High-resolution photoemission on Sr2RuO4 reveals correlation-enhanced effective spin-orbit coupling and dominantly local self-energies
A. Tamai, M. Zingl, E. Rozbicki, E. Cappelli, S. Ricco, A. de la Torre, S. McKeown Walker, F. Y. Bruno, P.D.C. King, W. Meevasana, M. Shi, M. Radovic, N.C. Plumb, A.S. Gibbs, A.P. Mackenzie, C. Berthod, H. Strand, M. Kim, A. Georges, F. Baumberger
HHigh-resolution photoemission on Sr RuO reveals correlation-enhanced effectivespin-orbit coupling and dominantly local self-energies A. Tamai, M. Zingl, E. Rozbicki, E. Cappelli, S. Ricc`o, A. de la Torre, S. McKeown-Walker, F. Y. Bruno, P.D.C. King, W. Meevasana, M. Shi, M. Radovi´c, N.C. Plumb, A.S. Gibbs, ∗ A.P. Mackenzie,
6, 3
C. Berthod, H. Strand, M. Kim,
7, 8
A. Georges,
9, 2, 8, 1 and F. Baumberger
1, 5 Department of Quantum Matter Physics, University of Geneva,24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom School 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 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany Department of Physics and Astronomy, Rutgers,The State University of New Jersey, Piscataway, NJ 08854, USA Centre de Physique Th´eorique Ecole Polytechnique,CNRS, Universite Paris-Saclay, 91128 Palaiseau, France Coll`ege de 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 modelFermi liquid Sr RuO using laser-based angle-resolved photoemission spectroscopy. Our precisemeasurement of the Fermi surface confirms the importance of spin-orbit coupling in this materialand 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. Bytaking into account the multi-orbital composition of quasiparticle states, we determine self-energiesassociated with each orbital component directly from the experimental data. This analysis demon-strates that the perceived angular dependence does not imply momentum-dependent many-bodyeffects, but arises from a substantial orbital mixing induced by spin-orbit coupling. A comparisonto single-site dynamical mean-field theory further supports the notion of dominantly local orbitalself-energies, and provides strong evidence for an electronic origin of the observed non-linear fre-quency dependence of the self-energies, leading to ‘kinks’ in the quasiparticle dispersion of Sr RuO . I. INTRODUCTION
The layered perovskite Sr RuO is an important modelsystem for correlated electron physics. Its intriguing su-perconducting ground state, sharing similarities with su-perfluid He [1–3], has attracted much interest and con-tinues to stimulate advances in unconventional supercon-ductivity [4]. Experimental evidence suggest odd-parityspin-triplet pairing, yet questions regarding the proxim-ity of other order parameters, the nature of the pairingmechanism and the apparent absence of the predictededge currents remain open [3–9]. Meanwhile, the normalstate of Sr RuO attracts interest as one of the clean-est oxide Fermi liquids [10–13]. Its precise experimen-tal characterization is equally important for understand-ing the unconventional superconducting ground state ofSr RuO [1–9, 14–17], as it is for benchmarking quanti-tative many-body calculations [18–23].Transport, thermodynamic and optical data ofSr RuO display textbook Fermi-liquid behavior be-low a crossover temperature of T FL ≈
25 K [10–13]. ∗ present address: ISIS Facility, Rutherford Appleton Laboratory,Chilton, Didcot OX11 OQX, United Kingdom Quantum oscillation and angle-resolved photoemissionspectroscopy (ARPES) measurements [24–34] further re-ported a strong enhancement of the quasiparticle effec-tive mass over the bare band mass. Theoretical progresshas been made recently in revealing the important roleof the intra-atomic Hund’s coupling as a key sourceof correlation effects in Sr RuO [18, 20, 35]. In thiscontext, much attention was devoted to the intrigu-ing properties of the unusual state above T FL , whichdisplays metallic transport with no signs of resistivitysaturation at the Mott-Ioffe-Regel limit [36]. Dynam-ical mean-field theory (DMFT) [37] calculations haveproven successful in explaining several properties of thisintriguing metallic state, as well as in elucidating thecrossover from this unusual metallic state into the Fermi-liquid regime [13, 18, 20–23, 38]. Within DMFT, theself-energies associated with each orbital component areassumed to be local. On the other hand, the low-temperature Fermi-liquid state is known to display strongmagnetic fluctuations at specific wave-vectors, as re-vealed, e.g., by neutron scattering [39, 40] and nuclearmagnetic resonance spectroscopy (NMR) [41, 42]. Thesemagnetic fluctuations were proposed early on to be animportant source of correlations [2, 43, 44]. In this pic-ture, it is natural to expect strong momentum depen- a r X i v : . [ c ond - m a t . s t r- e l ] M a y dence of the self-energy associated with these spin fluctu-ations. Interestingly, a similar debate was raised long agoin the context of liquid He, with ‘paramagnon’ theoriesemphasizing ferromagnetic spin-fluctuations and ‘quasi-localized’ approaches `a la Anderson-Brinkman emphasiz-ing local correlations associated with the strong repulsivehard-core, leading to increasing Mott-like localization asthe liquid is brought closer to solidification (for a review,see Ref. [45]).In this work, we report on new insights into the na-ture of the Fermi-liquid state of Sr RuO . Analyzing acomprehensive set of laser-based ARPES data with im-proved resolution and cleanliness, we reveal a strong an-gular (i.e., momentum) dependence of the self-energiesassociated with the quasiparticle bands. We demonstratethat this angular dependence originates in the variationof the orbital content of quasiparticle states as a func-tion of angle, and can be understood quantitatively. In-troducing a new framework for the analysis of ARPESdata for multi-orbital systems, we extract the electronicself-energies associated with the three Ruthenium t g or-bitals with minimal theoretical input. We find that theseorbital self-energies have strong frequency dependence,but surprisingly weak angular (i.e., momentum) depen-dence, and can thus be considered local to a very goodapproximation. Our results provide a direct experimentaldemonstration that the dominant effects of correlationsin Sr RuO are weakly momentum-dependent and canbe understood from a local perspective, provided theyare considered in relation to orbital degrees of freedom.One of the novel aspect of our work is to directly putthe locality ansatz underlying DMFT to the experimen-tal test. We also perform a direct comparison betweenDMFT calculations and our ARPES data, and find goodagreement with the measured quasiparticle dispersionsand angular dependence of the effective masses.The experimentally determined real part of the self-energy displays strong deviations from the low-energyFermi-liquid behavior Σ (cid:48) ( ω ) ∼ ω (1 − /Z ) + · · · forbinding energies | ω | larger than ∼
20 meV. These de-viations are reproduced by our DMFT calculations sug-gesting that the cause of these non-linearities are localelectronic correlations. Our results thus call for a revi-sion of earlier reports of strong electron-lattice couplingin Sr RuO [29–31, 46–50]. We finally quantify the ef-fective spin-orbit coupling (SOC) strength and confirmits enhancement due to correlations predicted theoreti-cally [21, 23, 51].This article is organized as follows. In Sec. II, webriefly present the experimental method and report ourmain ARPES results for the Fermi surface and quasipar-ticle dispersions. In Sec. III, we introduce the theoret-ical framework on which our data analysis is based. InSec. IV, we use our precise determination of the Fermisurface to reveal the correlation-induced enhancement ofthe effective SOC. In Sec. V we proceed with a directdetermination of the self-energies from the ARPES data.Sec. VI presents the DMFT calculations in comparison to experiments. Finally, our results are critically discussedand put in perspective in Secs. VII, VIII. II. EXPERIMENTAL RESULTSA. Experimental methods
The single crystals of Sr RuO used in our experi-ments were grown by the floating zone technique andshowed a superconducting transition temperature of T c = 1 .
45 K. ARPES measurements were performed withan MBS electron spectrometer and a narrow bandwidth11 eV(113 nm) laser source from Lumeras that was op-erated at a repetition rate of 50 MHz with 30 ps pulselength of the 1024 nm pump [52]. All experiments wereperformed at T ≈ ≈ . ◦ . In order to suppress the intensity of the sur-face layer states on pristine Sr RuO [54], we exposed thecleaved surfaces to ≈ . ≈
120 K. Under these conditions, CO preferentiallyfills surface defects and subsequently replaces apical oxy-gen ions to form a Ru – COO carboxylate in which the Cend of a bent CO binds to Ru ions of the reconstructedsurface layer [55]. B. Experimental Fermi surface and quasiparticledispersions
In Fig. 1 we show the Fermi surface and selected con-stant energy surfaces in the occupied states of Sr RuO .The rapid broadening of the excitations away from theFermi level seen in the latter is typical for ruthenatesand implies strong correlation effects on the quasipar-ticle properties. At the Fermi surface, one can read-ily identify the α , β and γ sheets that were reportedearlier [25, 27, 28]. However, compared with previousARPES studies we achieve a reduced line width and im-proved suppression of the surface layer states giving cleanaccess to the bulk electronic structure. This is partic-ularly evident along the Brillouin zone diagonal (ΓX)where we can clearly resolve all band splittings.In the following, we will exploit this advance to quan-tify the effects of SOC in Sr RuO and to provide newinsight into the renormalization of the quasiparticle ex-citations using minimal theoretical input only. To thisend we acquired a set of 18 high resolution dispersionplots along radial k -space lines (as parameterized by theangle θ measured from the ΓM direction). The subsetof data shown in Fig. 2 (a) immediately reveals a richbehavior with a marked dependence of the low-energydispersion on the Fermi surface angle θ . Along the ΓM Γ MX α β γ E F -10 meV-20 meV-30 meV-40 meV (a) (b) Γ M (cid:1) minmax FIG. 1. (a) Fermi surface of Sr RuO . The data were acquired at 5 K on a CO passivated surface with a photon energy of 11 eVand p -polarization for measurements along the ΓX symmetry line. The sample tilt around the ΓX axis used to measure the fullFermi surface results in a mixed polarization for data away from this symmetry axis. The Brillouin zone of the reconstructedsurface layer is indicated by diagonal dashed lines. Surface states and final state umklapp processes are suppressed to near thedetection limit. A comparison with ARPES data from a pristine cleave is shown in appendix A. The data in (a) have beenmirror-symmetrized for clarity. Original measured data span slightly more than a quadrant of the Brillouin zone. (b) Constantenergy surfaces illustrating the progressive broadening of the quasiparticle states away from the Fermi level E F . high-symmetry line our data reproduce the large differ-ence in Fermi velocity v β,γF for the β and γ sheet, whichis expected from the different cyclotron masses deducedfrom quantum oscillations [12, 25, 26] and was reportedin earlier ARPES studies [33, 56]. Our systematic data,however, reveal that this difference gradually disappearstowards the Brillouin zone diagonal ( θ = 45 ◦ ), where allthree bands disperse nearly parallel to one another. InSec. IV we will show that this equilibration of the Fermivelocity can be attributed to the strong effects of SOCaround the zone diagonal.To quantify the angle dependence of v β,γF from exper-iment, we determine the maxima k ν max ( ω ) of the mo-mentum distribution curves (MDCs) over the range of2-6 meV below the Fermi level E F and fit these k -spaceloci with a second-order polynomial. We then define theFermi velocity as the derivative of this polynomial at E F .This procedure minimizes artifacts due to the finite en-ergy resolution of the experiment. As shown in Fig. 2 (c),the Fermi velocities v β,γF obtained in this way show anopposite trend with azimuthal angle for the two Fermisheets. For the β band we observe a gentle decrease of v F as we approach the ΓX direction, whereas for γ the veloc-ity increases by more than a factor of two over the samerange [57]. This provides a first indication for a strongmomentum dependence of the self-energies Σ (cid:48) β,γ , whichwe will analyze quantitatively in Sec.V. Here, we limitthe discussion to the angle dependence of the mass en-hancement v b /v F , which we calculate from the measured quasiparticle Fermi velocities of Fig. 2 (c) and the cor-responding bare velocities v b of a reference Hamiltonianˆ H defined in Sec. IV. As shown in Fig. 2 (d), this con-firms a substantial many-body effect on the anisotropy ofthe quasiparticle dispersion. Along ΓM, we find a strongdifferentiation with mass enhancements of ≈ γ sheet and ≈ . β , whereas v b /v F approaches a com-mon value of ≈ . k F and velocities deter-mined from our data on a dense grid along the entireFermi surface, we can compute the cyclotron masses mea-sured by dHvA experiments, without relying on the ap-proximation of circular Fermi surfaces and/or isotropicFermi velocities used in earlier studies [33, 49, 56, 58].Expressing the cyclotron mass m ∗ as m ∗ = (cid:126) π ∂A F S ∂(cid:15) = (cid:126) π (cid:90) π k F ( θ ) ∂(cid:15)/∂k ( θ ) dθ , (1)where A F S is the Fermi surface volume, and using thedata shown in Fig. 2 (c), we obtain m ∗ γ = 17 . . m e and m ∗ β = 6 . . m e , in quantitative agreement withthe values of m ∗ γ = 16 m e and m ∗ β = 7 m e found indHvA experiments [12, 25, 26]. We thus conclude thatthe quasiparticle states probed by our experiments arerepresentative of the bulk of Sr RuO [59]. v b / v F θ (deg) βγ v F ( e V Å ) β γ θΓ MX αβ γ θ = 45º αγβ θ = 42º0.80.6k // ( Å -1 ) θ = 36º-80-400 E - E F ( m e V ) θ = 30º θ = 24º θ = 18º θ = 12º θ = 6º-80-400 E - E F ( m e V ) Γ M, θ = 0º γβ (a) (b) (c) (d) minmax 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 v F along the β and γ Fermi surface sheets. (d) Angular dependenceof the quasiparticle mass enhancement v b /v F . Here, v b is the bare velocity obtained from the single-particle Hamiltonianˆ H = ˆ H DFT + ˆ H SOC λ DFT +∆ λ defined in Sec. IV and v F is the quasiparticle Fermi velocity measured by ARPES. Error bars areobtained by propagating the experimental uncertainty on v F . Thin lines are guides to the eye. III. THEORETICAL FRAMEWORK
In order to define the electronic self-energy and as-sess the effect of electronic correlations in the spectralfunction measured by ARPES, we need to specify a one-particle Hamiltonian ˆ H as a reference point. At thisstage, we keep the presentation general. The particularchoice of ˆ H will be a focus of Sec. IV. The eigenstates | ψ ν ( k ) (cid:105) of ˆ H ( k ) at a given quasi-momentum k and thecorresponding eigenvalues ε ν ( k ) define the ‘bare’ bandstructure of the system, with respect to which the self-energy Σ νν (cid:48) ( ω, k ) is defined in the standard way from theinteracting Green’s function G − νν (cid:48) ( ω, k ) = [ ω + µ − ε ν ( k )] δ νν (cid:48) − Σ νν (cid:48) ( ω, k ) . (2)In this expression ν and ν (cid:48) label the bands and ω denotesthe energy counted from E F . The interacting value ofthe chemical potential µ sets the total electron number.Since µ can be conventionally included in ˆ H , we shallomit it in the following.The Fermi surfaces and dispersion relations of thequasiparticles are obtained as the solutions ω = 0 and ω = ω qp ν ( k ) ofdet [( ω − ε ν ( k )) δ νν (cid:48) − Σ (cid:48) νν (cid:48) ( ω, k )] = 0 . (3)In the above equation Σ (cid:48) denotes the real part of theself-energy. Its imaginary part Σ (cid:48)(cid:48) has been neglected, i.e. , we assume that quasiparticles are coherent with alifetime longer than 1 /ω qp . Our data indicate that thisis indeed the case up to the highest energies analyzedhere. At very low frequency, the lifetime of quasiparticlescannot be reliably tested by ARPES, since the intrinsicquasiparticle width is masked by contributions of the ex-perimental resolution, impurity scattering and inhomo-geneous broadening. However, the observation of strong quantum oscillations in the bulk provides direct evidencefor well-defined quasiparticles in Sr RuO down to thelowest energies [25, 27].It is important to note that the Green’s function G , theself-energy Σ and the spectral function A are in generalnon-diagonal matrices. This has been overlooked thus farin self-energy analyses of ARPES data but is essentialto determine the nature of many-body interactions inSr RuO , as we will show in Sec. V. A. Localized orbitals and electronic structure
Let us recall some of the important aspects of the elec-tronic structure of Sr RuO . As shown in Sec. II B, threebands, commonly denoted ν = { α, β, γ } , cross the Fermilevel. These bands correspond to states with t g symme-try deriving from the hybridization between localized Ru-4 d ( d xy , d yz , d xz ) orbitals and O-2 p states. Hence, we in-troduce a localized basis set of t g -like orbitals | χ m (cid:105) , withbasis functions conveniently labeled as m = { xy, yz, xz } .In practice, we use maximally localized Wannier func-tions [60, 61] constructed from the Kohn-Sham eigenbasisof a non-SOC density functional theory (DFT) calcula-tion (see appendix B 1 for details). We term the cor-responding Hamiltonian ˆ H DFT . It is important to notethat the choice of a localized basis set is not unique andother ways of defining these orbitals are possible (see,e.g., Ref. [62]).In the following this set of orbitals plays two importantroles. First, they are atom-centered and provide a setof states localized in real-space | χ m ( R ) (cid:105) . Secondly, theunitary transformation matrix to the band basis | ψ ν ( k ) (cid:105) U mν ( k ) = (cid:104) χ m ( k ) | ψ ν ( k ) (cid:105) , (4)allows us to define an ‘orbital’ character of each band ν as | U mν ( k ) | . In the localized-orbital basis the one-particleHamiltonian is a non-diagonal matrix, which readsˆ H mm (cid:48) ( k ) = (cid:88) ν U mν ( k ) ε ν ( k ) U ∗ m (cid:48) ν ( k ) . (5)The self-energy in the orbital basis is expressed asΣ mm (cid:48) ( ω, k ) = (cid:88) νν (cid:48) U mν ( k ) Σ νν (cid:48) ( ω, k ) U ∗ m (cid:48) ν (cid:48) ( k ) , (6)and conversely in the band basis asΣ νν (cid:48) ( ω, k ) = (cid:88) mm (cid:48) U ∗ mν ( k ) Σ mm (cid:48) ( ω, k ) U m (cid:48) ν (cid:48) ( k ) . (7) B. Spin-orbit coupling
We treat SOC as an additional term to ˆ H DFT , whichis independent of k in the localized-orbital basis, butleads to a mixing of the individual orbitals. The single-particle SOC term for atomic d -orbitals projected to the t g -subspace reads [63]ˆ H SOC λ = λ (cid:88) mm (cid:48) (cid:88) σσ (cid:48) c † mσ ( l mm (cid:48) · σ σσ (cid:48) ) c m (cid:48) σ (cid:48) , (8)where l are the t g -projected angular momentum matri-ces, σ are Pauli matrices and λ will be referred to in thefollowing as the SOC coupling constant. As documentedin appendix B 1, the eigenenergies of a DFT+SOC calcu-lation are well reproduced by ˆ H DFT + ˆ H SOC λ with λ DFT =100 meV.
IV. ENHANCED EFFECTIVE SPIN-ORBITCOUPLING AND SINGLE-PARTICLEHAMILTONIAN
The importance of SOC for the low-energy physics ofSr RuO has been pointed out by several authors [8, 17,21, 23, 31, 64–68]. SOC lifts degeneracies found in itsabsence and causes a momentum dependent mixing ofthe orbital composition of quasiparticle states, which hasnon-trivial implications for superconductivity [8, 17, 68].Signatures of SOC have been detected experimentally onthe Fermi surface of Sr RuO in the form of a small pro-trusion of the γ sheet along the zone diagonal [31, 66]and as a degeneracy lifting at the band bottom of the β sheet [68]. These studies reported an overall goodagreement between the experimental data and the ef-fects of SOC calculated within DFT [31, 66, 68]. Thisis in apparent contrast to more recent DMFT studiesof Sr RuO , which predict large but frequency indepen-dent off-diagonal contributions to the local self-energythat can be interpreted as a contribution ∆ λ to the ef-fective coupling strength λ eff = λ DFT + ∆ λ [21, 23], This is also consistent with general perturbation-theory con-siderations [51], which show a Coulomb-enhancement ofthe level splitting in the J basis, similar to a Coulomb-enhanced crystal-field splitting [69].In the absence of SOC, DFT yields a quasi-crossingbetween the β and γ Fermi surface sheets a few degreesaway from the zone diagonal, as displayed on Fig. 3 (a).Near such a point we expect the degeneracy to be liftedby SOC, leading to a momentum splitting ∆ k = λ eff /v and to an energy splitting of ∆ E = Zλ eff [23], as de-picted schematically in Fig. 3 (e). In these expressions, v ≡ √ v β v γ , with v β and v γ the bare band velocities in theabsence of SOC and correlations, and Z ≡ (cid:112) Z β Z γ in-volves the quasiparticle residues Z ν associated with eachband (also in the absence of SOC).It is clear from these expressions that a quantitativedetermination of λ eff is not possible from experimentalone. Earlier studies on Sr RuO [68] and iron-basedsuperconductors [70], have interpreted the energy split-ting ∆ E at avoided crossings as a direct measure of theSOC strength λ eff . However, in interacting systems ∆ E is not a robust measure of SOC since correlations canboth enhance ∆ E by enhancing λ eff and reduce it viathe renormalization factor Z . We thus quantify the en-hancement of SOC from the momentum splitting ∆ k ,which is not renormalized by the quasiparticle residue Z . The experimental splitting at the avoided crossingbetween the β and γ Fermi surface sheets indicated inFig. 3 (a) is ∆ k QP = 0 . − whereas DFT pre-dicts ∆ k DFT+ λ DFT = 0 .
046 ˚A − . We thus obtain an ef-fective SOC strength λ eff = λ DFT ∆ k QP / ∆ k DFT+ λ DFT =205(20) meV, in quantitative agreement with the predic-tions in Refs. [21, 23]. We note that despite this largeenhancement of the effective SOC, the energy splittingremains smaller than λ DFT as shown in Fig. 3 (f). Whendeviations from linearity in band dispersions are small,the splitting ∆ E is symmetric around the E F and canthus be determined from the occupied states probed inexperiment. Direct inspection of the data in Fig. 3 (f)yields ∆ E ≈
70 meV, which is about 2 / λ DFT andthus clearly not a good measure of SOC.The experimental splitting is slightly larger than thatexpected from the expression ∆ E = Zλ eff and our the-oretical determination of Z β and Z γ at the Fermi sur-face (which will be described in Sec. VI). This can beattributed to the energy dependence of Z , which, inSr RuO , is not negligible over the energy scale of SOC.Note that the magnitude of the SOC-induced splitting ofthe bands at the Γ point reported in Ref. [68] can alsobe explained by the competing effects of enhancement bycorrelations and reduction by the quasiparticle weight asshown in Ref. [23]. We also point out that the equili-bration of quasiparticle velocities close to the diagonal,apparent from Figs. 2 (a,c) and 3 (f) is indeed the be-havior expected close to an avoided crossing [23].Including the enhanced SOC determined from thisnon-crossing gap leads to a much improved theoreti-cal description of the entire Fermi surface [71]. As FIG. 3. Correlation enhanced effective SOC. (a) Quadrant of the experimental Fermi surface with a DFT calculation withoutSOC ( ˆ H DFT ) at the experimental k z ≈ . π/c (grey lines). (b,c) Same as (a) with calculations including SOC (DFT+ λ DFT )and enhanced SOC ( λ DFT + ∆ 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 ofa SOC induced degeneracy lifting. Here, Z = √ Z ν Z ν (cid:48) , where ν, ν (cid:48) labels the two bands and v = √ v ν v ν (cid:48) where v ν , v ν (cid:48) are barevelocities 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+ λ DFT + ∆ λ eigenstates along the Fermi surface. shown in Fig. 3 (b), our high-resolution experimentalFermi surface deviates systematically from a DFT cal-culation with SOC. Most notably, ˆ H DFT + ˆ H SOC λ DFT under-estimates the size of the γ sheet and overestimates the β sheet. Intriguingly, this is almost completely correctedin ˆ H DFT + ˆ H SOC λ DFT +∆ λ , with λ DFT + ∆ λ = 200 meV, asdemonstrated in Fig. 3 (c). Indeed, a close inspectionshows that the remaining discrepancies between experi-ment and ˆ H DFT + ˆ H SOC λ DFT +∆ λ break the crystal symmetry,suggesting that they are dominated by experimental ar-tifacts. A likely source for these image distortions is im-perfections in the electron optics arising from variationsof the work function around the electron emission spot onthe sample. Such distortions can presently not be fullyeliminated in low-energy photoemission from cleaved sin-gle crystals.Importantly, the change in Fermi surface sheet volumewith the inclusion of ∆ λ is not driven by a change in thecrystal-field splitting between the xy and xz, yz orbitals(see appendix B). The volume change occurs solely be-cause of a further increase in the orbital mixing inducedby the enhanced SOC. As shown in Fig. 3 (g), this mix-ing is not limited to the vicinity of the avoided crossingbut extends along the entire Fermi surface (see Fig. 9in appendix B 1 for the orbital character without SOC).For λ DFT + ∆ λ we find a minimal d xy and d xz,yz mixingfor the γ and β bands of 20 /
80% along the ΓM directionwith a monotonic increase to ≈
50% along the Brillouinzone diagonal ΓX. We note that this mixing varies withthe perpendicular momentum k z . However, around the experimental value of k z ≈ . π/c the variation is weak[72]. The analysis presented here and in Secs. III A andVI is thus robust with respect to a typical uncertainty in k z . These findings suggest that a natural reference single-particle Hamiltonian is ˆ H = ˆ H DFT + ˆ H SOC λ DFT +∆ λ . Thischoice ensures that the Fermi surface of ˆ H is very closeto that of the interacting system. From Eq. 3, this im-plies that the self-energy matrix approximately vanishesat zero binding energy: Σ (cid:48) νν (cid:48) ( ω = 0 , k ) (cid:39)
0. We chooseˆ H in this manner in all the following. Hence, from nowon | ψ ν ( k ) (cid:105) and ε ν ( k ) refer to the eigenstates and bandstructure of ˆ H = ˆ H DFT + ˆ H SOC λ DFT +∆ λ . We point out thatalthough ˆ H is a single-particle Hamiltonian, the effec-tive enhancement ∆ λ of SOC included in ˆ H is a corre-lation effect beyond DFT. V. EXPERIMENTAL DETERMINATION OFSELF-ENERGIESA. Self-energies in the quasiparticle/band basis
Working in the band basis, i.e., with the eigenstatesof ˆ H , the maximum of the ARPES intensity for agiven binding energy ω (maximum of the MDCs) corre-sponds to the momenta k which satisfy (following Eq. 3): ω − ε ν ( k ) − Σ (cid:48) νν ( ω, k ) = 0. Hence, for each binding en-ergy, each azimuthal cut, and each sheet of the quasi-particle dispersions, we fit the MDCs and determine the FIG. 4. Self-energies extracted in the band and orbital bases. (a) Real part of the self-energy Σ (cid:48) νν in the band basis (solidsymbols) in 9 ◦ steps of the Fermi surface angle θ . (b) Illustration of the relation between Σ (cid:48) νν , the bare bands given by ˆ H (thinlines) and the quasiparticle peak positions (solid symbols). (c) Compilation of Σ (cid:48) νν from panel (a). (d) Real part of theself-energy Σ (cid:48) mm in the orbital basis. momentum k ν max ( ω ) at their maximum. Using the valueof ε ν ( k ν max ( ω )) at this momentum yields the followingquantity ω − ε ν ( k ν max ( ω )) = Σ (cid:48) νν ( ω, k ν max ( ω )) ≡ Σ (cid:48) ν ( ω, θ ) . (9)This equation corresponds to the simple constructionillustrated graphically in Fig. 4 (b), and it is a standardway of extracting a self-energy from ARPES, as used inprevious works on several materials [32, 48, 73–75]. Wenote that this procedure assumes that the off-diagonalcomponents Σ (cid:48) ν (cid:54) = ν (cid:48) ( ω, k ) can be neglected for states closeto the Fermi surface (i.e., for small ω and k close to aFermi crossing). This assumption can be validated, asshown in appendix C. When performing this analysis, weonly include the α sheet for θ = 45 ◦ . Whenever the con-straint Σ (cid:48) ν ( ω → → H Fermiwave-vectors because we attribute these differences pre-dominantly to experimental artifacts.The determined self-energies for each band ν = α, β, γ and the different values of θ are depicted in Fig. 4 (a,c).For the β and γ sheets they show a substantial de-pendence on the azimuthal angle. Around ΓM we findthat Σ (cid:48) γ exceeds Σ (cid:48) β by almost a factor of two (at ω = −
50 meV), whereas they essentially coincide along the zone diagonal (ΓX). This change evolves as a functionof θ and occurs via a simultaneous increase in Σ (cid:48) β anda decrease in Σ (cid:48) γ for all energies as θ is increased from0 ◦ (ΓM) to 45 ◦ (ΓX). In order to better visualize thisangular dependence, a compilation of Σ (cid:48) ν ( ω ) for differentvalues of θ is displayed in Fig. 4 (c). B. Accounting for the angular dependence: localself-energies in the orbital basis
In this section, we introduce a different procedure forextracting self-energies from ARPES, by working in theorbital basis | χ m ( k ) (cid:105) . We do this by making two keyassumptions:1. We assume that the off-diagonal components arenegligible, i.e., Σ (cid:48) m (cid:54) = m (cid:48) (cid:39)
0. Let us note that inSr RuO even a k -independent self-energy has non-zero off-diagonal elements if ˆ H DFT + ˆ H SOC λ DFT is con-sidered. Using DMFT, these off-diagonal elementshave been shown to be very weakly dependent onfrequency in this material [23], leading to the no-tion of a static correlation enhancement of the ef-fective SOC (∆ λ ). In the present work, these off-diagonal frequency-independent components are al-ready incorporated into ˆ H (see Sec. IV), and thusthe frequency-dependent part of the self-energy is(approximately) orbital diagonal by virtue of thetetragonal crystal structure.2. We assume that the diagonal components of theself-energy in the orbital basis depend on the mo-mentum k only through the azimuthal angle θ k :Σ (cid:48) mm ( ω, k ) (cid:39) Σ (cid:48) m ( ω, θ k ). We neglect the depen-dence on the momentum which is parallel to theangular cut.Under these assumptions, the equation determiningthe quasiparticle dispersions readsdet (cid:104) ( ω − Σ (cid:48) m ( ω, θ k )) δ mm (cid:48) − ˆ H mm (cid:48) ( k ) (cid:105) = 0 . (10)In this equation, we have neglected the lifetime effectsassociated with the imaginary part Σ (cid:48)(cid:48) m . In order to ex-tract the functions Σ (cid:48) m ( ω, θ k ) directly from the ARPESdata, we first determine the peak positions k ν max ( ω, θ )for MDCs at a given angle θ and binding energy ω . Wethen compute (for the same ω and θ ) the matrix A mm (cid:48) ≡ ωδ mm (cid:48) − ˆ H mm (cid:48) ( k α max ( ω, θ k )) and similarly B mm (cid:48) , G mm (cid:48) for the β and γ band MDCs, k β max ( ω, θ ) and k γ max ( ω, θ ),respectively. In terms of these matrices, the quasiparticleequations (10) readdet[ A mm (cid:48) − Σ (cid:48) m δ mm (cid:48) ] = det[ B mm (cid:48) − Σ (cid:48) m δ mm (cid:48) ] == det[ G mm (cid:48) − Σ (cid:48) m δ mm (cid:48) ] = 0 . (11)However, when taking symmetry into account, the self-energy has only two independent components: Σ (cid:48) xy andΣ (cid:48) xz = Σ (cid:48) yz . Hence, we only need two of the above equa-tions to solve for the two unknown components of the self-energy. This means that we can also extract a self-energyin the directions where only two bands ( β and γ ) arepresent in the considered energy range of ω (cid:46)
100 meV,e.g., along ΓM. The resulting functions Σ (cid:48) m ( ω, θ k ) deter-mined at several angles θ are displayed in Fig. 4 (d).It is immediately apparent that, in contrast to Σ (cid:48) νν , theself-energies in the orbital basis do not show a strong an-gular (momentum) dependence, but rather collapse intotwo sets of points, one for the xy orbital and one for the xz/yz orbitals. Thus, we reach the remarkable conclu-sion that the angular dependence of the self-energy inthe orbital basis is negligible, within the range of bind-ing energies investigated here: Σ (cid:48) m ( ω, θ k ) (cid:39) Σ (cid:48) m ( ω ). Thisimplies that a good approximation of the full momentumand energy dependence of the self-energy in the band(quasiparticle) basis is given byΣ νν (cid:48) ( ω, k ) = (cid:88) m U ∗ mν ( k ) Σ m ( ω ) U mν (cid:48) ( k ) . (12)The physical content of this expression is that the an-gular (momentum) dependence of the quasiparticle self-energies emphasized above is actually due to the matrixelements U mν ( k ) defined in Eq. 4. In Sr RuO the an-gular dependence of these matrix elements is mainly dueto the SOC, as seen from the variation of the orbital content of quasiparticles in Fig. 3 (g). In appendix Dwe show the back-transform of Σ (cid:48) m ( ω, θ k = 18 ◦ ) intoΣ (cid:48) ν ( ω, θ k = 0 , , ◦ ). The good agreement with Σ (cid:48) νν directly extracted from experiment further justifies theabove expression and also confirms the validity of theapproximations made throughout this section.Finally, we stress that expression (12) precisely co-incides with the ansatz made by DMFT: within thistheory, the self-energy is approximated as a local( k -independent) object when expressed in a basis of lo-calized orbitals , while it acquires momentum dependencewhen transformed to the band basis. VI. COMPARISON TO DYNAMICALMEAN-FIELD THEORY
In this section we perform an explicit comparison ofthe measured quasiparticle dispersions and self-energiesto DMFT results. The latter are based on the Hamilto-nian ˆ H DFT , to which the Hubbard-Kanamori interactionwith on-site interaction U = 2 . J = 0 . A ( ω, k ) displayed as a color-intensity map. Clearly,the theoretical results are in near quantitative agreementwith the data: both the strong renormalization of theFermi velocity and the angular-dependent curvature ofthe quasiparticle bands are very well reproduced by thepurely local, and thus momentum-independent DMFTself-energies. This validates the assumption of no mo-mentum dependence along the radial k -space directionof the self-energy made in Sec. V B for the k/ω rangestudied here. The small deviations in Fermi wave vec-tors discernible in Fig. 5 are consistent with Fig. 3(c)and the overall experimental precision of the Fermi sur-face determination.In Fig. 6 (a), we compare the experimental self-energiesfor each orbital with the DMFT results. The overallagreement is notable. At low-energy, the self-energiesare linear in frequency and the agreement is excellent.The slope of the self-energies in this regime controls theangular-dependence of the effective mass renormalisa-tion. Using the local ansatz (12) into the quasiparticledispersion equation, and performing an expansion around E F , we obtain v b v νF ( θ ) = (cid:88) m Z m | U mν ( θ ) | , Z m ≡ − ∂ Σ (cid:48) m ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 . (13)In Fig. 6 (b), we show v b /v νF ( θ ) for the β and γ bandsusing the DMFT values Z xy = 0 . ± .
01 and Z xz/yz = Γ X, θ = 45º γβ α minmax-80-400 E - E F ( m e V ) Γ M, θ = 0º β γ θ = 9º 0.70.5 θ = 18º 0.70.5 θ = 27º 0.90.7 θ = 36º k // ( Å -1 ) FIG. 5. Comparison of experimental quasiparticle dispersions (markers) with DMFT spectral functions (color plots) calculatedfor different Fermi surface angles θ . . ± .
01 obtained at 29 K (appendix B 2). The over-all angular dependence and the absolute value of the γ band mass enhancement is very well captured by DMFT,while the β band is a bit overestimated. Close to thezone-diagonal ( θ = 45 ◦ ), the two mass enhancements areapproximately equal, due to the strong orbital mixinginduced by the SOC.Turning to larger binding energies, we see that the the-oretical Σ (cid:48) xy is in good agreement with the experimentaldata over the full energy range of 2-80 meV covered in ourexperiments. Both the theoretical and experimental self-energies deviate significantly from the linear regime downto low energies ( ∼
20 meV), causing curved quasiparticlebands with progressively steeper dispersion as the en-ergy increases (Fig. 5). In contrast, the agreement be-tween theory and experiment for the xz/yz self-energy issomewhat less impressive at binding energies larger than ∼
30 meV. Our DMFT self-energy Σ (cid:48) xz/yz overestimatesthe strength of correlations in this regime (by 20 − U which is the same for all or-bitals. Earlier cRPA calculations have suggested thatthis on-site interaction is slightly larger for the xy orbital( U xy = 2 . U xz/yz = 2 . RuO [21]. Another possible explanationis that this discrepancy is actually a hint of some mo-mentum dependent contribution to the self-energy, es-pecially dependence on momentum perpendicular to theFermi surface. We note in this respect that the discrep-ancy is larger for the α, β sheets which have dominant xz/yz character. These orbitals have, in the absence ofSOC, a strong one-dimensional character, for which mo-mentum dependence is definitely expected and DMFTis less appropriate. Furthermore, these FS sheets arealso the ones associated with nesting and spin-densitywave correlations, which are expected to lead to an ad- ditional momentum-dependence of the self-energy. Wefurther discuss possible contributions of spin fluctuationsin Sec. VIII. VII. KINKS
The self-energies Σ (cid:48) ( ω ) shown in Figs. 4 and 6 dis-play a fairly smooth curvature, rather than pronounced‘kinks’. Over a larger range, however, Σ (cid:48) xy from DMFTdoes show an energy scale marking the crossover fromthe strongly renormalized low-energy regime to weaklyrenormalized excitations. This is illustrated in the insetto Fig. 6 (a). Such purely electronic kinks were reportedin DMFT calculations of a generic system with Mott-Hubbard sidebands [76] and have been abundantly docu-mented in the theoretical literature since then [13, 20, 77–80]. In Sr RuO they are associated with the crossoverfrom the Fermi-liquid behavior into a more incoherentregime [18, 20]. The near quantitative agreement of thefrequency dependence of the experimental self-energiesΣ (cid:48) m ( ω ) and our single-site DMFT calculation providesstrong evidence for the existence of such electronic kinksin Sr RuO . In addition, it implies that the local DMFTtreatment of electronic correlations is capturing the dom-inant effects.Focusing on the low-energy regime of our experi-mental data, we find deviations from the linear formΣ (cid:48) ( ω ) = ω (1 − /Z ) characteristic of a Fermi liquid for | ω | >
20 meV, irrespective of the basis. However, this isonly an upper limit for the Fermi-liquid energy scale inSr RuO . Despite the improved resolution of our exper-iments, we cannot exclude an even lower crossover en-ergy to non-Fermi-liquid like excitations. We note thatthe crossover temperature of T FL ≈
25 K reported fromtransport and thermodynamic experiments [10–12] in-deed suggests a cross-over energy scale that is signifi-cantly below 20 meV.The overall behavior of Σ (cid:48) ( ω ) including the energyrange where we find strong changes in the slope agreeswith previous photoemission experiments, which were in-terpreted as evidence for electron-phonon coupling [29–31, 46, 48]. Such an interpretation, however, relies on0 v b / v F θ (deg) β band γ band Γ X2001000 Σ ' m ( m e V ) -90 -60 -30 0 ω (meV) d xz,yz d xy (a) (b) FIG. 6. (a) Average of the self-energies Σ (cid:48) xz/yz , Σ (cid:48) xy shown inFig. 4 (d) compared with DMFT self-energies calculated at29 K. The self-energies are shifted such that Σ (cid:48) m ( ω = 0) = 0.The inset shows the DMFT self-energies over a larger en-ergy range. Linear fits at low and high energy of Σ (cid:48) xy fromDMFT are shown as solid and dashed black lines, respectively.(b) Angle dependence of the mass enhancement v b /v F fromARPES (markers) and DMFT (solid line). The range indi-cated by the shaded areas corresponds to mass enhancementscalculated from the numerical data by using different meth-ods (see appendix B). Error bars on the experimental dataare obtained from propagating the estimated uncertainty ofthe Fermi velocities shown in Fig. 2 (c). a linear Fermi-liquid regime of electronic correlationsover the entire phonon bandwidth of ≈
90 meV [81],which is inconsistent with our DMFT calculations. More-over, attributing the entire curvature of Σ (cid:48) m in our datato electron-phonon coupling would result in unrealisticcoupling constants far into the polaronic regime, whichis hard to reconcile with the transport properties ofSr RuO [10–12]. We also note that a recent scanningtunneling microscope (STM) study reported very strongkinks in the β and γ sheets of Sr RuO [49], which is in-consistent with our data. We discuss the reason for thisdiscrepancy in appendix A. VIII. DISCUSSION AND PERSPECTIVES
In this article, we have reported on high-resolutionARPES measurements which allow for a determinationof the Fermi surface and quasiparticle dispersions ofSr RuO with unprecedented accuracy. Our data revealan enhancement (by a factor of about two) of the splittingbetween Fermi surface sheets along the zone diagonal, incomparison to the DFT value. This can be interpreted asa correlation-induced enhancement of the effective SOC,an effect predicted theoretically [21, 23, 51] and demon-strated experimentally here for this material, for the firsttime.Thanks to the improved cleanliness of our data, wehave been able to determine the electronic self-energiesdirectly from experiment, using both a standard proce-dure applied in the band (quasiparticle) basis as well asa novel procedure, introduced in the present article, inthe orbital basis. Combining these two approaches, wehave demonstrated that the large angular (momentum)dependence of the quasiparticle self-energies and disper-sions can be mostly attributed to the fact that quasipar-ticle states have an orbital content which is strongly an-gular dependent, due to the SOC. Hence, assuming self-energies which are frequency-dependent but essentiallyindependent of angle (momentum), when considered inthe orbital basis, is a very good approximation. Thisprovides a direct experimental validation of the DMFT ansatz .The key importance of atomic-like orbitals in corre-lated insulators is well established [82, 83]. The presentwork demonstrates that orbitals retain a considerablephysical relevance even in a metal in the low-temperatureFermi-liquid regime. Although the band and orbitalbasis used here are equivalent, our analysis shows thatthe underlying simplicity in the nature of correlationsemerges only when working in the latter and takinginto account the orbital origin of quasiparticles. BeyondSr RuO , this is an observation of general relevance tomultiband metals with strong correlations such as theiron-based superconductors [84, 85].Notwithstanding its success, the excellent agreement ofthe DMFT results with ARPES data does raise puzzlingquestions. Sr RuO is known to be host to strong mag-netic fluctuations [39–42], with a strong peak in its spinresponse χ ( Q ) close to the spin-density wave (SDW) vec-tor Q ∼ (2 π/ , π/ Q = 0.Indeed, tiny amounts of substitutional impurities inducelong-range magnetic order in this material, of either SDWof ferromagnetic type [86, 87]. Hence, it is a prominentopen question to understand how these long-wavelengthfluctuations affect the physics of quasiparticles in theFermi-liquid state. Single-site DMFT does not capturethis feedback, and the excellent agreement with the over-all quasiparticle physics must imply that these effectshave a comparatively smaller magnitude than the domi-nant local effect of correlations (on-site U and especially1Hund’s J ) captured by DMFT. A closely related ques-tion is how much momentum dependence is present inthe low-energy (Landau) interactions between quasipar-ticles. These effects are expected to be fundamental forsubsequent instabilities of the Fermi liquid, into eitherthe superconducting state in pristine samples or magneticordering in samples with impurities. Making progress onthis issue is also key to the understanding of the super-conducting state of Sr RuO , for which the precise na-ture of the pairing mechanism as well as symmetry of theorder parameter are still outstanding open questions [4]. ACKNOWLEDGMENTS
The experimental work has been supported by the Eu-ropean Research Council (ERC), the Scottish FundingCouncil, the UK EPSRC and the Swiss National ScienceFoundation (SNSF). Theoretical work was supported bythe ERC grant ERC-319286-QMAC and by the SNSF(NCCR MARVEL). The Flatiron Institute is a divisionof the Simons Foundation. AG and MZ gratefully ac-knowledge useful discussions with Gabriel Kotliar, An-drew J. Millis and Jernej Mravlje.
Appendix A: Bulk and surface electronic structureof Sr RuO FIG. 7. Bulk and surface electronic structure of Sr RuO .Left half: Fermi surface map of a CO passivated surface shownin the main text. Right half: Fermi surface acquired on apristine cleave at 21 eV photon energy. Intense surface statesare evident in addition to the bulk bands observed in the leftpanel. -80-400 E - E F ( m e V ) surface bulk β γ Γ M QPI-80-400 E - E F ( m e V ) // (Å -1 ) Γ X αγβ QPI (a) // (Å -1 ) γβ QPI Γ M (b) FIG. 8. Comparison with the STM data from Ref. [49]. (a)ΓM high-symmetry cut on a pristine cleave. The bulk β and γ and the surface β bands are labeled. (b) Laser-ARPES datafrom CO passivated surface showing the bulk band dispersionalong ΓX and ΓM. The dispersion obtained from quasiparticleinterference in Ref. [49] is overlaid with red markers. In Fig. 7 we compare the data presented in the maintext 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 andsurface bands by Shen et al. [54]. In particular, wefind that the larger β sheet has bulk character. Thisband assignment is used by the vast majority of sub-sequent ARPES publications [29–31, 33, 46, 56, 88, 89],except for Ref. [47], which reports a dispersion with muchlower Fermi velocity and a strong kink at 15 meV for thesmaller β sheet that we identify as a surface band.Wang et al. [49] have recently probed the low-energyelectronic structure of Sr RuO by STM. Analyzingquasiparticle interference patterns along the ΓX andΓM high-symmetry directions, they obtained band dis-persions with low Fermi velocities and strong kinks at10 meV and 37 meV. In Fig. 8 we compare the band dis-persion reported by Wang et al. with our ARPES data.Along both high-symmetry directions, we find a clear dis-crepancy with our data, which are in quantitative agree-ment with bulk de Haas van Alphen measurements, asdemonstrated in the main text. On the other hand, wefind a striking similarity between the STM data alongΓM and the band commonly identified as the surface β band [30, 54]. We thus conclude that the experiments re-ported in Ref. [49] probed the surface states of Sr RuO .This is fully consistent with the strong low-energy kinksand overall enhanced low-energy renormalization seen byARPES in the surface bands [50, 90].2 -3-2-101 ε k ( e V ) DFT (GGA-PBE) DFT (Wannier90)-3-2-101 ε k ( e V ) Γ M X Γ Z DFT + SOC (GGA-PBE) DFT + λ DFT (Wannier90)
FIG. 9. DFT band structure along the high-symmetrypath ΓMXΓZ compared to the eigenstates of our maximally-localized Wannier Hamiltonian ˆ H DFT for the three t g bands.Top: DFT (GGA-PBE) and eigenstates of ˆ H DFT . Bottom:DFT+SOC (GGA-PBE) and eigenstates of ˆ H DFT + ˆ H SOC λ DFT with a local SOC term (Eq. 8) and a coupling strength of λ DFT = 100 meV.
Appendix B: Computational details1. DFT and model Hamiltonian
These orbitals are centered on the Ru atoms and have t g symmetry, but are indeed linear combinations of Ru- d and O- p states. We do not add Wannier functions cen-tered on the oxygen atoms, because the resulting threeorbital Wannier model already accurately reproduces thethree bands crossing the Fermi energy, as demonstratedin Fig. 9. Also note that the Wannier function construc-tion allows to disentangle the γ band from the bands withdominantly O- p character below − H DFT with a maximally-localized Wannier function [60, 61] con-struction of t g -like orbitals for the three bands cross-ing the Fermi surface. These Wannier orbitals are ob-tained on a 10 × × k grid based on a non-SOCDFT calculation using WIEN2k [91] with the GGA-PBEfunctional [92], wien2wannier [93] and Wannier90 [94].The DFT calculation is performed with lattice parame-ters from Ref. [95] (measured at 100 K) and convergedwith twice as many k -points in each dimension.The eigenenergies of the resulting Wannier Hamilto-nian, ˆ H DFT , accurately reproduce the DFT band struc-ture (Fig. 9 top). Note that in the absence of SOC,
FIG. 10. Orbital character of the DFT FS without SOC at k z = 0 . π/c (left). The orbital character of the DFT+ λ DFT +∆ λ eigenstates at the same k z is reproduced on the right fromFig. 3. λ = 0 meV λ = 100 meV λ = 200 meV λ = 300 meV Δε CF = - 80 meV Δε CF = - 40 meV Δε CF = 0 meV Δε CF = 40 meV Δε CF = 80 meV ARPES M X
FIG. 11. Fermi surface of Sr RuO for λ = 0 meV (top left), λ = 100 meV (top right), λ = 200 meV (bottom right) and λ = 300 meV (bottom left) compared to ARPES (dashedblack line). λ = 100 meV corresponds to a DFT+SOC cal-culation and λ = 200 meV to an effective SOC enhanced byelectronic correlations (see main text). The different shadesof red indicate additional crystal-field splittings ∆ (cid:15) cf addedto (cid:15) cf = 85 meV of ˆ H DFT . the eigenstates retain pure orbital character, as shownin Fig. 10. To take SOC into account, we add the localsingle-particle term ˆ H SOC λ , as given in Eq. 8, with cou-pling constant λ . In the bottom panel of Fig. 9 we showthat the eigenenergies of ˆ H DFT + ˆ H SOC λ are in nearly per-fect agreement with the DFT+SOC band structure at avalue of λ DFT = 100 meV.Our model Hamiltonian provides the reference pointto which we define a self-energy, but it is also a per-3 Σ ' m ( m e V ) -100 -50 0 ω (meV) Pade xz Pade xy Beach xz Beach xy TRIQS/maxent xz TRIQS/maxent xy analytic continuation FIG. 12. Real part of DFT+DMFT self-energy in the consid-ered energy range obtained with three different analytic con-tinuation methods: Pad´e approximants (using TRIQS [96]),Stochastic continuation (after Beach [97]) and Maximum En-tropy (using TRIQS/maxent [98]). The difference betweenthe analytic continuation methods is smaller than the exper-imental error. fect playground to study the change in the Fermi surfaceunder the influence of SOC and the crystal-field split-ting between the xy and xz/yz orbitals. In the fol-lowing, we will confirm that the best agreement withthe experimental Fermi surface is found with an effec-tive SOC of λ eff = λ DFT + ∆ λ = 200 meV, but atthe same time keeping the DFT crystal-field splitting of (cid:15) cf = (cid:15) xz/yz − (cid:15) xy = 85 meV unchanged. We compare inFig. 11 the experimental Fermi surface (dashed lines) tothe one of ˆ H DFT + ˆ H SOC λ DFT . The Fermi surfaces for addi-tionally introduced crystal-field splittings ∆ (cid:15) cf between −
80 and 80 meV are shown with solid lines in differentshades of red. In contrast to the Fermi surface withoutSOC ( λ = 0 meV, top left panel), the Fermi surfaces withthe DFT SOC of λ = 100 meV (top right panel) resemblesthe overall structure of the experimental Fermi surface.However, the areas of the α and β sheets are too largeand the γ sheet is too small. Importantly, the agreementcannot be improved by adding ∆ (cid:15) cf . For example, alongΓM a ∆ (cid:15) cf of −
40 meV would move the Fermi surfacecloser to the experiment, but, on the other hand, alongΓX a ∆ (cid:15) cf of 80 meV would provide the best agreement.The situation is different if we consider an enhanced SOCof λ = 200 meV (bottom right panel). Then, we find anearly perfect agreement with experiment without anyadditional crystal-field splitting (∆ (cid:15) cf = 0 meV). At aneven higher SOC of λ = 300 meV (bottom left panel)we see again major discrepancies, but with an oppositetrend: The α and β sheets are now too small and the γ sheet is too large. Like in the case of λ = 100 meV thiscan not be cured by an adjustment of (cid:15) cf .
2. DMFT
We perform single-site DMFT calculation with theTRIQS/DFTTools [99] package for ˆ H DFT and Hubbard-Kanamori interactions with a screened Coulomb repul-sion U = 2 . J = 0 . −
100 to 0 meV the difference in the resulting self-energies (Fig. 12) is below the experimental uncertainty .The averaged quasiparticle renormalizations (of the threecontinuations) are: Z xy = 0 . ± .
01 and Z xz = Z yz =0 . ± .
01. For all other results presented in the maintext the Pad´e solution has been used.Our calculations at a temperature of 29 K use ˆ H DFT ,as the sign problem prohibits reaching such low temper-atures with SOC included. Nevertheless, calculationswith SOC were successfully carried out at a tempera-ture of 290 K using CT-INT [21] and at 230 K usingCT-HYB with a simplified two-dimensional tight-bindingmodel [23]. These works pointed out that electronic cor-relations in Sr RuO lead to an enhanced SOC. To bemore precise, Kim et al. [23] observed that electronic cor-relations in this material are described by a self-energywith diagonal elements close to the ones without SOCplus, to a good approximation, frequency-independentoff-diagonal elements, which can be absorbed in a staticeffective SOC strength of λ eff = λ DFT + ∆ λ (cid:39)
200 meV– this is the approach followed in the present article.In addition to the enhancement of SOC it was ob-served that low-energy many-body effect also lead to anenhancement of the crystal-field splitting [21, 23]. Inour DFT+DMFT calculation without SOC this resultsin a orbital-dependent splitting in the real part of theself-energies (∆ (cid:15) cf = 60 meV), which would move the γ sheet closer to the van Hove singularity and consequentlyworsen the agreement with the experimental Fermi sur-face along the ΓM direction (see bottom right panel ofFig. 11). Different roots of this small discrepancy arepossible, ranging from orbital-dependent double count-ing corrections to, in general, DFT being not perfectas ‘non-interacting’ reference point for DMFT. Zhang etal. [21] showed that by considering the anisotropy of theCoulomb tensor the additional crystal-field splitting issuppressed and consequently the disagreement betweentheory and experiment can be cured. We point out thatin comparision to the present work a large enhancement4 -40 -20 0 ω (meV) θ = 45° α band Γ X120600 Σ ' νν ( m e V ) -40 -20 0 ω (meV) θ = 0° γ band β band ~ Γ M -40 -20 0 ω (meV) θ = 18°-40 -20 0 ω (meV) Γ X, γ sheet Σ ' γγ Σ ' ββ Σ ' αα | Σ ' γβ | | Σ ' αγ | | Σ ' αβ |120600 Σ ' νν ( m e V ) -40 -20 0 ω (meV) Γ M, β sheet (a) (b) FIG. 13. Reconstructed self-energies in the quasiparticle basis. (a) Full matrix Σ (cid:48) νν (cid:48) ( ω, k ) calculated with Eq. 12 using theDMFT 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 θ = 45 ◦ (right). (b) Directly extracted Σ (cid:48) νν ( ω, k ν max ( ω )) (Sec. V A) compared to the Σ (cid:48) m ( ω, θ = 18 ◦ ) (Sec. V B) transformed toΣ (cid:48) νν ( ω, k ν max ( ω )) at 0, 18, and 45 ◦ . of the crystal-field splitting was observed in Ref. [21],presumably due to the larger interactions employed.Based on these considerations we calculate the corre-lated spectral function A ( ω, k ) (shown in Fig. 5 (a) of themain text) using the Hamiltonian with enhanced SOC( ˆ H DFT + ˆ H SOC λ DFT +∆ λ ) in combination with the frequency-dependent part of the non-SOC (diagonal) self-energy,but neglect the additional static part introduced byDMFT. Appendix C: Off-diagonal elements of Σ (cid:48) νν (cid:48) In Sec. V A we extracted Σ (cid:48) νν under the assumptionthat the off-diagonal elements can be neglected. To ob-tain insights about the size of the off-diagonal elementswe use Eq. 12 to calculate the full matrix Σ (cid:48) νν (cid:48) ( ω, k ) inthe band basis from the DMFT self-energy in the orbitalbasis Σ (cid:48) m ( ω ), as shown in Fig. 6 (a). This allows us toobtain the full self-energy matrix Σ (cid:48) νν (cid:48) ( ω, k ) at one spe-cific combination of k and ω . Note that for the resultspresented in Fig. 4 and Fig. 13 (b) this is not the case,because the extracted self-energies for each band corre-spond to different k ν max , which are further defined by theexperimental MDCs.In Fig. 13 (a) we show the result for two selected k points: on the β sheet for θ = 0 and on the γ sheetfor θ = 45 ◦ . For these k points the largest off-diagonalelement is Σ (cid:48) γβ , which is about 10 −
20% of the size of thediagonal elements. A scan performed for the whole k z =0 . π/c plane further confirms that | Σ (cid:48) ν (cid:54) = ν (cid:48) | is smaller than20 meV.However, when neglect the off-diagonal elements it isalso important to have a large enough energy separationof the bands. This can be understood by considering asimplified case of two bands ( ν, ν (cid:48) ) and rewriting Eq. 3, which determines the quasiparticle dispersion ω qp ν ( k ), as ω − ε ν ( k ) − Σ (cid:48) νν ( ω, k ) − Σ (cid:48) νν (cid:48) ( ω, k ) Σ (cid:48) ν (cid:48) ν ( ω, k ) ω − ε ν (cid:48) ( k ) − Σ (cid:48) ν (cid:48) ν (cid:48) ( ω, k ) = 0 . (C1)Setting the last term to zero, i.e., using the proceduredescribed in Sec. V A to extract Σ (cid:48) νν , is justified at ω = ω qp ν ( k ) as long asΣ (cid:48) νν ( ω, k ) (cid:29) Σ (cid:48) νν (cid:48) ( ω, k ) Σ (cid:48) ν (cid:48) ν ( ω, k ) ω − ε ν (cid:48) ( k ) − Σ (cid:48) ν (cid:48) ν (cid:48) ( ω, k ) (C2)In this condition the already small off-diagonal elementsenter quadratically, but also the denominator is not asmall quantity, because the energy separation of the barebands ( ε ν ( k ) − ε ν (cid:48) ( k )) is larger than the difference ofthe diagonal self-energies.By using the generalized version of Eq. C2 for all threebands, we find that the right-hand side of this equationis indeed less than 1 .
2% of Σ (cid:48) νν ( ω, k ν max ( ω )) for all ex-perimentally determined k ν max ( ω ). This means that forSr RuO treating each band separately when extractingΣ (cid:48) νν is well justified in the investigated energy range. Appendix D: Reconstruction of Σ (cid:48) νν In order to further test the validity of the local ansatz(Eq. 12) and establish the overall consistency of the twoprocedures used to extract the self-energy in Sec. V, weperform the following ‘reconstruction procedure’. We usethe Σ (cid:48) m ( ω, θ k ) (from Sec. V B) at one angle, e.g., θ = 18 ◦ ,and transform it into Σ (cid:48) νν ( ω, k ν max ( ω )) for other mea-sured angles, using Eq. 12. The good agreement betweenthe self-energy reconstructed in this manner (thin linesin Fig. 13 (b)) and its direct determination followingthe procedure of Sec. V A (dots) confirms the validityof the approximations used throughout Sec. V. 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