Unveiling the origin of the basal-plane antiferromagnetism in the spin–orbit Mott insulator Ba2IrO4: a density functional and model Hamiltonian study

Based on the density functional theory and model Hamiltonian, we studied the basal-plane antiferromagnetism in the spin–orbit Mott insulator Ba2IrO4. By comparing the magnetic properties of the bulk Ba2IrO4 with those of the single-layer Ba2IrO4, we demonstrate unambiguously that the basal-plane antiferromagnetism is caused by the intralyer magnetic interactions rather than by the previously proposed interlayer ones. Aiming at revealing the origin of the basal-plane antiferromagnetism, we add the single ion anisotropy and pseudo-quadrupole interactions into the general bilinear pseudo-spin Hamiltonian. The obtained magnetic interaction parameters indicate that the single ion anisotropy and pseudo-quadrupole interactions are unexpectedly strong. Systematical Monte Carlo simulations demonstrate that the basal-plane antiferromagnetism is caused by isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions. On the basis of this study the single ion anisotropy and pseudo-quadrupole interactions could play a role in explaining magnetic interactions in other iridates.


Introduction
Recently, iridium oxides have been extensively studied both experimentally and theoretically because many exotic and emerging phenomena are proposed to be realized in such iridium oxides [1,2]. For example, quantum spin Hall effect has been predicted in Na 2 IrO 3 [3,4]. Topological Mott insulator [5,6], Weyl semimetal and axion insulator [7] are theoretically shown to exist in the pyrochlore iridium oxides. Most excitingly, iridium oxides with the honeycomb lattice are greatly expected to realize the long-sought Kitaev model, which has an exactly solvable quantum spin liquid ground state [8][9][10][11]. Furthermore, some theoretical and experimental studies suggested that the prototypical iridium oxide Sr 2 IrO 4 (SIO) could display an unconventional high-T C superconductivity [12][13][14][15]. However, the appreciable ferromagnetic (FM) moment arising from the rotationally distorted IrO 6 octahedron in SIO may easily break the spin-singlet Cooper pairs [16,17]. Therefore other iridium oxides need to be explored. It is found that the square-lattice antiferromagnet Ba 2 IrO 4 (BIO) crystallizes in the space group I4/mmm [18] and is a closer analogue to the layered La 2 CuO 4 than SIO [19]. It is therefore expected that BIO is suitable for realizing the high-T C superconductivity.
However, there remains a fundamental and important issue regarding the basal-plane antiferromagnetism in BIO to be clarified. Experimentally, it is found that the magnetic moments of Ir 4+ ions are antiferromagnetically coupled and point along the [110] direction [18][19][20][21]. Katukuri et al theoretically investigated [22] how the [110]-pointing antiferromagnetism is established. Based on the general bilinear pseudo-spin Hamiltonian, they proposed that the experimentally observed basal-plane antiferromagnetism could be accounted for by including the additional interlayer magnetic interactions and the associated order-bydisorder quantum-mechanical effects [22]. However, the interlayer magnetic interactions in BIO may be too weak to play a dominant role in establishing the experimentally observed basal-plane antiferromagnetism, because the shortest distance between the interlayer Ir 4+ ions is about 7.3 Å [18] and the interlayer Ir 4+ ions are separated by two BaO layers. Therefore, it will be of great importance to reinvestigate the origin of the experimentally observed basal-plane antiferromagnetism in BIO.
In this paper, we put forward a new mechanism to explain the experimentally observed basal-plane antiferromagnetism in BIO, based on the density functional theory (DFT), model Hamiltonian, and Monte Carlo (MC) simulations. Firstly, the comparison of the magnetic properties, especially the basal-plane magnetocrystalline anisotropy energy (MAE), between the bulk and single-layer BIO clearly demonstrates that the experimentally observed basal-plane antiferromagnetism is caused by the intralyer magnetic interactions rather than by the previously proposed interlayer ones [22]. Next, it is proved that the calculated basal-plane MAE cannot be explained by the general bilinear pseudo-spin Hamiltonian. To bring to light the origin of the experimentally observed basal-plane antiferromagnetism, we add the single ion anisotropy (SIA) and pseudoquadrupole interaction into the general bilinear pseudo-spin Hamiltonian. By computing the magnetic interaction parameters in the model Hamiltonian, we find that the SIA and pseudo-quadrupole interaction are unexpectedly strong in BIO. Based on the systematical MC simulations, we demonstrate that the experimentally observed basal-plane antiferromagnetism can be explained by the isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions. On the basis of this study the SIA and pseudo-quadrupole interactions could play a role in explaining magnetic interactions in other iridates.

Computational methods
Our first-principles calculations are based on the DFT and performed within the generalized gradient approximation according to the Perdew-Burke-Ernzerhof parameterization as implemented in Vienna Ab initio Simulation Package [23]. In this work, we use the projector-augmented wave method [24] with an energy cutoff of 450 eV and a gamma-centered 7×7×3 k-point mesh grid. Besides, the experimental measured structure [18] and the´2 2 1supercell containing 4 Ir 4+ ions to accommodate the antiferromagnetic (AFM) state are adopted. To describe the electron correlation associated with the 5d electron of Ir 4+ ions, we use the rotationally invariant DFT+U scheme introduced by Liechtenstein et al [25]. The on-site Coulomb energy = U 2.5 eV and the Hund coupling = J 0.5 eV H [2,26,27] are applied to the Ir 5d electrons. Because Ir is a kind of heavy element and its relativistic spin-orbit coupling (SOC) ( ) l = 0.45 eV [28,29] is rather strong, SOC is taken into consideration in calculating the initial magnetic ground state, basal-plane MAE and the magnetic interactions parameters. The efficient exchange MC method is employed for finding the magnetic ground state [30,31].

Result and discussion
In this section, we first demonstrate that the basal-plane antiferromagnetism in BIO is caused by the intralayer magnetic interactions instead of the previously proposed interlayer ones [22]. We then prove the basal-plane MAE of BIO can be phenomenologically accounted for by adding the SIA and pseudo-quadrupole interactions into the general bilinear pseudo-spin Hamiltonian. Finally, we demonstrate the experimentally observed basalplane antiferromagnetism of BIO can be explained by the isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions.

DFT study on the basal-plane antiferromagnetism in BIO
Bulk BIO is a spin-orbit Mott insulator and has the quasi-two-dimensional lattice structure and tetragonal symmetry [18]. It consists of single-layer BIO perovskites and these single-layer BIO perovskites stack along the z-axis, i.e., the [001] direction (see figure 1(a)). There are two single-layer BIO perovskites in the unit cell of bulk BIO. In the tetragonal xy-plane, Ir 4+ ions are located at the center of the corner-sharing oxygen octahedrons which have a 7% tetragonal distortion along the z-axis [18]. The IrO 2 square-lattice planes are separated by two successive BaO layers and the Ir 4+ ions in the neighboring IrO 2 square-lattice planes are spatially separated as far as about 7.3 Å [18] (see figure 1(a)). Recent experiments show that the crystal field splitting associated with the tetragonal distortion of the IrO 6 octahedron is small so that BIO is a close realization of the spin-orbit Mott insulator [32].
Based on the DFT+U+SOC calculation, we first confirm the magnetic ground state of the bulk BIO to be the AFM state with the magnetic moment of Ir 4+ ions pointing along the [110] direction, which is in consistent with the experimental observation [18]. Here we take into account three representative AFM states: the AFM state with the magnetic moment of the Ir 4+ ions pointing along the [001] direction (denoted as AFM-001), along the [100] direction (denoted as AFM-100) and along the [110] direction (denoted as AFM-110) (see figure 1(b)).
Although the total energy of the AFM-100 state is lower by 1.7 meV/Ir than that of the AFM-001 state, it is slightly higher by 0.19 meV/Ir than that of the AFM-110 state. It is worth of noting that if the initial state is the FM state, it cannot be stabilized and the resulting state is the paramagnetic state and has a higher total energy than the AFM-001, AFM-100 and AFM-110 states.
Then we investigate in details the basal-plane MAE of the bulk BIO and find that the basal-plane MAE can be fitted to a formula which includes the four-order contribution. Since the AFM state cannot be accommodated by the unit cell of the bulk BIO which has only one Ir 4+ ion per IrO 2 square-lattice plane, we use the´2 2 1 supercell containing two Ir 4+ ions per IrO 2 square-lattice plane. To obtain the basal-plane MAE, we constrain the magnetic moments of the antiferromagnetically coupled Ir 4+ ions in the tetragonal xy-plane and rotate them by an angle f with respect to the [100] direction (see the inset of figure 1(c)). Figure 1(c) shows the dependence of the DFT+U+SOC calculated total energy on the rotated angle f. As is expected, the minimum of the total energy occurs at f =  45 , i.e., at the [110] direction. Because of the tetragonal symmetry, MAE can be approximated in the form of [33]  Note that the constant K 1 and K 2 in equation (1) are absorbed into the constant E . 0 Based on equation (2), the fitted MAE parameter K 3 is´-2.8 10 erg cm , 6 3 one order of magnitude larger than that of the body-center cubic iron [34].
To make clear the effect of the interlayer magnetic interactions on the basal-plane antiferromagnetism, we investigate the magnetic properties of the single-layer BIO. The lattice structure of the single-layer BIO is shown in the figure 2(a). It consists of one of the single-layer BIO perovskite and vacuum layers. The thickness of the vacuum region is 9.02 A. Because the Ir 4+ ions in the IrO 2 square-lattice planes are isolated by the thick enough vacuum layers, the interlayer magnetic interactions existing in the bulk BIO vanish in the single-layer BIO. Note that although the single-layer BIO (space group P4/mmm) has a slightly lower symmetry than the bulk BIO (space group I4/mmm), it preserves the vital tetragonal symmetry. Surprisingly, we find the interlayer magnetic interactions have no significant effect on the basal-plane antiferromagnetism, because the single-layer BIO almost reproduces the magnetic properties of the bulk BIO. First of all, the magnetic ground state of the singlelayer BIO is the AFM-110 state. Secondly, the AFM-100 state has the higher total energy than the AFM-110 state but has the lower total energy than the AFM-001 state. Most importantly, the single-layer BIO has almost the same basal-plane MAE as the bulk BIO. Using the same method applied to the calculation of the basal-plane MAE of the bulk BIO, we obtain the basal-plane MAE of the single-layer BIO and show it in the figure 2(b). By comparing the figures 1(c) and 2(b), one can come to a conclusion that the profile of the basal-plane MAE of the single-layer BIO is almost the same as that of the bulk BIO. Based on equation (2), the fitted MAE constant K 3 of the single-layer BIO is´-2.9 10 erg cm , 6 3 very close to that of the bulk BIO. Therefore, it is clearly demonstrated that the experimentally observed basal-plane antiferromagnetism is caused by the intralayer magnetic interactions instead of the interlayer ones. It also implies that the interlayer magnetic interactions are negligible compared with the intralayer ones, which will be discussed below (see table 1). Note that our result is distinctly different from the previous one which argued that interlayer magnetic interactions could have significant effect on the experimentally observed basal-plane antiferromagnetism [22].

Model Hamiltonian study on the basal-plane antiferromagnetism in BIO
Here we first recall the Hamiltonian employed by the previous studies on the iridium oxides [8,22,35,36]. The Ir 4+ ions yields an effective isospin Kramers-doublet ground states due to the strong octahedral crystal field and SOC [8,20,32]. The Kramers-doublet ground states are the pseudo-spin states |  ñ  and |  ñ  (denoted as S to be distinguished from the usual spin S) which are in the form of [8]: In equations (3.1) and (3.2), the angle a parameterizes the relative strength of the tetragonal crystal-field splitting and ( ) ( ) a l l = -D tan 2 2 2 (D and l are the tetragonal crystal-field splitting and SOC parameter, respectively). For a pair of pseudo-spins S i and S , j the general bilinear pseudo-spin Hamiltonian can be cast in the form of [8,22,35,36] In equation (4), r ij is the unit vector along the ij bond and g is anyone of the x, y and z. The first term is the isotropic Heisenberg interaction. The second term is the celebrated bond-dependent Kitaev interaction (see the [8] for the details). The tetragonal symmetry of BIO requires that only the J S S ij i z j z K term may not vanish [8], when the z-axis of the global coordinate system is set to be perpendicular to the IrO 2 square-lattice plane (see the global coordinate system plotted in the figure 1(a)). The third term is the Dzyaloshinskii-Moriya (DM) interactions but it vanishes in BIO. The last term is the pseudo-dipolar interaction. For convenience, hereafter, the magnitude of the pseudo-spin is absorbed into the magnetic interactions parameters J ,  We show here that the general pseudo-spin Hamiltonian defined by equation (4) cannot explain the calculated basal-plane MAE of the bulk BIO. Because the interlayer magnetic interactions are demonstrated to be negligible compared to the intralayer ones, they are left out of consideration here. We consider an arbitrary AFM state with the magnetic moments of the Ir 4+ ions pointing along the direction ( ) q f , in the´2 2 1 supercell and denote such state as ( ) q f AFM , . Firstly, the isotropic Heisenberg interaction has the SO(3) symmetry so it makes no contribution to the MAE. Secondly, the contribution from the nearest neighboring (NN) and next nearest neighboring (NNN) Kitaev interactions to the MAE is (see appendix A) In equation (5), J 1 K and J 2 K are the NN and NNN bond-dependent Kitaev interactions parameters, respectively. Thirdly, the contributions from the NN and NNN pseudo-dipole interactions to the MAE are (see the appendix A) In equation (6), J 1 DP and J 2 DP are the NN and NNN pseudo-dipole interactions parameters, respectively. Therefore, the contribution from the NN and NNN bond-dependent Kitaev and pseudo-dipole interactions to the MAE of an arbitrary ( ) q f AFM , state is only dependent on the azimuthal angle q and independent on the polar angle f. We can further demonstrate that even if the negligible bond-dependent Kitaev and pseudo-dipole interactions of the Ir-Ir pairs farther than the NNN Ir-Ir one are taken into consideration, their contribution to the MAE still only depends on the azimuthal angle q (see the equation (A.3) in the appendix A). In summary, one can conclude based on the general pseudo-spin Hamiltonian defined by equation (4) that the basal-plane AFM state ( ) q p = 2 is isotropic, which contradicts with the experimentally observed [110]-pointing AFM state [20].
In this work, we add the SIA and pseudo-quadrupole interaction into the general pseudo-spin Hamiltonian defined by equation (4). Although it is commonly believed that magnetic systems made up of transition-metal ions with one unpaired electron have no magnetic anisotropy arising from SOC [37], our recent work showed that most such transition-metal ions did have the SIA [38,39]. Because the pseudo-spin S state is an analogy to the spin state in the transition-metal ion with one unpaired electron, it is natural to generalize that the pseudospin S does have the SIA which is in the form of ( ) A S . z SIA 2 Note that the non-zero SIA has been pointed out to be present in Na 2 IrO 3 [40] and SIO [41] (See the theoretical and numerical demonstration and discussion of the SIA of the pseudo-spin state in the appendix B.) For the same reason, we further generalize that the quadrupolequadrupole coupling proposed by Van Vleck [42] can also exist in the pseudo-spin S systems: In equation (7), J ij QP and r ij are the pseudo-quadrupole interaction parameter and the unit vector along the ij bond, respectively. Now let us demonstrate the calculated basal-plane MAE can be well accounted for on the basis of the above model Hamiltonian. For an arbitrary ( ) q f AFM , state in the´2 2 1supercell, the contributions from the NN and NNN pseudo-quadrupole interactions to the MAE is (see the appendix A) In equation (8), J 1 QP and J 2 QP are the NN and NNN pseudo-quadrupole interaction parameters, respectively. Since the pseudo-quadrupole interaction is a kind of short-range interaction, it is reasonable that only the NN and NNN pseudo-quadrupole interactions are taken into account. Taking into consideration the isotropic Heisenberg, bond-dependent Kitaev, pseudo-dipole and pseudo-quadrupole interactions of the NN and NNN Ir-Ir pairs and the SIA, we obtain their contribution to the MAE of an arbitrary ( ) q f AFM , state as follows: sin sin sin sin cos . 9 The coefficients in equation (9) are which is the same as equation (2) and indicates that the basal-plane AFM state is anisotropic instead of isotropic. Furthermore, if the coefficient K 3 is negative, the energy of the basal-plane AFM state reaches its minimum when the magnetic moments of the Ir 4+ ions point along the [110] direction. So the experimentally observed basal-plane antiferromagnetism can be phenomenologically explained by adding the pseudo-quadrupole interactions and the SIA into the generalized pseudo-spin Hamiltonian defined by equation (4).
Therefore the model Hamiltonian which well describes the magnetic behavior of BIO should be in the form of In equation (10), the first and second terms describe the magnetic interactions of the NN and NNN Ir-Ir pairs in the tetragonal xy-plane, respectively. Note that the isotropic Heisenberg, bond-dependent Kitaev, pseudo-dipole and pseudo-quadrupole interactions are all taken into consideration for the NN and NNN Ir-Ir pairs in the tetragonal xy-plane. The last term is the SIA generalized in our present work. Because the distances of the third nearest neighboring (NNNN) Ir-Ir pairs in the tetragonal xy-plane and the interlayer Ir-Ir pairs are 8.1 Å and 7.3 Å, respectively, their Heisenberg interactions should be rather weak. This is confirmed by our calculations (see table 1). Compared to the isotropic Heisenberg interaction, their bond-dependent Kitaev, pseudo-dipole and pseudo-quadrupole interactions are much weaker and thereby left out of consideration hereafter. The considered magnetic interactions and SIA parameters are obtained based on the DFT+U+SOC calculations (see appendix C for the detailed calculation) and listed in the table 1. It is shown that the dominant magnetic interactions are the NN AFM and NNN FM Heisenberg interactions. So it is undoubted that the magnetic ground state should be AFM. Note that our calculated NN AFM Heisenberg interaction is well consistent with the previously calculated one [22,43]. Besides, there are several important points being worth of noting. Firstly, it is beyond expectation that the NN bond-dependent Kitaev interaction has a much smaller magnitude than the NNN bond-dependent Kitaev interaction. Actually, this is reasonable and understandable. The NN bond-dependent Kitaev interaction is realized through the 180°Ir-O-Ir bond so it should be rather weak [8], as the calculated one. However, it has been shown that there exists a large Kitaev interaction in slightly distorted 90°Ir-O-Ir bond of Na 2 IrO 3 [8,44]. Therefore it is expected that the four distorted Ir-O-Ir bonds (116.5°) facilitate and result in the unexpectedly large NNN Kitaev interaction (see the inset of figure 3(a)). Secondly, the pseudo-quadrupole interaction and the SIA are both strong, which is a manifestation of the strong SOC in iridium oxides. Finally, the NNNN and interlayer Heisenberg interactions are very weak, as expected. Note that the former and the latter are smaller by one order and by two orders than the NN Heisenberg interaction, respectively. An important reason for such weak interlayer Heisenberg interaction is that two BaO layers embed into the neighboring IrO 2 planes prevent the electron from hopping between the interlayer Ir 4+ ions. Because the NNNN and interlayer Heisenberg interactions are much weaker than the NN and NNN Heisenberg interactions, they will be left out of consideration in the below discussion.

S S S r r S S r r S S S S r r S S r r S
In order to make clear how the bond-dependent Kitaev, pseudo-dipole and pseudo-quadrupole interactions and the SIA affect the antiferromagnetism of BIO, we investigate the dependence of the energy differences while keeping other magnetic interaction parameters unchanged. The similar method is applied to calculate the dependence of ΔE1 and ΔE2 on the pseudo-dipole and pseudo-quadrupole interactions and the SIA parameters. Figure 3(a) shows the energy difference ΔE1 gets larger and larger as the bond-dependent Kitaev interaction becomes stronger and stronger. This indicates the bond-dependent Kitaev interaction does not favor the magnetic moments of the Ir 4+ ions to point along the z-axis. In the figures 3(b)-(d), the energy difference ΔE1 gets smaller and smaller as the corresponding pseudo-dipole, pseudo-quadrupole interactions and the SIA become stronger and stronger. This indicates that the pseudo-dipole, pseudo-quadrupole  Finally, we show that the experimentally observed AFM-110 state is caused by the isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions, based on the systematical MC simulations (see table 2). Because the isotropic Heisenberg interaction is the dominant magnetic interaction, it is always included in the MC simulations. However, the bond-dependent Kitaev, pseudo-dipole and pseudo-quadrupole interactions and the SIA are optionally included in the MC simulations. By comparing the Case 1 and 2 listed in the table 2, one can obtain that, for the antiferromagnetically coupled Ir 4+ ions, the bond-dependent Kitaev interaction makes their magnetic moments lie at the tetragonal xy-plane with the continuous degeneracy. However, Case 7 clearly shows the pseudo-quadrupole interaction can break the above-mentioned continuous degeneracy and the resulting magnetic structure is just the experimentally observed AFM-110 state. By carefully investigating the table 2, we can obtain that: (1) the magnetic ground state is the basal-plane AFM state with the magnetic moments of the Ir 4+ ions lying in the tetragonal xy-plane as long as the Heisenberg and Kitaev interactions are taken into account; (2) the magnetic ground state is the experimentally observed AFM-110 state as long as the isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions are taken into account; (3) although the pseudo-dipole, pseudo-quadrupole interactions, and the SIA favor the magnetic moments of the Ir 4+ ions to point along the z-axis (see the Case 3, 4 and 5 in the table 2), they cannot destroy the basal-plane AFM state established by the isotropic Heisenberg and bond-dependent Kitaev interactions. To sum up, it is clearly shown that the experimentally observed AFM-110 state can be explained by the isotropic Heisenberg, bond-dependent Kitaev and pseudo-quadrupole interactions.

Summary
In summary, we have studied the basal-plane antiferromagnetism in the novel spin-orbit Mott insulator Ba 2 IrO 4 . We show the basal-plane antiferromagnetism is caused by the intralayer magnetic interactions. We add the SIA and pseudo-quadrupole interaction into the general bilinear pseudo-spin Hamiltonian to unveil the origin of the basal-plane antiferromagnetism. We find that the SIA and pseudo-quadrupole interaction are unexpectedly strong. Based on the systematical MC simulations, we reveal the experimentally observed basalplane antiferromagnetism can be explained by the isotropic Heisenberg, bond-dependent Kitaev and pseudoquadrupole interactions. On the basis of this study the SIA and pseudo-quadrupole interactions could play a role in explaining magnetic interactions in other iridates.

Acknowledgments
This paper was partially supported by the National Natural Science Foundation of China, and the Research Program of Shanghai Municipality, the Ministry of Education, and Fok Ying Tung Education Foundation. We are grateful to Dr J H Yang, Y L Zhang for valuable discussions.
Appendix A. The dependence of MAE on the Kitaev, pseudo-dipole and pseudoquadrupole interactions In this appendix, we provide some details of the study on the MAE of the bond-dependent Kitaev, pseudo-dipole and pseudo-quadruple interactions. Here we consider an arbitrary ( ) q f AFM , state (see the definition in the main text) in the´2 2 1supercell. For a given Ir-Ir pair denoted as pair i, the magnetic moments of this Ir-Ir pair are coupled either antiferromagnetically or ferromagnetically. For example, the magnetic moments of the NN Ir-Ir pair are antiferromagnetically coupled while the magnetic moments of the NNN Ir-Ir pair are ferromagnetically coupled. The polar angle of a given Ir-Ir pair i with respect to the x-axis is denoted as j .
i For example, the polar angle j NN of the NN Ir-Ir pair is p 4. As a result of the four-fold rotation symmetry around the z-axis, there exist other three symmetrically equal Ir-Ir pairs with respect to a given Ir-Ir pair i. The polar angles of these four symmetrically equal Ir-Ir pairs are j , i j p + 2, i j p + i and j p + 3 2, i respectively. The contribution to the MAE from the Kitaev interaction of a given Ir-Ir pair i is In equation (A.1.0), J i K is the Kitaev interaction parameter and C i is ( ) -+ 1 1 if the given Ir-Ir pair is antiferromagnetically (ferromagnetically) coupled. Especially, the contributions from the NN and NNN Ir-Ir pairs are Table 2. Magnetic ground states obtained from systematical MC simulations in which different magnetic interactions are included. Plus sign (+) means the corresponding magnetic interaction is included in the MC simulations while the minus sign (−) means the corresponding magnetic interaction is not included in the MC simulations. The two-fold degeneracy of the AFM-001 state results from the time reversal invariance and the eight-fold degeneracy of the AFM-110 state results from the time reversal invariance and the four-fold rotation symmetry around the z-axis. respectively. Summing over all the Ir-Ir pairs farther than the NNN Ir-Ir pairs, we can obtain As for the contribution from the pseudo-dipole interaction of a given Ir-Ir pair i to the MAE, it is respectively. Therefore, the contribution from the pseudo-dipole interaction of the Ir-Ir pairs farther than the NNN Ir-Ir pairs is Based on the equations (A.1) and (A.2), we obtain the contribution from the Kitaev and pseudo-dipole interactions of the Ir-Ir pairs farther than the NNN Ir-Ir pairs to the MAE is As for the contribution from the pseudo-quadrupole interaction of the NNN Ir-Ir pair to the MAE, it is    states are equal, so the pure pseudo-spin J eff =1/2 states in the cubic crystal field have no SIA. Therefore the lack of SIA of the pseudo-spin J eff =1/2 states is protected by the cubic symmetry.
In the tetragonal crystal field, the Hamiltonian is read as In this case, the operators J , no matter the tetragonal distortion is elongation ( ) D > 0 or compression ( ) D < 0 along the z-axis, which indicates that the easy axis is the z-axis. Therefore the pseudo-spin » J 1 2 eff / state in Ba 2 IrO 4 with the tetragonal symmetry has SIA and its easy axis is the z-axis, which is consistent with our DFT calculated results.  Figure B2. The dependence on the constraint parameter a of the expectation 〈J x 〉 (read circle), 〈J z 〉 (read star) and energy difference ΔEx=E (J x )-E (J z =1/2) (blue circle), ΔEz=E (J z )-E (J z =1/2) (blue star) of the selected hole states based on the Hamiltonian equation (B.5) in the tetragonal crystal field splitting D < 0 (a) and D > 0 (b). The selected hole state is denoted by | ñ 6 . Here we set the tetragonal crystal field splitting D to be 50 meV according to the [32].