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

TL;DR

This study uses laser-based photoemission spectroscopy to show that electron correlations significantly enhance spin-orbit coupling and that the self-energies in Sr2RuO4 are predominantly local, with orbital mixing explaining angular dependencies.

Contribution

It provides the first direct experimental evidence that correlations enhance spin-orbit coupling and that self-energies are mainly local, clarifying the role of orbital mixing in Sr2RuO4.

Findings

Spin-orbit coupling is enhanced by electronic correlations.

Self-energies are mainly local and orbital-dependent.

Orbital mixing explains angular dependence of self-energies.

Abstract

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

Figures

Equations

$$m^{\ast }=\frac{{\hslash }^2}{2\pi }\frac{\partial A_{FS}}{\partial ϵ}=\frac{{\hslash }^2}{2\pi }{\int }_0^{2\pi }\frac{k_F(\theta )}{\partial ϵ/\partial k(\theta )}d\theta ,$$

$$G_{\nu {\nu }^{\prime }}^{-1}(\omega ,\mathbf{k})=\left[\omega +\mu -{\epsilon }_{\nu }\left(\mathbf{k}\right)\right]{\delta }_{\nu {\nu }^{\prime }}-{\Sigma }_{\nu {\nu }^{\prime }}(\omega ,\mathbf{k}).$$

$$\det \left[\left(\omega -{\epsilon }_{\nu }\left(\mathbf{k}\right)\right){\delta }_{\nu {\nu }^{\prime }}-{\Sigma }_{\nu {\nu }^{\prime }}^{\prime }(\omega ,\mathbf{k})\right]=0.$$

$$U_{m\nu }\left(\mathbf{k}\right)=⟨{\chi }_m\left(\mathbf{k}\right)⟩{\psi }_{\nu }\left(\mathbf{k}\right),$$

$${\hat{H}}_{m{m}^{\prime }}^0(\mathbf{k})=\sum _{\nu }{U}_{m\nu }\left(\mathbf{k}\right){\epsilon }_{\nu }\left(\mathbf{k}\right)U_{m^{\prime }\nu }^{\ast }\left(\mathbf{k}\right).$$

$${\Sigma }_{m{m}^{\prime }}(\omega ,\mathbf{k})=\sum _{\nu {\nu }^{\prime }}{U}_{m\nu }\left(\mathbf{k}\right){\Sigma }_{\nu {\nu }^{\prime }}(\omega ,\mathbf{k})U_{m^{\prime }{\nu }^{\prime }}^{\ast }\left(\mathbf{k}\right),$$

$${\Sigma }_{\nu {\nu }^{\prime }}(\omega ,\mathbf{k})=\sum _{m{m}^{\prime }}{U}_{m\nu }^{\ast }\left(\mathbf{k}\right){\Sigma }_{m{m}^{\prime }}(\omega ,\mathbf{k})U_{m^{\prime }{\nu }^{\prime }}\left(\mathbf{k}\right).$$

$${\hat{H}}_{\lambda }^{\mathrm{SOC}}=\frac{\lambda }{2}\sum _{m{m}^{\prime }}\sum _{\sigma {\sigma }^{\prime }}{c}_{m\sigma }^{†}\left({\mathbf{l}}_{m{m}^{\prime }}\cdot {\text{boldmathσ}}_{\sigma {\sigma }^{\prime }}\right)c_{m^{\prime }{\sigma }^{\prime }},$$

$$\omega -{\epsilon }_{\nu }\left({\mathbf{k}}_{\max }^{\nu }(\omega )\right)={\Sigma }_{\nu \nu }^{\prime }\left(\omega ,{\mathbf{k}}_{\max }^{\nu }(\omega )\right)\equiv {\Sigma }_{\nu }^{\prime }(\omega ,\theta ).$$

$$\det \left[(\omega -{\Sigma }_m^{\prime }(\omega ,{\theta }_{\mathbf{k}})){\delta }_{m{m}^{\prime }}-{\hat{H}}_{m{m}^{\prime }}^0(\mathbf{k})\right]=0.$$

$$\det [A_{m{m}^{\prime }}-{\Sigma }_m^{\prime }{\delta }_{m{m}^{\prime }}]=\det [B_{m{m}^{\prime }}-{\Sigma }_m^{\prime }{\delta }_{m{m}^{\prime }}]=$$

$$=\det [G_{m{m}^{\prime }}-{\Sigma }_m^{\prime }{\delta }_{m{m}^{\prime }}]=0.$$

$${\Sigma }_{\nu {\nu }^{\prime }}(\omega ,\mathbf{k})=\sum _m{U}_{m\nu }^{\ast }\left(\mathbf{k}\right){\Sigma }_m(\omega )U_{m{\nu }^{\prime }}\left(\mathbf{k}\right).$$

$$\frac{v_b}{v_F^{\nu }(\theta )}=\sum _m\frac{1}{Z_m}|U_{m\nu }(\theta ){|}^2,\frac{1}{Z_m}\equiv 1-{\frac{\partial {\Sigma }_m^{\prime }}{\partial \omega }|}_{\omega =0}.$$

$$\omega -{\epsilon }_{\nu }\left(\mathbf{k}\right)-{\Sigma }_{\nu \nu }^{\prime }(\omega ,\mathbf{k})-\frac{{\Sigma }_{\nu {\nu }^{\prime }}^{\prime }(\omega ,\mathbf{k}){\Sigma }_{{\nu }^{\prime }\nu }^{\prime }(\omega ,\mathbf{k})}{\omega -{\epsilon }_{{\nu }^{\prime }}\left(\mathbf{k}\right)-{\Sigma }_{{\nu }^{\prime }{\nu }^{\prime }}^{\prime }(\omega ,\mathbf{k})}=0.$$

$${\Sigma }_{\nu \nu }^{\prime }(\omega ,\mathbf{k})\gg \frac{{\Sigma }_{\nu {\nu }^{\prime }}^{\prime }(\omega ,\mathbf{k}){\Sigma }_{{\nu }^{\prime }\nu }^{\prime }(\omega ,\mathbf{k})}{\omega -{\epsilon }_{{\nu }^{\prime }}\left(\mathbf{k}\right)-{\Sigma }_{{\nu }^{\prime }{\nu }^{\prime }}^{\prime }(\omega ,\mathbf{k})}$$

Full text

I Introduction

The layered perovskite \ceSr2RuO4 is an important model system for correlated electron physics. Its intriguing superconducting ground state, sharing similarities with superfluid 3He 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 odd-parity spin-triplet 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 Black-Schaffer (2017); Steppke et al. (2017), as it is for benchmarking quantitative many-body 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 Fermi-liquid behavior below a crossover temperature of $T_{\mathrm{FL}}\approx 25$ K Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003); Stricker et al. (2014). Quantum oscillation and angle-resolved 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 intra-atomic 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 $T_{\mathrm{FL}}$, which displays metallic transport with no signs of resistivity saturation at the Mott-Ioffe-Regel limit Tyler et al. (1998). Dynamical mean-field 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 Fermi-liquid 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 self-energies associated with each orbital component are assumed to be local. On the other hand, the low-temperature Fermi-liquid state is known to display strong magnetic fluctuations at specific wave-vectors, as revealed, e.g., by neutron scattering Sidis et al. (1999); Steffens et al. (2019) 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 self-energy associated with these spin fluctuations. Interestingly, a similar debate was raised long ago in the context of liquid 3He, with ‘paramagnon’ theories emphasizing ferromagnetic spin-fluctuations and ‘quasi-localized’ approaches à la Anderson-Brinkman emphasizing local correlations associated with the strong repulsive hard-core, leading to increasing Mott-like localization as the liquid is brought closer to solidification (for a review, see Ref. Vollhardt (1984)).

In this work, we report on new insights into the nature of the Fermi-liquid state of \ceSr2RuO4. Analyzing a comprehensive set of laser-based ARPES data with improved resolution and cleanliness, we reveal a strong angular (i.e., momentum) dependence of the self-energies 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 multi-orbital systems, we extract the electronic self-energies associated with the three Ruthenium $t_{2g}$ orbitals with minimal theoretical input. We find that these orbital self-energies 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 momentum-dependent 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 self-energy displays strong deviations from the low-energy Fermi-liquid behavior $\Sigma^{\prime}(\omega)\sim\omega(1-1/Z)+\cdots$ for binding energies $|\omega|$ larger than $\sim$20\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$. These deviations are reproduced by our DMFT calculations suggesting that the cause of these non-linearities are local electronic correlations. Our results thus call for a revision of earlier reports of strong electron-lattice 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); Akebi et al. (2019). We finally quantify the effective spin-orbit 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 correlation-induced enhancement of the effective SOC. In Sec. V we proceed with a direct determination of the self-energies 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 $T_{c}=$1.45\text{,}\mathrm{K}$$. ARPES measurements were performed with an MBS electron spectrometer and a narrow bandwidth $11\text{\,}\mathrm{e}\mathrm{V}$($113\text{\,}\mathrm{n}\mathrm{m}$) laser source from Lumeras that was operated at a repetition rate of $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ with $30\text{\,}\mathrm{p}\mathrm{s}$ pulse length of the $1024\text{\,}\mathrm{n}\mathrm{m}$ pump He et al. (2016). All experiments were performed at $T\approx 5$ K using a cryogenic 6-axes sample goniometer, as described in Ref. Hoesch et al. (2017). A combined energy resolution of $3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ was determined from the width of the Fermi-Dirac distribution measured on a polycrystalline Au sample held at 4.2 K. The angular resolution was $\approx 0.2^{\circ}$. In order to suppress the intensity of the surface layer states on pristine \ceSr2RuO4 Shen et al. (2001), we exposed the cleaved surfaces to $\approx 0.5$ Langmuir \ceCO at a temperature of $\approx 120$ K. Under these conditions, \ceCO preferentially fills surface defects and subsequently replaces apical oxygen ions to form a \ceRu-COO 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 $\alpha$, $\beta$ and $\gamma$ 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 ($\Gamma$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 ${\bm{k}}$-space lines (as parameterized by the angle $\theta$ measured from the $\Gamma$M direction). The subset of data shown in Fig. 2 (a) immediately reveals a rich behavior with a marked dependence of the low-energy dispersion on the Fermi surface angle $\theta$. Along the $\Gamma$M high-symmetry line our data reproduce the large difference in Fermi velocity $v_{F}^{\beta,\gamma}$ for the $\beta$ and $\gamma$ 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 $(\theta=45^{\circ})$, 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 $v_{F}^{\beta,\gamma}$ from experiment, we determine the maxima ${\bm{k}}_{\mathrm{max}}^{\nu}(\omega)$ of the momentum distribution curves (MDCs) over the range of $26\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ below the Fermi level $E_{F}$ and fit these ${\bm{k}}$-space loci with a second-order polynomial. We then define the Fermi velocity as the derivative of this polynomial at $E_{F}$. This procedure minimizes artifacts due to the finite energy resolution of the experiment. As shown in Fig. 2 (c), the Fermi velocities $v_{F}^{\beta,\gamma}$ obtained in this way show an opposite trend with azimuthal angle for the two Fermi sheets. For the $\beta$ band we observe a gentle decrease of $v_{F}$ as we approach the $\Gamma$X direction, whereas for $\gamma$ the velocity increases by more than a factor of two over the same range 111We note that radial ${\bm{k}}$-space cuts are not exactly perpendicular to the Fermi surface. This can cause the velocities $v_{F}$ and $v_{b}$ given here to deviate by up to 10% from the Fermi velocity. However, since we evaluate the experimental and theoretical dispersion along the same ${\bm{k}}$-space cut, this effect cancels in Fig. 2 (d) and does not affect the self-energy determination in Sec. V. This provides a first indication for a strong momentum dependence of the self-energies $\Sigma^{\prime}_{\beta,\gamma}$, which we will analyze quantitatively in Sec.V. Here, we limit the discussion to the angle dependence of the mass enhancement $v_{b}/v_{F}$, which we calculate from the measured quasiparticle Fermi velocities of Fig. 2 (c) and the corresponding bare velocities $v_{b}$ of a reference Hamiltonian $\hat{H}^{0}$ defined in Sec. IV. As shown in Fig. 2 (d), this confirms a substantial many-body effect on the anisotropy of the quasiparticle dispersion. Along $\Gamma$M, we find a strong differentiation with mass enhancements of $\approx 5$ for the $\gamma$ sheet and $\approx 3.2$ for $\beta$, whereas $v_{b}/v_{F}$ approaches a common value of $\approx 4.4$ for both sheets along the Brillouin zone diagonal.

Before introducing the theoretical framework used to quantify the anisotropy of the self-energy and the effects of SOC, we compare our data quantitatively to bulk sensitive quantum oscillation measurements. Using the experimental Fermi wave vectors ${\bm{k}}_{F}$ 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 $m^{*}$ as

[TABLE]

where $A_{FS}$ is the Fermi surface volume, and using the data shown in Fig. 2 (c), we obtain $m^{*}_{\gamma}=17.3(2.0)$ $m_{e}$ and $m^{*}_{\beta}=6.1(1.0)$ $m_{e}$, in quantitative agreement with the values of $m^{*}_{\gamma}=16$ $m_{e}$ and $m^{*}_{\beta}=7$ $m_{e}$ 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 \ceSr2RuO4222We 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 $v_{F}$ difficult. The much lower value of $v_{F}^{\beta}$ along $\Gamma$M reported in Ref. Wang et al. (2017) corresponds to the surface $\beta$ band, as shown in appendix A..

III Theoretical Framework

In order to define the electronic self-energy and assess the effect of electronic correlations in the spectral function measured by ARPES, we need to specify a one-particle Hamiltonian $\hat{H}^{0}$ as a reference point. At this stage, we keep the presentation general. The particular choice of $\hat{H}^{0}$ will be a focus of Sec. IV. The eigenstates $|\psi_{\nu}\left({{\bm{k}}}\right)\rangle$ of \hat{H}^{0}$$\left({\bm{k}}\right) at a given quasi-momentum ${\bm{k}}$ and the corresponding eigenvalues $\varepsilon_{\nu}\left({\bm{k}}\right)$ define the ‘bare’ band structure of the system, with respect to which the self-energy $\Sigma_{\nu\nu^{\prime}}(\omega,{\bm{k}})$ is defined in the standard way from the interacting Green’s function

[TABLE]

In this expression $\nu$ and $\nu^{\prime}$ label the bands and $\omega$ denotes the energy counted from $E_{F}$. The interacting value of the chemical potential $\mu$ sets the total electron number. Since $\mu$ can be conventionally included in $\hat{H}^{0}$, we shall omit it in the following.

The Fermi surfaces and dispersion relations of the quasiparticles are obtained as the solutions $\omega=0$ and $\omega=\omega^{\mathrm{qp}}_{\nu}({\bm{k}})$ of

[TABLE]

In the above equation $\Sigma^{\prime}$ denotes the real part of the self-energy. Its imaginary part $\Sigma^{\prime\prime}$ has been neglected, i.e., we assume that quasiparticles are coherent with a lifetime longer than $1/\omega^{\mathrm{qp}}$. Our data indicate that this is indeed the case up to the highest energies analyzed here. At very low frequency, the lifetime of quasiparticles cannot be reliably tested by ARPES, since the intrinsic quasiparticle width is masked by contributions of the experimental resolution, impurity scattering and inhomogeneous broadening. However, the observation of strong quantum oscillations in the bulk provides direct evidence for well-defined quasiparticles in \ceSr2RuO4 down to the lowest energies Mackenzie et al. (1996b); Bergemann et al. (2000).

It is important to note that the Green’s function $G$, the self-energy $\Sigma$ and the spectral function $A$ are in general non-diagonal matrices. This has been overlooked thus far in self-energy analyses of ARPES data but is essential to determine the nature of many-body interactions in \ceSr2RuO4, as we will show in Sec. V.

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 $\nu=\{\alpha,\beta,\gamma\}$, cross the Fermi level. These bands correspond to states with $t_{2g}$ symmetry deriving from the hybridization between localized Ru-$4d$ ($d_{xy},d_{yz},d_{xz}$) orbitals and O-$2p$ states. Hence, we introduce a localized basis set of $t_{2g}$-like orbitals $|\chi_{m}\rangle$, with basis functions conveniently labeled as $m=\{xy,yz,xz\}$. In practice, we use maximally localized Wannier functions Marzari and Vanderbilt (1997); Souza et al. (2001) constructed from the Kohn-Sham eigenbasis of a non-SOC density functional theory (DFT) calculation (see appendix B.1 for details). We term the corresponding Hamiltonian $\hat{H}^{\mathrm{DFT}}$. 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 atom-centered and provide a set of states localized in real-space $|\chi_{m}\left({\mathbf{R}}\right)\rangle$. Secondly, the unitary transformation matrix to the band basis $|\psi_{\nu}\left({{\bm{k}}}\right)\rangle$

[TABLE]

allows us to define an ‘orbital’ character of each band $\nu$ as $|U_{m\nu}\left({\bm{k}}\right)|^{2}$. In the localized-orbital basis the one-particle Hamiltonian is a non-diagonal matrix, which reads

[TABLE]

The self-energy in the orbital basis is expressed as

[TABLE]

and conversely in the band basis as

[TABLE]

III.2 Spin-orbit coupling

We treat SOC as an additional term to $\hat{H}^{\mathrm{DFT}}$, which is independent of ${\bm{k}}$ in the localized-orbital basis, but leads to a mixing of the individual orbitals. The single-particle SOC term for atomic $d$-orbitals projected to the $t_{2g}$-subspace reads 333By restricting the Hamiltonian to the $t_{2g}$-subspace we neglect the $e_{g}$-$t_{2g}$ coupling terms of $\hat{H}^{\mathrm{SOC}}_{\lambda}$. This approximation is valid as long as the $e_{g}$-$t_{2g}$ crystal-field splitting is large in comparison to $\lambda$, which is the case for \ceSr2RuO4.

[TABLE]

where $\mathbf{l}$ are the $t_{2g}$-projected angular momentum matrices, $\sigma$ are Pauli matrices and $\lambda$ 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 $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda}$ with $\lambda_{\mathrm{DFT}}=$100\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$.

IV Enhanced effective spin-orbit coupling and single-particle Hamiltonian

The importance of SOC for the low-energy 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 non-trivial 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 $\gamma$ sheet along the zone diagonal Haverkort et al. (2008); Iwasawa et al. (2010) and as a degeneracy lifting at the band bottom of the $\beta$ 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 off-diagonal contributions to the local self-energy that can be interpreted as a contribution $\Delta\lambda$ to the effective coupling strength $\lambda^{\mathrm{eff}}=\lambda_{\mathrm{DFT}}+\Delta\lambda$ Zhang et al. (2016); Kim et al. (2018), This is also consistent with general perturbation-theory considerations Liu et al. (2008), which show a Coulomb-enhancement of the level splitting in the $J$ basis, similar to a Coulomb-enhanced crystal-field splitting Poteryaev et al. (2007).

In the absence of SOC, DFT yields a quasi-crossing between the $\beta$ and $\gamma$ 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 $\Delta k=\lambda^{\mathrm{eff}}/v$ and to an energy splitting of $\Delta E=Z\lambda^{\mathrm{eff}}$ Kim et al. (2018), as depicted schematically in Fig. 3 (e). In these expressions, $v\equiv\sqrt{v_{\beta}v_{\gamma}}$, with $v_{\beta}$ and $v_{\gamma}$ the bare band velocities in the absence of SOC and correlations, and $Z\equiv\sqrt{Z_{\beta}Z_{\gamma}}$ involves the quasiparticle residues $Z_{\nu}$ associated with each band (also in the absence of SOC).

It is clear from these expressions that a quantitative determination of $\lambda_{\mathrm{eff}}$ is not possible from experiment alone. Earlier studies on \ceSr2RuO4 Veenstra et al. (2014) and iron-based superconductors Borisenko et al. (2015), have interpreted the energy splitting $\Delta E$ at avoided crossings as a direct measure of the SOC strength $\lambda^{\mathrm{eff}}$. However, in interacting systems $\Delta E$ is not a robust measure of SOC since correlations can both enhance $\Delta E$ by enhancing $\lambda_{\mathrm{eff}}$ and reduce it via the renormalization factor $Z$. We thus quantify the enhancement of SOC from the momentum splitting $\Delta k$, which is not renormalized by the quasiparticle residue $Z$. The experimental splitting at the avoided crossing between the $\beta$ and $\gamma$ Fermi surface sheets indicated in Fig. 3 (a) is $\Delta k^{\mathrm{QP}}=0.094(9)$ Å*-1* whereas DFT predicts $\Delta k^{\mathrm{DFT+\lambda_{\mathrm{DFT}}}}=0.046$ Å*-1*. We thus obtain an effective SOC strength $\lambda^{\mathrm{eff}}=\lambda_{\mathrm{DFT}}\Delta k^{\mathrm{QP}}/\Delta k^{\mathrm{DFT+\lambda_{\mathrm{DFT}}}}=205(20)$ 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 $\lambda_{\mathrm{DFT}}$ as shown in Fig. 3 (f). When deviations from linearity in band dispersions are small, the splitting $\Delta E$ is symmetric around the $E_{F}$ and can thus be determined from the occupied states probed in experiment. Direct inspection of the data in Fig. 3 (f) yields $\Delta E\approx$70\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$, which is about $2/3$ of $\lambda_{\mathrm{DFT}}$ and thus clearly not a good measure of SOC.

The experimental splitting is slightly larger than that expected from the expression $\Delta E=Z\lambda^{\mathrm{eff}}$ and our theoretical determination of $Z_{\beta}$ and $Z_{\gamma}$ at the Fermi surface (which will be described in Sec. VI). This can be attributed to the energy dependence of $Z$, which, in \ceSr2RuO4, is not negligible over the energy scale of SOC. Note that the magnitude of the SOC-induced splitting of the bands at the $\Gamma$ 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 non-crossing gap leads to a much improved theoretical description of the entire Fermi surface 444Ref. Rozbicki et al. (2011) noted that an enhance effective SOC also improves the description of de Haas van Alphen data.. As shown in Fig. 3 (b), our high-resolution experimental Fermi surface deviates systematically from a DFT calculation with SOC. Most notably, $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}}$ underestimates the size of the $\gamma$ sheet and overestimates the $\beta$ sheet. Intriguingly, this is almost completely corrected in $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}+\Delta\lambda}$, with $\lambda_{\mathrm{DFT}}+\Delta\lambda=$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$, as demonstrated in Fig. 3 (c). Indeed, a close inspection shows that the remaining discrepancies between experiment and $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}+\Delta\lambda}$ break the crystal symmetry, suggesting that they are dominated by experimental artifacts. A likely source for these image distortions is imperfections in the electron optics arising from variations of the work function around the electron emission spot on the sample. Such distortions can presently not be fully eliminated in low-energy photoemission from cleaved single crystals.

Importantly, the change in Fermi surface sheet volume with the inclusion of $\Delta\lambda$ is not driven by a change in the crystal-field splitting between the $xy$ and $xz,yz$ 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 (see Fig. 9 in appendix B.1 for the orbital character without SOC). For $\lambda_{\mathrm{DFT}}+\Delta\lambda$ we find a minimal $d_{xy}$ and $d_{xz,yz}$ mixing for the $\gamma$ and $\beta$ bands of $20/80$% along the $\Gamma$M direction with a monotonic increase to $\approx 50$% along the Brillouin zone diagonal $\Gamma$X. We note that this mixing varies with the perpendicular momentum $k_{z}$. However, around the experimental value of $k_{z}\approx 0.4~{}\pi/c$ the variation is weak 555Using the free electron final state approximation, we obtain $k_{z}=(2m_{e}\hbar^{-2}(h\nu+U-\Phi)-k_{\parallel}^{2})^{1/2}\approx 0.4(0.12)\pi/c$ assuming an inner potential relative to $E_{F}$ of $U-\Phi=8.5(1.0)$ eV.. The analysis presented here and in Secs. III.1 and VI is thus robust with respect to a typical uncertainty in $k_{z}$. These findings suggest that a natural reference single-particle Hamiltonian is $\hat{H}^{0}$=$\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}+\Delta\lambda}$. This choice ensures that the Fermi surface of $\hat{H}^{0}$ is very close to that of the interacting system. From Eq. 3, this implies that the self-energy matrix approximately vanishes at zero binding energy: $\Sigma^{\prime}_{\nu\nu^{\prime}}(\omega=0,{\bm{k}})\simeq 0$. We choose $\hat{H}^{0}$ in this manner in all the following. Hence, from now on $|\psi_{\nu}\left({{\bm{k}}}\right)\rangle$ and $\varepsilon_{\nu}\left({\bm{k}}\right)$ refer to the eigenstates and band structure of $\hat{H}^{0}$=$\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}+\Delta\lambda}$. We point out that although $\hat{H}^{0}$ is a single-particle Hamiltonian, the effective enhancement $\Delta\lambda$ of SOC included in $\hat{H}^{0}$ is a correlation effect beyond DFT.

V Experimental determination of self-energies

V.1 Self-energies in the quasiparticle/band basis

Working in the band basis, i.e., with the eigenstates of $\hat{H}^{0}$, the maximum of the ARPES intensity for a given binding energy $\omega$ (maximum of the MDCs) corresponds to the momenta ${\bm{k}}$ which satisfy (following Eq. 3): $\omega-\varepsilon_{\nu}\left({\bm{k}}\right)-\Sigma^{\prime}_{\nu\nu}(\omega,{\bm{k}})=0$. Hence, for each binding energy, each azimuthal cut, and each sheet of the quasiparticle dispersions, we fit the MDCs and determine the momentum ${\bm{k}}_{\mathrm{max}}^{\nu}(\omega)$ at their maximum. Using the value of $\varepsilon_{\nu}\left({\bm{k}}_{\mathrm{max}}^{\nu}(\omega)\right)$ at this momentum yields the following quantity

[TABLE]

This equation corresponds to the simple construction illustrated graphically in Fig. 4 (b), and it is a standard way of extracting a self-energy 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 off-diagonal components $\Sigma^{\prime}_{\nu\neq\nu^{\prime}}(\omega,{\bm{k}})$ can be neglected for states close to the Fermi surface (i.e., for small $\omega$ and ${\bm{k}}$ close to a Fermi crossing). This assumption can be validated, as shown in appendix C. When performing this analysis, we only include the $\alpha$ sheet for $\theta=45^{\circ}$. Whenever the constraint $\Sigma^{\prime}_{\nu}(\omega\rightarrow 0)\rightarrow 0$ on the self-energy 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 $\hat{H}^{0}$ Fermi wave-vectors because we attribute these differences predominantly to experimental artifacts.

The determined self-energies for each band $\nu=\alpha,\beta,\gamma$ and the different values of $\theta$ are depicted in Fig. 4 (a,c). For the $\beta$ and $\gamma$ sheets they show a substantial dependence on the azimuthal angle. Around $\Gamma$M we find that $\Sigma^{\prime}_{\gamma}$ exceeds $\Sigma^{\prime}_{\beta}$ by almost a factor of two (at $\omega=$-50\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$), whereas they essentially coincide along the zone diagonal ($\Gamma$X). This change evolves as a function of $\theta$ and occurs via a simultaneous increase in $\Sigma^{\prime}_{\beta}$ and a decrease in $\Sigma^{\prime}_{\gamma}$ for all energies as $\theta$ is increased from $0^{\circ}$ ($\Gamma$M) to $45^{\circ}$ ($\Gamma$X). In order to better visualize this angular dependence, a compilation of $\Sigma^{\prime}_{\nu}(\omega)$ for different values of $\theta$ is displayed in Fig. 4 (c).

V.2 Accounting for the angular dependence: local self-energies in the orbital basis

In this section, we introduce a different procedure for extracting self-energies from ARPES, by working in the orbital basis $|\chi_{m}\left({\bm{k}}\right)\rangle$. We do this by making two key assumptions:

We assume that the off-diagonal components are negligible, i.e., $\Sigma^{\prime}_{m\neq m^{\prime}}\simeq 0$. Let us note that in \ceSr2RuO4 even a ${\bm{k}}$-independent self-energy has non-zero off-diagonal elements if $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}}$ is considered. Using DMFT, these off-diagonal 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 ($\Delta\lambda$). In the present work, these off-diagonal frequency-independent components are already incorporated into $\hat{H}^{0}$ (see Sec. IV), and thus the frequency-dependent part of the self-energy is (approximately) orbital diagonal by virtue of the tetragonal crystal structure. 2. 2.

We assume that the diagonal components of the self-energy in the orbital basis depend on the momentum ${\bm{k}}$ only through the azimuthal angle $\theta_{\bm{k}}$: $\Sigma^{\prime}_{mm}(\omega,{\bm{k}})\simeq\Sigma^{\prime}_{m}(\omega,\theta_{\bm{k}})$. We neglect the dependence on the momentum which is parallel to the angular cut.

Under these assumptions, the equation determining the quasiparticle dispersions reads

[TABLE]

In this equation, we have neglected the lifetime effects associated with the imaginary part $\Sigma^{\prime\prime}_{m}$. In order to extract the functions $\Sigma^{\prime}_{m}(\omega,\theta_{\bm{k}})$ directly from the ARPES data, we first determine the peak positions ${\bm{k}}_{\mathrm{max}}^{\nu}(\omega,\theta)$ for MDCs at a given angle $\theta$ and binding energy $\omega$. We then compute (for the same $\omega$ and $\theta$) the matrix $A_{mm^{\prime}}\equiv\omega\delta_{mm^{\prime}}-\hat{H}^{0}_{mm^{\prime}}\left({\bm{k}}_{\mathrm{max}}^{\alpha}(\omega,\theta_{{\bm{k}}})\right)$ and similarly $B_{mm^{\prime}}$, $G_{mm^{\prime}}$ for the $\beta$ and $\gamma$ band MDCs, ${\bm{k}}_{\mathrm{max}}^{\beta}(\omega,\theta)$ and ${\bm{k}}_{\mathrm{max}}^{\gamma}(\omega,\theta)$, respectively. In terms of these matrices, the quasiparticle equations (10) read

[TABLE]

However, when taking symmetry into account, the self-energy has only two independent components: $\Sigma^{\prime}_{xy}$ and $\Sigma^{\prime}_{xz}=\Sigma^{\prime}_{yz}$. Hence, we only need two of the above equations to solve for the two unknown components of the self-energy. This means that we can also extract a self-energy in the directions where only two bands ($\beta$ and $\gamma$) are present in the considered energy range of $\omega\lesssim$100\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$, e.g., along $\Gamma$M. The resulting functions $\Sigma^{\prime}_{m}(\omega,\theta_{\bm{k}})$ determined at several angles $\theta$ are displayed in Fig. 4 (d). It is immediately apparent that, in contrast to $\Sigma^{\prime}_{\nu\nu}$, the self-energies in the orbital basis do not show a strong angular (momentum) dependence, but rather collapse into two sets of points, one for the $xy$ orbital and one for the $xz/yz$ orbitals. Thus, we reach the remarkable conclusion that the angular dependence of the self-energy in the orbital basis is negligible, within the range of binding energies investigated here: $\Sigma^{\prime}_{m}(\omega,\theta_{\bm{k}})\simeq\Sigma^{\prime}_{m}(\omega)$. This implies that a good approximation of the full momentum and energy dependence of the self-energy in the band (quasiparticle) basis is given by

[TABLE]

The physical content of this expression is that the angular (momentum) dependence of the quasiparticle self-energies emphasized above is actually due to the matrix elements $U_{m\nu}\left({\bm{k}}\right)$ defined in Eq. 4. 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 back-transform of $\Sigma^{\prime}_{m}\left(\omega,\theta_{\bm{k}}=18^{\circ}\right)$ into $\Sigma^{\prime}_{\nu}\left(\omega,\theta_{\bm{k}}=0,18,45^{\circ}\right)$. The good agreement with $\Sigma^{\prime}_{\nu\nu}$ 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 (12) precisely coincides with the ansatz made by DMFT: within this theory, the self-energy is approximated as a local (${\bm{k}}$-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 Mean-Field Theory

In this section we perform an explicit comparison of the measured quasiparticle dispersions and self-energies to DMFT results. The latter are based on the Hamiltonian $\hat{H}^{\mathrm{DFT}}$, to which the Hubbard-Kanamori interaction with on-site interaction $U=$2.3\text{,}\mathrm{e}\mathrm{V} and Hund’s coupling $J=$0.4\text{\,}\mathrm{e}\mathrm{V} 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 $A\left(\omega,{\bm{k}}\right)$ displayed as a color-intensity map. Clearly, the theoretical results are in near quantitative agreement with the data: both the strong renormalization of the Fermi velocity and the angular-dependent curvature of the quasiparticle bands are very well reproduced by the purely local, and thus momentum-independent DMFT self-energies. This validates the assumption of no momentum dependence along the radial $k$-space direction of the self-energy made in Sec. V.2 for the $k/\omega$ range studied here. 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 self-energies for each orbital with the DMFT results. The overall 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 angular-dependence of the effective mass renormalisation. Using the local ansatz (12) into the quasiparticle dispersion equation, and performing an expansion around $E_{F}$, we obtain

[TABLE]

In Fig. 6 (b), we show $v_{b}/v_{F}^{\nu}(\theta)$ for the $\beta$ and $\gamma$ bands using the DMFT values $Z_{xy}=0.18\pm 0.01$ and $Z_{xz/yz}=0.30\pm 0.01$ obtained at $29\text{\,}\mathrm{K}$ (appendix B.2). The overall angular dependence and the absolute value of the $\gamma$ band mass enhancement is very well captured by DMFT, while the $\beta$ band is a bit overestimated. Close to the zone-diagonal ($\theta=45^{\circ}$), 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 $\Sigma^{\prime}_{xy}$ is in good agreement with the experimental data over the full energy range of $280\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ covered in our experiments. Both the theoretical and experimental self-energies deviate significantly from the linear regime down to low energies ($\sim$20\text{,}\mathrm{m}\mathrm{e}\mathrm{V}), causing curved quasiparticle bands with progressively steeper dispersion as the energy increases (Fig. [5](#S5.F5)). In contrast, the agreement between theory and experiment for the $xz/yz$ self-energy is somewhat less impressive at binding energies larger than $\sim$30\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}. Our DMFT self-energy $\Sigma^{\prime}_{xz/yz}$ overestimates the strength of correlations in this regime (by $20-25\%$), 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 self-energy, we note that the present DMFT calculation is performed with an on-site value of $U$ which is the same for all orbitals. Earlier cRPA calculations have suggested that this on-site interaction is slightly larger for the $xy$ orbital ($U_{xy}=$2.5\text{,}\mathrm{e}\mathrm{V} and $U_{xz/yz}=$2.2\text{\,}\mathrm{e}\mathrm{V}) 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 self-energy, especially dependence on momentum perpendicular to the Fermi surface. We note in this respect that the discrepancy is larger for the $\alpha,\beta$ sheets which have dominant $xz/yz$ character. These orbitals have, in the absence of SOC, a strong one-dimensional 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 spin-density wave correlations, which are expected to lead to an additional momentum-dependence of the self-energy. We further discuss possible contributions of spin fluctuations in Sec. VIII.

VII Kinks

The self-energies $\Sigma^{\prime}\left(\omega\right)$ shown in Figs. 4 and 6 display a fairly smooth curvature, rather than pronounced ‘kinks’. Over a larger range, however, $\Sigma^{\prime}_{xy}$ from DMFT does show an energy scale marking the crossover from the strongly renormalized low-energy 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 Mott-Hubbard sidebands Byczuk et al. (2007) and have been abundantly documented in the theoretical literature 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 Fermi-liquid 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 self-energies $\Sigma^{\prime}_{m}\left(\omega\right)$ and our single-site 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 low-energy regime of our experimental data, we find deviations from the linear form $\Sigma^{\prime}\left(\omega\right)=\omega(1-1/Z)$ characteristic of a Fermi liquid for $|\omega|>$20\text{,}\mathrm{m}\mathrm{e}\mathrm{V}, irrespective of the basis. However, this is only an upper limit for the Fermi-liquid energy scale in \ceSr2RuO4. Despite the improved resolution of our experiments, we cannot exclude an even lower crossover energy to non-Fermi-liquid like excitations. We note that the crossover temperature of $T_{\mathrm{FL}}\approx$25\text{\,}\mathrm{K} reported from transport and thermodynamic experiments Mackenzie et al. (1996a); Maeno et al. (1997); Bergemann et al. (2003) indeed suggests a cross-over energy scale that is significantly below $20\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$.

The overall behavior of $\Sigma^{\prime}\left(\omega\right)$ including the energy range where we find strong changes in the slope agrees with previous photoemission experiments, which were interpreted as evidence for electron-phonon 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 Fermi-liquid regime of electronic correlations over the entire phonon bandwidth of $\approx$90\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$ Braden et al. (2007), which is inconsistent with our DMFT calculations. Moreover, attributing the entire curvature of $\Sigma^{\prime}_{m}$ in our data to electron-phonon 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 $\beta$ and $\gamma$ 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 high-resolution 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 correlation-induced 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 improved cleanliness of our data, we have been able to determine the electronic self-energies directly from experiment, 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 self-energies 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 self-energies which are frequency-dependent 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.

The key importance of atomic-like orbitals in correlated insulators is well established Kugel and Khomskii (1982); Tokura and Nagaosa (2000). The present work demonstrates that orbitals retain a considerable physical relevance even in a metal in the low-temperature Fermi-liquid regime. Although the band and orbital basis used here are equivalent, our analysis shows that the underlying simplicity in the nature of correlations emerges only when working in the latter and taking into account the orbital origin of quasiparticles. Beyond \ceSr2RuO4, this is an observation of general relevance to multiband metals with strong correlations such as the iron-based superconductors Miao et al. (2016); Sprau et al. (2017).

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. (2019); Ishida et al. (2001); Imai et al. (1998), with a strong peak in its spin response $\chi({\mathbf{Q}})$ close to the spin-density wave (SDW) vector $\mathbf{Q}\sim(2\pi/3,2\pi/3)$, as well as quasi-ferromagnetic fluctuations which are broader in momentum around $\mathbf{Q}=0$. Indeed, tiny amounts of substitutional impurities induce long-range 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 long-wavelength fluctuations affect the physics of quasiparticles in the Fermi-liquid state. Single-site 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 (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 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 outstanding open questions Mackenzie et al. (2017).

Acknowledgements.

The experimental work has been supported by the European Research Council (ERC), the Scottish Funding Council, the UK EPSRC and the Swiss National Science Foundation (SNSF). Theoretical work was supported by the ERC grant ERC-319286-QMAC and by the SNSF (NCCR MARVEL). The Flatiron Institute is a division 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 $h\nu=$21\text{,}\mathrm{e}\mathrm{V}$$ 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 $\beta$ 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 $15\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for the smaller $\beta$ sheet that we identify as a surface band.

Wang et al. Wang et al. (2017) have recently probed the low-energy electronic structure of \ceSr2RuO4 by STM. Analyzing quasiparticle interference patterns along the $\Gamma$X and $\Gamma$M high-symmetry directions, they obtained band dispersions with low Fermi velocities and strong kinks at $10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ and $37\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. In Fig. 8 we compare the band dispersion reported by Wang et al. with our ARPES data. Along both high-symmetry directions, we find a clear discrepancy with our data, which are in quantitative agreement with bulk de Haas van Alphen measurements, as demonstrated in the main text. On the other hand, we find a striking similarity between the STM data along $\Gamma$M and the band commonly identified as the surface $\beta$ band Shen et al. (2001); Ingle et al. (2005). 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 strong low-energy kinks and overall enhanced low-energy renormalization seen by ARPES in the surface bands Kondo et al. (2016); Akebi et al. (2019).

Appendix B Computational details

B.1 DFT and model Hamiltonian

These orbitals are centered on the Ru atoms and have $t_{2g}$ symmetry, but are indeed linear combinations of Ru-$d$ and O-$p$ states. We do not add Wannier functions centered on the oxygen atoms, because the resulting three orbital Wannier model already accurately reproduces the three bands crossing the Fermi energy, as demonstrated in Fig. 9. Also note that the Wannier function construction allows to disentangle the $\gamma$ band from the bands with dominantly O-$p$ character below $-2\text{\,}\mathrm{e}\mathrm{V}$.

We generate our theoretical model Hamiltonian $\hat{H}^{\mathrm{DFT}}$ with a maximally-localized Wannier function Marzari and Vanderbilt (1997); Souza et al. (2001) construction of $t_{2g}$-like orbitals for the three bands crossing the Fermi surface. These Wannier orbitals are obtained on a $10\times 10\times 10$ ${\bm{k}}$ grid based on a non-SOC DFT calculation using WIEN2k Blaha et al. (2001) with the GGA-PBE 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 $100\text{\,}\mathrm{K}$) and converged with twice as many ${\bm{k}}$-points in each dimension.

The eigenenergies of the resulting Wannier Hamiltonian, $\hat{H}^{\mathrm{DFT}}$, accurately reproduce the DFT band structure (Fig. 9 top). Note that in the absence of SOC, the eigenstates retain pure orbital character, as shown in Fig. 10. To take SOC into account, we add the local single-particle term $\hat{H}^{\mathrm{SOC}}_{\lambda}$, as given in Eq. 8, with coupling constant $\lambda$. In the bottom panel of Fig. 9 we show that the eigenenergies of $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda}$ are in nearly perfect agreement with the DFT+SOC band structure at a value of $\lambda_{\mathrm{DFT}}=$100\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$.

Our model Hamiltonian provides the reference point to which we define a self-energy, 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 $xy$ and $xz/yz$ orbitals. In the following, we will confirm that the best agreement with the experimental Fermi surface is found with an effective SOC of $\lambda^{\mathrm{eff}}=\lambda_{\mathrm{DFT}}+\Delta\lambda=$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V}, but at the same time keeping the DFT crystal-field splitting of $\epsilon_{\mathrm{cf}}$ $=\epsilon_{xz/yz}-\epsilon_{xy}=$85\text{\,}\mathrm{m}\mathrm{e}\mathrm{V} unchanged. We compare in Fig. 11 the experimental Fermi surface (dashed lines) to the one of $\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}}$. The Fermi surfaces for additionally introduced crystal-field splittings $\Delta\epsilon_{\mathrm{cf}}$ between $-80$ and $80\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ are shown with solid lines in different shades of red. In contrast to the Fermi surface without SOC ($\lambda=$0\text{,}\mathrm{m}\mathrm{e}\mathrm{V}, top left panel), the Fermi surfaces with the DFT SOC of $\lambda=$100\text{\,}\mathrm{m}\mathrm{e}\mathrm{V} (top right panel) resembles the overall structure of the experimental Fermi surface. However, the areas of the $\alpha$ and $\beta$ sheets are too large and the $\gamma$ sheet is too small. Importantly, the agreement cannot be improved by adding $\Delta\epsilon_{\mathrm{cf}}$. For example, along $\Gamma$M a $\Delta\epsilon_{\mathrm{cf}}$ of $-40\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ would move the Fermi surface closer to the experiment, but, on the other hand, along $\Gamma$X a $\Delta\epsilon_{\mathrm{cf}}$ of $80\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ would provide the best agreement. The situation is different if we consider an enhanced SOC of $\lambda=$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V} (bottom right panel). Then, we find a nearly perfect agreement with experiment without any additional crystal-field splitting ($\Delta\epsilon_{\mathrm{cf}}=0\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$). At an even higher SOC of \lambda=300\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ (bottom left panel) we see again major discrepancies, but with an opposite trend: The \alpha$and$\beta$sheets are now too small and the$\gamma$sheet is too large. Like in the case of$\lambda=100\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ this can not be cured by an adjustment of \epsilon_{\mathrm{cf}}$.

B.2 DMFT

We perform single-site DMFT calculation with the TRIQS/DFTTools Aichhorn et al. (2016) package for $\hat{H}^{\mathrm{DFT}}$ and Hubbard-Kanamori interactions with a screened Coulomb repulsion $U=$2.3\text{,}\mathrm{e}\mathrm{V} and a Hund’s coupling $J=$0.4\text{\,}\mathrm{e}\mathrm{V} based on previous works Mravlje et al. (2011); Kim et al. (2018). The impurity problem is solved on the imaginary-time axis with the TRIQS/CTHYB Seth et al. (2016) solver at a temperature of $29\text{\,}\mathrm{K}$. The employed open-source software tools are based on the TRIQS library Parcollet et al. (2015). We assume an orbital-independent 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 self-energy to the real-frequency 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 $-100$ to $0\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ the difference in the resulting self-energies (Fig. 12) is below the experimental uncertainty . The averaged quasiparticle renormalizations (of the three continuations) are: $Z_{xy}=0.18\pm 0.01$ and $Z_{xz}=Z_{yz}=0.30\pm 0.01$. For all other results presented in the main text the Padé solution has been used.

Our calculations at a temperature of $29\text{\,}\mathrm{K}$ use $\hat{H}^{\mathrm{DFT}}$, as the sign problem prohibits reaching such low temperatures with SOC included. Nevertheless, calculations with SOC were successfully carried out at a temperature of $290\text{\,}\mathrm{K}$ using CT-INT Zhang et al. (2016) and at $230\text{\,}\mathrm{K}$ using CT-HYB with a simplified two-dimensional tight-binding 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 self-energy with diagonal elements close to the ones without SOC plus, to a good approximation, frequency-independent off-diagonal elements, which can be absorbed in a static effective SOC strength of $\lambda^{\mathrm{eff}}=\lambda_{\mathrm{DFT}}+\Delta\lambda\simeq$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$ – this is the approach followed in the present article.

In addition to the enhancement of SOC it was observed that low-energy many-body effect also lead 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 a orbital-dependent splitting in the real part of the self-energies ($\Delta\epsilon_{\mathrm{cf}}$ $=$60\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$), which would move the $\gamma$ sheet closer to the van Hove singularity and consequently worsen the agreement with the experimental Fermi surface along the $\Gamma$M direction (see bottom right panel of Fig. 11). Different roots of this small discrepancy are possible, ranging from orbital-dependent double counting corrections to, in general, DFT being not perfect as ‘non-interacting’ reference point for DMFT. 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. We point out that in comparision to the present work a large enhancement of the crystal-field splitting was observed in Ref. Zhang et al. (2016), presumably due to the larger interactions employed.

Based on these considerations we calculate the correlated spectral function $A\left(\omega,{\bm{k}}\right)$ (shown in Fig. 5 (a) of the main text) using the Hamiltonian with enhanced SOC ($\hat{H}^{\mathrm{DFT}}$+$\hat{H}^{\mathrm{SOC}}_{\lambda_{\mathrm{DFT}}+\Delta\lambda}$) in combination with the frequency-dependent part of the non-SOC (diagonal) self-energy, but neglect the additional static part introduced by DMFT.

Appendix C Off-diagonal elements of $\Sigma^{\prime}_{\nu\nu^{\prime}}$

In Sec. V.1 we extracted $\Sigma^{\prime}_{\nu\nu}$ under the assumption that the off-diagonal elements can be neglected. To obtain insights about the size of the off-diagonal elements we use Eq. 12 to calculate the full matrix $\Sigma^{\prime}_{\nu\nu^{\prime}}(\omega,{\bm{k}})$ in the band basis from the DMFT self-energy in the orbital basis $\Sigma^{\prime}_{m}(\omega)$, as shown in Fig. 6 (a). This allows us to obtain the full self-energy matrix $\Sigma^{\prime}_{\nu\nu^{\prime}}(\omega,{\bm{k}})$ at one specific combination of ${\bm{k}}$ and $\omega$. Note that for the results presented in Fig. 4 and Fig. 13 (b) this is not the case, because the extracted self-energies for each band correspond to different ${\bm{k}}^{\nu}_{\text{max}}$, which are further defined by the experimental MDCs.

In Fig. 13 (a) we show the result for two selected ${\bm{k}}$ points: on the $\beta$ sheet for $\theta=0$ and on the $\gamma$ sheet for $\theta=45^{\circ}$. For these ${\bm{k}}$ points the largest off-diagonal element is $\Sigma^{\prime}_{\gamma\beta}$, which is about $10-20\%$ of the size of the diagonal elements. A scan performed for the whole $k_{z}=0.4~{}\pi/c$ plane further confirms that $|\Sigma^{\prime}_{\nu\neq\nu^{\prime}}|$ is smaller than $20\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$.

However, when neglect the off-diagonal 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 ($\nu,\nu^{\prime}$) and rewriting Eq. 3, which determines the quasiparticle dispersion $\omega^{\mathrm{qp}}_{\nu}({\bm{k}})$, as

[TABLE]

Setting the last term to zero, i.e., using the procedure described in Sec. V.1 to extract $\Sigma^{\prime}_{\nu\nu}$, is justified at $\omega=\omega^{\mathrm{qp}}_{\nu}({\bm{k}})$ as long as

[TABLE]

In this condition the already small off-diagonal elements enter quadratically, but also the denominator is not a small quantity, because the energy separation of the bare bands ($\varepsilon_{\nu}\left({\bm{k}}\right)-\varepsilon_{\nu^{\prime}}\left({\bm{k}}\right)$) is larger than the difference of the diagonal self-energies.

By using the generalized version of Eq. 15 for all three bands, we find that the right-hand side of this equation is indeed less than $1.2\%$ of $\Sigma^{\prime}_{\nu\nu}(\omega,{\bm{k}}_{\mathrm{max}}^{\nu}(\omega))$ for all experimentally determined ${\bm{k}}_{\mathrm{max}}^{\nu}(\omega)$. This means that for \ceSr2RuO4 treating each band separately when extracting $\Sigma^{\prime}_{\nu\nu}$ is well justified in the investigated energy range.

Appendix D Reconstruction of $\Sigma^{\prime}_{\nu\nu}$

In order to further test the validity of the local ansatz (Eq. 12) and establish the overall consistency of the two procedures used to extract the self-energy in Sec. V, we perform the following ‘reconstruction procedure’. We use the $\Sigma^{\prime}_{m}(\omega,\theta_{\bm{k}})$ (from Sec. V.2) at one angle, e.g., $\theta=18^{\circ}$, and transform it into $\Sigma^{\prime}_{\nu\nu}\left(\omega,{\bm{k}}_{\mathrm{max}}^{\nu}(\omega)\right)$ for other measured angles, using Eq. 12. The good agreement between the self-energy reconstructed in this manner (thin lines in Fig. 13 (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 $\Sigma^{\prime}_{\nu\nu}$ is almost entirely due to the momentum dependence of the orbital content of quasiparticle states, i.e., of $U_{m\nu}\left({\bm{k}}\right)=\langle\chi_{m}\left({\bm{k}}\right)|\psi_{\nu}\left({{\bm{k}}}\right)\rangle$. In \ceSr2RuO4 the momentum dependence of these matrix elements is mainly due to the SOC.