Square Lattice Iridates

TL;DR

This paper reviews the electronic properties of Sr$_2$IrO$_4$, highlighting its similarities to cuprates, discussing potential for high-temperature superconductivity, and exploring how spin-orbit coupling influences its low-energy physics.

Contribution

It provides a comprehensive analysis of iridates' electronic structures and their relation to cuprates, emphasizing the role of spin-orbit coupling in high-temperature superconductor research.

Findings

Iridates exhibit cuprate-like phenomena such as pseudogaps and Fermi arcs.

Strong spin-orbit coupling shapes the low-energy physics of Sr$_2$IrO$_4$.

Potential pathways for high-temperature superconductivity in iridates are discussed.

Abstract

Over the last few years, Sr$_2$IrO$_4$, a single-layer member of the Ruddlesden-Popper series iridates, has received much attention as a close analog of cuprate high-temperature superconductors. Although there is not yet firm evidence for superconductivity, a remarkable range of cuprate phenomenology has been reproduced in electron- and hole-doped iridates including pseudogaps, Fermi arcs, and $d$-wave gaps. Further, a number of symmetry breaking orders reminiscent of those decorating the cuprate phase diagram have been reported using various experimental probes. We discuss how the electronic structures of Sr$_2$IrO$_4$ through strong spin-orbit coupling leads to the low-energy physics that had long been unique to cuprates, what the similarities and differences between cuprates and iridates are, and how these advance the field of high-temperature superconductivity by isolating essential ingredients of superconductivity from a rich array of phenomena that surround it. Finally, we comment on the prospect of finding a new high-temperature superconductor based on the iridate series.

Equations

$$|\tilde{\uparrow }⟩$$

$$|\tilde{\downarrow }⟩$$

$$H_{ij}=J\overrightarrow{S_i}\overrightarrow{S_j}+{\Gamma }_1{S}_i^z{S}_j^z\pm {\Gamma }_2(S_i^x{S}_j^x-S_i^y{S}_j^y),$$

Full text

1. I INTRODUCTION

2. II MAPPING ONTO CUPRATE PHYSICS

3. II.1 Spin-orbit Induced Mott Insulator

4. II.2 Magnetism

5. II.2.1 Superexchange and Magnetic Anisotropy

6. II.2.2 Overview of Experimental Results

7. II.3 Single-band Model

8. III ANALOGY AWAY FROM MOTT INSULATING PHASE

9. IV EMERGENT ELECTRONIC ORDERS

10. IV.1 Pseudogap and Competing Orders

11. IV.1.1 Electron doping

12. IV.1.2 Hole doping

13. IV.2 Superconductivity

14. V SUMMARY AND OUTLOOK

I INTRODUCTION

Since the discovery of the spin-orbit induced Mott state in 2008 Kim et al. (2008, 2009), iridates have been at the center of an intensive search for novel phenomena that arise from the combined influence of spin-orbit coupling (SOC) and electron correlations Pesin and Balents (2010); Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016); Winter et al. (2017); Hermanns et al. (2018). The new Mott phase has attracted much interest from both the topological insulator and the strongly correlated electron system communities. From the former perspective, the interactions open possibilities for novel topological phases of matter that arise in the presence of magnetic order, such as Weyl semimetals with surface Fermi arcs or Axion insulators with unusual electromagnetic responses. Here, pyrochlore iridates are prime candidates for the realization of novel topological phases Wan et al. (2011); Witczak-Krempa and Kim (2012). In the context of correlated electron physics, SOC leads to anisotropic and highly frustrated spin interactions, which manifest themselves in extraordinary manners when they dominate over the isotropic interactions that usually dictate the properties of conventional magnets. A prominent example is the possible realization of the Kitaev spin liquid with fractional excitations and statistics. Honeycomb iridates have been extensively investigated for their relevance to the Kitaev model Jackeli and Khaliullin (2009); Choi et al. (2012); Chaloupka et al. (2013); Chun et al. (2015); Chaloupka and Khaliullin (2015), along with a recent extension of the materials scope to the $4d$ transition-metal compound RuCl3 Plumb et al. (2014); Banerjee et al. (2016). The above two topics are discussed in recent reviews Witczak-Krempa et al. (2014); Rau et al. (2016); Schaffer et al. (2016); Hermanns et al. (2018).

The main topic of this review, Sr2IrO4, is distinct from the above threads in that it has a counterpart in weakly spin-orbit coupled systems, namely, cuprate high temperature superconductors (HTSC). Despite being central to the establishment of the spin-orbit Mott insulating phase, the effect of SOC is not very conspicuous in the physical properties of Sr2IrO4. The details of this rather exotic mechanism are reviewed in Section 2.

The essence of the cuprate physics is believed to be contained in its unique realization of a single-band spin-1/2 Heisenberg antiferromagnet (AF) on a quasi-two-dimensional (2D) lattice Lee et al. (2006), with strong exchange couplings J$$\sim130 meV Hayden et al. (1991). To the best of our knowledge, Sr2IrO4 is the first material outside of the cuprate family that also realizes this condition (Fig. 1) Kim et al. (2012a). This naturally leads to the expectation that Sr2IrO4 can be a platform for unconventional superconductivity Wang and Senthil (2011); Watanabe et al. (2013); Meng et al. (2014); Yang et al. (2014); Gao et al. (2015). Of course, no two materials share an identical electronic structure, and in fact the microscopic details of cuprates and Sr2IrO4 are quite different in many aspects.

To what extent does the analogy between cuprate and Sr2IrO4 physics remain valid? Is Sr2IrO4 as strongly correlated as cuprates, and are the spin dynamics indeed well described by spin-1/2 Heisenberg interactions? We address these issues in Section 2. Section 3 deals with more vital questions: does the analogy continue to hold away from the Mott insulating phase as charge carriers are introduced? Namely, is a single-band Fermi surface formed upon doping? Are there strong magnetic fluctuations dressing the quasiparticles and driving their mutual interactions?

While the above questions remain under investigation, experiments to date show that electron- and hole-doped square lattice iridates reproduce a remarkable range of phenomenology of cuprates. In Section 4, we review the nature of the pseudogap and $d$-wave gap observed by angle-resolved photoemission (ARPES) and scanning tunneling spectroscopy (STS), and evidence for symmetry breaking orders that possibly underlie the pseudogap.

II MAPPING ONTO CUPRATE PHYSICS

Soon after the discovery of HTSC in the cuprates, many complex oxides with a K2NiF4 structure (isostructural to La2CuO4) and its variants were searched for signs of superconductivity. Among these were Sr2RhO4 and Sr2IrO4 Subramanian et al. (1994), which are “one-hole” systems, albeit with a $t_{2g}$-hole in the low-spin $d^{5}$ configuration as opposed to an $e_{g}$-hole in cuprates. However, it was quickly apparent that $4d$ and $5d$ systems tend to be rather weakly correlated. In fact, Sr2RhO4 is a Fermi liquid metal Nagai et al. (2010); Perry et al. (2006) with a Fermi surface understood qualitatively in terms of a band structure calculated using density functional theory within the local density approximation (LDA) Kim et al. (2006); Baumberger et al. (2006). Interestingly, the Fermi surfaces of Sr2RhO4 and Sr2IrO4, calculated without SOC, are near indistinguishable due to their almost identical crystal structures with less than 1% difference in their lattice parameters Subramanian et al. (1994). A closer inspection, however, reveals that LDA does not accurately replicate the measured Fermi surface of Sr2RhO4, due to the presence of SOC in the $4d$ system Liu et al. (2008). In $5d$ Sr2IrO4, SOC plays a far more dramatic role Kim et al. (2008).

Early transport and magnetic studies noted that Sr2IrO4 shows a significant deviation from standard Fermi liquid behavior Crawford et al. (1994); Cao et al. (1998). However, the Mott insulating nature of Sr2IrO4 went unnoticed for many years, partly due to the insufficient sample quality that obscured the intrinsic physical properties Sung et al. (2016). Today, it is well established that Sr2IrO4 is a canted AF insulator with a magnetic structure shown in Fig. 1. In this section, we review the mechanism of the SOC driven Mott transition, the dynamics of spin and orbital degrees of freedom governing the low-energy physics below the charge gap, and the process by which strong SOC makes it possible to map the system onto the cuprate physics.

II.1 Spin-orbit Induced Mott Insulator

Depicted in Fig. 2 is the established picture of the SOC driven Mott transition in Sr2IrO4. In the absence of SOC, the partially filled bands of predominantly $t_{2g}$ orbital character would lead to a metallic ground state, which is precisely the case for Sr2RhO4 Nagai et al. (2010); Perry et al. (2006). Replacing the Rh4+ ions for Ir4+ has a very minor effect on the band structure, as long as SOC is not taken into account Martins et al. (2011). Compared to Rh however, Ir SOC is approximately three times stronger (\zeta$$\sim0.4 eV), leading to a notable modification from the metallic ground state predicted by pure LDA Kim et al. (2008). Introducing SOC splits off a narrow band near the Fermi level, for which a moderate Coulomb interaction $U$ ($\sim$2 eV) is sufficient to open a charge gap. Without SOC, this would require an unphysical U$$\sim5 eV. Indeed, the LDA+SOC+$U$ solution captures the salient features of the insulating state of Sr2IrO4, reproducing the electronic band structure measured by ARPES Kim et al. (2008) and the canted AF magnetic structure measured by resonant x-ray scattering (RXS) Kim et al. (2009).

From these calculations, it is clear that the insulating gap opens only as a result of the combined effects of SOC and $U$. However, there has been much debate over the correlation effects that drive the gap opening and whether Sr2IrO4 can be classified as a genuine Mott insulator. Although insulating above the Néel transition, this usual criterion for Mott insulators is not valid for 2D systems due to sizable short-range magnetic correlations that persist to much higher temperatures Fujiyama et al. (2012a); Vale et al. (2015). Some have argued that Sr2IrO4 is instead a Slater insulator Arita et al. (2012), where the gap opens primarily as a result of magnetic order. The observation of a significant gap reduction across TN, a typical Slater behavior, supports this viewpoint Moon et al. (2009); Li et al. (2013). On the other hand, the Mott picture is supported by the fact that the charge gap is an order of magnitude larger Moon et al. (2009); Nichols et al. (2014); Dai et al. (2014); Yan et al. (2015) than the magnetic energy scale J$$\sim60 meV Kim et al. (2012a).

In the end, many theoretical and experimental studies conclude that Sr2IrO4 has a mixed Slater and Mott character Hsieh et al. (2012); Carter and Kee (2013); Watanabe et al. (2014); Solovyev et al. (2015). In fact, there is no sharp border between a Mott and a Slater insulator and indeed it is not easy to establish any 2D magnetic system as a Mott insulator. Here instead we focus on a more pragmatic question: Are the electron correlations in Sr2IrO4 strong enough that strong coupling theories accurately capture their physics? And vitally, do they result in the formation of exotic phenomena? In the following, we argue that many aspects of the magnetism of Sr2IrO4 can be understood in a localized picture using formalisms developed to address the cuprate physics.

II.2 Magnetism

Apart from a rather large canting angle ($\sim$12∘) that follows the staggered in-plane rotation of the IrO6 octahedra Boseggia et al. (2013), the magnetic structure of Sr2IrO4 (Fig. 1) is very similar to that of La2CuO4 Vaknin et al. (1987). In cubic ($O_{h}$) symmetry, the ground state of the low-spin $d^{5}$ configuration is $J_{\mathrm{eff}}$=1/2 Kramers doublet, which is separated from $J_{\mathrm{eff}}$=3/2 quartet by 3/2$\zeta$ Abragam and Bleaney (1970), where $\zeta$ is the one-electron SOC. In tetragonal ($D_{4h}$) symmetry, the ground state remains a doublet, but their wave functions are modified through mixing with the $J_{\mathrm{eff}}$=3/2 quartet. In terms of $\ket{L_{z},S_{z}}$ basis with $\ket{L_{z}=0}$ = $\ket{xy}$ and $\ket{Lz=\pm 1}$ = $\mp\frac{1}{\sqrt{2}}$($\ket{yz}$ $\pm$ $i\ket{zx})$, the doublet is expressed as

[TABLE]

The angle $\theta$ parametrizes the distortion through $\tan 2\theta$ = $2\sqrt{2}\zeta/(\zeta+2\Delta_{\textrm{tet}})$ by the tetragonal crystal field splitting $\Delta_{\textrm{tet}}$. For example, $\theta$=$\frac{1}{2}\arctan 2\sqrt{2}$ in the cubic limit where $\Delta_{\textrm{tet}}$=0.

The energy level diagram is as shown in Fig. 3a. The doublet plays the role of spin-1/2 moments in cuprates and is referred to as ‘pseudospin’ hereafter. It is interesting to note that the “orbital” degeneracy is completely lifted by the very presence of SOC, in the sense that there are no other degrees of freedom remaining apart from spin-orbit entangled pseudospin-1/2. This mechanism of orbital degeneracy lifting through SOC is very different from the Jahn-Teller effect in the cuprates, but formally they both lead to spin-1/2 dynamics in a reduced Hilbert space. The single-orbital and strongly interacting spin-1/2 moments on a quasi-2D lattice is generally regarded as an essential ingredient to HTSC and a significant effort has focused upon the search for such systems. As it turns out, this task is extremely challenging, and to the best of our knowledge Sr2IrO4 is the only successful materialization of these physics outside of the cuprate family.

What is the nature of the magnetism expressed by the pseudospin-1/2 moments? First, we note that the orbital contribution to the magnetic moment is larger than spin Kim et al. (2008); Fujiyama et al. (2014), which is reflected by a minus sign in the $g$-factor. There are two conventions used in literature: one is $g$=(2,2,-2), which smoothly connects to the $g$-factor of pure $S$=1/2 moments as SOC is continuously tuned to zero Abragam and Bleaney (1970); the other is $g$=(-2,-2,-2), which has a more symmetric form and convenient in the strong SOC limit Thornley (1968); Chaloupka and Khaliullin (2016). The latter convention is consistent with the wave functions defined above. Given that orbital-dominated magnetism is highly sensitive to bond geometry, one may expect a strong deviation from the usual Heisenberg interactions Khaliullin (2005). Indeed, in a honeycomb lattice with 90*∘* bonds isotropic interactions can be strongly suppressed, potentially leading to the Kitaev spin liquid state Jackeli and Khaliullin (2009). This is not the case for a square lattice with bonds close to 180*∘*, where isotropic interactions are the leading terms. This becomes clear in the analysis of the superexchange interactions discussed in the following section.

II.2.1 Superexchange and Magnetic Anisotropy

In Mott insulators, magnetic interactions arise from virtual charge hopping back and forth between two neighboring sites. The charge transfer processes are spin independent, which means that the total spin on a given bond is conserved. The effective spin Hamiltonian can thus be written as an inner product of the two spins; i.e. the interaction is isotropic. On the other hand, pseudospin generally need not be conserved during a charge transfer process because pseudospins have mixed spin-up and spin-down character. However, in a 180∘ bonding geometry the hopping arises only between orbitals of same symmetry, i.e. $xz$ to $xz$ etc., which implies that not only spin but also the orbital moment, and hence the total pseudospin, are conserved during the exchange process. The IrO6 rotations break the inversion symmetry and lead to antisymmetric exchange, but this interaction can be gauged out via a suitable rotation of the basis Jackeli and Khaliullin (2009); Shekhtman et al. (1992). As a result, the single band interactions between $J_{\mathrm{eff}}$=1/2 pseudospins on a square lattice are entirely isotropic, just as in case of pure spins.

However, virtual excitations to $J_{\mathrm{eff}}$=3/2 states do allow spin flip and can lead to anisotropic interactions. This mechanism requires a nonzero Hund’s coupling ($J_{\textrm{H}}$) Jackeli and Khaliullin (2009); Igarashi and Nagao (2014a), through which the pseudospin in the excited level becomes sensitive to the orientation of the pseudospin in the ground state (Fig. 3b). Thus, one arrives Jackeli and Khaliullin (2009) at a pseudospin Hamiltonian of the form

[TABLE]

where $\Gamma_{1,2}$ are functions of $\theta$ and $J_{\textrm{H}}/U$. Note that the last term has different signs for bonds along x and y axes and sum up to zero classical energy, and hence does not contribute to the anisotropy unless quantum fluctuations are taken into account. Thus, $\Gamma_{1}$ is chiefly responsible for the anisotropy, and depending on its sign it leads to either XY- or Ising-type anisotropy. Calculations show that $\Gamma_{1}$ changes sign as a function of $\theta$ at $\theta$=$\pi/4$, or $\Delta_{\textrm{tet}}\approx$ 190 meV Jackeli and Khaliullin (2009). Thus, when $\Gamma_{1}$ is small, a near-ideal Heisenberg AF can be realized Zhang et al. (2013). However, this requires a fine tuning of the relevant parameters. In fact, experiments Kim et al. (2014a); Moretti Sala et al. (2014a) and calculations Katukuri et al. (2012); Bogdanov et al. (2015); Agrestini et al. (2017) consistently estimate \Delta_{\textrm{tet}}$$\sim -140 meV for Sr2IrO4 and \Delta_{\textrm{tet}}$$\sim 50 meV for Ba2IrO4  suggesting a sizable anisotropy term $\Gamma_{1}$.

II.2.2 Overview of Experimental Results

In parallel with the progress in our theoretical understanding of the magnetism deriving from spin-orbit entangled moments, significant advancements have been made in modern RXS techniques capable of testing the validity of the $J_{\mathrm{eff}}$=1/2 model. In particular, hard x-ray inelastic RXS (RIXS) has been instrumental in the elucidation of the spin-orbital dynamics of square lattice iridates, much in the same way that inelastic neutron scattering proved invaluable to the development of the comprehensive 2D Heisenberg description of cuprates.

These studies reveal the pseudospin dynamics in Sr2IrO4 that are remarkably similar to spin dynamics in La2CuO4, even as SOC is enhanced by an order of magnitude and correlations relatively weakened for the $5d$-electrons. Only the low energy dynamics around the spin wave gap mark the presence of strong SOC, indicating that the pseudospin interactions are well described by strong coupling theories. The most significant deviation from pure Heisenberg physics is an out-of-plane spin wave gap, which is a measure of the XY-type anisotropy. A smaller in-plane gap is chiefly due to pseudospin-lattice coupling, which leads to a slight structural orthorhombicity coupled to the moment orientation.

II.2.2.1 Probing spin-orbit entangled states

In an elastic RXS (REXS) measurement, an incident synchrotron x-ray wavelength is tuned through the Ir $L_{2,3}$ (2$p$-5$d$) absorption edge to amplify the magnetic signal specific to the iridium ion. The resonant scattering is a second order process arising from quantum interference among all intermediate states accessible from the ground state on a given site. In this way, the REXS signal encodes information regarding the wave function for valence electrons. Calculations of the x-ray scattering matrix elements show that resonant enhancement is expected only at the $L_{3}$ edge for the $J_{\mathrm{eff}}$=1/2 state, whereas equal intensities at the $L_{2}$ and $L_{3}$ edges would arise in a $S$=1/2 system. In the first REXS study of Sr2IrO4, indeed a strong magnetic Bragg reflection signal was observed only at the $L_{3}$ edge, consistent with the $J_{\mathrm{eff}}$=1/2 model Kim et al. (2009). However, it was pointed out that the vanishing $L_{2}$ intensity does not necessarily imply the $J_{\mathrm{eff}}$=1/2 state if the pseudospin lies in the $ab$-basal plane Chapon and Lovesey (2011); Moretti Sala et al. (2014b); Kim and Khaliullin (2017), which is the case for Sr2IrO4. Only the $z$ component of the pseudospin is sensitive to distortions away from the cubic limit, which means that for Sr2IrO4, measurement of dynamic fluctuations of pseudospins out of the $ab$-plane through RIXS at the $L_{2}$ edge can verify the $J_{\mathrm{eff}}$=1/2 state.

REXS and neutron diffraction studies confirm that antiferromagnetic long range order forms below TN$\simeq$ 230 K with a wavevector $k$=$(1\,1\,1)$ and pseudospin aligned in-plane Kim et al. (2009); Dhital et al. (2013); Ye et al. (2013); Boseggia et al. (2013). Magnetic reflections at $(1\,0\,4n+2)$ and $(0\,1\,4n)$ arise from the AF component of the magnetic structure shown in Fig. 1, while reflections at $(1\,0\,4n)$ and $(0\,1\,4n+2)$ correspond to its twin domain. We note that the two domains have different sets of reflections because the stacking pattern is correlated with the pseudospin direction. For example, if the pseudospin are oriented along the $a$-axis, then pseudospins in (010) planes are in phase. The canted component is confined to the $ab$-plane, forming a net ferromagnetic moment within each layer, which orders in an $+--+$ pattern along the $c$-axis.

Above a critical field H_{C}$$\approx2 kOe the stacking pattern rearranges in a ferromagnetic $++++$ pattern as $(1\,0\,\textrm{even})$ reflections disappear and $(1\,0\,\textrm{odd})$ emerge Kim et al. (2009). Although the stacking patterns along the $c$-axis is governed by small energy scale interlayer couplings Takayama et al. (2016), the complete magnetic structure vitally defines the space and point group symmetries, with consequences to selection rules in optical transitions. This is relevant to the optical second harmonic generation experiment discussed in Section 4.

II.2.2.2 Spin waves

Direct confirmation of the predominantly isotropic nature of the exchange interactions present in Sr2IrO4 have come from a comprehensive mapping of the full excitation spectrum using RIXS. The first published data Kim et al. (2012a), with an energy resolution of $\sim$130 meV, are shown in Fig. 4. Energy scans throughout the Brillouin zone reveal a single magnon mode with bandwidth of $\sim$200 meV dispersing from the magnetic zone center ($\pi$,$\pi$). Above this spin wave branch is a distinctive feature in a range 0.4–0.8 eV that lies on top of the electron-hole continuum formed from excitations across the Mott gap. This feature is an intra-site excitation of a hole from $J_{\mathrm{eff}}$=1/2 to 3/2 states (Fig. 3a) Kim et al. (2012a, b, 2014a); Igarashi and Nagao (2014b); Yang et al. (2015); Lu et al. (2018). In other words, an electron-hole pair is created across the spin-orbit split bands, which form a bound state and hence is called ‘spin-orbit exciton’. The magnitude of the $\sim$0.6 eV splitting of this lowest $d$-$d$ excitation can be compared to the $\sim$2 eV splitting of $e_{g}$ bands in the cuprates Hozoi et al. (2011) (Fig. 3a); below this energy scale system can be modeled as a single band system (considering only Cu $d$ states). Similarly, excitations to the $J_{\mathrm{eff}}$=3/2 level are clearly separated from the single electron band and the spin wave energy scales, despite the relatively smaller splitting magnitude.

The Heisenberg-like dispersion of the spin wave is made explicit by comparing the energy and intensity of the spin wave dispersion throughout the Brillouin zone with La2CuO4, as shown in Fig 5. The overall similarity in both the dispersion and intensity between Sr2IrO4 and La2CuO4 is remarkable. The Sr2IrO4 magnon branch has a bandwidth of $\sim$200 meV, as compared to $\sim$300 meV in La2CuO4 Coldea et al. (2001); Headings et al. (2010), consistent with reduced energy scales of $t$ and $U$ [see section 2.3]. Both systems are fitted with the phenomenological $J-J^{\prime}-J"-J_{c}$ model, which takes into account respective first, second, third-neighbor isotropic exchange terms and four-spin cyclic exchange. Notably, in both systems the downturn from ($\pi$,0) to ($\pi/2$,$\pi/2$), equivalent to (1/2,0) to (3/4,1/4) in crystallographic notation, can be accounted for by either a ferromagnetic $J^{\prime}$ or $J_{c}$, which can not be differentiated from spin wave dispersions. This dispersion is more pronounced in Sr2IrO4 implying that the further neighbor interactions are relatively stronger as compared to La2CuO4, which reflects the more extended nature of 5$d$ orbitals. To date, fitting to the highest resolution published data gives $J$=57(7) meV, $J^{\prime}$=-18(3) meV and $J"$=14(2) meV, with $J_{c}$ set to zero Vale et al. (2015).

II.2.2.3 Magnetic anisotropy

A study of the low energy spin dynamics provides more insight into the magnitude and nature of anisotropic interactions, and thus the extent to which the $S$=1/2 and $J_{\mathrm{eff}}$=1/2 system can be compared. One approach is to probe the critical behavior by using magnetic diffuse scattering. Here, however, two separate studies present differing conclusions; one study finds critical behavior at the onset of magnetic order consistent with a Heisenberg universality class Fujiyama et al. (2012a), while another indicates a significant deviation from it Vale et al. (2015).

A more direct approach is to look at the magnitude of the spin-wave gap. RIXS measurements conducted with $\approx$50 meV energy resolution points to a partially resolved $\approx$30 meV spin-wave gap at the $\Gamma$ point Kim et al. (2014a); Vale et al. (2015). On the other hand, Raman scattering Gim et al. (2016) and electron spin resonance Bahr et al. (2014) measurements reveal a sharp mode at 1$\sim$2.5 meV that develops below the Néel transition. Recently, this low energy feature has been resolved via high-resolution RIXS and inelastic neutron scattering Porras (2018), both of which indicate a spin-wave gap between 2 and 3 meV at the magnetic zone center ($\pi,\pi$). While these results initially appeared incompatible, it is now clear that they correspond to different spin wave modes associated with spins rotating in the $ab$-plane and out of the plane. From the polarization dependence of the RXS process, the 30 meV gap, measured in normal incidence, can be assigned to the out-of-plane mode. This is significantly larger than the $\approx$2.5 meV out-of-plane gap in La2CuO4 Peters et al. (1988). The in-plane gap of $\approx$2.5 meV, on the other hand, is comparable to $\approx$1 meV in La2CuO4 Peters et al. (1988).

Although an in-plane gap is expected from interlayer couplings and/or quantum fluctuations associated with $\Gamma_{2}$ Katukuri et al. (2014), it turns out that the in-plane gap is primarily set by pseudospin-lattice coupling Liu and Khaliullin (2018), which is essentially a Jahn-Teller effect. The magnetic ordering breaks the tetragonal symmetry of the lattice as the AF component of the pseudospins align along either $a$- or $b$ axis, which with a non-zero pseudospin-lattice coupling leads to a small orthorhombic distortion of the lattice. Thus, the magnitude of the in-plane spin-wave gap is a measure of the magnetoelastic coupling. Based upon the observed magnitude of the gap, an orthorhombic distortion of $\epsilon\approx 10^{-4}$ is expected below $T_{\textrm{N}}$ Liu and Khaliullin (2018), which is smaller than the resolution limits of most experimental techniques outside of Larmor diffraction Náfrádi et al. (2016) or dilatometry. There are other experimental signs, however. The pseudospin-lattice coupling is, for example, manifested as anomalous broadening of the linewidth of some Raman-active phonon modes Gretarsson et al. (2016a, 2017) and hardening of optical phonons across $T_{\textrm{N}}$ Moon et al. (2009). The rotational behavior of pseudospins under external magnetic fields, reported in magnetometry Takayama et al. (2016), torque magnetometry Nauman et al. (2017), and magnetoresistance Wang et al. (2014) studies, is consistent with the presence of pseudospin-lattice coupling. Further, the most recent detailed analysis shows that the behavior is incompatible with four-fold symmetric anisotropy expected for tetragonal symmetry and instead is well described by a pseudospin-lattice coupling  Porras (2018); Liu and Khaliullin (2018).

The out-of-plane gap, significantly larger than that of cuprates, is a deviation from an ideal Heisenberg AF assumed in many theories of Sr2IrO4. The implications of XY anisotropy for emergent electronic orders that form through doping are currently unknown, as it is generally not found in weak SOC systems and has not been studied in any detail. We note in passing that the bilayer compound Sr3Ir2O7 has a strong Ising anisotropy Kim et al. (2012c, d); Fujiyama et al. (2012b); Boseggia et al. (2012), which leads to pseudospin dynamics distinct from the present case and thus may have different effects on the effective interactions between doped carriers.

II.3 Single-band Model

As we have seen from Fig. 3 and Eq. 1, when the symmetry is lowered from cubic symmetry the pseudospins acquire anisotropic shape through mixing between $J_{\mathrm{eff}}$=1/2 and $J_{\mathrm{eff}}$=3/2 states, which is chiefly responsible for the magnetic anisotropy Jackeli and Khaliullin (2009); Mohapatra et al. (2017). In other words, interactions among pseudospins-1/2 are renormalized by virtual excitations to $J_{\mathrm{eff}}$=3/2 states, but otherwise the latter disappear in the effective low-energy description of Sr2IrO4. As a result, one arrives at a single-band model Wang and Senthil (2011); Jin et al. (2009); Watanabe et al. (2010), akin to the so-called $t-J$ model for cuprates, or the more extended $t-t^{\prime}-t^{\prime\prime}-J$ model.

The analogy to cuprates is made explicit by defining the pseudospins on a local basis taking into account the staggered rotation of IrO6 octahedra, which gives effective hopping terms $t\approx 0.26$ eV, $t^{\prime}\approx t/4$ and $t^{\prime\prime}\approx t/10$. Note that the nearest neighbor (NN) hopping $t$ is roughly only 60% of that in the cuprates despite the fact that 5$d$ orbitals are more spatially extended. This is because the pseudospins use orbitals of $t_{2g}$ symmetry with lobes pointing away from NN bonds, and mixing of these orbitals by SOC further reduces the overlap between NN pseudospins. However, $U$ is also smaller in Sr2IrO4 by a similar factor, so that the ratio $t/U$ is not very different from cuprates. As a result, Sr2IrO4 can be thought of as cuprates with the parameters scaled down to $\sim$60%.

One difference from the cuprates is the sign of next NN hopping $t^{\prime}$, which may be responsible for the particle-hole asymmetry of the cuprate phase diagram. However, the sign can be changed by a particle-hole transformation, meaning that the hole-doped (electron-doped) side of cuprates maps onto electron-doped (hole-doped) side of Sr2IrO4 Wang and Senthil (2011). Indeed, as we shall see in the following sections, electron-doped Sr2IrO4 reproduces many features of hole-doped cuprates.

III ANALOGY AWAY FROM MOTT INSULATING PHASE

Having discussed the analogy between Sr2IrO4 and cuprates in their Mott insulating phases, we now proceed to compare the electronic structures in doped phases, summarized in Fig. 6. First, we note that in both cases adding an electron makes a closed shell with no internal structure. This contrasts with more complex structure on the other side with one electron removed. For cuprates, it is well known that an added hole goes into the O 2$p_{x/y}$ orbital rather than into the Cu 3$d_{x^{2}-y^{2}}$ orbital, which classifies cuprates as charge transfer insulators in the Zaanen-Sawatzky-Allen scheme Zaanen et al. (1985). Thus, the ground state is a bound state of two-holes, whose two $S$=1/2 moments combine to make a so-called Zhang-Rice singlet Zhang and Rice (1988).

On the other hand, Sr2IrO4 is a Mott insulator in the sense that the charge gap opens between $d$ bands (see Fig. 2), meaning that the doped hole goes into the $d$ band; i.e., $d^{5}\rightarrow d^{4}$ in a localized picture. Interestingly, the ground state of the low-spin $d^{4}$ configuration is $J_{\mathrm{eff}}$= 0 singlet with antiparallel spin and orbital Khaliullin (2013). Although this intra-site singlet formed by SOC has a different microscopic structure from that of the Zhang-Rice singlet, they share some formal similarities having identical (zero-spin) quantum numbers.

In the cuprates, the Zhang-Rice mapping reduces the multi-orbital physics to an effective single-band physics, and its stability has been discussed in connection with the validity of the single-band model. Although the single-band model is widely accepted for cuprates, whether it contains the essential physics of HTSC continues to be debated Varma (1997). For Sr2IrO4, the validity of the single-band model is crucial for its analogy with cuprates. It is not a priori clear if and to how far the single-band picture continues to be valid away from the Mott insulating phase, because once orbital moments are quenched and SOC becomes ineffective as electrons become itinerant, the system reverts to a three-band system (as is the case for Sr2RhO4). This is an issue pertinent not only to the hole-doping case Sohn et al. (2016) but also to the electron-doping case Cheng et al. (2016).

In the case of electron doping, a RIXS study on La-doped Sr2IrO4, shown in Fig. 7, has revealed that paramagnon excitations persist well into the metallic phase (\sim$$10\% electron doping) that has a clear Fermi surface Gretarsson et al. (2016b). The excitations within $J_{\mathrm{eff}}$=1/2 states (magnons) and to $J_{\mathrm{eff}}$=3/2 states (spin-orbit excitons) remain well separated and visible (Fig. 7a), indicating that pseudospin-1/2 moments persist despite rapid metallic charge fluctuations. Although heavily damped, their dispersions can be traced to reveal anisotropic softening along (0,0)-($\pi$,$\pi$) direction (Fig. 7b), which is in stark contrast to the hardening of the magnons in electron-doped cuprates Lee et al. (2014), but resembles the results seen in holed-doped cuprates Le Tacon et al. (2011); Dean et al. (2013); Guarise et al. (2014). This validates the single-band picture for electron-doped Sr2IrO4 and their analogy to hole-doped cuprates.

Pseudospin dynamics in hole-doped Sr2IrO4 has not been studied in any detail, but there are experimental indications that they map onto electron-doped cuprates. It has been shown that hole doping through replacing Rh3+ for Ir4+ results in suppression of the magnetic order at a much higher critical doping of $\sim$17% Qi et al. (2012); Clancy et al. (2014) as compared to $\sim$3% in La-doped Sr2IrO4 Gretarsson et al. (2016b); Chen et al. (2015, 2018), pointing to electron-hole asymmetry as in the cuprates but with electron-hole conjugation. This is further corroborated by ARPES studies, which reveal fermiology of Rh-doped Sr2IrO4 reminiscent of electron-doped cuprates Cao et al. (2016). While promising, the inference to cuprate electron-hole asymmetry is based mostly on studies of La-doped and Rh-doped Sr2IrO4, and the lack of a broader range of material families precludes making a general conclusion.

Besides chemical doping, in situ deposition of potassium atoms on a cleaved sample surface provides a clean method of electron doping Kim et al. (2014b, 2016), and thus has been instrumental in studying the electronic evolution of Sr2IrO4, to be discussed in the next Section. However, this method only works at low temperatures in ultra-high vacuum and doping is limited to few surface layers, and thus can only be used in combination with surface-sensitive techniques such as ARPES and STS.

IV EMERGENT ELECTRONIC ORDERS

Ultimately, the analogy with cuprates is only justified by the presence of electronic orders that emerge when the magnetic order is suppressed as charge carriers are introduced— most prominently $d$-wave superconductivity, but also phenomena surrounding the superconducting dome in the pseudogap regime. To date, the most enlightening results have come from studies of the electronic structures using ARPES Kim et al. (2014b, 2016) and STS Yan et al. (2015) with in situ surface electron doping. These studies reveal a pseudogap and Fermi arcs that closely follow cuprate phenomenology. Significantly, a $d$-wave gap emerges at low temperatures potentially signaling the superconducting state. In this section, we review selected experimental results that most vividly reveal such emergent phases in doped Sr2IrO4.

IV.1 Pseudogap and Competing Orders

In general, pseudogap refers to a partial suppression of spectral weight in certain regions of the Fermi surface. In the cuprate phase diagram, the pseudogap phase forms around the superconducting dome from the underdoped to slightly overdoped regions and up to relatively high temperatures. Due to its close proximity to the superconducting dome, it is widely believed that the pseudogap is key to understanding of HTSC. However, despite decades of exhaustive investigation, the origin of the pseudogap and its relation to the superconducting state remains unclear.

One thought is that the pseudogap is a precursor to superconducting gap and Fermi arcs result from broadening of the spectra that exhibit the d-wave gap Reber et al. (2012). In this picture, phase incoherent Cooper pairs are formed above the superconducting critical temperature ($T_{c}$), which is much suppressed relative to the energy scale of pair formation due to fluctuations enhanced by the reduced dimensionality. Another idea is that the pseudogap is a magnetic gap that arises due to short-range AF fluctuations Wang et al. (2015). A yet another possibility is that the pseudogap represents an electronic order associated with a symmetry-broken phase in competition with superconductivity. While the question remains hotly contested, it is widely believed that the underlying physics is contained in minimal models of doped spin-1/2 Mott insulator on a square lattice.

Being the first non-cuprate material to realize this condition, Sr2IrO4 offers a unique opportunity to verify this hypothesis. We recall that Sr2IrO4 does not have O 2$p$ band near its Fermi level and thus in this sense is a ‘cleaner’ realization of the single-band model, especially in the electron-doped side where only the $J_{\mathrm{eff}}$=1/2 states are relevant.

IV.1.1 Electron doping

We start our discussion with electron-doped Sr2IrO4, which reproduces a remarkable range of cuprate phenomenology. In La-doped Sr2IrO4 Ge et al. (2011), as electrons are introduced a subtle change starts to be detected below 4% doping. A RXS study shows that the long-range AF order has a slightly lower $T_{\mathrm{N}}$ and a reduced $c$-axis correlation Chen et al. (2018). STS detects no change in the gap region in this doping range, suggesting that electrons are bound to dopant atoms Battisti et al. (2017). As the electron concentration is further increased, a sudden heterogeneous collapse of the Mott gap occurs and nano-scale pseudogap puddles start to appear with a gap value around 70$\sim$300 meV (Fig. 8a) Chen et al. (2015); Battisti et al. (2017). In momentum space, the pseudogap opens around ($\pi$,0), leaving a broken Fermi surface referred to as ‘Fermi arcs’ in the cuprate literature (Fig. 8b) de la Torre et al. (2015); Brouet et al. (2015); Terashima et al. (2017).

Because the Fermi arcs require a radical rethinking of the notion of the topological integrity of the band, a simpler explanation has been given that they are part of the Fermi pocket that forms around ($\pi$/2,$\pi/2$) as a result of the backfolding of the bands by the ($\pi$,$\pi$) magnetic order, but only one side of it being visible due to interactions and ARPES matrix element effects He et al. (2015). Notably, In La-doped Sr2IrO4 there is an additional ($\pi$,$\pi$) scattering by the superstructure of IrO6 rotation, which leads to an apparent Fermi pocket. This feature is not a Fermi pocket, however, as the gap is already open away from the zone boundary (Fig. 8b); i.e. there is no closed zero-energy contour.

Evidence that the Fermi arcs have a nontrivial many-body origin is provided by their strong temperature and doping dependence. Using in-situ surface electron doping, achieved by depositing alkali metal atoms on a cleaved surface of Sr2IrO4, the electronic evolution over the entire range from the Mott insulating phase to the Fermi-liquid metal phase has been measured Kim et al. (2016). At one monolayer coverage of potassium (K), corresponding to $\sim$9% electron doping, a circular Fermi surface encloses an area larger than one-half of the 2D Brillouin zone (Fig. 9a). However, at 0.5 monolayer, the Fermi surface breaks up into arcs as the gap opens in the region near ($\pi$,0). Furthermore, the underlying Fermi surface now encloses the an area less than half of the 2D Brillouin zone, which is inconsistent with electron doping. This implies that the pseudogap is more than merely opening of a gap as the assumption of the underlying Fermi surface violates the Luttinger sum rule.

A similar behavior is seen as a function of temperature (Fig. 9b). At a constant doping ($\sim$0.7 monolayer), as the system is cooled from T=110 K to T=30 K pseudogap opening around ($\pi$,0) is clearly seen. Coming back to La-doped Sr2IrO4, optical conductivity below a threshold frequency of about 17 meV shows a gradual suppression of the spectral weight below $\sim$100 K Seo et al. (2017).

Intriguingly, a RXS study conducted in the pseudogap phase Chen et al. (2018) has uncovered an incommensurate magnetic scattering reminiscent of unidirectional spin density waves observed in the hole-doped cuprate La2-xSrxCuO4 Fujita et al. (2002); Drachuck et al. (2014), supporting the idea that the pseudogap is associated with a symmetry-breaking order. In a similar vein, a $d$-wave spin-orbit density wave order with a circulating staggered pseudospin current was proposed to underlie the pseudogap Zhou et al. (2017). It is claimed that the order is already present in the undoped insulating phase, manifest as the splitting of the bands at ($\pi$,0) whose two-fold degeneracy is otherwise protected by certain lattice symmetries. It is interesting to note that a magneto-electric loop-current order has been inferred from a second harmonic generation experiment Zhao et al. (2016) performed on the parent and hole-doped Sr2IrO4, although it is symmetry-wise distinct from the spin-orbit density wave.

IV.1.2 Hole doping

As discussed in Section 3, a doped hole has a different electronic structure than that of an electron, and is more likely to involve multi-orbital physics. This means more possibilities for emergent electronic phases Meng et al. (2014). Known methods for hole doping of Sr2IrO4 include K doping Ge et al. (2011) (not to be confused with K deposition discussed above) and Rh substitution in place of Ir  Qi et al. (2012); Clancy et al. (2014); Brouet et al. (2015); Ye et al. (2015); Cao et al. (2016); Chikara et al. (2017). Although Rh and Ir are in the same column of the periodic table, their different electron affinities lead to a charge transfer from Rh to Ir, resulting in an effective hole doping of Ir sites ($d^{4}$) and dilution of magnetism by nonmagnetic Rh*+3* ($d^{6}$). Because Rh doping creates disorders in the IrO2 layers Chikara et al. (2015), it would be more favorable to use K doping with dopants in SrO layers.

Not much is known about the physical properties of K-doped Sr2IrO4. Although the authors of Ref. Ge et al. (2011) do not provide a detailed analysis, the resistivity at $\sim$4% doping seems to show a linear temperature dependence over a wide temperature range 60$\sim$350 K. On the other hand, Rh-doped Sr2IrO4 exhibit incoherent charge transport characterized by the absence of quasiparticles, a momentum-independent pseudogap Louat et al. (2018), and localization of carriers at low temperatures Qi et al. (2012). In both K- and Rh-doped Sr2IrO4, the static magnetic order survives up to a much higher doping concentration ($\sim$17%) as compared to electron doping.

A recent study using optical second harmonic generation on the parent and Rh-doped Sr2IrO4 reported an electronic order that breaks the spatial inversion and rotational symmetries Zhao et al. (2016). The technique exploits a nonlinear optical response of a material and is highly sensitive to changes in the point group symmetry. The onset of the signal nearly coincides with the Néel temperature in the parent compound, but the two temperatures diverge with increasing Rh-doping as the AF transition is more quickly suppressed. A symmetry analysis shows that the data is consistent with an orbital loop-current order, which was proposed to underlie the pseudogap in cuprates Varma (1997); Fauqué et al. (2006). Assuming that the time-reversal symmetry is also broken, magnetic space groups consistent with the data are narrowed down to $2^{\prime}/m$ or $m1^{\prime}$, for which different loop-current patterns can be constructed (Fig. 10). Subsequently, a polarized neutron diffraction measurement found a peak in a spin-flip scattering channel whose onset temperature matches that of the second harmonic generation, supporting the orbital loop-current interpretation of the second harmonic generation experiment Jeong et al. (2017).

Loop current-ordered phases, however, are based on a three-band model including O 2$p$ states—a feature not present in a hole- or electron-doped Sr2IrO4. There are, in fact, simpler explanations for the second harmonic signal Di Matteo and Norman (2016). The $2^{\prime}/m$ magnetic space group is also consistent with the $-+-+$ stacking pattern that can readily form with minor perturbation through disorder, doping, or laser pump. It has been pointed out that the inversion symmetry can also be broken locally at a boundary between two magnetic domains through magneto-elastic coupling Liu and Khaliullin (2018), and particularly in the paramagnetic phase with short-range magnetic fluctuations the lattice can appear inversion symmetry broken depending on the timescale of the measurement.

IV.2 Superconductivity

Although there is no experimental evidence of superconductivity in electron or hole-doped Sr2IrO4 at the time of writing, tantalizing indications of $d$-wave superconductivity have been glimpsed in in situ electron-doped Sr2IrO4. An ARPES measurement shows that below $\sim$30 K the Fermi arcs shrink to point nodes as a $d$-wave-symmetric gap opens along the arc (Fig. 11a) Kim et al. (2016). A fit to the $d$-wave form factor, however, reveals a prominent deviation near the antinode (Fig. 11b). A linear extrapolation of the near-nodal gap returns a gap maximum in the range 16-30 meV, depending on the doping concentration (or the surface potassium coverage). This is smaller by a factor of two than the actual gaps $2\Delta^{*}$ measured at the anti-node, pointing to the possibility that the pseudogap and the $d$-wave gap might have two distinct origins. In fact, this is a feature observed universally in underdoped cuprates Yoshida et al. (2012); Hashimoto et al. (2014), and thus Sr2IrO4 can shed new light onto the elusive relationship between the pseudogap and the $d$-wave gap.

In real space, the pseudogap and the $d$-wave gap show anticorrelations Yan et al. (2015), again in much similarity with the STS measurements on the cuprates. The $d$I/$d$V spectra exhibit a strong spatial inhomogeneity with high-energy-gap regions separated from low-energy-gap regions. The low-energy gap opens symmetrically about the zero bias energy below a doping-dependent temperature, ranging between 20$\sim$50 K for the surface coverage between 0.5$\sim$0.7 monolayers, with the gap magnitude in the range 10$\sim$28 meV. We note that the results for 0.7 monolayer (which is the only overlapping data point) are in good agreement with the ARPES results (Fig. 11c).

Based on the above results it is tempting to associate the $d$-wave gap with superconductivity. Although theoretically $d$-wave gap is not exclusive to superconductivity Marston and Affleck (1989); Chakravarty et al. (2001), no other phase of matter has been observed to exhibit a $d$-wave gap symmetric about the Fermi level. However, establishing superconductivity requires macroscopic evidence, i.e., Meissner effect or zero-resistance, which are very difficult to measure on in situ doped samples. We note that La-doped samples only partially reproduce these results de la Torre et al. (2015); high-temperature pseudogaps are observed but no there are no $d$-wave gaps, but this may only be due to insufficient sample quality Sung et al. (2016).

V SUMMARY AND OUTLOOK

Over the last decade, the field of iridates has grown to a size that a single review article can no longer cover all of the exciting physics they encompass. In this review, we focused on one material system, the square lattice iridate Sr2IrO4, for which the central issue has been whether it would display HTSC like cuprates. Even staying within this somewhat narrow scope, limited space did not allow us to touch upon different approaches to metallize Sr2IrO4 through epitaxial thin films Rayan Serrao et al. (2013); Nichols et al. (2013), high pressure Haskel et al. (2012); Zocco et al. (2014), gating Lu et al. (2015); Ravichandran et al. (2016), and other chemical routes Okabe et al. (2013). So far, the most promising results have been obtained from in situ doping used in combination with ARPES and STS, but the extreme surface sensitivity of these methods is a serious impediment to further scientific progress.

What is the physical origin of the low temperature $d$-wave gap observed in in situ electron-doped Sr2IrO4, and why is it not reproduced in La-doped Sr2IrO4? These are important outstanding questions, because if the $d$-wave gap is not due to superconductivity, it at the very least represents a new phase of matter likely in competition with $d$-wave superconductivity. One hint is that the $d$-wave gap phase may be very fragile against disorder. It is important to note that even the parent compound Sr2IrO4—much more so for the La-doped samples—is difficult to grow in high quality as it is prone to contamination by oxygen vacancies Guevara et al. (2018), which may mask the intrinsic properties of parent Sung et al. (2016) and possibly destroy the coherence of $d$-wave gap in electron-doped Sr2IrO4. In fact, a well-defined quasiparticle is only present in the ARPES spectra of in situ doped Sr2IrO4 but not in La-doped Sr2IrO4. In the former, close to one monolayer coverage the system is almost disorder free, and the off-plane charge reservoir also acts toward screening any charged impurities near the sample surface where the $d$-wave gap is observed. The destructive effect of poorly screened Coulomb impurities on electronic quasiparticles is manifested in the insulating Sr2IrO4 by charge-neutral (spin-orbit) excitons, which, despite their much higher excitation energies, are much more long-lived than fermionic quasiparticles Kim et al. (2014a).

These results suggest that methods beyond traditional chemical doping, such as digital doping through heterostructures or hydrogenation may be the way to proceed. The relatively simpler chemistry of the cuprates, also the pnictides for that matter, fueled the explosive growth of the HTSC field. But the chemistry and physics differences between iridates and cuprates may be used to our advantage in solving the HTSC conundrum, by separating material specifics and issues idiosyncratic to cuprates from essentials of HTSC. In particular, although both iridates and cuprates are single-band systems at low energies, they are derived from Mott and charge-transfer insulating parent compounds, respectively. The results from electron-doped Sr2IrO4 already tell us that this distinction seems not to be essential for the emergence of pseudogap and formation of $d$-wave gapped Fermi surface, and that these phenomena can be captured by a single-band model interacting with short-range AF fluctuations. Future studies of square-lattice iridates may provide new insights on whether Mott versus charge-transfer dichotomy is essential for HTSC itself.

In solving these problems, one can use material design and experimental probes not applicable to cuprates. For example, the fact that the bilayer compound Sr3Ir2O7 with strong Ising anisotropy does not show any sign of unconventional charge dynamics when doped de la Torre et al. (2014); Hogan et al. (2015) suggest that a clever engineering of spin dynamics can lead to novel tunable material properties. Such materials design can be readily tested on a small single crystal using hard x-ray RIXS with its access to spin dynamics over the full Brillouin zone up to very high energies with energy resolutions now getting better than 10 meV Kim et al. (2018). Although not discussed in this review, iridates are also one of the major playgrounds for spin liquids Okamoto et al. (2007); Kitagawa et al. (2018), which has a historical connection to HTSC Anderson (1973). Beyond a cuprate analog, iridates are thus interesting in their own right and at the moment we are probably seeing only the tip of the iceberg.

We acknowledge insightful discussions with B. Keimer, G. Jackeli, H. Takagi, H. Gretarsson, M. Le Tacon, J. van den Brink, J. H.Kim, Y.-B. Kim, H.-Y. Kee, C. Kim, J.-H.Park, Y. Bang, and E. G. Moon.