To take SOC into account, we add the local ˆ SOC functional [92], wien2wannier [93] and Wannier90 [94]. single-particle term Hλ , as given in Eq.8, with cou- The DFT calculation is performed with lattice parame- pling constant λ. In the bottom panel of Fig.9 we show ˆ DFT ˆ SOC ters from Ref. [95] (measured at 100 K) and converged that the eigenenergies of H +Hλ are in nearly per- with twice as many k-points in each dimension. fect agreement with the DFT+SOC band structure at a The eigenenergies of the resulting Wannier Hamilto- value of λDFT = 100 meV. nian, Hˆ DFT, accurately reproduce the DFT band struc- Our model Hamiltonian provides the reference point ture (Fig.9 top). Note that in the absence of SOC, to which we define a self-energy, but it is also a per- 13

2. DMFT 200 analytic continuation We perform single-site DMFT calculation with the TRIQS/DFTTools [99] package for Hˆ DFT and Hubbard- Kanamori interactions with a screened Coulomb repul- sion U = 2.3 eV and a Hund’s coupling J = 0.4 eV based on previous works [20, 23]. The impurity prob- (meV)

m 100 ' lem is solved on the imaginary-time axis with the Σ Pade xz TRIQS/CTHYB [100] solver at a temperature of 29 K. Pade xy The employed open-source software tools are based Beach xz Beach xy on the TRIQS library [96]. We assume an orbital- TRIQS/maxent xz independent double counting, and hence it can be ab- TRIQS/maxent xy sorbed into an effective chemical potential, which is ad- 0 justed such that the filling is equal to four electrons. -100 -50 0 For the analytic continuation of the self-energy to the ω (meV) real-frequency axis we employ three different methods: Padéapproximants (using TRIQS [96]), Stochastic con- FIG. 12. Real part of DFT+DMFT self-energy in the consid- tinuation (after Beach [97]) and Maximum Entropy (us- ered energy range obtained with three different analytic con- ing TRIQS/maxent [98]). In the relevant energy range tinuation methods: Padéapproximants (using TRIQS [96]), from −100 to 0 meV the difference in the resulting self- Stochastic continuation (after Beach [97]) and Maximum En- tropy (using TRIQS/maxent [98]). The difference between energies (Fig. 12) is below the experimental uncertainty . the analytic continuation methods is smaller than the exper- The averaged quasiparticle renormalizations (of the three imental error. continuations) are: Zxy = 0.18 ± 0.01 and Zxz = Zyz = 0.30 ± 0.01. For all other results presented in the main text the Padésolution has been used. Our calculations at a temperature of 29 K use Hˆ DFT, as the sign problem prohibits reaching such low temper- fect playground to study the change in the Fermi surface atures with SOC included. Nevertheless, calculations under the influence of SOC and the crystal-field split- with SOC were successfully carried out at a tempera- ting between the xy and xz/yz orbitals. In the fol- ture of 290 K using CT-INT [21] and at 230 K using lowing, we will confirm that the best agreement with CT-HYB with a simplified two-dimensional tight-binding the experimental Fermi surface is found with an effec- model [23]. These works pointed out that electronic cor- eff tive SOC of λ = λDFT + ∆λ = 200 meV, but at relations in Sr2RuO4 lead to an enhanced SOC. To be the same time keeping the DFT crystal-field splitting of more precise, Kim et al. [23] observed that electronic cor- cf = xz/yz − xy = 85 meV unchanged. We compare in relations in this material are described by a self-energy Fig. 11 the experimental Fermi surface (dashed lines) to with diagonal elements close to the ones without SOC the one of Hˆ DFT+Hˆ SOC . The Fermi surfaces for addi- plus, to a good approximation, frequency-independent λDFT tionally introduced crystal-field splittings ∆cf between off-diagonal elements, which can be absorbed in a static eff −80 and 80 meV are shown with solid lines in different effective SOC strength of λ = λDFT + ∆λ ' 200 meV shades of red. In contrast to the Fermi surface without – this is the approach followed in the present article. SOC (λ = 0 meV, top left panel), the Fermi surfaces with In addition to the enhancement of SOC it was ob- the DFT SOC of λ = 100 meV (top right panel) resembles served that low-energy many-body effect also lead to an the overall structure of the experimental Fermi surface. enhancement of the crystal-field splitting [21, 23]. In However, the areas of the α and β sheets are too large our DFT+DMFT calculation without SOC this results and the γ sheet is too small. Importantly, the agreement in a orbital-dependent splitting in the real part of the cannot be improved by adding ∆cf . For example, along self-energies (∆cf = 60 meV), which would move the γ ΓM a ∆cf of −40 meV would move the Fermi surface sheet closer to the van Hove singularity and consequently closer to the experiment, but, on the other hand, along worsen the agreement with the experimental Fermi sur- ΓX a ∆cf of 80 meV would provide the best agreement. face along the ΓM direction (see bottom right panel of The situation is different if we consider an enhanced SOC Fig. 11). Different roots of this small discrepancy are of λ = 200 meV (bottom right panel). Then, we find a possible, ranging from orbital-dependent double count- nearly perfect agreement with experiment without any ing corrections to, in general, DFT being not perfect additional crystal-field splitting (∆cf = 0 meV). At an as ‘non-interacting’ reference point for DMFT. Zhang et even higher SOC of λ = 300 meV (bottom left panel) al. [21] showed that by considering the anisotropy of the we see again major discrepancies, but with an opposite Coulomb tensor the additional crystal-field splitting is trend: The α and β sheets are now too small and the γ suppressed and consequently the disagreement between sheet is too large. Like in the case of λ = 100 meV this theory and experiment can be cured. We point out that can not be cured by an adjustment of cf . in comparision to the present work a large enhancement 14

(a) ΓM, β sheet ΓX, γ sheet (b) ΓM ΓX 120 Σ'γγ 120 Σ'ββ γ band Σ'αα |Σ'γβ| (meV) (meV) |Σ'αγ| νν νν ' ' 60 |Σ'αβ| 60 Σ Σ

~ β band α band

θ = 0° θ = 18° θ = 45° 0 0 -40 -20 0 -40 -20 0 -40 -20 0 -40 -20 0 -40 -20 0 ω (meV) ω (meV) ω (meV) ω (meV) ω (meV)

0 FIG. 13. Reconstructed self-energies in the quasiparticle basis. (a) Full matrix Σνν0 (ω, k) calculated with Eq. 12 using the DMFT self-energy (shown in Fig.6) at two selected k points: on the β sheet for θ = 0◦ (left) and on the γ sheet for ◦ 0 ν 0 ◦ θ = 45 (right). (b) Directly extracted Σνν (ω, kmax(ω)) (Sec.VA) compared to the Σ m(ω, θ = 18 ) (Sec.VB) transformed to 0 ν ◦ Σνν (ω, kmax(ω)) at 0, 18, and 45 .

qp of the crystal-field splitting was observed in Ref. [21], which determines the quasiparticle dispersion ων (k), as presumably due to the larger interactions employed. 0 0 0 Σνν0 (ω, k)Σν0ν (ω, k) ω − εν (k) − Σνν (ω, k) − 0 = 0 . Based on these considerations we calculate the corre- ω − εν0 (k) − Σν0ν0 (ω, k) lated spectral function A (ω, k) (shown in Fig.5 (a) of the (C1) main text) using the Hamiltonian with enhanced SOC Setting the last term to zero, i.e., using the procedure ˆ DFT ˆ SOC 0 (H +Hλ +∆λ) in combination with the frequency- described in Sec.VA to extract Σ νν , is justified at DFT qp dependent part of the non-SOC (diagonal) self-energy, ω = ων (k) as long as but neglect the additional static part introduced by 0 0 0 Σνν0 (ω, k)Σν0ν (ω, k) DMFT. Σνν (ω, k) 0 (C2) ω − εν0 (k) − Σν0ν0 (ω, k) In this condition the already small off-diagonal elements enter quadratically, but also the denominator is not a small quantity, because the energy separation of the bare

0 0 bands (εν (k) − εν (k)) is larger than the difference of Appendix C: Off-diagonal elements of Σ 0 νν the diagonal self-energies. By using the generalized version of Eq. C2 for all three In Sec.VA we extracted Σ 0 under the assumption bands, we find that the right-hand side of this equation νν 0 ν that the off-diagonal elements can be neglected. To ob- is indeed less than 1.2% of Σνν (ω, kmax(ω)) for all ex- ν tain insights about the size of the off-diagonal elements perimentally determined kmax(ω). This means that for 0 Sr2RuO4 treating each band separately when extracting we use Eq. 12 to calculate the full matrix Σ 0 (ω, k) in νν 0 the band basis from the DMFT self-energy in the orbital Σνν is well justified in the investigated energy range. 0 basis Σm(ω), as shown in Fig.6 (a). This allows us to 0 obtain the full self-energy matrix Σ 0 (ω, k) at one spe- νν Appendix D: Reconstruction of Σ0 cific combination of k and ω. Note that for the results νν presented in Fig.4 and Fig. 13 (b) this is not the case, because the extracted self-energies for each band corre- In order to further test the validity of the local ansatz ν (Eq. 12) and establish the overall consistency of the two spond to different kmax, which are further defined by the experimental MDCs. procedures used to extract the self-energy in Sec.V, we perform the following ‘reconstruction procedure’. We use In Fig. 13 (a) we show the result for two selected k 0 ◦ the Σm(ω, θk) (from Sec.VB) at one angle, e.g., θ = 18 , points: on the β sheet for θ = 0 and on the γ sheet and transform it into Σ0 (ω, kν (ω)) for other mea- ◦ νν max for θ = 45 . For these k points the largest off-diagonal sured angles, using Eq. 12. The good agreement between 0 element is Σγβ, which is about 10−20% of the size of the the self-energy reconstructed in this manner (thin lines diagonal elements. A scan performed for the whole kz = in Fig. 13 (b)) and its direct determination following 0 0.4 π/c plane further confirms that |Σν6=ν0 | is smaller than the procedure of Sec.VA (dots) confirms the validity 20 meV. of the approximations used throughout Sec.V. It also However, when neglect the off-diagonal elements it is shows that the origin of the strong momentum depen- 0 also important to have a large enough energy separation dence of Σνν is almost entirely due to the momentum of the bands. This can be understood by considering a dependence of the orbital content of quasiparticle states, simplified case of two bands (ν, ν0) and rewriting Eq.3, i.e., of Umν (k) = hχm (k) |ψν (k)i. In Sr2RuO4 the momentum dependence of these matrix elements is mainly due to the SOC. 15

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