Spectral functions of Sr${}_2$IrO${}_4$: theory versus experiment

The spin-orbit Mott insulator Sr${}_2$IrO${}_4$ has attracted a lot of interest in recent years from theory and experiment due to its close connection to isostructural high-temperature copper oxide superconductors. Despite of not being superconducting its spectral features closely resemble those of the cuprates, including Fermi surface and pseudogap properties. In this article, we review and extend recent work in the theoretical description of the spectral function of pure and electron-doped Sr${}_2$IrO${}_4$ based on a cluster extension of dynamical mean-field theory ("oriented-cluster DMFT") and compare it to available angle-resolved photoemission data. Current theories provide surprisingly good agreement for pure and electron-doped Sr${}_2$IrO${}_4$, both in the paramagnetic and antiferromagnetic phases. Most notably, one obtains simple explanations for the experimentally observed steep feature around the $M$ point and the pseudo-gap-like spectral feature in electron-doped Sr${}_2$IrO${}_4$.


Introduction
Electronic Coulomb correlations are well-established as one of the main players determining the excitations and response properties of 3d transition metal oxides or f-electron compounds. The relatively modest spatial extension of the orbitals forming the valence states in these materials leads to comparably large effective local Coulomb interactions. They are in particular large enough to compete with the kinetic energy even when several partially filled valence states are involved. It is well-known from studies of the multi-orbital Hubbard model that the critical interaction strength needed to localize the electrons and induce a Mott insulating state grows with the orbital degeneracy [1,2], translating the enhanced kinetic energy in the multi-orbital context.
Multi-orbital physics is rather the generic situation in transition metal oxides [3] and even more so in f-electron compounds. Examples include t 2g -systems such as titanates [4] or vanadates [5], or Mn-, Fe-or Ni-based e g -oxides [6]. A notable exception are high-temperature superconducting cuprates, where a single hole in an otherwise completely filled 3d-shell can be dominantly attributed to an orbital of d x 2 −y 2 character, corresponding -in the parent compounds -to a nominally half-filled configuration. Most of the exotic properties of cuprates are directly or indirectly attributed to this specific situation [7,8,9] ‡.
Iridium oxides have added interesting twists to these quite general considerations [12]. First, the rather extended nature of the Ir 5d orbitals would suggest correlations to play a minor role in these systems if indeed all three t 2g orbitals were contributing to the low-energy physics around the Fermi level in a more or less degenerate way. The effective local Coulomb interactions have rather modest values of the order of 2 eV [13,14,15], comparable to the t 2g -electron bandwidth, and would -in a degenerate threeorbital model -at most lead to a modest mass enhancement in a metallic state [13]. A radically different situation is realized in nature: focussing on the prototypical Sr 2 IrO 4 , one is dealing with an insulating material with a strongly temperature-dependent gap, both in its low-temperature magnetic and high-temperature paramagnetic phase, as confirmed by x-ray photoelectron [16,17,18,19,20], angle-resolved photoemission [12,21,20,22,23,24,25,26,27,28,29,30], and scanning tunneling [31,32] spectroscopy, optical conductivity measurements [33,12,34,35,36,37,38] as well as magneto [39,40], thermo [41], and electrical [42,43,44,45,46] transport measurements.
The system does not explore the full space of orbital fluctuations corresponding to the nominal t 5 2g configuration, but rather adopts cuprate-like single orbital physics. The microscopic reason can be found in a subtle interplay between spin-orbit interactions, structural distortions and Coulomb correlations [13,15]. The former two result in a rearrangement of the electronic states in such a way that -at the one-particle leveltwo nearly filled bands (usually labeled by an effective quantum number j ef f = 3/2) and a half-filled (j ef f = 1/2) band can be identified. The role of the Coulomb correlations is then two-fold: first, to enhance the spin-orbital polarization of this state such as to leave only the now exactly half-filled j ef f = 1/2 at the Fermi level, and second, to localize the hole in this state. Energy (eV) Figure 1. Left: Wannier-projected j eff =1/2 (turquoise) and j eff =3/2 bands (|m j | = 1/2 in magenta, |m j | = 3/2 in yellow) within DFT+SOC. Right: After applying Coulomb interactions, the j eff =1/2 manifold splits into an upper and lower Hubbard band. Level scheme adopted from Kim et al. 2008 [12].
The net result is thus an insulating state that bears some resemblance to the one of the parent compounds of the cuprates: it is realized within a layered crystal structure that deviates from the one of the celebrated La 2 CuO 4 cuprate only by rotations of the oxygen octahedra around the crystallographic c-axis; it is of single-orbital nature, and it is accompanied by antiferromagnetic order in the low-temperature regime. It has been noted that Sr 2 IrO 4 could be described by a twisted Hubbard model [52], bridging to the most popular model for high-temperature superconducting cuprates. In both cases, the antiferromagnetic order is driven by superexchange constants, which are of the same order of magnitude for Sr 2 IrO 4 (nearest neighbor exchange J N N between ∼ 0.06eV [53,54] and ∼ 0.1eV [17]) and its related cuprate La 2 CuO 4 (J N N ∼ 0.14eV [55]). In Sr 2 IrO 4 , a canted antiferromagnetic ground state is formed below the Néel temperature of T N ∼ 240K, which amounts to a weak ferromagnetic moment in c-direction [56].
Doping by chemical substitutions or vacancies leads in the cuprates to by now wellcharacterized but still intriguing superconducting phases. In iridates, doping has been achieved by La- [57,42], Rh- [42], or Co-substitutions [58], O-vacancies [59] and surfacedoping [20,29]. From those, in particular Lanthanum and Rhodium substitutions have been used in recent years to systematically study electron-and hole-doped Sr 2 IrO 4 respectively. In both cases, the substitutions primarily act like electronic doping and do not change much the lattice constant of Sr 2 IrO 4 [42,57]. Yet, any direct observation of superconductivity remains elusive. In the cuprates, the normal state is itself anything but normal: the most intriguing feature is probably the appearance of a pseudogap in the low-energy electronic states in immediate vicinity to the Fermi level [60], which has continued to stimulate experimental [61,62,63,64] and theoretical [65,7,8] studies.
Over the last decades, spectroscopic techniques have evolved into a most powerful tool to characterize low-energy excitations in correlated materials in a direct fashion (for a review, see e.g. [66]). Up to matrix elements translating modulations induced by light-matter coupling, angle-resolved photoemission measures the spectral function A(k, ω) related to the one-particle Green's function by The latter is in principle accessible by advanced many-body techniques, such as dynamical mean field theory (for reviews see [67,68]) and its extensions. Recent advances using combined density functional theory dynamical mean field schemes [69,70,71] have made substantial progress in describing spectral properties of correlated materials in an ab initio (i.e. materials-specific) fashion (see e.g. [72,73,74,75,76,77,78]). Iridates have been in the focus since 2010 [79,13,14,48,47], with most recent advances giving a detailed account of spectral properties both in the pure and doped samples based on an extension of DMFT [30].
In this paper, we review and extend recent theoretical results using the cluster extension of DMFT proposed in Ref. [30] in comparison to available experimental data to assess the current state of the art in the field.
The paper is organized as follows: In the next section the oriented cluster dynamical mean-field theory of Ref. [30] is reviewed. Sections 3 and 4 focus on the paramagnetic high-temperature and antiferromagnetic low-temperature phase of undoped Sr 2 IrO 4 respectively. We compare calculated spectra to experimentally measured ones in both cases. Electron-doped Sr 2 IrO 4 is discussed in Sec. 5 with a focus on Fermi surface and pseudogap features of the spectral function. A summary and outlook conclude the paper in Sec.6.

Technique: Oriented Cluster Dynamical Mean-Field Theory
In dynamical mean-field theory (DMFT) [67,68] the interacting many-electron problem is mapped onto a single impurity site that is coupled dynamically to a non-interacting bath. Since this impurity represents an arbitrary site in the lattice it amounts to a purely local description and correlation effects, which are taken into account via the impurity self-energy Σ imp , are therefore local in nature, too. The momentum-dependent lattice Green's function is constructed using Σ imp via In the case of Sr 2 IrO 4 , the one-particle hamiltonian H(k) is obtained from a tightbinding fit of the j eff =1/2 band, see Appendix A. The Green's function of the impurity Figure 2. Crystal structure of Sr 2 IrO 4 . Sr 2 IrO 4 crystallizes in a I4 1 /acd structure (A) and can be described by a unit cell which comprises two Sr 2 IrO 4 layers (I) and (II) in c-direction (B) with a total of four Ir atoms (two per layer). Within such a layer, the environments of neighboring Ir atoms differ since the surrounding oxygen octahedra are rotated by ∼ 11 • in opposite directions. Due to these rotations of the oxygen octahedra the unit cell is doubled and the respective Brillouin zone takes only half of the size of an undistorted geometry (C). Panels (B) and (C) adopted from Martins et al. [30].
problem is then equal to the local Green's function of the lattice problem, which is ensured by the DMFT self-consistency condition which determines the bath and restores the translational invariance. In general, the system is of multi-orbital nature with multiple correlated shells a per unit cell. For each correlated site, the self-energy of the effective impurity site has to be rotated into the local reference frame via Here, Σ imp is a matrix in orbital space, i.e. a N × N matrix for a system consisting of N correlated orbitals per correlated shell. If we assume that -due to crystallographic symmetries-the N a correlated sites are equivalent, the overall self-energy matrix Σ is of size N · N a × N · N a . Moreover, the self-energy is block-diagonal in the correlated shells, i.e.
Since all shells are rotated into their local coordinate system, using the self-energy Σ aa (ω) in Eq.(2) leads to a lattice Green's function, which respects all point group symmetries of the crystal. In cluster extensions of DMFT [80] the original lattice is tiled into (identical) clusters, which can break the point group symmetry of the crystal. The location x Figure 3. Comparison of DMFT and OC-DMFT: Since more than one cluster orientation is allowed, the self-consistency condition includes an orientation average which restores the point-group symmetry. In this case, the momenta k have to be taken out of the Brillouin zone I α corresponding to the respective orientation.
of each site can then be represented by specifying the origin of a clusterx and the site within the cluster X. Using the self-consistency condition of Eq.(3) only restores the translational invariance, but does not remove possible violations of the point group symmetry. Instead of limiting ourselves to clusters, which have the full point group symmetry of the original lattice, we recently proposed to use oriented clusters and to restore the point group symmetry via a modified embedding condition [30]. Choosing an oriented two-atom dimer as a cluster, which represents two nearest-neighbor Ir sites, one is left with two possible (perpendicular) cluster orientations. Therefore, we can define a momentum-resolved Green's function evaluated with an orientation dependent self-energy: The momenta K are taken out of the Brillouin zone corresponding to the Bravais lattice of the cluster locationsx [80]. As a consequence, when constructing the local Green's function, the sum now runs over the two possible orientations α = x, y and over their corresponding momenta K ∈ I α : The momentum-resolved spectral function A(k, ω) to be compared with the ARPES spectra later on is obtained from the k-resolved Green function Here, k is the momentum-vector in the crystallographic Brillouin zone of the lattice. For Sr 2 IrO 4 neighboring Ir sites already differ in the rotation of their oxygen octahedra and the crystallographic unit cell consists of four Ir atoms. A possible choice for the unit cell amounts to taking two neighboring Ir sites in two stacked a-b layers in c-direction, see Fig. 2(A). Choosing two dimers as the building block for a cluster tiling of the lattice therefore does not change the Brillouin zone and keeps the same crystallographic cell.
Since we use the minimal cluster that incorporates non-local effects, it is possible to gain insight into the physical processes which cause certain features in the spectrum. For instance, inter-site fluctuations of the dimer are of antiferromagnetic spin nature and are encoded in the non-local part of the cluster self-energy. The latter can be seen as energy-dependent modulations of the inter-site hopping processes that enter in the Green's function G α . In Sr 2 IrO 4 this selection of an orientation α causes a x − ysymmetry breaking in G α . This shows the importance of the orientational average in Eq. (7), which restores this symmetry. Most interestingly, signatures of the fluctuations can be seen in terms of tilted lens-shaped Fermi surfaces in electron-doped Sr 2 IrO 4 as discussed later in Sec. 5.   illustrates schematically how including non-local fluctuations via the dimer self-energy Σ α within OC-DMFT affects the spectral function. The starting point of the calculations is a tight-binding fit of the DFT j eff =1/2 bands, which defines the bare dispersion (k). Since the unit cell comprises two layers in c-direction with a dimer each, we take the weak inter-layer hopping into account and obtain two slightly different downfolded inter-layer bands as discussed in Appendix A. For each of the two dispersions we then calculate the lattice Green's function for the two possible dimer orientations.
Choosing one dimer configuration changes the non-interacting spectrum of the band shown in Fig. 4(A) into the interacting spectral function A α (k, ω) shown in (B). Besides the splitting of the band into a bonding and an antibonding band, additional satellite bands emerge at larger energy. For comparison, the bonding and antibonding poles of the dimer Green's function are indicated by orange, the position of their satellites by green lines.
We construct a Hamiltonian to model the j eff =1/2 band which amounts to a oneband Hubbard model of interaction strength U eff = 1.1eV using a tight-binding modelling of the dispersion. This effective interaction U eff takes into account the large nearestneighbor interaction, which is due to the extended nature of the 5d orbitals. On the level of our dimer solver, the value of 1.1eV is in good agreement with recent constrained random phase approximation (cRPA) [81] estimates [13,82] § and produces a gap size of ∆ ≈ 0.3eV consistent with experiment [34]. Furthermore, if we assume strong coupling and do a rough estimation of the exchange coupling via J = 4t 2 /U ef f , using the two different dispersions we obtain values of J = 0.09eV and J = 0.13eV, which are close to the values determined from experiment [53,54,17].
To solve the cluster problem, we generalized to a two-site cluster the Hubbard-I solver [83]. Hubbard-I consists in neglecting the hybridization in the calculation of the self-energy and is therefore appropriate for materials in the strong-coupling limit. For this reason it has for example been used for rare-earth elements and materials [70,84]. Here, our philosophy is different: We will show in the following that the dimer Hubbard-I approximation gives an excellent account of the spectral properties of Sr 2 IrO 4 despite its proximity to the Mott transition. For more details on the technique itself, the cluster Hubbard-I solver used for the calculations and how to determine a thermodynamically consistent charge, we refer the reader to the supplementary material of Ref. [30] and references therein. Further details on the Hubbard dimer can be found in Ref. [85].

Sr 2 IrO 4 in the Paramagnetic High-Temperature Phase
At high temperatures, Sr 2 IrO 4 is a paramagnetic spin-orbit Mott insulator [12]. In the following section we discuss the spectral function in this paramagnetic phase, both calculated from OC-DMFT and measured using ARPES. Figure 5 shows the spectral function as well as their j eff =3/2 and j eff =1/2 parts along the path Γ−M −X −Γ. The model shows a gapped spectrum with the highest occupied states residing at the X point, while the lowest unoccupied states are at the M point.

Spectra from Theory
In the j eff =1/2 manifold, one can identify four separate peaks out of which the bonding and antibonding ones have largest spectral weight. Furthermore, one sees § For a dimer, the nearest-neighbor interaction V reduces the on-site interaction U to U eff = U − V . After averaging the cRPA interaction values over the three t 2g orbitals, one arrives at values of U eff = 1.3eV [13] or U eff = 1.0eV [82]. small satellites in the electron-and hole-spectrum, as can be seen in Fig. 5(C). The bonding-antibonding splitting of two neighboring iridate layers in c-direction is much smaller such that it does not give rise to further splitting into separable peaks.
The j eff =3/2 band does not cross the Fermi level and is completely filled. Still, it is responsible for spectral features at low energies such as the highest occupied states at and close to the Γ point. Away from the Γ point, the j eff =3/2 manifold mainly contributes spectral weight at energies E < −1eV.

Experimental Spectra
In angle-resolved photoemission spectroscopy it is difficult to investigate insulators due to the build-up of surface charge. However, for insulators with a small gap like Sr 2 IrO 4 it is feasible to measure the spectral function at moderate temperatures (in case of Sr 2 IrO 4 without further ado for synchrotron radiation down to T ∼ 50K [24,23,28]). Most of the photoemission studies on Sr 2 IrO 4 focussed on the low-temperature phase, which shows antiferromagnetic long-range order of the J = 1/2 pseudospins. Up to our knowledge there is only one recent study presenting photoemission spectra in the paramagnetic high-temperature phase [30].   Fig. 2(B). Although the rather high temperature causes considerable thermal broadening, different bands can still be identified. In particular, the spectra in the first and second Brillouin zone differ drastically due to matrix element effects: In the first BZ spectral weight of the j eff =1/2 band is enhanced whereas the second BZ is clearly dominated by contributions of the j eff =3/2 manifold. Consequently, the first removal state at Γ, which is of j eff =3/2 character, is found in the second BZ and the first removal state at X in the first BZ.
In Fig. 6(C) the calculated spectrum in the PM phase is shown along the same path in the first BZ. Due to the rotations of the oxygen octahedra the unit cell is doubled as compared to the undistorted structure (see, e.g., Ba 2 IrO 4 [86]) and the bands are backfolded at the boundary of the (reduced) Brillouin zone. In the theory, by definition, there is no difference between first and second BZ. A quantitative comparison would require a calculation of the photoemission matrix elements which is beyond the scope of the present work. Nevertheless it is clear that a meaningful qualitative comparison can be obtained by assuming that features of the spectral function should show up in experiment either in the first or the second BZ.
Comparing with the calculated spectra, the spectral feature in the first BZ at the bonding energy E B ∼ 1eV between M and X can be clearly attributed to be of j eff =3/2 character. More precisely, the features of j eff =3/2 character in the first BZ stems from the m j = 3/2 band whereas the spectrum in the second BZ originates from the m j = 1/2 band.
The stark differences between spectra in the first and second BZ indicate that the symmetry breaking potential which doubles the unit cell is rather weak: In case of strong symmetry breaking that reduces the symmetry from I 4 /mmm to I 4 /acd, the two Brillouin zones would show very similar spectra. Here, however, the distortions of the oxygen octahedra introduce only a small Fourier component of the corresponding potential. In addition, the polarization of the photons naturally leads to matrix element effects that enhance or suppress spectral weight in an orbital-selective manner [87,88].
In contrast to single-site DMFT [30] or LDA+U+SOC [21] spectra, the present OC-DMFT spectrum correctly captures the absence of a weakly dispersive branch of j eff =1/2 character at the Γ point with an onset at −0.2eV below the Fermi energy. As can be seen from figure 6(B) the onset of spectral weight at the Γ point is only at −0.5eV and should have rather j eff =3/2 character showing a drastic difference to the spectrum calculated in DMFT. The reason of this improvement lies in the explicit inclusion of inter-site fluctuations: Antiferromagnetic short-range fluctuations within the dimer provide the previously missing ingredient. The energy scale of this intersite fluctuations can be estimated as described in Ref. [30] as 108meV (78meV) for the interlayer (anti)bonding j eff =1/2 states. This values compare well with an experimental estimate for the magnetic exchange coupling in the paramagnetic phase J ∼ 0.1eV obtained from the temperature dependence of the in-plane correlation length within resonant magnetic x-ray scattering [17]. PURE  (1BZ), restoring the translational invariance of the solid after calculating its self-energy from the quantum dimer problem, and the sum over α restores its point group. More details on the general philosophy and practical implementation can be found in the Supplemental Material [68].
In a metal, electronic screening is drastically enhanced as compared to an insulator with profound consequences for the spectra [69,70]. This is even more true in 5d compounds, where the relatively extended nature of the 5d orbitals induces interatomic interactions that are relatively large as compared to the local ones [71]. For this reason, the effective Hubbard interactions are expected to be smaller in the La-doped compound than in the pure sample [72]. We mimic this effect here by using a smaller on-site Hubbard interaction for the doped case (U eff = 0.6 eV) than for the undoped one (U eff = 1.1 eV).
Results for undoped Sr 2 IrO 4 . We have measured ARPES spectra of Sr 2 IrO 4 under the experimental conditions described in the Supplemental Material [68]. Figure 1 displays the resulting spectra along the -M-X-path in the first BZ and the second BZ (2BZ). Due to matrix element effects, these results display characteristic differences, with spectra in the 1BZ amplifying the j eff = 1/2 contribution, while in the 2BZ the j eff = 3/2 contribution is dominant. This strong matrix element effect expresses the fact that the corresponding Fourier component of the potential (lowering the symmetry from I 4 /mmm to I 4 /acd ) is weak. In agreement with ARPES spectra in the literature [28,29], we find the first removal state at to be of j eff = 3/2 character, while the j eff = 1/2 states are strongly dispersive, with a maximum at X. In our theoretical calculations, we do not address matrix element effects, which would differentiate results between different BZs. For this reason, the theoretical spectral function is compared to the sum of the experimental spectra. Most intriguingly, in single-site DMFT calculations (see the Supplemental Material [68]), the j eff = 1/2 states form a very weakly dispersive feature with an onset of spectral weight at − 0.2 eV below the Fermi level, in strong disagreement with the experimental spectra in Figs. 1(c) and 1(e). Figure 1(d) displays the results of our present calculations that include nonlocal many-body correlations within our new OC-DMFT scheme. The comparison to the experimental spectra yields impressive agreement, demonstrating that nonlocal many-body effects were indeed the missing ingredients for assessing spectral properties of this compound. The OC-DMFT treatment effectively includes inter-iridium site fluctuations in the half-filled j eff = 1/2 manifold, and especially includes the intersite magnetic exchange of energy scale 4t 2 /U into the description. With the present parameters we obtain a value of ∼108 meV (78 meV) for states of interlayer (anti)bonding nature, which coincides with the experimental estimate of the magnetic exchange coupling J = 4t 2 /U ∼ 100 meV [44]. As a consequence of the antiferromagnetic fluctuations, the dominated by the j eff = 3/2 bands. Our calculations show that even within the j eff = 3/2 manifold the experimental signal is selective with respect to the m j quantum number. The main contribution to the spectral weight seen in the 2BZ comes from the m j = 1/2 band whereas the m j = 3/2 contributes to the 1BZ.
The left panel of Fig. 2(a) shows an ARPES spectrum acquired at X at − 0.25 eV at room temperature. According to ARPES measurements of the magnetic phase, the lowest energy excitations disperse up to the X point and never cross the Fermi level. This spectrum is qualitatively similar to the one measured below T N in previous works [2,25], demonstrating that spectral properties are largely insensitive to the presence  [30]. The high-symmetry points of the Brillouin zone are indicated by blue points. The first and second Brillouin zone are the same in the calculation, but differ due to matrix-element effects in experiment.
To compare the ARPES spectra further with calculated spectral functions, figure 7 shows equal energy maps at E B = 0.25eV and E B = 0.5eV. The constant energy maps are very similar to those taken below T N in the antiferromagnetic phase [12,21,28], which shows that the photoemission spectra are not sensitive to long-range magnetic order. At E = −0.25eV, the peak at X due to the j eff =1/2 band is well visible in both experiment and theory, whereas the weak precursor at the Γ point is barely visible in the experiment and shows intensity modulations between different BZs. For E = −0.5eV, the intense blobs at Γ and much weaker peaks at Γ show strong matrix element effects in the ARPES spectrum. Apart from these matrix element effects, both spectra are again in good agreement.

Sr 2 IrO 4 in the Antiferromagnetic Low-Temperature Phase
So far we discussed the paramagnetic phase of pure Sr 2 IrO 4 at temperatures above the Néel temperature of T N ∼ 230K [12]. Below T N the j eff =1/2 pseudospins of Sr 2 IrO 4 order and a phase transition towards an antiferromagnetic phase occurs [43]. Since the moments are slightly tilted out of plane, the ground state is rather a canted antiferromagnet [56,89]. In this section we will discuss the spectral properties of the antiferromagnetically ordered low-temperature phase of Sr 2 IrO 4 .

Spectra from Theory
Starting from the same effective one-band Hubbard model, but allowing for antiferromagnetic ordering within the dimers of the different planes leads in the OC-DMFT treatment to an AF ground state at low temperatures. In our cluster treatment we assume an antiferromagnetic alignment of the j eff = 1/2 moments on neighboring Ir spins. This is done by including an antiferromagnetic Weiss field on the cluster, whose strength is determined via a variational principle using self-energy functional theory [90,91]. One should note that the experimentally measured magnetic configuration shows additional spin canting and leads to an effective weak ferromagnetic moment [56]. From a modeling point of view this can be included via a twisted Hubbard model [52]. Since we focus here rather on spectral than on magnetic properties we keep an antiferromagnetic alignment of the moments, which -in principle-slightly overestimates the magnetization. The corresponding spectral function for pure Sr 2 IrO 4 in this AF configuration is shown in figure 8.
The antiferromagnetic spectrum is very similar to a paramagnetic spectrum calculated for the same temperature. This similarity is due to antiferromagnetic fluctuations, which are the essential ingredient for a precise description of the spectral function of paramagnetic Sr 2 IrO 4 . Compared to the paramagnetic spectrum, the splitting between (anti)bonding band and (upper) lower Hubbard satellite band is a bit enhanced. As a consequence, the two different contributions to the hole part of the spectrum at Γ can now be identified, whereas the paramagnetic spectrum of Fig. 5(C) shows only one single peak at ∼ 0.6eV. Furthermore, some spectral weight is transferred from the (anti)bonding bands to the satellites. Although it is only a minor change, the antiferromagnetic ordering leads thereby to additional spectral weight between X and M at ∼ −1eV.

Experimental Spectra
Using angle-resolved photoemission spectroscopy to investigate pure Sr 2 IrO 4 is limited to moderate temperatures due to surface charging effects that prevent an investigation of the sample at very small temperatures. Nevertheless, spectra for temperatures down to T = 20K have been measured [25], which is deep in the antiferromagnetic phase.
Although the data quality is not as high as for doped samples, which are metallic and do not have this problem, the spectra allow for the extraction of dispersion that can be compared to our calculated spectrum, see Fig. 9. Overall, the calculated spectrum is in good agreement with the different experimental data sets. Since the experimental data shown were taken under different experimental conditions like temperature, photon energy or polarization, the dispersions vary a bit. This is in particular visible at the upper band edge close to the X point, where the position of the j eff =1/2 band maximum varies by several 100meV between the different works.
In order to compare the spectra more quantitatively, we plot the calculated spectral function at the high-symmetry points Γ, M, and X with the energy distribution curves (EDC) of different experimental studies in figure 10. Again, the calculated spectral function reproduces all features of the experimental EDCs at least qualitatively. Modulations at larger bonding energies E ≤ −1eV are more pronounced in the calculated spectra than in experiment, but the low-energy region compares favorably. At Γ and X the onset of spectral weight and the position of maximal intensity agrees well between calculation and experiment. However, at M the peak position in the calculated spectrum is at smaller energy than in experiment. One might think of two reasons for this deviation. First, it is known that antiferromagnetism is overestimated within a dimer cluster and therefore leads here to a stronger splitting between bonding and antibonding band. Secondly, the ground state of Sr 2 IrO 4 at low temperatures is a canted antiferromagnet whereas on the cluster level the system adopts a purely antiferromagnetic configuration with larger net staggered magnetization. Quite generally one observes that the spectra in PM and AF phases are very similar . This is consistent with our finding that our dimer approach, which introduces AF fluctuations, provides an extremely good description of the PM. It also explains the success of AF DMFT calculations even in its single-site version to describe both phases. Note that an AF ordered dimer is well described by a mean-field description and the AF ordered insulator can -in this sense-be considered a band insulator. This effect explains the apparent success of DFT+U-like descriptions of spectra of Sr 2 IrO 4 [12,47], which however strongly differ from experiment at the Γ and M point.
Interestingly, a similar trend has also been found in the undoped cuprate Sr 2 CuO 2 Cl 2 , where ARPES spectra in PM and AF phases are much akin [93].

Constant Energy Cuts
In figure 11 we compare the calculated constant energy maps of Sr 2 IrO 4 in the antiferromagnetic phase at two different binding energies with ARPES spectra from Ref. [25]. Qualitatively, the features are well captured by the OC-DMFT calculation: At E = −200meV only the j eff =1/2 bands at the X points have non-zero spectral weight and theory and experiment only differ in the x − y anisotropy of spectral intensity in experiment, which stems from matrix-element effects that are not taken into account in the calculation. Due to linear polarization of the photon beam in k y direction the experimental spectra have larger spectral weight around X-points in k y -than in k xdirection, whereas the calculated spectra still show x − y symmetry. Compared to Fig. 11A, at a binding energy of E = 400meV both experiment and theory show in addition some spectral weight at the Γ point due to the j eff =3/2 band, which is peaked at slightly larger binding energies. The main contribution of spectral weight still stems from the j eff =1/2 band, however, in contrast to the measurement the calculated spectrum is not peaked at the X points, but rather around them. Good agreement with the j eff =1/2 part of the measured spectrum would rather be obtained by assuming that the j eff =1/2 part of the experimental spectrum corresponds in fact to slightly smaller binding energies than indicated. This can be seen in the third column of Fig. 11, where the j eff =1/2 part of the calculated spectrum is in addition shifted by −0.15eV. This is consistent with the comparison to other ARPES works shown in Fig. 9, where one sees that the dispersion of Ref. [25] is located at the X point at ∼ 0.1eV www.nature.com/scientificreports/ that has been reached is Sr 2 IrO 4 being a typical system where the Mott-and slater-type behaviors coexist. In addition, Sr 2 IrO 4 has attracted much attention because it exhibits a number of similarities to the parent compound La 2 CuO 4 of high temperature cuprate superconductors in the crystal structure, electronic structure, magnetic structure, and even possible high temperature superconductivity that is predicted in doped Sr 2 IrO 4 23,24 .
Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to directly probe the low energy electronic structures of solid materials 25 . While the ARPES results on Sr 2 IrO 4 and related compounds within a relatively narrow energy window agree with the J eff = 1/2 Mott insulator model 7,13-15,18 , little work has been done in studying its high energy electronic behaviors. In this paper, we report the observation of unusual high energy bands in Sr 2 IrO 4 . Our comprehensive angle-resolved photoemission study over a wide energy window reveals for the first time nearly-vertical bands in Sr 2 IrO 4 . Such exotic bands cannot be understood in terms of the band structure calculations; they cannot be understood within the J eff = 1/2 Mott insulator model either. The observed high energy anomaly resemble the unusual high energy waterfall bands discovered in the high temperature cuprate superconductors [26][27][28][29][30][31][32][33][34][35] . These observations point to the significant role of the strong spin-orbit coupling, together with a moderate electron correlation, in giving rise to new high energy excitations in the 5d transition metal oxides. Figure 1 shows the constant energy contours of Sr 2 IrO 4 at different binding energies. No spectral weight is present at the Fermi level (not shown in Fig. 1), consistent with the insulating nature of Sr 2 IrO 4 [5][6][7] . At a binding energy of 0.2 eV, the spectral weight appears first as a circular spot around the X(π, 0) and its equivalent locations (Fig. 1a). Further increase of the binding energy to 0.4 eV results in the enlargement of the spot into a square-shape and the emergence of spectral weight near the Γ (0, 0) point (Fig. 1b). When the binding energy increases to 0.8 eV, the strong spectral weight near X points vanishes with a formation of a few disconnected patches around X, while the spectral weight near Γ exhibits a petal-like shape with four leaves (Fig. 1c). The measured constant energy contours at low binding energy (0 ~ 0.4 eV) are consistent with those reported before 7,14 . The constant energy contours at an intermediate binding energy (e.g., 0.4 eV) are also consistent with the band structure calculations (Fig. 1d) that include both the on-site Coulomb repulsion U and the spin-orbit coupling 7 . In terms of the spin-orbit-coupling-driven Mott insulator model 7 , the unoccupied states are mainly the J eff = 1/2 state, while the occupied states are a mixture of the J eff = 1/2 and 3/2 states. Due to the strong spin-orbit coupling, the topmost low energy valence state at X is more with J eff = 1/2 character (β sheet near X in Fig. 1d), while the topmost low energy valence state at Γ is more with J eff = 3/2 character (α sheet near Γ in Fig. 1d) 7 . The consistency of the low energy electronic structure with the previous reports and the band structure calculations lays a foundation for our following investigation of high binding energy electronic structure in Sr 2 IrO 4 .
At high binding energies, we find that the electronic structure of Sr 2 IrO 4 is quite unusual. Figure 2 shows band structure along two high-symmetry momentum cuts covering a large energy range till ~6 eV: one cut is across Γ (Fig. 2a-d), the other is across X (Fig. 2e-h) (for more momentum cuts, see Fig. S1, S2 and S3 in Supplementary Materials). Here we show both the original data (Fig. 2a,e), and their corresponding momentum- (Fig. 2b,f) and energy-second-derivative (Fig. 2c,g) images. The second-derivative images help to highlight the band structure more clearly although many features are ng a typical system where the Mott-and slater-type behaviors coexist. uch attention because it exhibits a number of similarities to the parperature cuprate superconductors in the crystal structure, electronic even possible high temperature superconductivity that is predicted in spectroscopy (ARPES) is a powerful tool to directly probe the low d materials 25 . While the ARPES results on Sr 2 IrO 4 and related comenergy window agree with the J eff = 1/2 Mott insulator model 7,[13][14][15]18 , ing its high energy electronic behaviors. In this paper, we report the bands in Sr 2 IrO 4 . Our comprehensive angle-resolved photoemission reveals for the first time nearly-vertical bands in Sr 2 IrO 4 . Such exotic rms of the band structure calculations; they cannot be understood model either. The observed high energy anomaly resemble the unuscovered in the high temperature cuprate superconductors [26][27][28][29][30][31][32][33][34][35] . These nt role of the strong spin-orbit coupling, together with a moderate to new high energy excitations in the 5d transition metal oxides. ergy contours of Sr 2 IrO 4 at different binding energies. No spectral el (not shown in Fig. 1), consistent with the insulating nature of 0.2 eV, the spectral weight appears first as a circular spot around the (Fig. 1a). Further increase of the binding energy to 0.4 eV results in square-shape and the emergence of spectral weight near the Γ (0, 0) energy increases to 0.8 eV, the strong spectral weight near X points disconnected patches around X, while the spectral weight near Γ ur leaves (Fig. 1c). The measured constant energy contours at low sistent with those reported before 7,14 . The constant energy contours (e.g., 0.4 eV) are also consistent with the band structure calculations site Coulomb repulsion U and the spin-orbit coupling 7 . In terms of 7 y contours of Sr 2 IrO 4 and its comparison with calculations. (a-c) f the spectral weight distribution for Sr 2 IrO 4 measured at ~20 K at eV, 0.4 eV, and 0.8 eV, respectively. (d) is the calculated constant energy V by including on-site Coulomb repulsion and spin-orbit coupling 7 . magnetic Brillouin zone boundary for the IrO 2 plane. ntificreports/ that has been reached is Sr 2 IrO 4 being a typical system where the Mott-and slater-type behaviors coexist. In addition, Sr 2 IrO 4 has attracted much attention because it exhibits a number of similarities to the parent compound La 2 CuO 4 of high temperature cuprate superconductors in the crystal structure, electronic structure, magnetic structure, and even possible high temperature superconductivity that is predicted in doped Sr 2 IrO 4 23,24 .
Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to directly probe the low energy electronic structures of solid materials 25 . While the ARPES results on Sr 2 IrO 4 and related compounds within a relatively narrow energy window agree with the J eff = 1/2 Mott insulator model 7,[13][14][15]18 , little work has been done in studying its high energy electronic behaviors. In this paper, we report the observation of unusual high energy bands in Sr 2 IrO 4 . Our comprehensive angle-resolved photoemission study over a wide energy window reveals for the first time nearly-vertical bands in Sr 2 IrO 4 . Such exotic bands cannot be understood in terms of the band structure calculations; they cannot be understood within the J eff = 1/2 Mott insulator model either. The observed high energy anomaly resemble the unusual high energy waterfall bands discovered in the high temperature cuprate superconductors [26][27][28][29][30][31][32][33][34][35] . These observations point to the significant role of the strong spin-orbit coupling, together with a moderate electron correlation, in giving rise to new high energy excitations in the 5d transition metal oxides. Figure 1 shows the constant energy contours of Sr 2 IrO 4 at different binding energies. No spectral weight is present at the Fermi level (not shown in Fig. 1), consistent with the insulating nature of Sr 2 IrO 4 [5][6][7] . At a binding energy of 0.2 eV, the spectral weight appears first as a circular spot around the X(π, 0) and its equivalent locations (Fig. 1a). Further increase of the binding energy to 0.4 eV results in the enlargement of the spot into a square-shape and the emergence of spectral weight near the Γ (0, 0) point (Fig. 1b). When the binding energy increases to 0.8 eV, the strong spectral weight near X points vanishes with a formation of a few disconnected patches around X, while the spectral weight near Γ exhibits a petal-like shape with four leaves (Fig. 1c). The measured constant energy contours at low binding energy (0 ~ 0.4 eV) are consistent with those reported before 7,14 . The constant energy contours at an intermediate binding energy (e.g., 0.4 eV) are also consistent with the band structure calculations (Fig. 1d) that include both the on-site Coulomb repulsion U and the spin-orbit coupling 7 . In terms of www.nature.com/scientificreports/ that has been reached is Sr 2 IrO 4 being a typical system where the Mott-and slater-type behaviors coexist. In addition, Sr 2 IrO 4 has attracted much attention because it exhibits a number of similarities to the parent compound La 2 CuO 4 of high temperature cuprate superconductors in the crystal structure, electronic structure, magnetic structure, and even possible high temperature superconductivity that is predicted in doped Sr 2 IrO 4 23,24 .
Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to directly probe the low energy electronic structures of solid materials 25 . While the ARPES results on Sr 2 IrO 4 and related compounds within a relatively narrow energy window agree with the J eff = 1/2 Mott insulator model 7,[13][14][15]18 , little work has been done in studying its high energy electronic behaviors. In this paper, we report the observation of unusual high energy bands in Sr 2 IrO 4 . Our comprehensive angle-resolved photoemission study over a wide energy window reveals for the first time nearly-vertical bands in Sr 2 IrO 4 . Such exotic bands cannot be understood in terms of the band structure calculations; they cannot be understood within the J eff = 1/2 Mott insulator model either. The observed high energy anomaly resemble the unusual high energy waterfall bands discovered in the high temperature cuprate superconductors [26][27][28][29][30][31][32][33][34][35] . These observations point to the significant role of the strong spin-orbit coupling, together with a moderate electron correlation, in giving rise to new high energy excitations in the 5d transition metal oxides. Figure 1 shows the constant energy contours of Sr 2 IrO 4 at different binding energies. No spectral weight is present at the Fermi level (not shown in Fig. 1), consistent with the insulating nature of Sr 2 IrO 4 [5][6][7] . At a binding energy of 0.2 eV, the spectral weight appears first as a circular spot around the X(π, 0) and its equivalent locations (Fig. 1a). Further increase of the binding energy to 0.4 eV results in the enlargement of the spot into a square-shape and the emergence of spectral weight near the Γ (0, 0) point (Fig. 1b). When the binding energy increases to 0.8 eV, the strong spectral weight near X points vanishes with a formation of a few disconnected patches around X, while the spectral weight near Γ exhibits a petal-like shape with four leaves (Fig. 1c). The measured constant energy contours at low binding energy (0 ~ 0.4 eV) are consistent with those reported before 7,14 . The constant energy contours at an intermediate binding energy (e.g., 0.4 eV) are also consistent with the band structure calculations (Fig. 1d) that include both the on-site Coulomb repulsion U and the spin-orbit coupling 7 . In terms of ntificreports/ that has been reached is Sr 2 IrO 4 being a typical system where the Mott-and slater-type behaviors coexist. In addition, Sr 2 IrO 4 has attracted much attention because it exhibits a number of similarities to the parent compound La 2 CuO 4 of high temperature cuprate superconductors in the crystal structure, electronic structure, magnetic structure, and even possible high temperature superconductivity that is predicted in doped Sr 2 IrO 4 23,24 .
Angle-resolved photoemission spectroscopy (ARPES) is a powerful tool to directly probe the low energy electronic structures of solid materials 25 . While the ARPES results on Sr 2 IrO 4 and related compounds within a relatively narrow energy window agree with the J eff = 1/2 Mott insulator model 7,[13][14][15]18 , little work has been done in studying its high energy electronic behaviors. In this paper, we report the observation of unusual high energy bands in Sr 2 IrO 4 . Our comprehensive angle-resolved photoemission study over a wide energy window reveals for the first time nearly-vertical bands in Sr 2 IrO 4 . Such exotic bands cannot be understood in terms of the band structure calculations; they cannot be understood within the J eff = 1/2 Mott insulator model either. The observed high energy anomaly resemble the unusual high energy waterfall bands discovered in the high temperature cuprate superconductors [26][27][28][29][30][31][32][33][34][35] . These observations point to the significant role of the strong spin-orbit coupling, together with a moderate electron correlation, in giving rise to new high energy excitations in the 5d transition metal oxides. Figure 1 shows the constant energy contours of Sr 2 IrO 4 at different binding energies. No spectral weight is present at the Fermi level (not shown in Fig. 1), consistent with the insulating nature of Sr 2 IrO 4 [5][6][7] . At a binding energy of 0.2 eV, the spectral weight appears first as a circular spot around the X(π, 0) and its equivalent locations (Fig. 1a). Further increase of the binding energy to 0.4 eV results in the enlargement of the spot into a square-shape and the emergence of spectral weight near the Γ (0, 0) point (Fig. 1b). When the binding energy increases to 0.8 eV, the strong spectral weight near X points vanishes with a formation of a few disconnected patches around X, while the spectral weight near Γ exhibits a petal-like shape with four leaves (Fig. 1c). The measured constant energy contours at low binding energy (0 ~ 0.4 eV) are consistent with those reported before 7,14 . The constant energy contours at an intermediate binding energy (e.g., 0.4 eV) are also consistent with the band structure calculations (Fig. 1d) that include both the on-site Coulomb repulsion U and the spin-orbit coupling 7 . In terms of the spin-orbit-coupling-driven Mott insulator model 7 , the unoccupied states are mainly the J eff = 1/2 state, while the occupied states are a mixture of the J eff = 1/2 and 3/2 states. Due to the strong spin-orbit coupling, the topmost low energy valence state at X is more with J eff = 1/2 character (β sheet near X in Fig. 1d), while the topmost low energy valence state at Γ is more with J eff = 3/2 character (α sheet near Γ in Fig. 1d) 7 . The consistency of the low energy electronic structure with the previous reports and the band structure calculations lays a foundation for our following investigation of high binding energy electronic structure in Sr 2 IrO 4 .
At high binding energies, we find that the electronic structure of Sr 2 IrO 4 is quite unusual. Figure 2 shows band structure along two high-symmetry momentum cuts covering a large energy range till ~6 eV: one cut is across Γ (Fig. 2a-d), the other is across X (Fig. 2e-h) (for more momentum cuts, see Fig. S1, S2 and S3 in Supplementary Materials). Here we show both the original data (Fig. 2a,e), and their corresponding momentum- (Fig. 2b,f) and energy-second-derivative (Fig. 2c,g) images. The second-derivative images help to highlight the band structure more clearly although many features are larger binding energies. We note that the dispersion at the Γ point is not affected, but rather consistent between the different ARPES data shown in Fig. 9. Therefore, the origin of the apparent slight difference of the energetic difference between j eff =1/2 and j eff =3/2 bands in the spectra of Liu et al. as compared to theory remains a puzzle.

Electron-doped Sr 2 IrO 4 at Low Temperatures
The similarity of its electronic structure with hole-doped high-temperature cuprate superconductors has sparked interest in electron-doped Sr 2 IrO 4 and triggered experimental and theoretical investigations in recent years. In contrast to the cuprates, no superconductivity has been found in Sr 2 IrO 4 , it rather becomes a paramagnetic metal down to lowest temperatures for sufficiently strong doping. Nevertheless, spectral properties like pseudogap features inherent to cuprates have also been proposed for doped Sr 2 IrO 4 [52,24]. In this section, we will discuss the spectra of electrondoped Sr 2 IrO 4 at low temperatures calculated from theory and compare it to ARPES measurements of La-doped samples.

Theoretical Spectra
In order to calculate the spectral function of electron-doped Sr 2 IrO 4 we apply OC-DMFT to the j eff =1/2 band at an electron filling of n = 0.9. It should be noted that the j eff =3/2 bands are still completely filled and therefore not much affected by the change in the electron filling n, see Fig. 12.
For the j eff =1/2 band the change in electron filling renders the system metallic. In a metal, electronic screening is much enhanced as compared to an insulator and as a consequence, the effective local Coulomb interaction strength is reduced [94]. Here, we take the enhanced screening into account by using an effectively reduced Coulomb interaction strength of U eff = 0.6eV, which leads to good agreement of the calculated spectra with ARPES data. The corresponding spectral function A(k, ω) is plotted in Fig. 12. Due to the electron doping, the Fermi surface now has pockets around the M point and the lower j eff =1/2 branch is close to E F at M . Note that we have not recalculated the position of the j eff =3/2 states in the doped case from DMFT. Instead, we estimate an electrostatic shift of ∆E Hartree = U ∆n, which was applied to the bands calculated in the undoped case.

Experimental Spectra
Most experimental works on electron-doped Sr 2 IrO 4 focus on the low-energy region at and close to the Fermi level E F . Before discussing two spectral features close to E F in more detail in the following two subsections, we compare in Fig. 13 the calculated spectrum to available ARPES data of La-doped samples (Sr 1−x La x ) 2 IrO 4 . ¶ The spectral feature of j eff =3/2 character at the Γ point is correctly captured, as well as the dispersion of the upper j eff =1/2 branch around the M point, which leads to pocket structures in the Fermi surface. Along the Γ−X direction the j eff =1/2 dispersion is captured with small deviations from the ARPES data of Ref. [24]. However, the calculated spectrum of the lower j eff =1/2 branch deviates close to M from the measured photoemission spectrum and is discussed in the next subsection.

Spectra close to the M point
In this subsection, we focus on spectral properties around the M point, see Fig. 14.
In Ref. [24] de la Torre et al. suggested that the bands with nearly linear dispersion close to the M point might extrapolate to a Dirac point at E ∼ −0.1eV before they continue still with a quasi-linear dispersion up to the Fermi energy (Fig. 14(B)). For a slightly less La-doped sample, Brouet et al. found a very similar quasi-linear feature close to the M point [23], which is much reduced in intensity, but non-negligible. However, they interpret this feature as an artifact of the fitting procedure applied to the momentum distribution curves (MDC): Since the j eff =1/2 energy distribution curve has a quite broad peak at lower energy the tail of this peak can lead to a signal with quasi vertical dispersion that is picked up in the MDC analysis. For the case of larger La-doping, where the states close to E F clearly form a pocket around M , this suggests also a different interpretation of the signal for binding energies E B > 0.1eV. Instead of a Dirac point of two connected bands, the region below E ∼ −0.1eV could be gapped and the spectral weight just be picked up by the broad tail of the lower j eff =1/2 band at lower energy.
To investigate this question we plot the j eff =1/2 part of the calculated spectral function for 10% electron doping in Fig. 14(C1,2) once in a large energy region and once in close vicinity of the Fermi energy. As can be seen from (C1) the OC-DMFT spectrum is gapped in the region −0.35eV ≤ E ≤ −0.1eV and has large spectral weight at M and E ∼ −0.4eV. Along roughly half the distance M − Γ the spectral features are in good agreement with (A), close to Γ spectral contributions from j eff =3/2 render the comparison more difficult, but main features are qualitatively still comparable to (A). When comparing to (B) one notices good agreement for the band forming the pocket in the immediate vicinity of the Fermi energy. Also for energies E ≤ −0.35eV the spectrum agrees well with (B) suggesting a reinterpretation of the "gapped" region −0.35eV ≤ E ≤ −0.1eV in terms of Ref. [23].

Fermi Surface, Electron Pockets and Pseudogap
For La-doping x ≥ 0.05 the system is a paramagnetic metal with a Fermi surface with well-defined electron pockets. In Fig. 15 we compare ARPES spectra at the Fermi energy of 5% La-doped Sr 2 IrO 4 with the calculated Fermi surface at 10% electron doping. The  [53], and at T = 8K (C), adopted from de la Torre et al. [24]. The calculated Fermi surface at 10% electrondoped Sr 2 IrO 4 is shown in (A). electron pockets around the M points are lens-shaped with their elongated axis pointing towards the X points. Comparing position, shape and size of the pockets in (A) with the experimental observations leads to overall good agreement. Most spectral weight of the pockets is found at their tips, consistent with experiment ((B) and (C)). By symmetry, both sides of the pockets carry equal spectral weight in the theoretical spectral function. In the ARPES spectra, the spectral weight in the second BZ is enhanced as compared to the backfolded part of the pocket.
In experiments, resistivity measurements of La-doped samples display unusual metallic behavior at low temperatures [23,24] and for x = 0.05 doped samples an anti-nodal pseudogap was found using ARPES [24], shown in Fig. 16(B). Since in ARPES the hole-part of the spectrum cannot be accessed, one often uses the particlehole symmetrized spectral function to extract pseudogap features. To compare with experimental findings, we plot this symmetrised spectral function in Fig. 16. Along the remnant Fermi surface, which in this case is a nearly circular path in k-space, one then extracts the "gap" ∆(α) of the symmetrised spectral function as a function of angle α. This gap is defined by the peak position ω * of this spectral function closest to the Fermi level at a given k-point, ∆(α) = 2ω * (k α ). Figure 16. Calculated ∆(α) of 10% electron-doped Sr 2 IrO 4 (A) and measured pseudogap (B) of a 5% La-doped sample at T = 8K. In both cases, the quantities were extracted along the remnant Fermi surface as discussed in the text. Panel (b) is adopted from de la Torre et al. [24], BF denotes back-folded.
Comparing the calculated ∆(α) in Fig. 16A with the pseudogap (PG) extracted in Ref. [24] from ARPES measurements (Fig. 16B) shows qualitatively similar features. Both ∆(α) and the PG emerge at the tips and outside of the elliptical pocket along the circular k-path. In addition, within the calculations even the pocket structure itself at small angles α shows regions of non-zero ∆(ω). These spectral features can be tracked back to the average over the two perpendicular cluster orientations used within the OC-DMFT calculation [30]. To observe them separately would, however, require a very high resolution in experiment. Furthermore, the origin of the large ∆(ω) values close to the antinodal point can be identified in the calculation by comparing to the full spectral function including the hole part. In contrast to the pseudo-gap-like features close to the tips of the pockets, the antinodal point shows a true gap since the spectral peaks closest to the Fermi energy belong to the j eff =3/2 band.

Conclusion and Outlook
In this work we gave an overview of cluster-DMFT-based calculations of spectral functions of pure and electron-doped Sr 2 IrO 4 . For undoped Sr 2 IrO 4 we obtained good agreement with ARPES spectra both in the antiferromagnetic and paramagnetic phase, and in the electron-doped system pocket structures of the Fermi surface and prominent pseudogap features of the spectrum could be reproduced.
As most theoretical works on Sr 2 IrO 4 , also our work here is based on effective oneorbital descriptions of the correlated j eff =1/2 bands and lead to good agreement with experiment. However, both ab initio calculations [79] and angle-resolved photoemission spectra [88,87] revealed a strong k-dependence of the t 2g contributions that form the effective j eff =1/2 band. One of the challenges to future studies is to extend the current description to polarization-dependent ARPES spectra, where effects not captured in a local j eff =1/2 picture could enter into play.
Furthermore, a question that has been raised recently is whether hidden orders such as the time-reversal symmetry breaking hidden order found in neutron scattering experiments [95] or the (different) hidden order investigated in Ref. [44] might influence spectral properties.
The best fit was obtained with ε 0 = −0.237 eV,t 1 = 0.171 eV, t 2 = 0.0935 eV, t 3 = 0.0169 eV and ε 1 = 0.026 eV. Due to the inter-layer hopping t 3 , one obtains two bands: An inter-layer bonding and an inter-layer antibonding band. + One can also downfold this effect and consider each layer independently. In this case, for the interlayer bonding band, one gets the following parameters: ε 0 = −0.217 eV,t 1 = 0.188 eV, t 2 = 0.0935 eV and ε 1 = 0.021 eV , while for the antibonding band, one gets ε 0 = −0.217 eV,t 1 = 0.154 eV, t 2 = 0.0935 eV and ε 1 = 0.029 eV, see  Inter-layer bonding band Inter-layer antibonding band Figure A1. Tight-binding fit of the DFT bands for the two-sheet unit cell consisting of two dimers per sheet, leading to an inter-layer bonding and inter-layer antibonding band in c-direction.
Just as in DFT calculations, the filling of these tight-binding dispersions is slightly larger than half-filling. However, three-orbital DMFT calculations showed that treating the full j eff =1/2 and j eff =3/2 manifold leads to a charge-transfer into the j eff =3/2 bands and results in a half-filled j eff =1/2 band [15]. To set our one-band effective model, we take into account this charge-transfer by adjusting the chemical potential µ such that the j eff =1/2 manifold is half-filled. + Note that the inter-layer bonding-/antibonding bands should not be confused with the bonding-/antibonding-splitting within the dimer, which we refer to in the main text.

Appendix B. Comparison of different ARPES studies on Sr 2 IrO 4
In Table B1 we list the temperatures and photon energies at which the experimental ARPES measurements that were used in this article have been performed.
Temperature T (K) Energy hν (eV) Kim [30] 300 80 Table B1. Comparison of the temperature and photon energies of the different ARPES studies on undoped Sr 2 IrO 4 of which data is shown here.