PHYSICAL REVIEW X 9, 021048 (2019)

High-Resolution Photoemission on Sr2RuO4 Reveals Correlation-Enhanced Effective Spin-Orbit Coupling and Dominantly Local Self-Energies

A. Tamai,1,* M. Zingl,2 E. Rozbicki,3 E. Cappelli,1 S. Riccò,1 A. de la Torre,1 S. McKeown Walker,1 F. Y. Bruno,1 † P. D. C. King,3 W. Meevasana,4 M. Shi,5 M. Radović,5 N. C. Plumb,5 A. S. Gibbs,3, A. P. Mackenzie,6,3 C. Berthod,1 H. U. R. Strand,2 M. Kim,7,8 A. Georges,9,2,8,1 and F. Baumberger1,5 1Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland 2Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA 3SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, United Kingdom 4School of Physics, Suranaree University of Technology and Synchrotron Light Research Institute, Nakhon Ratchasima, 30000, Thailand and Thailand Center of Excellence in Physics, CHE, Bangkok, 10400, Thailand 5Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 6Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany 7Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA 8Centre de Physique Th´eorique, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France 9Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France

(Received 16 December 2018; revised manuscript received 27 March 2019; published 6 June 2019)

We explore the interplay of electron-electron correlations and spin-orbit coupling in the model Fermi liquid Sr2RuO4 using laser-based angle-resolved photoemission spectroscopy. Our precise measurement of the Fermi surface confirms the importance of spin-orbit coupling in this material and reveals that its effective value is enhanced by a factor of about 2, due to electronic correlations. The self-energies for the β and γ sheets are found to display significant angular dependence. By taking into account the multi-orbital composition of quasiparticle states, we determine self-energies associated with each orbital component directly from the experimental data. This analysis demonstrates that the perceived angular dependence does not imply momentum-dependent many-body effects but arises from a substantial orbital mixing induced by spin-orbit coupling. A comparison to single-site dynamical mean-field theory further supports the notion of dominantly local orbital self-energies and provides strong evidence for an electronic origin of the observed nonlinear frequency dependence of the self-energies, leading to “kinks” in the quasiparticle dispersion of Sr2RuO4.

DOI: 10.1103/PhysRevX.9.021048 Subject Areas: Condensed Matter Physics, Strongly Correlated Materials

I. INTRODUCTION superconducting ground state, sharing similarities with superfluid 3He [1–3], has attracted much interest and The layered perovskite Sr2RuO4 is an important model continues to stimulate advances in unconventional super- system for correlated electron physics. Its intriguing conductivity [4]. Experimental evidence suggests odd-parity spin-triplet pairing, yet questions regarding the proximity of

* other order parameters, the nature of the pairing mechanism, Corresponding author. and the apparent absence of the predicted edge currents [email protected] † – Present address: ISIS Facility, Rutherford Appleton Labora- remain open [3 9]. Meanwhile, the normal state of Sr2RuO4 tory, Chilton, Didcot OX11 OQX, United Kingdom. attracts interest as one of the cleanest oxide Fermi liquids [10–13]. Its precise experimental characterization is Published by the American Physical Society under the terms of equally important for understanding the unconventional the Creative Commons Attribution 4.0 International license. superconducting ground state of Sr2RuO4 [1–9,14–17],as Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, it is for benchmarking quantitative many-body calculations and DOI. [18–23].

2160-3308=19=9(2)=021048(18) 021048-1 Published by the American Physical Society A. TAMAI et al. PHYS. REV. X 9, 021048 (2019)

Transport, thermodynamic, and optical data of Sr2RuO4 orbital degrees of freedom. One of the novel aspects of our display textbook Fermi-liquid behavior below a crossover work is to directly put the locality ansatz underlying DMFT ≈ 25 – temperature of TFL K [10 13]. Quantum oscillation to the experimental test. We also perform a direct com- and angle-resolved photoemission spectroscopy (ARPES) parison between DMFT calculations and our ARPES data, measurements [24–34] further reported a strong enhance- and we find good agreement with the measured quasipar- ment of the quasiparticle effective mass over the bare band ticle dispersions and angular dependence of the effective mass. Theoretical progress has been made recently in masses. revealing the important role of the intra-atomic Hund’s The experimentally determined real part of the self- coupling as a key source of correlation effects in Sr2RuO4 energy displays strong deviations from the low-energy [18,20,35]. In this context, much attention was devoted to Fermi-liquid behavior Σ0ðωÞ ∼ ωð1 − 1=ZÞþ for bind- ω the intriguing properties of the unusual state above TFL, ing energies j j larger than about 20 meV. These deviations which displays metallic transport with no signs of resis- are reproduced by our DMFT calculations, suggesting that tivity saturation at the Mott-Ioffe-Regel limit [36]. the cause of these nonlinearities are local electronic Dynamical mean-field theory (DMFT) [37] calculations correlations. Our results thus call for a revision of earlier have proven successful in explaining several properties reports of strong electron-lattice coupling in Sr2RuO4 of this intriguing metallic state, as well as in elucidating [29–31,46–50]. We finally quantify the effective spin-orbit the crossover from this unusual metallic state into the coupling (SOC) strength and confirm its enhancement due Fermi-liquid regime [13,18,20–23,38]. Within DMFT, to correlations predicted theoretically [21,23,51]. the self-energies associated with each orbital component This article is organized as follows. In Sec. II, we briefly are assumed to be local. On the other hand, the low- present the experimental method and report our main temperature Fermi-liquid state is known to display strong ARPES results for the Fermi surface and quasiparticle magnetic fluctuations at specific wave vectors, as revealed, dispersions. In Sec. III, we introduce the theoretical e.g., by neutron scattering [39,40] and nuclear magnetic framework on which our data analysis is based. In resonance (NMR) spectroscopy [41,42]. These magnetic Sec. IV, we use our precise determination of the Fermi fluctuations were proposed early on to be an important surface to reveal the correlation-induced enhancement of source of correlations [2,43,44]. In this picture, it is natural the effective SOC. In Sec. V, we proceed with a direct to expect strong momentum dependence of the self-energy determination of the self-energies from the ARPES data. associated with these spin fluctuations. Interestingly, a Section VI presents the DMFT calculations in comparison similar debate was raised long ago in the context of liquid to experiments. Finally, our results are critically discussed 3He, with “paramagnon” theories emphasizing ferromag- and put into perspective in Secs. VII and VIII. netic spin fluctuations and “quasilocalized” approaches ala` Anderson-Brinkman emphasizing local correlations asso- II. EXPERIMENTAL RESULTS ciated with the strong repulsive hard core, leading to increasing Mott-like localization as the liquid is brought A. Experimental methods closer to solidification (for a review, see Ref. [45]). The single crystals of Sr2RuO4 used in our experiments In this work, we report on new insights into the nature were grown by the floating zone technique and showed a of the Fermi-liquid state of Sr2RuO4. Analyzing a com- superconducting transition temperature of Tc ¼ 1.45 K. prehensive set of laser-based ARPES data with improved ARPES measurements were performed with a MBS elec- resolution and cleanliness, we reveal a strong angular (i.e., tron spectrometer and a narrow-bandwidth 11-eV (113-nm) momentum) dependence of the self-energies associated laser source from Lumeras that was operated at a repetition with the quasiparticle bands. We demonstrate that this rate of 50 MHz with a 30-ps pulse length of the 1024-nm angular dependence originates in the variation of the orbital pump [52]. All experiments were performed at T ≈ 5 K content of quasiparticle states as a function of angle, and it using a cryogenic 6-axes sample goniometer, as described can be understood quantitatively. Introducing a new frame- in Ref. [53]. A combined energy resolution of 3 meV was work for the analysis of ARPES data for multi-orbital determined from the width of the Fermi-Dirac distribution systems, we extract the electronic self-energies associated measured on a polycrystalline Au sample held at 4.2 K. with the three Ruthenium t2g orbitals with minimal theo- The angular resolution was approximately 0.2°. In order to retical input. We find that these orbital self-energies have suppress the intensity of the surface layer states on pristine strong frequency dependence but surprisingly weak angular Sr2RuO4 [54], we exposed the cleaved surfaces to approxi- (i.e., momentum) dependence, and they can thus be mately 0.5 Langmuir CO at a temperature of approximately considered local to a very good approximation. Our results 120 K. Under these conditions, CO preferentially fills provide a direct experimental demonstration that the surface defects and subsequently replaces apical oxygen dominant effects of correlations in Sr2RuO4 are weakly ions to form a Ru–COO carboxylate in which the C end of a momentum dependent and can be understood from a local bent CO2 binds to Ru ions of the reconstructed surface perspective, provided they are considered in relation to layer [55].

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B. Experimental Fermi surface and [12,25,26] and was reported in earlier ARPES studies quasiparticle dispersions [33,56]. Our systematic data, however, reveal that this In Fig. 1, we show the Fermi surface and selected difference gradually disappears towards the Brillouin zone θ 45 constant energy surfaces in the occupied states of diagonal ð ¼ °Þ, where all three bands disperse nearly Sr2RuO4. The rapid broadening of the excitations away parallel to one another. In Sec. IV, we show that this from the Fermi level seen in the latter is typical for equilibration of the Fermi velocity can be attributed to the ruthenates and implies strong correlation effects on the strong effects of SOC around the zone diagonal. β;γ quasiparticle properties. At the Fermi surface, one can To quantify the angle dependence of vF from experi- kν ω readily identify the α, β, and γ sheets that were reported ment, we determine the maxima maxð Þ of the momentum earlier [25,27,28]. However, compared with previous distribution curves (MDCs) over the range of 2–6meV k ARPES studies, we achieve a reduced linewidth and below the Fermi level EF and fit these -space loci with improved suppression of the surface-layer states giving a second-order polynomial. We then define the Fermi clean access to the bulk electronic structure. This is velocity as the derivative of this polynomial at EF. This particularly evident along the Brillouin zone diagonal procedure minimizes artifacts due to the finite energy (ΓX), where we can clearly resolve all band splittings. resolution of the experiment. As shown in Fig. 2(c), the β;γ In the following, we exploit this advance to quantify the Fermi velocities vF obtained in this way show an opposite effects of SOC in Sr2RuO4 and to provide new insight into trend with azimuthal angle for the two Fermi sheets. For the the renormalization of the quasiparticle excitations using β band, we observe a gentle decrease of vF as we approach minimal theoretical input only. To this end, we acquired a the ΓX direction, whereas for γ, the velocity increases by set of 18 high-resolution dispersion plots along radial k- more than a factor of 2 over the same range [57]. This space lines (as parametrized by the angle θ measured from provides a first indication for a strong momentum depend- Γ Σ0 the M direction). The subset of data shown in Fig. 2(a) ence of the self-energies β;γ, which we analyze quantita- immediately reveals a rich behavior with a marked depend- tively in Sec. V. Here, we limit the discussion to the angle ence of the low-energy dispersion on the Fermi surface dependence of the mass enhancement vb=vF, which we angle θ. Along the ΓM high-symmetry line, our data calculate from the measured quasiparticle Fermi velocities β;γ reproduce the large difference in Fermi velocity vF for of Fig. 2(c) and the corresponding bare velocities vb of a the β and γ sheets, which is expected from the different reference Hamiltonian Hˆ 0 defined in Sec. IV. As shown in cyclotron masses deduced from quantum oscillations Fig. 2(d), this confirms a substantial many-body effect on

(a)X (b) M EF

-10 meV

M -20 meV

-30 meV

max -40 meV min

FIG. 1. (a) Fermi surface of Sr2RuO4. The data were acquired at 5 K on a CO passivated surface with a photon energy of 11 eV and p polarization for measurements along the ΓX symmetry line. The sample tilt around the ΓX axis used to measure the full Fermi surface results in a mixed polarization for data away from this symmetry axis. The Brillouin zone of the reconstructed surface layer is indicated by diagonal dashed lines. Surface states and final-state umklapp processes are suppressed to near the detection limit. A comparison with ARPES data from a pristine cleave is shown in the Appendix A. The data in panel (a) have been mirror symmetrized for clarity. Original measured data span slightly more than a quadrant of the Brillouin zone. (b) Constant energy surfaces illustrating the progressive broadening of the quasiparticle states away from the Fermi level EF.

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(a) (c) 0 0.8 ) max

(meV) -40 (eV F 0.4 F v E-E -80 M, min 0.0 (b) (d) 0 X 6

F 5 v (meV)

-40 / F b

v 4 E-E -80 3

0.6 0.8 0.6 0.8 0.6 0.8 0.6 0.8 M 0 45 90 -1 FS angle (deg) k// ( )

FIG. 2. (a) Quasiparticle dispersions measured with p-polarized light, for different azimuthal angles θ as defined in panel (b). (c) Angular dependence of the quasiparticle velocity vF along the β and γ Fermi surface sheets. (d) Angular dependence of the ˆ 0 ˆ DFT quasiparticle mass enhancement vb=vF. Here, vb is the bare velocity obtained from the single-particle Hamiltonian H ¼ H þ Hˆ SOC defined in Sec. IV, and v is the quasiparticle Fermi velocity measured by ARPES. Error bars are obtained by propagating the λDFTþΔλ F experimental uncertainty on vF. the anisotropy of the quasiparticle dispersion. Along ΓM, Hamiltonian Hˆ 0 as a reference point. At this stage, we keep we find a strong differentiation with mass enhancements the presentation general. The particular choice of Hˆ 0 will of about 5 for the γ sheet and 3.2 for β, whereas v =v ˆ 0 b F be a focus of Sec. IV. The eigenstates jψ νðkÞi of H ðkÞ at a approaches a common value of approximately 4.4 for both given quasimomentum k and the corresponding eigen- sheets along the Brillouin zone diagonal. values ενðkÞ define the “bare” band structure of the system, Before introducing the theoretical framework used to with respect to which the self-energy Σνν0 ðω; kÞ is defined quantify the anisotropy of the self-energy and the effects of in the standard way from the interacting Green’s function SOC, we compare our data quantitatively to bulk sensitive quantum oscillation measurements. Using the experimental k −1 ω k (ω μ − ε k )δ − Σ ω k Fermi wave vectors F and velocities determined from our Gνν0 ð ; Þ¼ þ νð Þ νν0 νν0 ð ; Þ: ð2Þ data on a dense grid along the entire Fermi surface, we can compute the cyclotron masses measured by de Haas-van In this expression, ν and ν0 label the bands and ω denotes Alphen experiments, without relying on the approximation the energy counted from E . The interacting value of the of circular Fermi surfaces and/or isotropic Fermi velocities F chemical potential μ sets the total electron number. Since μ used in earlier studies [33,49,56,58]. Expressing the cyclo- ˆ 0 tron mass m as can be conventionally included in H , we omit it in the following. Z2π The Fermi surfaces and dispersion relations of the ℏ2 ∂A ℏ2 k θ ω 0 FS Fð Þ θ quasiparticles are obtained as the solutions ¼ and m ¼ ¼ d ; ð1Þ QP 2π ∂ϵ 2π ∂ϵ=∂kðθÞ ω ¼ ων ðkÞ of 0

0 (ω − ε k )δ 0 − Σ ω k 0 where AFS is the Fermi surface volume, and using the det ½ νð Þ νν νν0 ð ; Þ ¼ : ð3Þ data shown in Fig. 2(c), we obtain mγ ¼ 17.3ð2.0Þme and Σ0 mβ ¼ 6.1ð1.0Þme, in quantitative agreement with the values In the above equation, denotes the real part of the self-energy. Its imaginary part Σ00 has been neglected; of mγ ¼ 16 me and mβ ¼ 7 me found in de Haas-van Alphen experiments [12,25,26]. We thus conclude that i.e., we assume that quasiparticles are coherent with a 1 ωQP the quasiparticle states probed by our experiments are lifetime longer than = . Our data indicate that this is indeed the case up to the highest energies analyzed here. representative of the bulk of Sr2RuO4 [59]. At very low frequency, the lifetime of quasiparticles cannot be reliably tested by ARPES since the intrinsic III. THEORETICAL FRAMEWORK quasiparticle width is masked by contributions of the In order to define the electronic self-energy and assess experimental resolution, impurity scattering and inho- the effect of electronic correlations in the spectral function mogeneous broadening. However, the observation of measured by ARPES, we need to specify a one-particle strong quantum oscillations in the bulk provides direct

021048-4 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019) evidence for well-defined quasiparticles in Sr2RuO4 term for atomic d orbitals projected to the t2g subspace down to the lowest energies [25,27]. reads [63] It is important to note that the Green’s function G, the Σ λ X X self-energy , and the spectral function A are, in general, ˆ SOC † Hλ ¼ c σðl 0 · σσσ0 Þc 0σ0 ; ð8Þ nondiagonal matrices. This aspects has been overlooked 2 m mm m 0 σσ0 thus far in self-energy analyses of ARPES data, but it is mm essential to determine the nature of many-body interactions where l are the t2g-projected angular-momentum matrices, in Sr2RuO4, as we show in Sec. V. σ are Pauli matrices, and λ will be referred to in the following as the SOC coupling constant. As documented A. Localized orbitals and electronic structure in Appendix B1, the eigenenergies of a DFT þ SOC ˆ DFT ˆ SOC Let us recall some of the important aspects of the calculation are well reproduced by H þ Hλ with λ 100 electronic structure of Sr2RuO4.AsshowninSec.II B, DFT ¼ meV. three bands, commonly denoted ν ¼fα; β; γg,crossthe Fermi level. These bands correspond to states with t2g IV. ENHANCED EFFECTIVE SPIN-ORBIT symmetry derived from the hybridization between COUPLING AND SINGLE-PARTICLE localized Ru-4d (dxy, dyz, dxz) orbitals and O-2p states. HAMILTONIAN Hence, we introduce a localized basis set of t2 -like g The importance of SOC for the low-energy physics orbitals jχ i, with basis functions conveniently labeled m of Sr2RuO4 has been pointed out by several authors as m ¼fxy; yz; xzg. In practice, we use maximally local- [8,17,21,23,31,64–68]. SOC lifts degeneracies found in ized Wannier functions [60,61] constructed from the its absence and causes a momentum-dependent mixing of Kohn-Sham eigenbasis of a non-SOC density functional the orbital composition of quasiparticle states, which has theory (DFT) calculation (see Appendix B1 for details). nontrivial implications for superconductivity [8,17,68]. ˆ DFT We term the corresponding Hamiltonian H .Itis Signatures of SOC have been detected experimentally on important to note that the choice of a localized basis the Fermi surface of Sr2RuO4 in the form of a small set is not unique, and other ways of defining these orbitals protrusion of the γ sheet along the zone diagonal [31,66] are possible (see, e.g., Ref. [62]). and as a degeneracy lifting at the band bottom of the β sheet In the following, this set of orbitals plays two important [68]. These studies reported an overall good agreement roles. First, they are atom centered and provide a set of between the experimental data and the effects of SOC χ R states localized in real space j mð Þi. Second, the unitary calculated within DFT [31,66,68]. This is in apparent ψ ν k transformation matrix to the band basis j ð Þi, contrast to more recent DMFT studies of Sr2RuO4, which predict large but frequency-independent off-diagonal con- k χ ψ Umνð Þ¼h kmj kνi; ð4Þ tributions to the local self-energy that can be interpreted as a contribution Δλ to the effective coupling strength “ ” ν allows us to define an orbital character of each band as λeff λ Δλ [21,23]. This result is also consistent with k 2 ¼ DFT þ jUmνð Þj . In the localized-orbital basis, the one-particle general perturbation-theory considerations [51], which Hamiltonian is a nondiagonal matrix, which reads show a Coulomb enhancement of the level splitting in X the J basis, similar to a Coulomb-enhanced crystal-field ˆ 0 k k ε k k Hmm0 ð Þ¼ Umνð Þ νð ÞUm0νð Þ: ð5Þ splitting [69]. ν In the absence of SOC, DFT yields a quasicrossing β γ The self-energy in the orbital basis is expressed as between the and Fermi surface sheets a few degrees away from the zone diagonal, as displayed in Fig. 3(a). X Near such a point, we expect the degeneracy to be lifted by Σ 0 ω; k U ν k Σνν0 ω; k U 0 0 k ; 6 mm ð Þ¼ m ð Þ ð Þ m ν ð Þ ð Þ Δ λeff νν0 SOC, leading to a momentum splitting k ¼ =v and to an energy splitting of ΔE ¼ Zλeff [23], as depicted sche- pffiffiffiffiffiffiffiffiffi and conversely in the band basis as matically in Fig. 3(e). In these expressions, v ≡ vβvγ, X where vβ and vγ are the bare band velocitiespffiffiffiffiffiffiffiffiffiffi in the absence Σνν0 ðω; kÞ¼ UmνðkÞΣmm0 ðω; kÞUm0ν0 ðkÞ: ð7Þ of SOC and correlations, and Z ≡ ZβZγ involves the mm0 quasiparticle residues Zν associated with each band (also in the absence of SOC). It is clear from these expressions that a quantitative B. Spin-orbit coupling λ determination of eff is not possible from experiment alone. ˆ DFT We treat SOC as an additional term to H , which is Earlier studies on Sr2RuO4 [68] and iron-based super- independent of k in the localized-orbital basis but leads to a conductors [70] have interpreted the energy splitting ΔE mixing of the individual orbitals. The single-particle SOC at avoided crossings as a direct measure of the SOC

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FIG. 3. Correlation enhanced effective SOC. (a) Quadrant of the experimental Fermi surface with a DFT calculation without SOC ˆ DFT (H ) at the experimental kz ≈ 0.4π=c (grey lines). (b,c) Same as panel (a), with calculations including SOC (DFT þ λDFT) and λ Δλ k enhanced SOC ( DFT þ ), respectively. For details, see the main text. (d) Comparison of the experimental MDC along the -space – cut indicated in panel (a) with different calculationsffiffiffiffiffiffiffiffiffiffiffi shown in panels (a) (c). (e) Schematic illustrationffiffiffiffiffiffiffiffiffiffi of the renormalization of a p 0 p SOC-induced degeneracy lifting. Here, Z ¼ ZνZν0 , where ν, ν label the two bands, and v ¼ vνvν0 , where vν, vν0 are bare velocities in the absence of SOC [23] (see text). (f) Experimental quasiparticle dispersion along the k-space cut indicated in panel (a). (g) Orbital character of the DFT þ λDFT þ Δλ eigenstates along the Fermi surface.

strength λeff. However, in interacting systems, ΔE is not a be explained by the competing effects of enhancement by robust measure of SOC since correlations can both enhance correlations and reduction by the quasiparticle weight as Δ λ E by enhancing eff and reduce it via the renormalization shown in Ref. [23]. We also point out that the equilibration factor Z. We thus quantify the enhancement of SOC from of quasiparticle velocities close to the diagonal, apparent the momentum splitting Δk, which is not renormalized from Figs. 2(a), 2(c), and 3(f), is indeed the behavior by the quasiparticle residue Z. The experimental splitting at expected close to an avoided crossing [23]. the avoided crossing between the β and γ Fermi surface Including the enhanced SOC determined from this −1 sheets indicated in Fig. 3(a) is ΔkQP ¼ 0.094ð9Þ Å , noncrossing gap leads to a much-improved theoretical DFT λ −1 description of the entire Fermi surface [71]. As shown in whereas DFT predicts Δk þ DFT ¼ 0.046 Å . We thus λeff λ Δ QP Fig. 3(b), our high-resolution experimental Fermi surface obtain an effective SOC strength ¼ DFT k = λ deviates systematically from a DFT calculation with SOC. Δ DFTþ DFT 205 20 k ¼ ð Þ meV, in quantitative agreement ˆ DFT ˆ SOC Most notably, H þ Hλ underestimates the size of the with the predictions in Refs. [21,23]. We note that despite DFT γ β this large enhancement of the effective SOC, the energy sheet and overestimates the sheet. Intriguingly, this is ˆ DFT ˆ SOC λ almost completely corrected in H þ Hλ Δλ, with splitting remains smaller than DFT, as shown in Fig. 3(f). DFTþ λ Δλ 200 When deviations from linearity in band dispersions are DFT þ ¼ meV, as demonstrated in Fig. 3(c). small, the splitting ΔE is symmetric around the EF and can Indeed, a close inspection shows that the remaining thus be determined from the occupied states probed in discrepancies between experiment and Hˆ DFT þ Hˆ SOC λDFTþΔλ experiment. Direct inspection of the data in Fig. 3(f) yields break the crystal symmetry, suggesting that they are Δ ≈ 70 2 3 λ E meV, which is about = of DFT and thus clearly dominated by experimental artifacts. A likely source for not a good measure of SOC. these image distortions is imperfections in the electron The experimental splitting is slightly larger than that optics arising from variations of the work function around expected from the expression ΔE ¼ Zλeff and our theo- the electron emission spot on the sample. Such distortions retical determination of Zβ and Zγ at the Fermi surface cannot presently be fully eliminated in low-energy photo- (which will be described in Sec. VI). This discrepancy can emission from cleaved single crystals. be attributed to the energy dependence of Z, which, in Importantly, the change in Fermi surface sheet volume Sr2RuO4, is not negligible over the energy scale of SOC. with the inclusion of Δλ is not driven by a change in the Note that the magnitude of the SOC-induced splitting of crystal-field splitting between the xy and xz, yz orbitals (see the bands at the Γ point reported in Ref. [68] can also Appendix B1). The volume change occurs solely because

021048-6 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019) of a further increase in the orbital mixing induced by the V. EXPERIMENTAL DETERMINATION enhanced SOC. As shown in Fig. 3(g), this mixing is not OF SELF-ENERGIES limited to the vicinity of the avoided crossing but extends A. Self-energies in the band basis along the entire Fermi surface (see Fig. 10 in Appendix B1 for the orbital character without SOC). For λ þ Δλ,we Working in the band basis, i.e., with the eigenstates of DFT ˆ 0 find a minimal dxy and dxz;yz mixing for the γ and β bands H , the maximum of the ARPES intensity for a given of 20%=80% along the ΓM direction with a monotonic binding energy ω (maximum of the MDCs) corresponds to increase to approximately 50% along the Brillouin zone the momenta k that satisfy [following Eq. (3)] ω − ενðkÞ − 0 diagonal ΓX. We note that this mixing varies with the Σννðω; kÞ¼0. Hence, for each binding energy, each perpendicular momentum kz. However, around the exper- azimuthal cut, and each sheet of the quasiparticle disper- imental value of k ≈ 0.4π=c, the variation is weak [73]. sions, we fit the MDCs and determine the momentum z kν ω ε (kν ω ) The analysis presented here and in Secs. II B and VI is maxð Þ at their maximum. Using the value of ν maxð Þ thus robust with respect to a typical uncertainty in kz. at this momentum yields the following quantity: These findings suggest that a natural reference single- 0 ω − ε (kν ω ) Σ0 (ω kν ω ) ≡ Σ0 ω θ particle Hamiltonian is Hˆ ¼ Hˆ DFT þ Hˆ SOC . This ν maxð Þ ¼ νν ; maxð Þ νð ; Þ: ð9Þ λDFTþΔλ ˆ 0 choice ensures that the Fermi surface of H is very close This equation corresponds to the simple construction to that of the interacting system. From Eq. (3), this implies illustrated graphically in Fig. 4(b), and it is a standard that the self-energy matrix approximately vanishes at zero way of extracting a self-energy from ARPES, as used in Σ0 ω 0 k ≃ 0 ˆ 0 binding energy: νν0 ð ¼ ; Þ . We choose H in this previous works on several materials [32,48,74–76].We manner in all of the following. Hence, from now on, note that this procedure assumes that the off-diagonal jψ νðkÞi and ενðkÞ refer to the eigenstates and band structure Σ0 ω k components ν≠ν0 ð ; Þ can be neglected for states close of Hˆ 0 ¼ Hˆ DFT þ Hˆ SOC . We point out that although Hˆ 0 λDFTþΔλ to the Fermi surface (i.e., for small ω and k close to a Fermi is a single-particle Hamiltonian, the effective enhancement crossing). This assumption can be validated, as shown in Δλ of SOC included in Hˆ 0 is a correlation effect Appendix C. When performing this analysis, we only beyond DFT. include the α sheet for θ ¼ 45°. Whenever the constraint

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

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Σ0 ω → 0 → 0 β γ νð Þ on the self-energy is not precisely obeyed, γ band MDCs, kmaxðω; θÞ and kmaxðω; θÞ, respectively. In a small shift is applied to set it to zero. We chose this terms of these matrices, the quasiparticle equations (10) procedure to correct for the minor differences between the read experimental and the reference Hˆ 0 Fermi wave vectors 0 0 because we attribute these differences predominantly to det½Amm0 − Σmδmm0 ¼det½Bmm0 − Σmδmm0 experimental artifacts. 0 ¼ det½G 0 − Σ δ 0 ¼0: ð11Þ The determined self-energies for each band ν ¼ α, β, γ mm m mm and the different values of θ aredepictedinFigs.4(a) β γ However, when taking symmetry into account, the self- and 4(c). For the and sheets, they show a substantial Σ0 Γ energy has only two independent components: xy and dependence on the azimuthal angle. Around M, we find 0 0 0 0 Σxz ¼ Σyz. Hence, we only need two of the above equations that Σγ exceeds Σβ by almost a factor of 2 (at ω ¼ −50 meV), to solve for the two unknown components of the self-energy. whereas they essentially coincide along the zone diagonal Γ θ This means that we can also extract a self-energy in the ( X). This change evolves as a function of and occurs via a β γ 0 0 directions where only two bands ( and ) are present in the simultaneous increase in Σ and a decrease in Σγ for all β considered energy range of ω ≲ 100 meV, e.g., along ΓM. energies as θ is increased from 0° (ΓM) to 45° (ΓX). In order 0 The resulting functions Σmðω; θkÞ determined at several to better visualize this angular dependence, a compilation of θ 0 angles are displayed in Fig. 4(d). It is immediately apparent ΣνðωÞ for different values of θ is displayed in Fig. 4(c). 0 that, in contrast to Σνν, the self-energies in the orbital basis do not show a strong angular (momentum) dependence, but B. Accounting for the angular dependence: rather, they collapse into two sets of points, one for the xy Local self-energies in the orbital basis orbital and one for the xz=yz orbitals. Thus, we reach the In this section, we introduce a different procedure for remarkable conclusion that the angular dependence of the extracting self-energies from ARPES, by working in the self-energy in the orbital basis is negligible, within the range Σ0 ω θ ≃ Σ0 ω orbital basis jχmðkÞi. We do this by making two key of binding energies investigated here: mð ; kÞ mð Þ. assumptions: This result implies that a good approximation of the full (1) We assume that the off-diagonal components are momentum and energy dependence of the self-energy in the Σ0 ≃0 band (quasiparticle) basis is given by negligible, i.e., m≠m0 . Let us note that in Sr2RuO4, even a k-independent self-energy has nonzero off- X ˆ DFT ˆ SOC Σνν0 ω; k U ν k Σ ω U ν0 k : 12 diagonal elements if H þ Hλ is considered. ð Þ¼ m ð Þ mð Þ m ð Þ ð Þ DFT m Using DMFT, these off-diagonal elements have been shown to be very weakly dependent on frequency in The physical content of this expression is that the angular this material [23], leading to the notion of a static (momentum) dependence of the quasiparticle self-energies Δλ correlation enhancement of the effective SOC ( ). emphasized above is actually due to the matrix elements In the present work, these off-diagonal frequency- UmνðkÞ definedinEq.(4).InSr2RuO4, the angular independent components are already incorporated into dependence of these matrix elements is mainly due to the ˆ 0 H (see Sec. IV), and thus the frequency-dependent SOC, as seen from the variation of the orbital content part of the self-energy is (approximately) orbital of quasiparticles in Fig. 3(g).InAppendixD, we show diagonal by virtue of the tetragonal crystal structure. 0 0 the back-transform of Σmðω; θk ¼ 18°Þ into Σνðω; θk ¼ 0 (2) We assume that the diagonal components of the self- 0; 18; 45°Þ. The good agreement with Σνν directly extracted energy in the orbital basis depend on the momentum k from experiment further justifies the above expression and θ Σ0 ω k ≃ only through the azimuthal angle k: mmð ; Þ also confirms the validity of the approximations made Σ0 ω θ mð ; kÞ. We neglect the dependence on the mo- throughout this section. mentum, which is parallel to the angular cut. Finally, we stress that expression (12) precisely coin- Under these assumptions, the equation determining the cides with the ansatz made by DMFT: Within this theory, quasiparticle dispersions reads the self-energy is approximated as a local (k-independent) object when expressed in a basis of localized orbitals, (ω − Σ0 ω θ )δ − ˆ 0 k 0 det½ mð ; kÞ mm0 Hmm0 ð Þ ¼ : ð10Þ while it acquires momentum dependence when transformed to the band basis. In this equation, we have neglected the lifetime effects 00 associated with the imaginary part Σm. In order to extract 0 VI. COMPARISON TO DYNAMICAL the functions Σmðω; θkÞ directly from the ARPES data, kν ω θ MEAN-FIELD THEORY we first determine the peak positions maxð ; Þ for θ ω MDCs at a given angle and binding energy . We then In this section, we perform an explicit comparison of ω θ ≡ ωδ − compute (for the same and ) the matrix Amm0 mm0 the measured quasiparticle dispersions and self-energies ˆ 0 (kα ω θ ) β Hmm0 maxð ; kÞ and similarly Bmm0, Gmm0 for the and to DMFT results. The latter are based on the Hamiltonian

021048-8 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019)

0

max

(meV) -40 F E-E -80 M, X, min 0.5 0.7 0.5 0.7 0.5 0.7 0.5 0.7 0.7 0.9 0.7 0.9 -1 k// ( )

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

ˆ DFT H , to which the Hubbard-Kanamori interaction with (a) on-site interaction U ¼ 2.3 eV and Hund’s coupling J ¼ dxy 0.4 eV [20] is added. For the details of the DMFT 200 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 þ DMFT calculations dxz,yz 100

in this context. Figure 5(a) shows the experimental quasi- (meV) 600 m particle dispersion extracted from our ARPES data (circles) ' on top of the DMFT spectral function Aðω; kÞ displayed as a color-intensity map. Clearly, the theoretical results are in 200 near quantitative agreement with the data: Both the strong 0 renormalization of the Fermi velocity and the angular- -200 dependent curvature of the quasiparticle bands are very -400 0 400 well reproduced by the purely local, and thus momentum- independent, DMFT self-energies. This result validates -90 -60 -30 0 (meV) the assumption of no momentum dependence along the radial k-space direction of the self-energy made in Sec. VB (b) X for the k=ω range studied here. The small deviations in 6 band Fermi wave vectors discernible in Fig. 5 are consistent with Fig. 3(c) and the overall experimental precision of the 5

Fermi surface determination. F v /

In Fig. 6(a), we compare the experimental self-energies b v for each orbital with the DMFT results. The overall 4 agreement is notable. At low energy, the self-energies are linear in frequency, and the agreement is excellent. The slope of the self-energies in this regime controls the 3 band angular dependence of the effective mass renormalization. 0 45 90 Using the local ansatz (12) in the quasiparticle dispersion FS angle (deg) equation and performing an expansion around EF,we obtain Σ0 Σ0 FIG. 6. (a) Average of the self-energies xz=yz, xy shown in Fig. 4(d) compared with DMFT self-energies calculated at 29 K. X 1 1 ∂Σ0 0 vb 2 m The self-energies are shifted such that Σmðω ¼ 0Þ¼0. The inset ν ¼ jUmνðθÞj ; ≡ 1 − : ð13Þ v ðθÞ Zm Zm ∂ω ω 0 shows the DMFT self-energies over a larger energy range. Linear F m ¼ 0 fits at low and high energies of Σxy from DMFT are shown as In Fig. 6(b), we show v =vν ðθÞ for the β and γ bands using solid and dashed black lines, respectively. (b) Angle dependence b F of the mass enhancement v =v from ARPES (markers) and the DMFT values Z 0.18 0.01 and Z 0.30 b F xy ¼ xz=yz ¼ DMFT (solid line). The range indicated by the shaded areas 0 01 . obtained at 29 K (Appendix B2). The overall angular corresponds to mass enhancements calculated from the numerical dependence and the absolute value of the γ band mass data by using different methods (see Appendix B2). Error bars on enhancement is very well captured by DMFT, while the β the experimental data are obtained from propagating the esti- band is a bit overestimated. Close to the zone diagonal mated uncertainty of the Fermi velocities shown in Fig. 2(c).

021048-9 A. TAMAI et al. PHYS. REV. X 9, 021048 (2019)

(θ ¼ 45°), the two mass enhancements are approximately behavior into a more incoherent regime [18,20]. The near equal, due to the strong orbital mixing induced by the SOC. quantitative agreement of the frequency dependence of the 0 Turning to larger binding energies, we see that the experimental self-energies ΣmðωÞ and our single-site 0 theoretical Σxy is in good agreement with the experimental DMFT calculation provides strong evidence for the exist- data over the full energy range of 2–80 meV covered in our ence of such electronic kinks in Sr2RuO4. In addition, it experiments. Both the theoretical and experimental self- implies that the local DMFT treatment of electronic energies deviate significantly from the linear regime down correlations captures the dominant effects. to low energies (∼20 meV), causing curved quasiparticle Focusing on the low-energy regime of our experimental bands with progressively steeper dispersion as the energy data, we find deviations from the linear form Σ0ðωÞ¼ increases (Fig. 5). In contrast, the agreement between ωð1 − 1=ZÞ characteristic of a Fermi liquid for jωj > theory and experiment for the xz=yz self-energy is some- 20 meV, irrespective of the basis. However, this is only what less impressive at binding energies larger than about an upper limit for the Fermi-liquid energy scale in Σ0 Sr2RuO4. Despite the improved resolution of our experi- 30 meV. Our DMFT self-energy xz=yz overestimates the strength of correlations in this regime (by 20%–25%), ments, we cannot exclude an even lower crossover energy with a theoretical slope larger than the experimental one. to non-Fermi-liquid-like excitations. We note that the ≈ 25 Correspondingly, the quasiparticle dispersion is slightly crossover temperature of TFL K reported from trans- – steeper in this regime than the theoretical result, as can also port and thermodynamic experiments [10 12] indeed be seen in Fig. 5. suggests a crossover energy scale that is significantly There may be several reasons for this discrepancy. below 20 meV. Σ0 ω Even while staying in the framework of a local self- The overall behavior of ð Þ including the energy energy, we note that the present DMFT calculation is range where we find strong changes in the slope agrees performed with an on-site value of U, which is the same with previous photoemission experiments, which were interpreted as evidence for electron-phonon coupling for all orbitals. Earlier cRPA calculations have suggested – that this on-site interaction is slightly larger for the xy [29 31,46,48]. Such an interpretation, however, relies on a linear Fermi-liquid regime of electronic correlations over orbital (U ¼ 2.5 eV and U ¼ 2.2 eV) [20],and xy xz=yz the entire phonon bandwidth of approximately 90 meV recent work has advocated the relevance of this for [82], which is inconsistent with our DMFT calculations. DFT þ DMFT calculations of Sr2RuO4 [21].Another Moreover, attributing the entire curvature of Σ0 in our data possible explanation is that this discrepancy is actually m to electron-phonon coupling would result in unrealistic a hint of some momentum-dependent contribution to the coupling constants far into the polaronic regime, which is self-energy, especially the dependence on the momentum hard to reconcile with the transport properties of Sr2RuO4 perpendicular to the Fermi surface. We note, in this – α β [10 12]. We also note that a recent scanning tunneling respect, that the discrepancy is larger for the , sheets microscope (STM) study reported very strong kinks in the that have dominant xz=yz character. These orbitals have, β and γ sheets of Sr2RuO4 [49], which is inconsistent with in the absence of SOC, a strong one-dimensional char- our data. We discuss the reason for this discrepancy in acter, for which momentum dependence is definitely Appendix A. expected and DMFT is less appropriate. Furthermore, these FS sheets are also the ones associated with nesting VIII. DISCUSSION AND PERSPECTIVES and spin-density wave correlations, which are expected to lead to an additional momentum dependence of the In this article, we have reported on high-resolution self-energy. We further discuss possible contributions of ARPES measurements that allow for a determination spin fluctuations in Sec. VIII. of the Fermi surface and quasiparticle dispersions of Sr2RuO4 with unprecedented accuracy. Our data reveal an enhancement (by a factor of about 2) of the splitting VII. KINKS between Fermi surface sheets along the zone diagonal, in The self-energies Σ0ðωÞ shown in Figs. 4 and 6(a) display comparison to the DFT value. This can be interpreted as a a fairly smooth curvature, rather than pronounced “kinks.” correlation-induced enhancement of the effective SOC, 0 Over a larger range, however, Σxy from DMFT does show an effect predicted theoretically [21,23,51] and demon- an energy scale marking the crossover from the strongly strated experimentally here for this material, for the renormalized low-energy regime to weakly renormalized first time. excitations. This is illustrated in the inset to Fig. 6(a). Such Thanks to the improved cleanliness of our data, we purely electronic kinks were reported in DMFT calcula- have been able to determine the electronic self-energies tions of a generic system with Mott-Hubbard sidebands directly from experiment, using both a standard procedure [77] and have been abundantly documented in the theo- applied in the band (quasiparticle) basis as well as a retical literature since then [13,20,78–81].InSr2RuO4, they novel procedure, introduced in the present article, in the are associated with the crossover from the Fermi-liquid orbital basis. Combining these two approaches, we have

021048-10 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019) demonstrated that the large angular (momentum) depend- Council, the UK EPSRC, and the Swiss National Science ence of the quasiparticle self-energies and dispersions can Foundation (SNSF). Theoretical work was supported by be mostly attributed to the fact that quasiparticle states have the ERC Grant No. ERC-319286-QMAC and by the SNSF an orbital content that is strongly angular dependent, due to (NCCR MARVEL). The Flatiron Institute is a division the SOC. Hence, assuming self-energies that are frequency of the Simons Foundation. A. G. and M. Z. gratefully dependent but essentially independent of angle (momen- acknowledge useful discussions with Gabriel Kotliar, tum), when considered in the orbital basis, is a very good Andrew J. Millis, and Jernej Mravlje. approximation. This provides a direct experimental vali- dation of the DMFT ansatz. APPENDIX A: BULK AND SURFACE The key importance of atomiclike orbitals in correlated Sr RuO insulators is well established [83,84]. The present work ELECTRONIC STRUCTURE OF 2 4 demonstrates that orbitals retain a considerable physical In Fig. 7, we compare the data presented in the main text relevanceeveninametalinthelow-temperatureFermi- with data from a pristine cleave taken with hν ¼ 21 eV at liquid regime. Although the band and orbital bases used here the SIS beam line of the Swiss Light Source. This are equivalent, our analysis shows that the underlying comparison confirms the identification of bulk and surface simplicity in the nature of correlations emerges only when bands by Shen et al. [54]. In particular, we find that the working in the latter and taking into account the orbital larger β sheet has bulk character. This band assignment is origin of quasiparticles. Beyond Sr2RuO4,thisisanobser- used by the vast majority of subsequent ARPES publi- vation of general relevance to multiband metals with strong cations [29–31,33,46,56,89,90], except for Ref. [47], which correlations such as the iron-based superconductors [85,86]. reports a dispersion with much lower Fermi velocity and a Notwithstanding its success, the excellent agreement of strong kink at 15 meV for the smaller β sheet that we the DMFT results with ARPES data does raise puzzling identify as a surface band. questions. Sr2RuO4 is known to be host to strong magnetic Wang et al. [49] have recently probed the low-energy – fluctuations [39 42], with a strong peak in its spin response electronic structure of Sr2RuO4 by STM. Analyzing χðQÞ close to the spin-density wave (SDW) vector quasiparticle interference patterns along the ΓX and ΓM Q ∼ ð2π=3; 2π=3Þ, as well as quasiferromagnetic fluctua- high-symmetry directions, they obtained band dispersions tions that are broader in momentum around Q ¼ 0. Indeed, with low Fermi velocities and strong kinks at 10 meV and tiny amounts of substitutional impurities induce long-range 37 meV. In Fig. 8, we compare the band dispersion reported magnetic order in this material, of either SDW of ferro- by Wang et al. with our ARPES data. Along both high- magnetic type [87,88]. Hence, it is a prominent open symmetry directions, we find a clear discrepancy with our question to understand how these long-wavelength fluctu- data, which are in quantitative agreement with bulk de Haas ations affect the physics of quasiparticles in the Fermi- liquid state. Single-site DMFT does not capture this Bulk Surface + Bulk feedback, and the excellent agreement with the overall X quasiparticle physics must imply that these effects have a comparatively smaller magnitude than the dominant local effect of correlations (on-site U and especially Hund’s J) captured by DMFT. A closely related question is how much momentum dependence is present in the low-energy (Landau) interactions between quasiparticles. These effects are expected to be fundamental for subsequent instabilities of the Fermi liquid, into either the superconducting state M 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 Sr2RuO4, for which the precise nature of the pairing mechanism as well as the symmetry of the order parameter are still outstanding open questions [4]. The research data supporting this publication can be accessed at the University of St Andrews Research h = 11 eV h = 21 eV Portal [102]. FIG. 7. Bulk and surface electronic structure of Sr2RuO4. Left ACKNOWLEDGMENTS half: Fermi surface map of a CO passivated surface shown in the main text. Right half: Fermi surface acquired on a pristine cleave The experimental work has been supported by the at 21 eV photon energy. Intense surface states are evident in European Research Council (ERC), the Scottish Funding addition to the bulk bands observed in the left panel.

021048-11 A. TAMAI et al. PHYS. REV. X 9, 021048 (2019)

(a) 1 0 0 (meV)

F -40 Surface -1 (eV) k E - -80 M Bulk QPI -2 0.4 0.6 0.8 DFT (GGA-PBE) (b) DFT (Wannier90) -3 0 1 (meV)

F -40 0

E - -80 X QPI M QPI -1 (eV) 0.6 0.8 0.4 0.6 0.8 k -1 -1 k// (Å ) k// (Å ) -2 DFT + SOC (GGA-PBE) Γ FIG. 8. Comparison with the STM data from Ref. [49]. (a) M DFT + DFT (Wannier90) high-symmetry cut on a pristine cleave. The bulk β and γ and the -3 surface β bands are labeled. (b) Laser-ARPES data from CO M X Z passivated surface showing the bulk band dispersion along ΓX FIG. 9. DFT band structure along the high-symmetry path and ΓM. The dispersion obtained from quasiparticle interference ΓMXΓZ compared to the eigenstates of our maximally localized in Ref. [49] is overlaid with red markers. ˆ DFT Wannier Hamiltonian H for the three t2g bands. Top panel: DFT (GGA-PBE) and eigenstates of Hˆ DFT. Bottom panel: DFT þ van Alphen measurements, as demonstrated in the main text. SOC (GGA-PBE) and eigenstates of Hˆ DFT Hˆ SOC, with a local þ λDFT λ 100 On the other hand, we find a striking similarity between the SOC term [Eq. (8)] and a coupling strength of DFT ¼ meV. STM data along ΓM and the band commonly identified β as the surface band [30,54]. We thus conclude that the allows us to disentangle the γ band from the bands with experiments reported in Ref. [49] probed the surface states of dominantly O-p character below −2 eV. Sr2RuO4. This is fully consistent with the strong low-energy Note that in the absence of SOC, the eigenstates retain kinks and overall enhanced low-energy renormalization seen pure orbital character, as shown in Fig. 10. To take SOC by ARPES in the surface bands [50,91]. ˆ SOC into account, we add the local single-particle term Hλ ,as given in Eq. (8), with coupling constant λ. In the bottom ˆ DFT APPENDIX B: COMPUTATIONAL DETAILS panel of Fig. 9, we show that the eigenenergies of H þ ˆ SOC Hλ are in nearly perfect agreement with the DFT þ SOC 1. DFT and model Hamiltonian λ 100 band structure at a value of DFT ¼ meV. We generate our theoretical model Hamiltonian Hˆ DFT Our model Hamiltonian provides the reference point to with a maximally localized Wannier function [60,61] which we define a self-energy, but it is also a perfect construction of t2g-like orbitals for the three bands crossing playground to study the change in the Fermi surface the Fermi surface. These Wannier orbitals are obtained on a under the influence of SOC and the crystal-field splitting 10 × 10 × 10 k grid based on a non-SOC DFT calculation between the xy and xz=yz orbitals. In the following, we using WIEN2k [92] with the GGA-PBE functional [93], confirm that the best agreement with the experimental wien2wannier [94], and Wannier90 [95]. The DFT calcu- Fermi surface is found with an effective SOC of λeff λ Δλ 200 lation is performed with lattice parameters from Ref. [96] ¼ DFT þ ¼ meV, but at the same time, we ϵ ϵ − ϵ (measured at 100 K) and converged with twice as many k keep the DFT crystal-field splitting of cf ¼ xz=yz xy ¼ points in each dimension. 85 meV unchanged. We compare in Fig. 11 the exper- The Wannier orbitals are centered on the Ru atoms and imental Fermi surface (dashed lines) to the one of ˆ DFT ˆ SOC have t2 symmetry but are indeed linear combinations of H þ H . The Fermi surfaces for additionally intro- g λDFT Δϵ −80 Ru-d and O-p states. We do not add Wannier functions duced crystal-field splittings cf between and centered on the oxygen atoms because the resulting three- 80 meV are shown with solid lines in different shades of orbital Wannier model already accurately reproduces the red. In contrast to the Fermi surface without SOC three bands crossing the Fermi energy, as demonstrated in (λ ¼ 0 meV, top-left panel), the Fermi surfaces with the Fig. 9. Also note that the Wannier function construction DFT SOC of λ ¼ 100 meV (top-right panel) resemble the

021048-12 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019)

perfect agreement with experiment without any additional Δϵ 0 crystal-field splitting ( cf ¼ meV). At an even higher SOC of λ ¼ 300 meV (bottom-left panel), we see again major discrepancies but with an opposite trend: The α and β sheets are now too small, and the γ sheet is too large. Like in the case of λ ¼ 100 meV, this cannot be cured by an ϵ adjustment of cf.

2. DMFT We perform single-site DMFT calculation with the TRIQS/DFTTools [97] package for Hˆ DFT and Hubbard- Kanamori interactions with a screened Coulomb repulsion U ¼ 2.3 eV and a Hund’s coupling J ¼ 0.4 eV based on previous works [20,23]. The impurity problem is solved on the imaginary-time axis with the TRIQS/CTHYB [98] FIG. 10. Orbital character of the DFT FS without SOC at solver at a temperature of 29 K. The employed open-source 0 4π λ kz ¼ . =c (left). The orbital character of the DFT þ DFT þ software tools are based on the TRIQS library [99].We Δλ eigenstates at the same kz is reproduced on the right from assume an orbital-independent double counting, and hence Fig. 3(g). it can be absorbed into an effective chemical potential, which is adjusted such that the filling is equal to four overall structure of the experimental Fermi surface. electrons. For the analytic continuation of the self-energy to However, the areas of the α and β sheets are too large, the real-frequency axis, we employ three different methods: and the γ sheet is too small. Importantly, the agreement Pad´e approximants (using TRIQS [99]), stochastic con- Δϵ cannot be improved by adding cf. For example, along tinuation (after Beach [100]), and maximum entropy (using Γ Δϵ 40 M, a cf of þ meV would move the Fermi surface TRIQS/maxent [101]). In the relevant energy range from closer to the experiment; on the other hand, along ΓX, a −100 to 0 meV, the difference in the resulting self-energies Δϵ −80 cf of meV would provide the best agreement. The (Fig. 12) is below the experimental uncertainty. The situation is different if we consider an enhanced SOC of averaged quasiparticle renormalizations (of the three con- λ ¼ 200 meV (bottom-right panel). Then, we find a nearly tinuations) are Zxy ¼ 0.180.01 and Zxz ¼ Zyz ¼ 0.30 0.01. For all other results presented in the main text, the M X Pad´e solution has been used.

200 Analytic continuation = 0 meV = 100 meV

CF = - 80 meV CF = - 40 meV CF = 0 meV

CF = 40 meV (meV)

m 100 ' CF = 80 meV ARPES Pade xz Pade xy = 300 meV = 200 meV Beach xz Beach xy TRIQS/maxent xz TRIQS/maxent xy 0 -100 -50 0 (meV) FIG. 11. Fermi surface of Sr2RuO4 for λ ¼ 0 meV (top left), λ ¼ 100 meV (top right), λ ¼ 200 meV (bottom right), and λ ¼ FIG. 12. The real part of DFT þ DMFTself-energyinthe 300 meV (bottom left) compared to ARPES (dashed black considered energy range obtained with three different analytic line). Note that λ ¼ 100 meV corresponds to a DFT þ SOC continuation methods: Pad´e approximants (using TRIQS [99]), calculation and λ ¼ 200 meV to an effective SOC enhanced by stochastic continuation (after Beach [100]), and maximum electronic correlations (see main text). The different shades of entropy (using TRIQS/maxent [101]).Thedifferencebetween Δϵ red indicate additional crystal-field splittings cf added to the analytic continuation methods is smaller than the exper- ϵ 85 ˆ DFT cf ¼ meV of H . imental error.

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Our calculations at a temperature of 29 K use Hˆ DFT,as but we neglect the additional static part introduced the sign problem prohibits reaching such low temperatures by DMFT. with SOC included. Nevertheless, calculations with SOC were successfully carried out at a temperature of 290 K using CT-INT [21] and at 230 K using CT-HYB with a APPENDIX C: OFF-DIAGONAL Σ0 simplified two-dimensional tight-binding model [23]. ELEMENTS OF νν0 These works pointed out that electronic correlations in 0 In Sec. VA, we extracted Σνν under the assumption that Sr2RuO4 lead to an enhanced SOC. To be more precise, the off-diagonal elements can be neglected. To obtain Kim et al. [23] observed that electronic correlations in insights about the size of the off-diagonal elements, we this material are described by a self-energy with diagonal 0 use Eq. (12) to calculate the full matrix Σ 0 ðω; kÞ in the elements close to the ones without SOC plus, to a good νν band basis from the DMFT self-energy in the orbital basis approximation, frequency-independent off-diagonal ele- 0 ΣmðωÞ, as shown in Fig. 6(a). This procedure allows us to ments, which can be absorbed in a static effective 0 obtain the full self-energy matrix Σ 0 ðω; kÞ at one specific SOC strength of λeff ¼ λ þ Δλ ≃ 200 meV—this is νν DFT combination of k and ω. Note that for the results presented the approach followed in the present article. in Figs. 4 and 13(b), this is not the case because the In addition to the enhancement of SOC, it was observed extracted self-energies for each band correspond to differ- that low-energy many-body effects also lead to an kν enhancement of the crystal-field splitting [21,23]. In our ent max, which are further defined by the experimental MDCs. DFT þ DMFT calculation without SOC, this results in a k orbital-dependent splitting in the real part of the self- In Fig. 13(a), we show the result for two selected Δϵ 60 γ points: on the β sheet for θ ¼ 0 and on the γ sheet for energies ( cf ¼ meV), which would move the sheet θ 45 k closer to the van Hove singularity and consequently worsen ¼ °. For these points, the largest off-diagonal Σ0 – the agreement with the experimental Fermi surface along element is γβ, which is about 10% 20% of the size of the ΓM direction (see bottom-right panel of Fig. 11). the diagonal elements. A scan performed for the whole 0 4π Σ0 Different roots of this small discrepancy are possible, kz ¼ . =c plane further confirms that j ν≠ν0 j is smaller ranging from orbital-dependent double-counting correc- than 20 meV. tions to, in general, DFT being not perfect as “noninteract- However, when neglecting the off-diagonal elements, it ing” reference point for DMFT. Zhang et al. [21] showed is also important to have a large enough energy separation that by considering the anisotropy of the Coulomb tensor, of the bands. This can be understood by considering a the additional crystal-field splitting is suppressed, and simplified case of two bands (ν; ν0) and rewriting Eq. (3), QP consequently, the disagreement between theory and experi- which determines the quasiparticle dispersion ων ðkÞ,as ment can be cured. We point out that in comparison to the present work, a large enhancement of the crystal-field 0 0 Σ 0 ðω; kÞΣ 0 ðω; kÞ ω − ε k − Σ0 ω k − νν ν ν 0 splitting was observed in Ref. [21], presumably due to the νð Þ ννð ; Þ 0 ¼ : ω − εν0 ðkÞ − Σ 0 0 ðω; kÞ larger interactions employed. ν ν Based on these considerations, we calculate the corre- ðC1Þ lated spectral function Aðω; kÞ [shown in Fig. 5 of the main text] using the Hamiltonian with enhanced SOC Setting the last term to zero, i.e., using the procedure ˆ DFT ˆ SOC 0 (H H ) in combination with the frequency- described in Sec. VA to extract Σνν, is justified at þ λDFTþΔλ QP dependent part of the non-SOC (diagonal) self-energy, ω ¼ ων ðkÞ as long as

(a) (b) M, sheet X, sheet M X 120 ' 120 ' band ' | ' |

| ' | (meV) vv ' ' (meV) 60 | ' | 60 band band

0 0 -40 -20 0 -40 -20 0 -40 -20 0 -40 -20 0 -40 -20 0 (meV) (meV) (meV) (meV) (meV)

Σ0 ω k FIG. 13. Reconstructed self-energies in the quasiparticle basis. (a) Full matrix νν0 ð ; Þ calculated with Eq. (12) using the DMFT self-energy (shown in Fig. 6(a)) at two selected k points: on the β sheet for θ ¼ 0° (left) and on the γ sheet for θ ¼ 45° (right). Σ0 ω kν ω Σ0 ω θ 18 (b) Directly extracted ννð ; max Þ (solid symbols) (Sec. VA) compared to the mð ; ¼ °Þ (Sec. VB) transformed to Σ0 ω kν ω ννð ; max Þ at 0, 18, and 45° (thin lines).

021048-14 HIGH-RESOLUTION PHOTOEMISSION ON Sr2RuO4 REVEALS … PHYS. REV. X 9, 021048 (2019)

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