Highresolution photoemission on \ceSr2RuO4 reveals correlationenhanced effective spinorbit coupling and dominantly local selfenergies
Abstract
We explore the interplay of electronelectron correlations and spinorbit coupling in the model Fermi liquid \ceSr2RuO4 using laserbased angleresolved photoemission spectroscopy. Our precise measurement of the Fermi surface confirms the importance of spinorbit coupling in this material and reveals that its effective value is enhanced by a factor of about two, due to electronic correlations. The selfenergies for the and sheets are found to display significant angular dependence. By taking into account the multiorbital composition of quasiparticle states, we determine selfenergies associated with each orbital component directly from the experimental data. This analysis demonstrates that the perceived angular dependence does not imply momentumdependent manybody effects, but arises from a substantial orbital mixing induced by spinorbit coupling. A comparison to singlesite dynamical meanfield theory further supports the notion of dominantly local orbital selfenergies, and provides strong evidence for an electronic origin of the observed nonlinear frequency dependence of the selfenergies, leading to ‘kinks’ in the quasiparticle dispersion of \ceSr2RuO4.
present address: ]ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, United Kingdom
I Introduction
The layered perovskite \ceSr2RuO4 is an important model system for correlated electron physics. Its intriguing superconducting ground state, sharing similarities with superfluid He Maeno et al. (1994); Rice and Sigrist (1995); Mackenzie and Maeno (2003), has attracted much interest and continues to stimulate advances in unconventional superconductivity Mackenzie et al. (2017). Experimental evidence suggest oddparity spintriplet pairing, yet questions regarding the proximity of other order parameters, the nature of the pairing mechanism and the apparent absence of the predicted edge currents remain open Ishida et al. (1998); Anwar et al. (2016); Hicks et al. (2014); Scaffidi et al. (2014); Kirtley et al. (2007); Mackenzie and Maeno (2003); Mackenzie et al. (2017). Meanwhile, the normal state of \ceSr2RuO4 attracts interest as one of the cleanest oxide Fermi liquids Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003); Stricker et al. (2014). Its precise experimental characterization is equally important for understanding the unconventional superconducting ground state of \ceSr2RuO4 Maeno et al. (1994); Rice and Sigrist (1995); Mackenzie and Maeno (2003); Mackenzie et al. (2017); Ishida et al. (1998); Anwar et al. (2016); Hicks et al. (2014); Scaffidi et al. (2014); Kirtley et al. (2007); Raghu et al. (2010); Huo et al. (2013); Scaffidi et al. (2014); Komendová and BlackSchaffer (2017); Steppke et al. (2017), as it is for benchmarking quantitative manybody calculations Georges et al. (2013); Liebsch and Lichtenstein (2000); Mravlje et al. (2011); Zhang et al. (2016); Mravlje and Georges (2016); Kim et al. (2018).
Transport, thermodynamic and optical data of \ceSr2RuO4 display textbook Fermiliquid behavior below a crossover temperature of K Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003); Stricker et al. (2014). Quantum oscillation and angleresolved photoemission spectroscopy (ARPES) measurements Oguchi (1995); Mackenzie et al. (1996b, 1998); Bergemann et al. (2000); Damascelli et al. (2000); Iwasawa et al. (2005); Ingle et al. (2005); Iwasawa et al. (2010, 2012); Zabolotnyy et al. (2013); Burganov et al. (2016) further reported a strong enhancement of the quasiparticle effective mass over the bare band mass. Theoretical progress has been made recently in revealing the important role of the intraatomic Hund’s coupling as a key source of correlation effects in \ceSr2RuO4 Mravlje et al. (2011); de’ Medici et al. (2011); Georges et al. (2013). In this context, much attention was devoted to the intriguing properties of the unusual state above , which displays metallic transport with no signs of resistivity saturation at the MottIoffeRegel limit Tyler et al. (1998). Dynamical meanfield theory (DMFT) Georges et al. (1996) calculations have proven successful in explaining several properties of this intriguing metallic state, as well as in elucidating the crossover from this unusual metallic state into the Fermiliquid regime Mravlje et al. (2011); Georges et al. (2013); Stricker et al. (2014); Mravlje and Georges (2016); Deng et al. (2016); Zhang et al. (2016); Kim et al. (2018). Within DMFT, the selfenergies associated with each orbital component are assumed to be local. On the other hand, the lowtemperature Fermiliquid state is known to display strong magnetic fluctuations at specific wavevectors, as revealed, e.g., by neutron scattering Sidis et al. (1999); Steffens et al. (2018) and nuclear magnetic resonance spectroscopy (NMR) Ishida et al. (2001); Imai et al. (1998). These magnetic fluctuations were proposed early on to be an important source of correlations Rice and Sigrist (1995); Lou et al. (2003); Kim et al. (2017). In this picture, it is natural to expect strong momentum dependence of the selfenergy associated with these spin fluctuations. Interestingly, a similar debate was raised long ago in the context of liquid He, with ‘paramagnon’ theories emphasizing ferromagnetic spinfluctuations and ‘quasilocalized’ approaches à la AndersonBrinkman emphasizing local correlations associated with the strong repulsive hardcore, leading to increasing Mottlike localization as the liquid is brought closer to solidification (for a review, see Ref. Vollhardt (1984)).
In this work, we report on new insights into the nature of the Fermiliquid state of \ceSr2RuO4. Analyzing a comprehensive set of laserbased ARPES data with improved resolution and cleanliness, we reveal a strong angular (i.e., momentum) dependence of the selfenergies associated with the quasiparticle bands. We demonstrate that this angular dependence originates in the variation of the orbital content of quasiparticle states as a function of angle, and can be understood quantitatively. Introducing a new framework for the analysis of ARPES data for multiorbital systems, we extract the electronic selfenergies associated with the three Ruthenium orbitals with minimal theoretical input. We find that these orbital selfenergies have strong frequency dependence, but surprisingly weak angular (i.e., momentum) dependence. and can thus be considered local to a very good approximation. Our results provide a direct experimental demonstration that the dominant effects of correlations in \ceSr2RuO4 are weakly momentumdependent and can be understood from a local perspective, provided they are considered in relation to orbital degrees of freedom. 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 DMFT calculations and our ARPES data, and find good agreement with the measured quasiparticle dispersions and angular dependence of the effective masses.
The experimentally determined real part of the selfenergy displays strong deviations from the lowenergy Fermiliquid behavior for binding energies larger than . These deviations are reproduced by our DMFT calculations suggesting that the cause of these nonlinearities are local electronic correlations. Our results thus call for a revision of earlier reports of strong electronlattice coupling in \ceSr2RuO4 Aiura et al. (2004); Iwasawa et al. (2005); Ingle et al. (2005); Iwasawa et al. (2010); Kim et al. (2011); Iwasawa et al. (2013); Wang et al. (2017). We finally quantify the effective spinorbit coupling (SOC) strength and confirm its enhancement due to correlations predicted theoretically Liu et al. (2008); Zhang et al. (2016); Kim et al. (2018).
This article is organized as follows. In Sec. II, we briefly present the experimental method and report our main ARPES results for the Fermi surface and quasiparticle dispersions. In Sec. III, we introduce the theoretical framework on which our data analysis is based. In Sec. IV, we use our precise determination of the Fermi surface to reveal the correlationinduced enhancement of the effective SOC. In Sec. V we proceed with a direct determination of the selfenergies from the ARPES data. Sec. VI presents the DMFT calculations in comparison to experiments. Finally, our results are critically discussed and put in perspective in Secs. VII, VIII.
Ii Experimental Results
ii.1 Experimental methods
The single crystals of \ceSr2RuO4 used in our experiments were grown by the floating zone technique and showed a superconducting transition temperature of . ARPES measurements were performed with an MBS electron spectrometer and a narrow bandwidth () laser source from Lumeras that was operated at a repetition rate of with pulse length of the pump He et al. (2016). All experiments were performed at K using a cryogenic 6axes sample goniometer, as described in Ref. Hoesch et al. (2017). A combined energy resolution of was determined from the width of the FermiDirac distribution measured on a polycrystalline Au sample held at 4.2 K. The angular resolution was . In order to suppress the intensity of the surface layer states on pristine \ceSr2RuO4 Shen et al. (2001), we exposed the cleaved surfaces to L \ceCO at a temperature of K. Under these conditions, \ceCO preferentially fills surface defects and subsequently replaces apical oxygen ions to form a \ceRuCOO carboxylate in which the \ceC end of a bent \ceCO2 binds to \ceRu ions of the reconstructed surface layer Stöger et al. (2014).
ii.2 Experimental Fermi surface and quasiparticle dispersions
In Fig. 1 we show the Fermi surface and selected constant energy surfaces in the occupied states of \ceSr2RuO4. The rapid broadening of the excitations away from the Fermi level seen in the latter is typical for ruthenates and implies strong correlation effects on the quasiparticle properties. At the Fermi surface, one can readily identify the , and sheets that were reported earlier Mackenzie et al. (1996b); Bergemann et al. (2000); Damascelli et al. (2000). However, compared with previous ARPES studies we achieve a reduced line width and improved suppression of the surface layer states giving clean access to the bulk electronic structure. This is particularly evident along the Brillouin zone diagonal (X) where we can clearly resolve all band splittings.
In the following, we will exploit this advance to quantify the effects of SOC in \ceSr2RuO4 and to provide new insight into the renormalization of the quasiparticle excitations using minimal theoretical input only. To this end we acquired a set of 18 high resolution dispersion plots along radial space lines (as parameterized by the angle measured from the M direction). The subset of data shown in Fig. 2 (a) immediately reveals a rich behavior with a marked dependence of the lowenergy dispersion on the Fermi surface angle . Along the M highsymmetry line our data reproduce the large difference in Fermi velocity for the and sheet, which is expected from the different cyclotron masses deduced from quantum oscillations Mackenzie et al. (1996b, 1998); Bergemann et al. (2003) and was reported in earlier ARPES studies Shen et al. (2007); Zabolotnyy et al. (2013). Our systematic data, however, reveal that this difference gradually disappears towards the Brillouin zone diagonal , where all three bands disperse nearly parallel to one another. In Sec. IV we will show that this equilibration of the Fermi velocity can be attributed to the strong effects of SOC around the zone diagonal.
To quantify the angle dependence of from experiment, we determine the maxima of the momentum distribution curves (MDCs) over the range of  below the Fermi level and fit these space loci with a secondorder polynomial. We then define the Fermi velocity as the derivative of this polynomial at . This procedure minimizes artifacts due to the finite energy resolution of the experiment.
As shown in Fig. 2 (c), the Fermi velocities obtained in this way show an opposite trend with azimuthal angle for the two Fermi sheets. For the band we observe a gentle decrease of as we approach the X direction, whereas for the velocity increases by more than a factor of two over the same range
Before introducing the theoretical framework used to quantify the anisotropy of the selfenergy and the effects of SOC, we compare our data quantitatively to bulk sensitive quantum oscillation measurements. Using the experimental Fermi wave vectors and velocities determined from our data on a dense grid along the entire Fermi surface, we can compute the cyclotron masses measured by dHvA experiments, without relying on the approximation of circular Fermi surfaces and/or isotropic Fermi velocities used in earlier studies Baumberger et al. (2006); Shen et al. (2007); Zabolotnyy et al. (2013); Wang et al. (2017). Expressing the cyclotron mass as
(1) 
where is the Fermi surface volume, and using the data shown in Fig. 2 (c), we obtain and , in quantitative agreement with the values of and found in dHvA experiments Mackenzie et al. (1996b, 1998); Bergemann et al. (2003). We thus conclude that the quasiparticle states probed by our experiments are representative of the bulk of \ceSr2RuO4
Iii Theoretical Framework
The goal of this work is to extract directly from the ARPES data some key information about quasiparticle properties. We are in particular interested in: (i) a precise determination of the Fermi surface, (ii) the quasiparticle velocities and their renormalization by electronic correlations, and (iii) a direct determination of the electronic selfenergy.
In order to define the latter and assess the effect of electronic correlations we need to specify a oneparticle Hamiltonian as a reference point. At this stage, we keep the presentation general, but the particular choice of will be a focus of Sec. IV. The eigenstates of at a given quasimomentum and the corresponding eigenvalues define the ‘bare’ band structure of the system, with respect to which the selfenergy is defined from the interacting Green’s function
(2) 
In this expression and label the bands and denotes the binding energy counted from , which we set to zero throughout the rest of this work. The interacting value of the chemical potential sets the total electron number. Since can be conventionally included in , we shall omit it in the following. The electronic spectral function is related to the Green’s function
(3) 
It is this quantity which is most directly related to the ARPES signal: within the sudden approximation, in the absence of finalstate interactions, the ARPES intensity is given by convolved with a function representing the experimental resolution and assuming an average over the polarization. Here, is a matrix element and is the Fermi function. It is important to note that , and are in general nondiagonal matrices.
The Fermi surface of the system is the locus of zeroenergy excitations, and hence corresponds to the momenta which are the solutions of
(4) 
The dispersion relations of the quasiparticles are obtained as the solutions of
(5) 
In the above equations denotes the real part of the selfenergy. Its imaginary part has been neglected, i.e., we assume that quasiparticles are coherent with a lifetime much longer than . This assumption gradually breaks down as one departs from the Fermi level or as temperature is increased.
iii.1 Localized orbitals and electronic structure
Let us recall some of the important aspects of the electronic structure of \ceSr2RuO4. As shown in Sec. II.2, three bands, commonly denoted , cross the Fermi level. These bands correspond to states with symmetry deriving from the hybridization between localized Ru () orbitals and O states. Hence, we introduce a localized basis set of like orbitals , with basis functions conveniently labeled as . In practice, we use maximally localized Wannier functions Marzari and Vanderbilt (1997); Souza et al. (2001) constructed from the KohnSham eigenbasis of a nonSOC density functional theory (DFT) calculation (see appendix B.1 for details). We term the corresponding Hamiltonian . It is important to note that the choice of a localized basis set is not unique and other ways of defining these orbitals are possible (see, e.g., Ref. Lechermann et al. (2006)).
In the following this set of orbitals plays two important roles. First, they are atomcentered and provide a set of states localized in realspace . Secondly, the unitary transformation matrix to the band basis
(6) 
allows us to define an ‘orbital’ character of each band as . In the localizedorbital basis the oneparticle Hamiltonian is a nondiagonal matrix, which reads
(7) 
The selfenergy in the orbital basis is expressed as
(8) 
and conversely in the band basis as
(9) 
iii.2 Spinorbit coupling
We treat SOC as an additional term to , which is
independent of in the localizedorbital basis, but leads to a mixing of the individual orbitals. The singleparticle SOC term for atomic orbitals projected to the subspace reads
(10) 
where are the projected angular momentum matrices, are Pauli matrices and will be referred to in the following as the SOC coupling constant. As documented in appendix B.1, the eigenenergies of a DFT+SOC calculation are well reproduced by + with .
Iv Enhanced effective spinorbit coupling and singleparticle Hamiltonian
The importance of SOC for the lowenergy physics of \ceSr2RuO4 has been pointed out by several authors Ng and Sigrist (2002); Eremin et al. (2002); Haverkort et al. (2008); Iwasawa et al. (2010); Puetter and Kee (2012); Veenstra et al. (2014); Scaffidi et al. (2014); Steppke et al. (2017); Zhang et al. (2016); Kim et al. (2018). SOC lifts degeneracies found in its absence and causes a momentum dependent mixing of the orbital composition of quasiparticle states, which has nontrivial implications for superconductivity Veenstra et al. (2014); Scaffidi et al. (2014); Steppke et al. (2017). Signatures of SOC have been detected experimentally on the Fermi surface of \ceSr2RuO4 in the form of a small protrusion of the sheet along the zone diagonal Haverkort et al. (2008); Iwasawa et al. (2010) and as a degeneracy lifting at the band bottom of the sheet Veenstra et al. (2014). These studies reported an overall good agreement between the experimental data and the effects of SOC calculated within DFT Haverkort et al. (2008); Iwasawa et al. (2010); Veenstra et al. (2014). This is in apparent contrast to more recent DMFT studies of \ceSr2RuO4, which predict large but frequency independent offdiagonal contributions to the local selfenergy that can be interpreted as a contribution to the effective coupling strength Zhang et al. (2016); Kim et al. (2018), consistent with general perturbationtheory considerations Liu et al. (2008).
In the absence of SOC, DFT yields a quasicrossing between the and Fermi surface sheets a few degrees away from the zone diagonal, as displayed on Fig. 3 (a). Near such a point we expect the degeneracy to be lifted by SOC, leading to a momentum splitting and to an energy splitting of Kim et al. (2018), as depicted schematically in Fig. 3 (e). In these expressions, , with and the bare band velocities in the absence of SOC and correlations, and involves the quasiparticle residues associated with each band (also in the absence of SOC).
It is clear from these expressions that a quantitative determination of is not possible from experiment alone. Earlier studies on \ceSr2RuO4 Veenstra et al. (2014) and ironbased superconductors Borisenko et al. (2015), have interpreted the energy splitting at avoided crossings as a direct measure of the SOC strength . However, in interacting systems is not a robust measure of SOC since correlations can both enhance by enhancing and reduce it via the renormalization factor . We thus quantify the enhancement of SOC from the momentum splitting , which is not renormalized by the quasiparticle residue . The experimental splitting at the avoided crossing between the and Fermi surface sheets indicated in Fig. 3 (a) is Å whereas DFT predicts Å. We thus obtain an effective SOC strength meV, in quantitative agreement with the predictions in Refs. Zhang et al. (2016); Kim et al. (2018). We note that despite this large enhancement of the effective SOC, the energy splitting remains smaller than as illustrated in Fig. 3 (e). When deviations from linearity in band dispersions are small, the splitting is symmetric around the and can thus be determined from the occupied states probed in experiment. Direct inspection of the data in Fig. 3 (f) yields , which is about of and thus clearly not a good measure of SOC.
The experimental splitting is slightly larger than that expected from the expression and our theoretical determination of and at the Fermi surface. This can be attributed to the energy dependence of , which, in \ceSr2RuO4, is not negligible over the energy scale of SOC. Note that the SOCinduced splitting of the bands at the point reported in Ref. Veenstra et al. (2014) can also be explained by the competing effects of enhancement by correlations and reduction by the quasiparticle weight as shown in Ref. Kim et al. (2018). We also point out that the equilibration of quasiparticle velocities close to the diagonal, apparent from Figs. 2 (a,c) and 3 (f) is indeed the behavior expected close to an avoided crossing Kim et al. (2018).
Including the enhanced SOC determined from this noncrossing gap leads to a much improved theoretical description of the entire Fermi surface
Importantly, the change in Fermi surface sheet volume with the inclusion of is not driven by a change in the crystal field splitting between the and orbitals (see appendix B). The volume change occurs solely because of a further increase in the orbital mixing induced by the enhanced SOC. As shown in Fig. 3 (g), this mixing is not limited to the vicinity of the avoided crossing but extends along the entire Fermi surface. For we find a minimal and mixing for the and bands of % along the M direction with a monotonic increase to % along the Brillouin zone diagonal X.
We note that this mixing varies with the perpendicular momentum . However, around the experimental value of the variation is weak
V Experimental determination of selfenergies
v.1 Selfenergies in the quasiparticle/band basis
Working in the band basis, i.e., with the eigenstates of , the maximum of the ARPES intensity for a given binding energy (maximum of the MDCs) corresponds to the momenta which satisfy (following Eq. 5): . Hence, for each binding energy, each azimuthal cut, and each sheet of the quasiparticle dispersions, we fit the MDCs and determine the momentum at their maximum. Using the value of at this momentum yields the following quantity
(11) 
This equation corresponds to the simple construction illustrated graphically in Fig. 4 (b), and it is a standard way of extracting a selfenergy from ARPES, as used in previous works on several materials Hengsberger et al. (1999); Lanzara et al. (2001); Iwasawa et al. (2012, 2013); Tamai et al. (2013). We note that this procedure assumes that the offdiagonal components can be neglected for states close to the Fermi surface (i.e., for small and close to a Fermi crossing). This assumption can be validated, as shown in appendix C. When performing this analysis, we only include the sheet for . Whenever the constraint on the selfenergy is not precisely obeyed, a small shift is applied to set it to zero. We chose this procedure to correct for the minor differences between the experimental and the reference Fermi wavevectors because we attribute these differences predominantly to experimental artifacts.
The determined selfenergies for each band and the different values of are depicted in Fig. 4 (a,c). For the and sheets they show a substantial dependence on the azimuthal angle. Around M we find that exceeds by almost a factor of two (at ), whereas they essentially coincide along the zone diagonal (X). This change evolves as a function of and occurs via a simultaneous increase in and a decrease in for all energies as is increased from (M) to (X). In order to better visualize this angular dependence, a compilation of for different values of is displayed in Fig. 4 (c).
v.2 Accounting for the angular dependence: local selfenergies in the orbital basis
In this section, we introduce a different procedure for extracting selfenergies from ARPES, by working in the orbital basis . We do this by making two key assumptions:

We assume that the offdiagonal components are negligible, i.e., . Let us note that in \ceSr2RuO4 even a independent selfenergy has nonzero offdiagonal elements if + is considered. Using DMFT, these offdiagonal elements have been shown to be very weakly dependent on frequency in this material (Kim et al., 2018), leading to the notion of a static correlation enhancement of the effective SOC (). In the present work, these offdiagonal frequencyindependent components are already incorporated into (see Sec. IV), and thus the frequencydependent part of the selfenergy is (approximately) orbital diagonal by virtue of the tetragonal crystal structure.

We assume that the diagonal components of the selfenergy in the orbital basis depend on the momentum only through the azimuthal angle : . We neglect the dependence on the momentum which is parallel to the angular cut.
Under these assumptions, the equation determining the quasiparticle dispersions reads
(12) 
In this equation, we have neglected the lifetime effects associated with the imaginary part . In order to extract the functions directly from the ARPES data, we first determine the peak positions for MDCs at a given angle and binding energy . We then compute (for the same and ) the matrix and similarly , for the and band MDCs, and , respectively. In terms of these matrices, the quasiparticle equations (12) read
(13) 
However, when taking symmetry into account, the selfenergy has only two independent components: and . Hence, we only need two of the above equations to solve for the two unknown components of the selfenergy. This means that we can also extract a selfenergy in the directions where only two bands ( and ) are present in the considered energy range of , e.g., along M. The resulting functions determined at several angles are displayed in Fig. 4 (d). It is immediately apparent that, in contrast to , the selfenergies in the orbital basis do not show a strong angular (momentum) dependence, but rather collapse into two sets of points, one for the orbital and one for the orbitals. Thus, we reach the remarkable conclusion that the angular dependence of the selfenergy in the orbital basis is negligible, within the range of binding energies investigated here: . This implies that a good approximation of the full momentum and energy dependence of the selfenergy in the band (quasiparticle) basis is given by
(14) 
The physical content of this expression is that the angular (momentum) dependence of the quasiparticle selfenergies emphasized above is actually due to the matrix elements defined in Eq. 6. In \ceSr2RuO4 the angular dependence of these matrix elements is mainly due to the SOC, as seen from the variation of the orbital content of quasiparticles in Fig. 3 (g). In appendix D we show the backtransform of into . The good agreement with directly extracted from experiment further justifies the above expression and also confirms the validity of the approximations made throughout this section.
Finally, we stress that expression (14) precisely coincides with the ansatz made by DMFT: within this theory, the selfenergy is approximated as a local (independent) object when expressed in a basis of localized orbitals, while it acquires momentum dependence when transformed to the band basis.
Vi Comparison to Dynamical MeanField Theory
In this section we perform an explicit comparison of the measured quasiparticle dispersions and selfenergies to DMFT results. The latter are based on the Hamiltonian , to which the HubbardKanamori interaction with onsite interaction and Hund’s coupling Mravlje et al. (2011) is added. For the details of the DMFT calculation and especially the treatment of SOC in this framework, we refer the reader to appendix B. There, we also comment on some of the limitations and shortcomings of the current state of the art for DFT+DMFT calculations in this context. Fig. 5 (a) shows the experimental quasiparticle dispersion extracted from our ARPES data (circles) on top of the DMFT spectral function displayed as a colorintensity map. Clearly, the theoretical results are in near quantitative agreement with the data: both the strong renormalization of the Fermi velocity and the angulardependent curvature of the quasiparticle bands are very well reproduced. The small deviations in Fermi wave vectors discernible in Fig. 5 are consistent with Fig. 3(c) and the overall experimental precision of the Fermi surface determination.
In Fig. 6 (a), we compare the experimental selfenergies for each orbital with the DMFT results. The overall agreement is notable. At lowenergy, the selfenergies are linear in frequency and the agreement is excellent. The slope of the selfenergies in this regime controls the angulardependence of the effective mass renormalisation. Using the local ansatz (14) into the quasiparticle dispersion equation, and performing an expansion around , we obtain
(15) 
In Fig. 6 (b), we show for the and bands using the DMFT values and obtained at (appendix B.2). The overall angular dependence and the absolute value of the band mass enhancement is very well captured by DMFT, while the band is a bit overestimated. Close to the zonediagonal (), the two mass enhancements are approximately equal, due to the strong orbital mixing induced by the SOC.
Turning to larger binding energies, we see that the theoretical is in remarkable agreement with the experimental data over the full energy range of  covered in our experiments. Both the theoretical and experimental selfenergies deviate significantly from the linear regime down to low energies (), causing curved quasiparticle bands with progressively steeper dispersion as the energy increases (Fig. 5). In contrast, the agreement between theory and experiment for the selfenergy is somewhat less impressive at binding energies larger than . Our DMFT selfenergy overestimates the strength of correlations in this regime (by ), with a theoretical slope larger than the experimental one. Correspondingly, the quasiparticle dispersion is slightly steeper in this regime than the theoretical result, as can be also seen in Fig. 5.
There may be several reasons for this discrepancy. Even while staying in the framework of a local selfenergy, we note that the present DMFT calculation is performed with an onsite value of which is the same for all orbitals. Earlier cRPA calculations have suggested that this onsite interaction is slightly larger for the orbital ( and ) Mravlje et al. (2011) and recent work has advocated the relevance of this for DFT+DMFT calculations of \ceSr2RuO4 Zhang et al. (2016). Another possible explanation is that this discrepancy is actually a hint of some momentum dependent contribution to the selfenergy, especially dependence on momentum perpendicular to the Fermi surface. We note in this respect that the discrepancy is larger for the sheets which have dominant character. These orbitals have, in the absence of SOC, a strong onedimensional character, for which momentum dependence is definitely expected and DMFT is less appropriate. Furthermore, these FS sheets are also the ones associated with nesting and spindensity wave correlations, which are expected to lead to an additional momentumdependence of the selfenergy. We further discuss possible contributions of spin fluctuations in Sec. VIII.
Vii Kinks
The selfenergies shown in Figs. 4 and 6 display a fairly smooth curvature, rather than pronounced ‘kinks’. Over a larger range, however, from DMFT does show an energy scale marking the crossover from the strongly renormalized lowenergy regime to weakly renormalized excitations. This is illustrated in the inset to Fig. 6 (a). Such purely electronic kinks were reported in DMFT calculations of a generic system with MottHubbard sidebands Byczuk et al. (2007) and have been abundantly documented since then Raas et al. (2009); Mravlje et al. (2011); Held et al. (2013); Deng et al. (2013); Žitko et al. (2013); Stricker et al. (2014). In \ceSr2RuO4 they are associated with the crossover from the Fermiliquid behavior into a more incoherent regime Mravlje et al. (2011); Georges et al. (2013). The near quantitative agreement of the frequency dependence of the experimental selfenergies and our singlesite DMFT calculation provides strong evidence for the existence of such electronic kinks in \ceSr2RuO4. In addition, it implies that the local DMFT treatment of electronic correlations is capturing the dominant effects.
Focusing on the lowenergy regime, we find deviations from the linear form characteristic of a Fermi liquid for , irrespective of the basis. However, this is only an upper limit for the Fermiliquid energy scale in \ceSr2RuO4. Despite the improved resolution of our experiments, we cannot exclude an even lower crossover energy to nonFermiliquid like excitations. We note that this is consistent with the crossover temperature of reported from transport and thermodynamic experiments Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003).
The overall behavior of including the energy range where we find strong changes in the slope agrees with previous photoemission experiments, which were interpreted as evidence for electronphonon coupling Aiura et al. (2004); Iwasawa et al. (2005); Ingle et al. (2005); Iwasawa et al. (2010, 2013). Such an interpretation, however, relies on a linear Fermiliquid regime of electronic correlations over the entire phonon bandwidth of Braden et al. (2007), which is inconsistent with our DMFT calculations. Moreover, attributing the entire curvature of in our data to electronphonon coupling would result in unrealistic coupling constants far into the polaronic regime, which is hard to reconcile with the transport properties of \ceSr2RuO4 Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003). We also note that a recent scanning tunneling microscope (STM) study reported very strong kinks in the and sheets of \ceSr2RuO4 Wang et al. (2017), which is inconsistent with our data. We discuss the reason for this discrepancy in appendix A.
Viii Discussion and Perspectives
In this article, we have reported on highresolution ARPES measurements which allow for a determination of the Fermi surface and quasiparticle dispersions of \ceSr2RuO4 with unprecedented accuracy. Our data reveal an enhancement (by a factor of about two) of the splitting between Fermi surface sheets along the zone diagonal, in comparison to the DFT value. This can be interpreted as a correlationinduced enhancement of the effective SOC, an effect predicted theoretically Liu et al. (2008); Zhang et al. (2016); Kim et al. (2018) and demonstrated experimentally here for this material, for the first time.
Thanks to the high resolution, we have been able to determine the electronic selfenergies directly from the ARPES data, using both a standard procedure applied in the band (quasiparticle) basis as well as a novel procedure, introduced in the present article, in the orbital basis. Combining these two approaches, we have demonstrated that the large angular (momentum) dependence of the quasiparticle selfenergies and dispersions can be mostly attributed to the fact that quasiparticle states have an orbital content which is strongly angular dependent, due to the SOC. Hence, assuming selfenergies which are frequencydependent but essentially independent of angle (momentum), when considered in the orbital basis, is a very good approximation. This provides a direct experimental validation of the DMFT ansatz, and indeed, the comparison between the ARPES data and DMFT calculations is found to be remarkably accurate. The key importance of atomiclike orbitals in correlated insulators is well established Kugel and Khomskii (1982); Tokura and Nagaosa (2000). However, the fact that orbitals retain such considerable physical relevance even in an metal in the lowtemperature Fermiliquid regime is a remarkable fact. Thinking only in terms of quasiparticles associated with a specific Fermi surface sheet is insufficient to unravel the physics of correlated metals such as \ceSr2RuO4. As the present work demonstrates, hidden simplicity in the nature of correlations is unraveled when instead considering the orbital degrees of freedom, and taking into account the mixed orbital content of each quasiparticle state. Beyond \ceSr2RuO4, this is an observation of general relevance to metals with strong correlations (see, e.g., Ref. Miao et al. (2016); Sprau et al. (2017) in the context of ironbased superconductors)
Notwithstanding its success, the excellent agreement of the DMFT results with ARPES data does raise puzzling questions. \ceSr2RuO4 is known to be host to strong magnetic fluctuations Sidis et al. (1999); Steffens et al. (2018); Ishida et al. (2001); Imai et al. (1998), with a strong peak in its spin response close to the spindensity wave (SDW) vector , as well as quasiferromagnetic fluctuations which are broader in momentum around . Indeed, tiny amounts of substitutional impurities induce longrange magnetic order in this material, of either SDW of ferromagnetic type Braden et al. (2002); Ortmann et al. (2013). Hence, it is a prominent open question to understand how these longwavelength fluctuations affect the physics of quasiparticles in the Fermiliquid state. Singlesite DMFT does not capture this feedback, and the excellent agreement with the overall quasiparticle physics must imply that these effects have a comparatively smaller magnitude than the dominant local effect of correlations (onsite and especially Hund’s ) captured by DMFT. A closely related question is how much momentum dependence is present in the lowenergy (Landau) interactions between quasiparticles. These effects are expected to be fundamental for subsequent instabilities of the Fermi liquid, into either the superconducting state in pristine samples or magnetic ordering in samples with impurities. Making progress on this issue is also key to the understanding of the superconducting state of \ceSr2RuO4, for which the precise nature of the pairing mechanism as well as symmetry of the order parameter are still open questions Mackenzie et al. (2017).
Acknowledgements.
The experimental work has been supported by the ERC, the Scottish Funding Council, the UK EPSRC and the SNSF. Theoretical work was supported by the European Research Council grant ERC319286QMAC and by the SNSF (NCCR MARVEL). The Flatiron Institute is a division of the Simons Foundation. AG and MZ gratefully acknowledge useful discussions with Gabriel Kotliar, Andrew J. Millis and Jernej Mravlje.Appendix A Bulk and surface electronic structure of \ceSr2RuO4
In Fig. 7 we compare the data presented in the main text with data from a pristine cleave taken with at the SIS beamline of the Swiss Light Source. This comparison confirms the identification of bulk and surface bands by Shen et al. Shen et al. (2001). In particular, we find that the larger sheet has bulk character. This band assignment is used by the vast majority of subsequent ARPES publications Shen et al. (2007); Aiura et al. (2004); Wang et al. (2004); Iwasawa et al. (2005); Ingle et al. (2005); Kidd et al. (2005); Iwasawa et al. (2010); Zabolotnyy et al. (2013), except for Ref. Kim et al. (2011), which reports a dispersion with much lower Fermi velocity and a strong kink at for the smaller sheet that we identify as a surface band.
Wang et al. Wang et al. (2017) have recently probed the lowenergy electronic structure of \ceSr2RuO4 by STM. Analyzing quasiparticle interference patterns along the X and M highsymmetry directions, they obtained band dispersions with low Fermi velocities and strong kinks at and . In Fig. 8 we compare the band dispersion reported by Wang et al. with our ARPES data. Along both highsymmetry 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 find a striking similarity between the STM data along M and the band commonly identified as the surface band. We thus conclude that the experiments reported in Ref. Wang et al. (2017) probed the surface states of \ceSr2RuO4. This is fully consistent with the enhanced lowenergy renormalization of the surface bands seen in ARPES Kondo et al. (2016).
Appendix B Computational details
b.1 DFT and model Hamiltonian
We generate our theoretical model Hamiltonian from maximallylocalized Wannier function Marzari and Vanderbilt (1997); Souza et al. (2001) for the three Ru orbitals. These are constructed on a grid based on a nonSOC DFT calculation using WIEN2k Blaha et al. (2001) with the GGAPBE functional Perdew et al. (1996), wien2wannier Kuneš et al. (2010) and Wannier90 Mostofi et al. (2008). The DFT calculation is performed with lattice parameters from Ref. Vogt and Buttrey (1995) (measured at ) and converged with twice as many points in each dimension.
The eigenenergies of the resulting Wannier Hamiltonian, , accurately reproduce the DFT band structure (Fig. 9 (a)). To take SOC into account, we add the local singleparticle term , as given in Eq. 10, with coupling constant . In Fig. 9 (b) we show that the eigenenergies of + are in nearly perfect agreement with the DFT+SOC band structure at a value of . Our model Hamiltonian provides the reference point to which we define a selfenergy, but it is also a perfect playground to study the change in the Fermi surface under the influence of SOC and the crystal field splitting between the and orbitals. Concerning the latter, the constructed has already a splitting of .
In the following, we will confirm that the best agreement with the experimental Fermi surface is found with an effective SOC of , but at the same time keeping the crystal field splitting unchanged. We compare in Fig. 10 the experimental Fermi surface (dashed lines) to the one of +. The Fermi surfaces for additionally introduced crystal field splittings between and are shown with solid lines in different shades of red. In contrast to the Fermi surface without SOC (, top left panel), the Fermi surfaces with the DFT SOC of (top right panel) resembles the overall structure of the experimental Fermi surface. However, the areas of the and sheets are too large and the sheet is too small. Importantly, the agreement cannot be improved by adding . For example, along M a of would move the Fermi surface closer to the experiment, but, on the other hand, along X a of would provide the best agreement. The situation is different if we consider an enhanced SOC of (bottom right panel). Then, we find a nearly perfect agreement with experiment without any additional crystal field splitting (). At an even higher SOC of (bottom left panel) we see again major discrepancies, but with an opposite trend: The and sheets are now too small and the sheet is too large. Like in the case of this can not be cured by an adjustment of .
b.2 Dmft
We perform singlesite DMFT calculation with the TRIQS/DFTTools Aichhorn et al. (2016) package for and HubbardKanamori interactions with a screened Coulomb repulsion and a Hund’s coupling based on previous works Mravlje et al. (2011); Kim et al. (2018). The impurity problem is solved on the imaginarytime axis with the TRIQS/CTHYB Seth et al. (2016) solver at a temperature of . The employed opensource software tools are based on the TRIQS library Parcollet et al. (2015). We assume an orbitalindependent double counting, and hence it can be absorbed into an effective chemical potential, which is adjusted such that the filling is equal to four electrons. For the analytic continuation of the selfenergy to the realfrequency axis we employ three different methods: Padé approximants (using TRIQS Parcollet et al. (2015)), Stochastic continuation (after Beach Beach (2004)) and Maximum Entropy (using TRIQS/maxent TRI ()). In the relevant energy range from to the difference in the resulting selfenergies is smaller than the experimental resolution (Fig. 11). The averaged quasiparticle renormalizations (of the three continuations) are: and . For all other results presented in the main text the Padé solution has been used.
Our calculations at a temperature of use , as the sign problem prohibits reaching such low temperatures with SOC included. Nevertheless, calculations with SOC were successfully carried out at a temperature of using CTINT Zhang et al. (2016) and at using CTHYB with a simplified twodimensional tightbinding model Kim et al. (2018). These works pointed out that electronic correlations in \ceSr2RuO4 lead to an enhanced SOC. To be more precise, Kim et al. Kim et al. (2018) observed that electronic correlations in this material are described by a selfenergy with diagonal elements close to the ones without SOC plus, to a good approximation, frequencyindependent offdiagonal elements, which can be absorbed in a static effective SOC strength of – this is the approach followed in the present article.
In addition to the enhancement of SOC it was observed that DFT+DMFT also leads to an enhancement of the crystal field splitting Zhang et al. (2016); Kim et al. (2018). In our DFT+DMFT calculation without SOC this results in , which would move the sheet nearly to van Hove filling and consequently worsen the agreement with the experimental Fermi surface (see bottom right panel of Fig. 10). Different roots of this discrepancy are possible, ranging from orbitaldependent double counting corrections to, in general, DFT being not ideal as ‘noninteracting’ reference point for DMFT. Importantly, Zhang et al. Zhang et al. (2016) showed that by considering the anisotropy of the Coulomb tensor the additional crystal field splitting is suppressed and consequently the disagreement between theory and experiment can be cured.
Based on these considerations we calculate the correlated spectral function (shown in Fig. 5 (a) of the main text) using the Hamiltonian with enhanced SOC (+) in combination with the frequencydependent part of the nonSOC (diagonal) selfenergy, but neglect the additional static part introduced by DMFT.
Appendix C Offdiagonal elements of
In Sec. V.1 we extracted under the assumption that the offdiagonal elements can be neglected. To obtain insights about the size of the offdiagonal elements we use Eq. 14 to calculate the full matrix in the band basis from the DMFT selfenergy in the orbital basis , as shown in Fig. 6 (a). This allows us to obtain the full selfenergy matrix at one specific combination of and . Note that for the results presented in Fig. 4 and Fig. 12 (b) this is not the case, because the extracted selfenergies for each band correspond to different , which are further defined by the experimental MDCs.
In Fig. 12 (a) we show the result for two selected points: on the sheet for and on the sheet for . For these points the largest offdiagonal element is , which is about of the size of the diagonal elements. A scan performed for the whole plane further confirms that is smaller than .
However, when neglect the offdiagonal elements it is also important to have a large enough energy separation of the bands. This can be understood by considering a simplified case of two bands () and rewriting Eq. 5, which determines the quasiparticle dispersion , as
(16) 
Setting the last term to zero, i.e., using the procedure described in Sec. V.1 to extract , is justified at as long as
(17) 
In this condition the already small offdiagonal elements enter quadratically, but also the denominator is not a small quantity, because the energy separation of the bare bands () is larger than the difference of the diagonal selfenergies.
By using the generalized version of Eq. 17 for all three bands, we find that the righthand side of this equation is indeed less than of for all experimentally determined . This means that for \ceSr2RuO4 treating each band separately when extracting is well justified in the investigated energy range.
Appendix D Reconstruction of
In order to further test the validity of the local ansatz (Eq. 14) and establish the overall consistency of the two procedures used to extract the selfenergy in Sec. V, we perform the following ‘reconstruction procedure’. We use the (from Sec. V.2) at one angle, e.g., , and transform it into for other measured angles, using Eq. 14. The good agreement between the selfenergy reconstructed in this manner (thin lines in Fig. 12 (b)) and its direct determination following the procedure of Sec. V.1 (dots) confirms the validity of the approximations used throughout Sec. V. It also shows that the origin of the strong momentum dependence of is almost entirely due to the momentum dependence of the orbital content of quasiparticle states, i.e., of . In \ceSr2RuO4 the momentum dependence of these matrix elements is mainly due to the SOC.
Footnotes
 We note that radial space cuts are not exactly perpendicular to the Fermi surface. This can cause the velocities and given here to deviate by up to 10% from the Fermi velocity. However, since we evaluate the experimental and theoretical dispersion along the same space cut, this effect cancels in Fig. 2 (d) and does not affect the selfenergy determination in Sec. V
 We attribute the higher Fermi velocities reported in some earlier studies Iwasawa et al. (2005, 2010, 2012); Burganov et al. (2016) to the lower energy resolution, which causes an extended range near the Fermi level where the dispersion extracted from fits to individual MDCs is artificially enhanced, rendering a precise determination of difficult. The much lower value of along M reported in Ref. Wang et al. (2017) corresponds to the surface band, as shown in appendix A.
 By restricting the Hamiltonian to the subspace we neglect the  coupling terms of . This approximation is valid as long as the  crystal field splitting is large in comparison to , which is the case for \ceSr2RuO4.
 Ref. Rozbicki et al. (2011) noted that an enhance effective SOC also improves the description of de Haas van Alphen data.
 Using the free electron final state approximation, we obtain assuming an inner potential relative to of eV.
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