#  Exploration of Trivial and Non-trivial electronic phases, and Collinear and
Non-collinear Magnetic phases in low-spin d5 perovskites

Amit Chauhan  B. R. K. Nanda<EMAIL_ADDRESS>Condensed Matter Theory and
Computational Lab, Department of Physics, IIT Madaras, Chennai-36, India

###### Abstract

The $4d$ and $5d$ transition metal oxides have become important members of the
emerging quantum materials family due to competition between onsite Coulomb
repulsion ($U$) and spin-orbit coupling (SOC). Specifically, the systems with
$d^{5}$ electronic configuration in an octahedral environment are found to be
capable of posessing invariant semimetallic state and perturbations can lead
to diverse magnetic phases. In this work, by formulating a multi-band Hubbard
model and performing SOC tunable DFT+$U$ calculations on a prototype SrIrO3
and extending the analysis to other iso-structural and isovalent compounds, we
present eight possible electronic and magnetic configurations in the $U$-SOC
phase diagram that can be observed in the family of low-spin $d^{5}$
perovskites. They include the protected Dirac semimetal state, metal and
insulator regimes, collinear and noncollinear spin ordering. The latter is
explained through connecting hopping interactions to the rotation and tilting
of the octahedra as observed in GdFeO3. Presence of several soft phase
boundaries makes the family of $d^{5}$ perovskites an ideal platform to study
electronic and magnetic phase transitions under external stimuli.

††preprint: APS

##  I Introduction

In the past decade the discovery that spin-orbit coupling leads to the
formation of symmetry protected conducting surface or edge states in a wide
range of topological class of compounds, has attracted a great deal of
attention among the researchers working in quantum many-body theory Hasan and
Kane (2010); Qi and Zhang (2011). The SOC is a relativistic effect and it is
treated as a small perturbation in solids. But for systems with heavy
elements, it varies with the square of the atomic number and therefore it has
an intriguing influence on the electronic and magnetic structure of such
systems. In the family of correlated oxide systems, where the onsite Coulomb
repulsion with other competing interactions such as Hund’s coupling
($J_{\mathrm{H}}$), crystal field ($\Delta$) and electron hopping strength
($t$) create a rich electronic and magnetic phase diagram, the presence of SOC
bridges the symmetry protected topologically invariant states and magnetism.
Moving down the periodic table from $3d$ to $4d$ to $5d$ elements, while the
SOC increases, the $d$ orbitals become more spatially extended to reduce the
correlation strength $U$. For the $5d$ families of oxides, both SOC and $U$
becomes comparable and these two competing interactions give rise to emergent
phenomena.

For the experimenters and theoreticians alike, the family of iridates, one of
the $5d$ oxide families, have become a test bed for exploring these emergent
phenomena that includes topological Mott insulator Pesin and Balents (2010),
fractional Chern insulator Bergholtz and Liu (2013), Kitaev’s celebrated spin
liquid phase Baskaran  _et al._ (2007); Takagi  _et al._ (2019); Okamoto  _et
al._ (2007), Dirac and Weyl semimetals Yan and Felser (2017); Vafek and
Vishwanath (2014), Axion insulator Wan  _et al._ (2012), and
high-$T_{\mathrm{c}}$ superconductivity. For example, pyrochlore iridates
exhibit topological semimetal and Mott insulating phases Wan  _et al._ (2011);
Witczak-Krempa  _et al._ (2013); Topp  _et al._ (2018). The honeycomb iridates
host unconventional magnetic and spin liquid phases Rau  _et al._ (2014);
Chaloupka  _et al._ (2013); Reuther  _et al._ (2014); Takahashi  _et al._
(2019); Kenney  _et al._ (2019). The monolayer and bilayer Ruddlesden-Popper
strontium iridates, Sr2IrO4 and Sr3Ir2O7, exhibits canted in-plane and
collinear out-of-plane antiferromagnetic (AF) insulating state Nauman  _et
al._ (2017); Kim  _et al._ (2008); Watanabe  _et al._ (2013); Yan  _et al._
(2015); Fujiyama  _et al._ (2012).

Among the family of iridates, the orthorhomic pervoskite iridates with
chemical formula AIrO3, constitutes a large class of anisotropic oxides with
exceptionally wide range of properties. In these oxides the IrO6 octahedra is
distorted as compared to the symmetric cubic structure. These class of oxides
with Pbnm space group symmetry was proposed to realize a new class of metal
dubbed topologically crystalline metal with flat surface state and phase
transition to other non-trivial phases in the presence of external field has
been theoretically proposed Chen  _et al._ (2015). For example, SrIrO3 and
CaIrO3 are paramagntic Dirac semimetals (DSM) Nie  _et al._ (2015); Fujioka
_et al._ (2017, 2019, 2021). Carter et al. Carter  _et al._ (2012) showed that
SrIrO3 undergoes a phase transition from semimetal to strong topological
insulator by breaking the mirror symmetry. BaIrO3 is a Pauli paramagnetic
metal (PM) with Fermi liquid ground state below 6 K Cheng  _et al._ (2013). In
comparison to the AIrO3, the other $3d$ and $4d$ perovskites with isolvalent
$d^{5}$ configuration, where the SOC is weak and electron correlation is
strong, have different electronic and magnetic phases in their ground state.
For example, SrCoO3 and CaCoO3 exhibits high-spin (HS) ferromagnetic metal
(FM) state whereas SrRhO3 and YRuO3 exhibits low-spin (LS) FM and canted
antiferromagnetic insulator (CAFI) ground state Xia  _et al._ (2017); Mathi
Jaya  _et al._ (1991); Singh (2003); Ji  _et al._ (2020).

For the perovskites, due to strong octahedral crystal field the $d$ orbital
degeneracy gets lifted giving rise to $t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$
manifold. In the presence of SOC the $t_{\mathrm{2g}}$ manifold splits into
spin-orbital entangled pseudo-spin $J_{\mathrm{eff}}$ = 3/2 and
$J_{\mathrm{eff}}$ = 1/2 states. In the case of a $d^{5}$ (Ir4+)
configuration, the $J_{\mathrm{eff}}$ = 3/2 states are completely occupied and
the ground state is constituted of the $J_{\mathrm{eff}}$ = 1/2 pseudo-spin.
These narrow $J_{\mathrm{eff}}$ = 1/2 bands formed due to strong SOC are more
susceptible to Mott localization by $U$, due to reduced bandwidth, which
implies that stronger the SOC is, weak critical $U$ ($U_{\mathrm{c}}$) will be
required to make a transition from metal to a spin-orbit coupled Mott
insulator. By carrying out first-principles density functional theory (DFT)
calculations within the framework of LDA+$U$ on the prototype system SrIrO3,
Zeb et al. Zeb and Kee (2012) presented an electronic phase diagram in the
$U$-SOC configuration space, and broadly suggested three domains. These are
magnetic metal in the weak to intermediate SOC and weak to high $U$ regime,
nonmagnetic metal (NM) or DSM in the weak to intermediate $U$ and weak to high
SOC regime, and magnetic insulator in the weak to high SOC and intermediate to
high $U$ regime.

The aforementioned discussion on the electronic structure of orthorhombic
perovskites suggest that a much more intricate phase diagram needs to be
constructed for the family of $d^{5}$ oxides in the $U$-SOC space in order to
explore the emerging quantum phases in this family. Firstly, it needs to be
revisited if in the absence of SOC, the system remains metallic even for very
high $U$. Secondly, in the absence of collinear spin-arrangement questions
arise whether sub-domains with non-trivial quantum states exist within the
four proposed broad domains. Thirdly, the role of structural distortions -
from the isotropic cubic configuration to symmetry lowered orthorhombic
configuration - needs to be examined in building the phase diagram as it is
known to be a key player in driving the electronic and magnetic phase
transitions in the perovskite family Torrance  _et al._ (1992); García-Muñoz
_et al._ (1992); Miyasaka  _et al._ (2003); Zhou  _et al._ (2005, 2010); Vogt
_et al._ (2003); Kim  _et al._ (2017); Liu  _et al._ (2015).

In this work we have employed a multi-band Hubbard model which is solved self-
consistently in the mean-field framework and validated the results with
DFT+$U$ calculations to establish a detailed phase diagram discovering the
earlier unexplored sub-domains. The present study establishes the complex
interplay among SOC, $U$, $\Delta$ and structural distortions to explain the
formation of collinear and non-collinear magnetism as well as metallic,
insulating, and nontrivial semiemtallic phases in the family of LS $d^{5}$
perovskites.

##  II Structural and Computational Details

Bulk SrIrO3 crystallizes in an orthorhombic pervoskite crystal structure
(space group Pbnm) with the GdFeO3-type lattice distortion as shown in Fig.
1(a). The IrO6 octahedra undergoes a staggered rotation about the $c$ axis by
an angle $\theta_{\mathrm{r}}$ = ${153.53}^{\circ}$, followed by another
rotation about the [110] direction by tilt angle $\theta_{\mathrm{t}}$ =
${156.52}^{\circ}$. Due to these tilt and rotations of the IrO6 octahedra, the
unit cell gets doubled in the ab as well as in the ac plane due to
$\sqrt{2}a_{\mathrm{0}}\times\sqrt{2}a_{\mathrm{0}}\times 2a_{\mathrm{0}}$
supercell geometry where $a_{\mathrm{0}}$ is the nearest-neighbour Ir-Ir
distance. Hence, SrIrO3 contains four formula units per unit cell leading to
four inequivalent Ir sublattices (A,B,C,D). The two signs on each Ir
sublattice represents the sense of the rotation and tilting of the octahedra,
clockwise (+) or anticlockwise (-). The experimental lattice parameters are
$a$ = 5.56 Å, $b$ = 5.59 Å and $c$ = 7.88 Å Zhao  _et al._ (2008).

Figure 1: (a) Crystal structure of orthorhombic SrIrO3.(b) Corresponding bulk
Brillouin zone with high symmetric k-points. (c) Bulk electronic structure of
SrIrO3. The Fermi energy is set to be 0. A Dirac node appears at the high
symmetric k-point U revealing the semimetallic nature of SrIrO3. (d) Schematic
illustration of the effect of crystal field, SOC and onsite Coulomb repulsion
on the Ir-$d$ states.

In order to study the effect of correlation and SOC on the electronic and
magnetic phases in bulk SrIrO3, we developed a multi-band Hubbard model and
solved it self-consistently as discussed in Sec. IV and also carried out
comprehensive DFT calculations in the $U$-SOC space. The calculations are
performed using plane-wave based projector augmented wave method (PAW) Kresse
and Joubert (1999); Blöchl (1994) as implemented in Vienna ab-initio
simulation package (VASP) Kresse and Furthmüller (1996) within the
Perdew$-$Burke$-$Ernzerhof generalized gradient approximation (GGA) for
exchange correlation functional. In this study, the SOC $\lambda$ is varied in
units of $\lambda_{\mathrm{0}}$, where $\lambda_{\mathrm{0}}$ is the real SOC
strength (= 0.43 eV) as obtained from the SCF calculations to achieve the
ground state. The Brillouin zone integrations are carried out using $8\times
8\times 4$ Monkhorost-Pack k-mesh which yields 256 k-points in the irreducible
part of the Brilluion zone. The kinetic energy cutoff for plane-wave basis set
was chosen to be $500$ eV. The strong correlation effect is incorporated via
an effective onsite correlation parameter $U_{\mathrm{eff}}$ = $U-J$ through
the rotationally invariant approach introduced by Dudarev Dudarev  _et al._
(1998).

##  III Ground State Electronic Structure of SrIrO3

The ground state electronic structure, as shown in Fig. 1(c), implies that
bulk SrIrO3 exhibits paramagnetic semimetallic ground state with a Dirac node
at the high symmetry point U. This is due to the combined effect of cubic
asymmetry, octahedral crystal field and SOC towards the removal of degeneracy
in the $d$ manifold as illustrated in Fig. 1(d). Due to strong octahedral
crystal field, the five-fold degenerate $d$ states split into a higher-energy
and unoccupied $e_{\mathrm{g}}$ doublet and a lower-energy $t_{\mathrm{2g}}$
triplet. While the $e_{\mathrm{g}}$ states are unaffected by the SOC, the
$t_{\mathrm{2g}}$ manifold is further SOC split into a $J_{\mathrm{eff}}$ =
1/2 doublet ($m_{\mathrm{j}}$ = $\pm$ 1/2) and a $J_{\mathrm{eff}}$ = 3/2
($m_{\mathrm{j}}$ = $\pm$ 3/2, $\pm$ 1/2) quartet, hence, forming three
Kramers pair. The expressions $\ket{J,m_{\mathrm{j}}}$ are given by,

$\displaystyle\ket{\frac{1}{2},\pm{\frac{1}{2}}}$
$\displaystyle=\frac{1}{\sqrt{3}}(\ket{yz,\bar{\sigma}}\pm{\ket{xy,\sigma}}\pm{i}\ket{xz,\bar{\sigma}})$
(1) $\displaystyle\ket{\frac{3}{2},\pm{\frac{1}{2}}}$
$\displaystyle=\frac{1}{\sqrt{6}}(\ket{yz,\bar{\sigma}}\mp
2{\ket{xy,\sigma}}\pm{i}\ket{xz,\bar{\sigma}})$ (2)
$\displaystyle\ket{\frac{3}{2},\pm{\frac{3}{2}}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{yz,\sigma}\pm{i}\ket{xz,{\sigma}})$ (3)

where $\pm$ corresponds to spin $\sigma$ = $\uparrow$/$\downarrow$,
respectively. With lowering in symmetry through $\theta_{\mathrm{r}}$ and
$\theta_{\mathrm{t}}$, two pairs of bands make linear crossing at the high
symmetry point U to create double Dirac nodes resulting in a DSM phase. It has
been reported that the two Dirac nodes may not be degenerate in energy and in
that case the cones may penetrate to form a nodal ring Zeb and Kee (2012).
Through our model Hamiltonian in the Sec. IV, we will attribute the formation
of the DSM phase due to the development of $t_{\mathrm{2g}}$-$e_{\mathrm{g}}$
interaction in the next-nearest neighborhood due to tilting and rotation of
octahedra. Besides the crystal field and SOC, the onsite Coulomb repulsion is
also deterministic of the electronic structure of the $d^{5}$ (Ir4+)
compounds. Four out of five $d$ electrons occupy the $J_{\mathrm{eff}}$ = 3/2
state, leaving one electron in the $J_{\mathrm{eff}}=1/2$ state (see Fig.
1(d)). The onsite repulsion further splits this degenerate state to create a
lower Hubbard (LHB) and upper Hubbard (UHB) subband. However, experimental
studies Longo  _et al._ (1971); Nie  _et al._ (2015); Fujioka  _et al._ (2017)
including ARPES measurements show that the SrIrO3 is a paramagnetic semimetal.
This demonstrates that the competition between SOC and $U$ can lead to a
plethora of electronic and magnetic phases for the $d^{5}$ perovskites.

Apart from the perovskite structure, several other iridates and iridate like
systems with $d^{5}$ configuration and with similar octahedral complexes,
exhibit wide range of magnetic phases with spin anisotropy Modic  _et al._
(2014); Nauman  _et al._ (2017); Boseggia  _et al._ (2013); Chaloupka and
Khaliullin (2016); Biffin  _et al._ (2014); Reuther  _et al._ (2014).
Therefore, we utilize SrIrO3 as prototype to build a platform so that the
emerging quantum phases in the $U$-SOC configuration space can be envisaged
and analyzed. The phase diagram will be discussed in detail in the Sec. V.1.

##  IV Multi-Band Hubbard Model

Across the iridates family, it has been shown that the competition between $U$
and $\lambda$ influences the eigen states substantially. To get a flavour of
it in the single pervoskite structure, we employ a mean-field based multi-band
Hubbard model Hamiltonian to explore the non-trivial/trivial phases emerging
in the weak/strong $U$-SOC regime. To start with, we first consider tight
binding (TB) model with SOC, where a minimal basis set formed by Ir-$d$
orbitals ($xy$,$yz$,$xz$,$x^{2}-y^{2}$,$3z^{2}-r^{2}$) has been considered.
The five orbital basis set, instead of $t_{\mathrm{2g}}$ based three orbital,
introduces $t_{\mathrm{2g}}$-$e_{\mathrm{g}}$ intermixing due to finite
$\theta_{\mathrm{r}}$ and $\theta_{\mathrm{t}}$ in the distorted frame leading
to significant altering of the band structure.

To extract the nearest-neighbour (NN), next-nearest-neighbour (NNN) $\sigma$
and $\pi$ hopping interactions and SOC strength, a TB model for undistorted
SrIrO3 is formulated and then using these parameters and transforming the
hopping matrices in the rotated basis (as explained in Appendix A), the effect
of distortion on the electronic and magnetic properties has been examined. In
the second quantization notation, the TB + SOC component of the Hamiltonian is
given by,

$\displaystyle H_{\mathrm{TB-SOC}}$ $\displaystyle=$
$\displaystyle\sum_{i,\alpha}\epsilon_{i,\alpha}c_{i,\alpha}^{\dagger}c_{i,\alpha}+$
(4)
$\displaystyle\sum_{i,j,\alpha,\beta,\sigma}t_{i,\alpha,j,\beta}(c_{i,\sigma,\alpha}^{\dagger}c_{j,\sigma,\beta}+h.c.)$
$\displaystyle+\lambda\sum_{\alpha,\beta,\sigma,\bar{\sigma}}\bra{\alpha\sigma}\bm{L}\cdot\bm{S}\ket{\beta\bar{\sigma}}c_{\alpha,\sigma}^{\dagger}c_{\beta,\bar{\sigma}}$

Here, i(j), $\alpha$($\beta$) are site and orbital indices, respectively. The
parameters $\epsilon_{\mathrm{i\alpha}}$ and $t_{\mathrm{i\alpha j\beta}}$
represents the onsite energy and strength of hopping integrals, respectively.
The SOC is added in the third term of the Hamiltonian with $\lambda$ denoting
its strength. In compact form, for a single formula unit as is the case with
cubic perovskite structure, H is given by

$H=\begin{pmatrix}H_{\uparrow\uparrow}^{5\times
5}&H_{\uparrow\downarrow}^{5\times 5}\\\\[5.69046pt]
H_{\downarrow\uparrow}^{5\times 5}&H_{\downarrow\downarrow}^{5\times
5}\end{pmatrix}$ (5)

Here, H↑↑ = H↓↓ and H↑↓ = (H${}_{\downarrow\uparrow})^{\dagger}$ to ensure the
time reversal (TR) invariance of the Hamiltonian. For the four formula unit
($\sqrt{2}a_{\mathrm{0}}\times\sqrt{2}a_{\mathrm{0}}\times 2a_{\mathrm{0}}$)
supercell, which build the primitive unit cell of the orthorhombioc phase, the
augmented Hamiltonian representing four-Ir (A,B,C,D) sublattices then takes
the shape of

$H=\begin{pmatrix}H_{\uparrow\uparrow}^{20\times
20}&H_{\uparrow\downarrow}^{20\times 20}\\\\[5.69046pt]
H_{\downarrow\uparrow}^{20\times 20}&H_{\downarrow\downarrow}^{20\times
20}\end{pmatrix}$ (6)

The TB component built with Slater-Koster formalism Slater and Koster (1954)
is further elaborated in the Appendix A.

For the distorted case, with finite $\theta_{\mathrm{r}}$ and
$\theta_{\mathrm{t}}$, as defined through Fig. 1(a), the hopping matrices are
obtained by using the following transformation

$\displaystyle\tilde{H}_{\mathrm{TB}}$ $\displaystyle=R^{T}H_{\mathrm{TB}}R$
(7)

where R is the transformation matrix and can be calculated using rotation
matrix for the cubic harmonics (L = 2) (Eq. A3 of Appendix A). For the
rotation and tilting, as appropriate for SrIrO3, it is respectively defined as

$R(\theta_{\mathrm{t}})=\begin{pmatrix}\frac{1}{2}(1+\cos^{2}{\theta_{\mathrm{t}}})&\frac{1}{2\sqrt{2}}\sin{2\theta_{\mathrm{t}}}&-\frac{1}{2\sqrt{2}}\sin{2\theta_{\mathrm{t}}}&0&-\sqrt{\frac{3}{4}}\sin^{2}{\theta_{\mathrm{t}}}\\\\[5.69046pt]
-\frac{1}{2\sqrt{2}}\sin{2\theta_{\mathrm{t}}}&\frac{1}{2}(2\cos^{2}\theta_{\mathrm{t}}+\cos{\theta_{\mathrm{t}}}-1)&-\frac{1}{2}(2\cos^{2}{\theta_{\mathrm{t}}}-\cos{\theta_{\mathrm{t}}}-1)&-\frac{1}{\sqrt{2}}\sin{\theta_{\mathrm{t}}}&-\sqrt{\frac{3}{8}}\sin{2\theta_{\mathrm{t}}}\\\\[5.69046pt]
\frac{1}{2\sqrt{2}}\sin{2\theta_{\mathrm{t}}}&-\frac{1}{2}(2\cos^{2}{\theta_{\mathrm{t}}}-\cos{\theta_{\mathrm{t}}}-1)&\frac{1}{2}(2\cos^{2}{\theta_{\mathrm{t}}}+\cos{\theta_{\mathrm{t}}}-1)&-\frac{1}{\sqrt{2}}\sin{\theta_{\mathrm{t}}}&\sqrt{\frac{3}{8}}\sin{2\theta_{\mathrm{t}}}\\\\[5.69046pt]
0&\frac{1}{\sqrt{2}}\sin{\theta_{\mathrm{t}}}&\frac{1}{\sqrt{2}}\sin{\theta_{\mathrm{t}}}&\cos{\theta_{\mathrm{t}}}&0\\\\[5.69046pt]
-\sqrt{\frac{3}{4}}\sin^{2}{\theta_{\mathrm{t}}}&\sqrt{\frac{3}{8}}\sin{2\theta_{\mathrm{t}}}&-\sqrt{\frac{3}{8}}\sin{2\theta_{\mathrm{t}}}&0&1-\frac{3}{2}\sin^{2}{\theta_{\mathrm{t}}}\end{pmatrix}$
(8)

$R(\theta_{\mathrm{r}})=\begin{pmatrix}\cos{2\theta_{\mathrm{r}}}&0&0&\sin{2\theta_{\mathrm{r}}}&0\\\\[5.69046pt]
0&\cos{\theta_{\mathrm{r}}}&\sin{\theta_{\mathrm{r}}}&0&0\\\\[5.69046pt]
0&-\sin{\theta_{\mathrm{r}}}&\cos{\theta_{\mathrm{r}}}&0&0\\\\[5.69046pt]
-\sin{2\theta_{\mathrm{r}}}&0&0&\cos{2\theta_{\mathrm{r}}}&0\\\\[5.69046pt]
0&0&0&0&1\\\\[5.69046pt] \end{pmatrix}$ (9)

The values of onsite energy ($\epsilon$), hopping strength ($t$) and SOC
strength ($\lambda_{\mathrm{0}}$) are obtained by fitting the TB+SOC bands
with the DFT+SOC bands for the undistorted case as shown in Figs. 2 (a,b). The
fitted values are listed in Table 1. TB bands, as shown in Fig. 2, captures
very well the essential features of the DFT band structure. In this LS
($t_{\mathrm{2g}}^{5}e_{\mathrm{g}}^{0}$) configuration, while the overlapping
of the $t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$ bands is prominent in the
undistorted frame (Figs. 2(a,b)), they are very well segregated in the
distorted frame (Fig. 2(c)) and this is very well captured in our TB model
(Fig. 2(d)). The distortion also introduces a DSM phase as linear bands cross
each other at the high symmetry point U in the vicinity of the Fermi energy
($E_{\mathrm{F}}$).

Figure 2: DFT (blue) and TB (red) band structures for undistorted (a,b) and distorted (c,d) SrIrO3 with $\lambda$ = 0.43 eV, respectively. Segregation of $t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$ states, as well as the formation of DSM phase, due to distortion of the octahedra. Table 1: Calculated values of onsite energy, interaction parameters and SOC strength in units of eV. The parameters $\epsilon_{\mathrm{1}}$ and $\epsilon_{\mathrm{2}}$ give the values of onsite energy for $t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$ states, respectively. For the undistorted case, we have considered identical hopping strengths, $t_{\mathrm{1}}$ (NN $\pi$), $t_{\mathrm{2}}$ (NNN $\sigma$) and $t_{\mathrm{3}}$ (NNN $\pi$) for $t_{\mathrm{2g}}$ \- $t_{\mathrm{2g}}$ interactions, $t_{\mathrm{4}}$ (NN $\sigma$), $t_{\mathrm{5}}$ (NNN $\sigma$), $t_{\mathrm{6}}$ (NNN $\pi$) for $e_{\mathrm{g}}$-$e_{\mathrm{g}}$ interactions and $t_{\mathrm{7}}$ (NNN $\sigma$) for $t_{\mathrm{2g}}$-$e_{\mathrm{g}}$ interactions, as relevant for the cubic symmetry. $\epsilon_{\mathrm{1}}$ | $\epsilon_{\mathrm{2}}$ | $t_{\mathrm{1}}$ | $t_{\mathrm{2}}$ | $t_{\mathrm{3}}$ | $t_{\mathrm{4}}$ | $t_{\mathrm{5}}$ | $t_{\mathrm{6}}$ | $t_{\mathrm{7}}$ | $\lambda$
---|---|---|---|---|---|---|---|---|---
-0.79 | 2.4 | -0.38 | -0.13 | 0.04 | -0.85 | -0.5 | 0.1 | 0.04 | 0.43

Having formulated the kinetic part of the Hamiltonian, we now consider the
interacting part of the Hamiltonian ($H_{\mathrm{int}}$) which is described in
terms of the multi-orbital Hubbard-Kanamori formalism Luo (2013).

$\begin{split}H_{\mathrm{int}}=U\sum_{i,\alpha}n_{\mathrm{i,\alpha,\uparrow}}n_{\mathrm{i,\alpha,\downarrow}}+(U^{\prime}-\frac{J_{\mathrm{H}}}{2})\sum_{i,\alpha<\beta}n_{\mathrm{i,\alpha}}n_{\mathrm{i,\beta}}\\\
-2J_{\mathrm{H}}\sum_{i,\alpha<\beta}S^{z}_{\mathrm{i,\alpha}}\cdot
S^{z}_{\mathrm{i,\beta}}=H_{1}+H_{2}+H_{3}\end{split}$ (10)

Here, the first term two terms gives the energy cost of having the electrons
in the same or different orbitals at the same lattice site. The third term
gives Hund’s rule coupling that favours the ferromagnetic alignment of spins
in the orbitals at the same lattice site. The relation $U^{\prime}$ = $U$ \-
$2J_{\mathrm{H}}$ between the Kanamori parameters Kanamori (1963) has been
used here.

In the Hartree approximation, $H_{\mathrm{1}}$, $H_{\mathrm{2}}$, and
$H_{\mathrm{3}}$ can be decoupled as

$\displaystyle H_{1}\approx
U\sum_{i,\alpha}(n_{\mathrm{i,\alpha,\uparrow}}\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle+n_{\mathrm{i,\alpha,\downarrow}}\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle$ $\displaystyle-\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle)$ (11)

$\displaystyle H_{2}\approx
U\sum_{i,\alpha<\beta}(n_{\mathrm{i,\alpha}}\langle
n_{\mathrm{i,\beta}}\rangle+n_{\mathrm{i,\beta}}\langle
n_{\mathrm{i,\alpha}}\rangle-\langle n_{\mathrm{i,\alpha}}\rangle\langle
n_{\mathrm{i,\beta}}\rangle)$ (12)

$\begin{split}H_{3}\approx-\frac{J_{H}}{2}\sum_{i,\alpha<\beta}(n_{\mathrm{i,\alpha,\uparrow}}\langle
n_{\mathrm{i,\beta,\uparrow}}\rangle+n_{\mathrm{i,\beta,\uparrow}}\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle\\\
-n_{\mathrm{i,\alpha,\uparrow}}\langle n_{\mathrm{i,\beta,\downarrow}}\rangle-
n_{\mathrm{i,\beta,\downarrow}}\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle\\\
-n_{\mathrm{i,\alpha,\downarrow}}\langle n_{\mathrm{i,\beta,\uparrow}}\rangle-
n_{\mathrm{i,\beta,\uparrow}}\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle\\\
+n_{\mathrm{i,\alpha,\downarrow}}\langle
n_{\mathrm{i,\beta,\downarrow}}\rangle+n_{\mathrm{i,\beta,\downarrow}}\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle\\\ -\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle\langle
n_{\mathrm{i,\beta,\uparrow}}\rangle+\langle
n_{\mathrm{i,\alpha,\uparrow}}\rangle\langle
n_{\mathrm{i,\beta,\downarrow}}\rangle\\\ +\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle\langle
n_{\mathrm{i,\beta,\uparrow}}\rangle-\langle
n_{\mathrm{i,\beta,\downarrow}}\rangle\langle
n_{\mathrm{i,\alpha,\downarrow}}\rangle)\end{split}$ (13)

where, $n_{\mathrm{i,\alpha}}$ = $n_{\mathrm{i,\alpha,\uparrow}}$ \+
$n_{\mathrm{i,\alpha,\downarrow}}$, is the total charge density of orbital
$\alpha$ at site i.

Hence, the total Hamiltonian is given by

$\displaystyle H=H_{\mathrm{TB-SOC}}+H_{\mathrm{int}}.$ (14)

It is solved self-consistently in the momentum space by employing Hartree
approximation as described above. The $U$-SOC space give rise to different
magnetic phases, as shown in Fig. 3, beyond the semimetallic paramagnetic
phase. We set $J_{\mathrm{H}}$ = 3$t$ and $t$ = $t_{\mathrm{1}}$ as the energy
scale unit. For $\lambda$ = 0, a FM phase (see Fig. 3(a)), with highly
dispersive bands (higher mobility) stabilizes. For even higher $U$ values, a
FM state remains stable for the undistorted case.

Figure 3: Electronic band structures obtained for the undistorted structure
(first column) and for the distorted structure (second coloumn) with $t$ =
0.38 eV defined as the energy scale unit. (a,b) without SOC and high $U$,
(c,d) with SOC and low $U$, and (e,f) with SOC and moderate $U$ value.

With SOC, the system shows intriguing magnetic phases where the
$J_{\mathrm{eff}}$ = 1/2 states determine the ground state. For lower $U$
values, a weak G-type antiferromagnetic metal (G-AFM) state (Fig. 3(c)) with
broken Dirac node forms the ground state. It is due to breaking of the TR
symmetry. Here, the SOC induces the spin degeneracy and strengthen the
localization and hence decreases the mobility. The weak G-AFM state has small
hole and electron pockets which disappears on further increasing $U$ and a gap
is opened in the $J_{\mathrm{eff}}$ = 1/2 spectrum to stabilize a G-type
antiferromagnetic insulator (G-AFI) phase (Fig. 3(d)). Finite distortion
induces anisotropy in the orbital ($t_{\mathrm{2g}}$) occupancies with onsite
Coulomb repulsion enhances the spin split and drives the system to a
ferromagnetic insulator (FI) phase (Fig. 3(b)). Hence, competition between
$U$, SOC and structural distortion alters the magnetic and electronic phase of
the system substantially. Recent studies on iridates and their superlattices
emphasize the crucial role of distortion induced anisotropic spin interactions
leading to non-collinear magnetic phases Mohapatra  _et al._ (2019); Kim  _et
al._ (2017); Liu  _et al._ (2015). However, the current model is based on
collinear magnetism and therefore cannot predict non-collinear phases due to
the absence of exchange terms in the Hamiltonian. Therefore, we carried out
all electron DFT calculations to explore the novel electronic and magnetic
states in the $U$-SOC domain and build the quantum phase diagram of $d^{5}$
pervoskites in the LS state by taking SrIrO3 as a prototype.

##  V DFT+$U$+SOC Electronic Structure

Through Fig. 4, we will analyze the electronic structure evolution as a
function of $U$ and SOC. In the ground state, we estimated the SOC strength
$\lambda_{\mathrm{0}}$ to be 0.43 eV by measuring the split between
$J_{\mathrm{eff}}$ = 1/2 and $J_{\mathrm{eff}}$ = 3/2 states which is equal to
3/2 $\lambda_{\mathrm{0}}$ (see Fig. 1(d)). In order to examine the role of
$\lambda$, we scaled it in units of $\lambda_{0}$. For $\lambda/\lambda_{0}$ =
$U$ = 0, it is pure octahedral crystal field effect where the $d$ orbitals
split into lower-lying and partially occupied three-fold degenerate
$t_{\mathrm{2g}}$ and upper-lying two-fold degenerate $e_{\mathrm{g}}$ states
as shown in Fig. 4(a,b). Due to imbalance in the population of the states in
two different spin channels the system becomes a Stoner ferromagnet. As
$\lambda/\lambda_{0}$ increases, real spin states of the $t_{\mathrm{2g}}$
manifold evolve and give rise to pseudo-spin states with the formation of
spin-orbit entangled $J_{\mathrm{eff}}$ = 1/2 states lying in the vicinity of
the $E_{\mathrm{F}}$ and $J_{\mathrm{eff}}$ = 3/2 state lying below in the
valence band. For weak $\lambda/\lambda_{0}$, the four $J_{\mathrm{eff}}$ =
1/2 pairs, corresponding to the four-Ir sublattices, create two set of four-
fold degenerate bands along the k-path U-R. These two sets merge at the high
symmetry point U with increasing SOC to form a DSM phase (see Figs. 4(e,g,i)).

With increasing correlation effect, say for $U$ = 3 eV and no SOC, as shown in
Figs. 4(c,d), the onsite repulsion $U$n↑n↓ increases the spin split to form a
gap which in turn makes the system a FI. This phase is very well captured by
the model Hamiltonian described in the previous section. As SOC competes with
$U$, Figs. 4(f,h,j), the four $J_{\mathrm{eff}}$ = 1/2 pairs create lower and
upper Hubbard subbands, which breaks the Dirac node to form a gap and
concurrently stabilizes the system in a CAFI state. The band gap rises from
0.23 eV at $\lambda$/$\lambda_{\mathrm{0}}=0$ to 0.93 eV at
$\lambda$/$\lambda_{\mathrm{0}}$ = 2, manifesting the amplified effect of
correlations as the SOC strength increases. The spin anisotropy and the
overall phases predicted in the $U$-SOC domain will be elaborated further in
the next subsection.

Figure 4: Evolution of bulk electronic structure as a function of $U$ and SOC.
The upper and lower rows represents the band structure without and with $U$
for four values of $\lambda$/$\lambda_{0}$, namely, 0 (a - d), 0.6 (e, f), 1
(g, h), and 2 (i, j). (Upper row) Mixing of up-spin and down-spin states with
increasing $\lambda$/$\lambda_{0}$ leading to the formation of DSM phase.
(Lower row) Amplified effect of correlations with increasing
$\lambda$/$\lambda_{0}$. Enhanced spin splitting leads to a gap opening and
stabilize the system in a ferromagnetic insulator state for $\lambda$ = 0 and
in a canted antiferromagnetic insulator state for finite
$\lambda$/$\lambda_{0}$.

###  V.1 BULK PHASE DIAGRAM

The phase diagram for the orthorhombic SrIrO3 spanned in $U$-SOC space, as
evaluated from the DFT+$U$+SOC calculations is presented in Fig. 5. It shows
five distinct phases: (i) collinear FM as shown in blue, (ii) collinear FI as
shown in maroon, (iii) canted antiferromagnet metal (CAFM) as shown in grey,
(iv) CAFI as shown in red, and (v) DSM as shown in green. Let us first examine
the weak SOC regime ($\lambda$/$\lambda_{o}$ $\leq$ 0.2). For U $\approx$
$2.5$ eV, the system remains a FM with collinear magnetic ordering as shown in
Fig. 6(a).

Figure 5: (Top row) The electronic phase diagram of orthorhombic SrIrO3 as a
function of $U$ and SOC. (Middle and bottom rows) The planar and normal
component of spin moment ($S_{\mathrm{p}}$ and $S_{\mathrm{z}}$) and orbital
moment ($L_{\mathrm{p}}$ and $L_{\mathrm{z}}$) respectively. The star-marks
represent the ground state of LS $d^{5}$ pervoskites. The green star-mark
represent CaIrO3, the blue star-mark represent LaRuO3 and SrRhO3, and the red
star-mark represent YRuO3, respectively.

This results from the imbalance of the population of partially occupied
$t_{\mathrm{2g}}$ states in the up-spin and down-spin channels as already
explained through Figs. 4(a,b) in the previous subsection. With further
increase in $U$, the system undergoes a transition from FM to FI because of
the fact that with enhanced spin split the earlier partially occupied band in
the up-spin channel is now occupied completely whereas in the down spin
channel it becomes empty (see Fig. 4(c,d)). The FI phase, as predicted from
our model and DFT calculations, was not captured in the earlier study Zeb and
Kee (2012) where the stability of the metallic phase for all $U$ in the non-
SOC regime was reported. The strength of local magnetic moment at Ir depends
on U. It increases from $0.27$ $\mu_{\text{B}}$ at U = $0$ eV to $0.72$
$\mu_{\text{B}}$ at U = $4$ eV. With the LS state, the saturated magnetic
moment is 1 $\mu_{\text{B}}$ which can be realized for further higher value of
$U$. The LS state favours the ferromagnetic order over the antiferrromagnetic
one.

Figure 6: Schematic illustrating evolution of magnetic ordering in bulk SrIrO3
at $U$ = 4 eV with (a) $\lambda$/$\lambda_{0}$ = 0, (b)
$\lambda$/$\lambda_{0}$ = 1, and (c) $\lambda$/$\lambda_{0}$ = 2,
respectively.

With finite $\lambda/\lambda_{0}$, the up-spin and down-spin states mix to
form spin-orbit entangled pseudo-spin $J_{\mathrm{eff}}$ = 1/2 and
$J_{\mathrm{eff}}$ = 3/2 states. The former occupy the $E_{\mathrm{F}}$ and
hence determine the electronic and magnetic phase of the system. For the
intermediate range (0.4 $\leq$ $\lambda$/$\lambda_{0}$ $\leq$ 1.4), the lower
value of U maintains the metallicity and at the same time weakens the magnetic
state with the formation of a DSM phase. It is important to note that the
electron hopping, SOC and onsite Coulomb repulsion are in the same energy
scale and hence any perturbation can give rise to a different quantum phase.
Here, we show that even weak U transforms the DSM to CAFM phase suggesting
this weakly correlated perovskite is near to the magnetic instability. The
onsite Coulomb repulsion facilitate the non-collinear ordering of the spin-
orbit entangled states as the planar-spin ($S_{\mathrm{p}}$) component
increase in magnitude in proportion to $\lambda$/$\lambda_{0}$ which can be
observed from the spin-intensity map shown in the middle panel of Fig. 5. The
onsite repulsion also strengthens the localization to stabilize the CAFI phase
as can be seen from the phase diagram.

For the large $\lambda$/$\lambda_{0}$ regime ($\lambda$/$\lambda_{0}$ $\geq$
1.4), the system exhibits either DSM for low $U$ or CAFI phase for lower and
higher $U$ values. In the latter case, the $S_{\mathrm{z}}$ component
gradually vanishes with SOC (see Fig. 5 middle panel) to create a transition
from the collinear along $z$ to non-coplanar and canted to pure co-planar spin
arrangement. This transition is schematically illustrated in Fig. 6. Earlier
studies have presented an unusual trend where it has been suggested that
higher value of $U$ is required to induce metal-to-insulator transition for
the higher value of SOC Zeb and Kee (2012). However, as expected, in this
study we observe that a lower $U$ value is sufficient to stabilize a CAFI
phase.

The 4$d$ pervoskites SrRhO3, LaRuO3 and YRuO3 and the $5d$ perovskites are
reported to be having LS $d^{5}$ electonic configuration and all of them
undergo GdFeO3-type distortion of varied order. Their electronic and magnetic
phases can be mapped to the phase diagram of Fig. 5. The compounds SrRhO3
Singh (2003) and LaRuO3 Kobayashi  _et al._ (1994), with weak SOC and moderate
correlation exhibits FM ground state and can be placed in the region (0.2
$\leq$ $\lambda$/$\lambda_{0}$ $\leq$ 0.4, 1 $\leq$ $U$ $\leq$ 2 eV). The
compound YRuO3 Ji  _et al._ (2020) exhibits CAFI phase and lies in (0.6 $\leq$
$\lambda$/$\lambda_{0}$ $\leq$ 0.8, 3 $\leq$ $U$ $\leq$ 4 eV) zone. Like
SrIrO3, CaIrO3 Fujioka  _et al._ (2019) stabilizes in paramagnetic DSM phase
and lies in (0.8 $\leq$ $\lambda$/$\lambda_{0}$ $\leq$ 1, 0 $\leq$ $U$ $\leq$
0.5 eV) zone of the phase diagram.

Figure 7: Electronic phase diagram of undistorted (cubic) SrIrO3 as a function
of $U$ and SOC. The star-mark with cyan color represents the ground state of
LS $d^{5}$ pervoskite BaIrO3, respectively

To identify the divison of role between SOC and structural distortions in
estabilishing quantum phases, here, we have computed the phase diagram for
undistorted (cubic) SrIrO3 in the $U$-SOC space (see Fig. 7). Most
significantly, we observe that the non-collinear spin-ordering is missing and
the entire phase diagram is spanned by collinear magnetic phases. There are
four distinct phases observed and these are: (i) FM, as shown in blue (ii)
G-AFM, as shown in magenta, (iii) G-AFI, as shown in lime and (iv) NM/PM, as
shown in cyan. In the weak SOC ($\lambda$/$\lambda_{0}$ $\leq$ 0.2) and weak
$U$ regime, the system stabilizes in the NM state contrary to the FM phase,
driven by small but finite moment, in the distorted structure. As the strength
of onsite Coloumb repulsion increases, unlike the case of distorted structure,
here no phase transition occurs and the system remains in the FM phase. The FM
phase, for higher $U$ values is well captured by our model Hamiltonian for the
undistorted structure.

For the intermediate SOC strength (0.4 $\leq$ $\lambda$/$\lambda_{0}$ $\leq$
1.6), there is a narrow domain in which G-AFM phase stabilizes. In the strong
$U$ limit, transition from G-AFM to G-AFI phase occurs where the
$U_{\mathrm{c}}$ value for metal-insulator transition is found to be high as
compared to the distorted case. For example, $U_{\mathrm{c}}$ varies from 1 to
2 eV, whereas, for the undistorted case, a higher $U_{\mathrm{c}}$ between 2
to 4 eV is required for metal-insulator transition. The bandwidth, a measure
of extent of localization, in a Hubbard model weakens the hopping integral
roughly by a factor of 1/$U$. The distortion also weakens the hopping
integral. Therefore, metal-insulator transition can be achieved with a lower
value of $U$ for the distorted structure. For the large SOC domain
($\lambda$/$\lambda_{0}$ $\geq$ 1.6), system exhibits either NM/PM phase for
lower $U$ values or G-AFI phase for intermediate and higher $U$ values. The
compound BaIrO$3$ Cheng  _et al._ (2013), like SrIrO3, exhibits Pauli
paramagnetic ground state and can be mapped to the region in (0.8 $\leq$
$\lambda$/$\lambda_{0}$ $\leq$ 1, 0 $\leq$ $U$ $\leq$ 1 eV) in the phase
diagram.

##  VI Origin of Non-Collinear Magnetism: Effect of Rotation and Tilting

Figure 8: Variation of planar and normal spin (first coloumn) and orbital
moments (second coloumn) with rotation and tilting of octahedra, respectively.

Absence of planar-spin component in the cubic SrIrO3 implies that distortion
is the key to the stabilization of non-collinear spin ordering. The effect of
distortions on magnetic ordering is also observed in iridates and their
superlattices. For example, Sr3Ir2O7 exhibits a robust c-axis collinear
antiferromagnetic ordering with negligible $\theta_{\mathrm{t}}$ = 179.5∘
Hogan  _et al._ (2016); Fujiyama  _et al._ (2012) but on the other side, the
bilayer superlattice 2SIO/1STO, exhibits c-axis canted AF ordering which is
attributed to the presence of finite $\theta_{\mathrm{t}}$ ($\approx$
$172^{\circ}$). It has been reported that further enhancement of
$\theta_{\mathrm{t}}$ beyond 172∘ even drives 2SIO/1STO through a quantum
critical point where out-of-plane collinear to $ab$-plane canted magnetic
phase transition occurs Meyers  _et al._ (2019). Therefore, to analyze the
effect of structural distortions on the LS state of $d^{5}$ pervoskites, we
have provided a quantitative measure of it by creating a geometrical design,
described in Appendix B where the $\theta_{\mathrm{r}}$ and
$\theta_{\mathrm{t}}$ can be varied smoothly.

For $\lambda$/$\lambda_{0}$ = 1 and $U$ = 4 eV, the planar and normal spin and
orbital moments are plotted as a function of $\theta_{\mathrm{r}}$ and
$\theta_{\mathrm{t}}$ in Fig. 8. In the absence of tilting,
$\theta_{\mathrm{t}}$ = 180∘, see Figs. 8(a,b), system always favours
collinear spin ordering with vanishing $S_{\mathrm{p}}$ and $L_{\mathrm{p}}$
components. As far as $S_{\mathrm{z}}$ is concerned, it increases with
$\theta_{\mathrm{r}}$, hence, increasing the total magnetization of the
system. Finite tilting (see Figs. 8(c,e,g)) leads to non-coplanar spin
arrangement where $S_{\mathrm{z}}$ decreases sharply with increasing
$S_{\mathrm{p}}$ component upto $\theta_{\mathrm{r}}$ = 170∘. With further
increase in $\theta_{\mathrm{r}}$, $S_{\mathrm{p}}$ component decreases
whereas $S_{\mathrm{z}}$ component increases and finally both components
become comparable for higher values of $\theta_{\mathrm{r}}$. More or less the
$L_{\mathrm{p}}$ component also follow the same trend as the $S_{\mathrm{p}}$
with increase in $\theta_{\mathrm{r}}$ but with magnitude larger as compare to
the $S_{\mathrm{p}}$ (see Figs. 8(d,f,h)). However, for $\theta_{\mathrm{t}}$
= 150∘, the $L_{\mathrm{z}}$ component vanishes completely suggesting
quenching of $L_{\mathrm{z}}$ component for higher $\theta_{\mathrm{t}}$. The
isospin reorientation with tilting and rotation that we get from this first-
principles study is attributed to the orbital mixing hoppings arising in the
distorted frame. Many body models are being designed to relate such spin
anisotropy to the Kitaev-type interactions developed due to octahedral tilting
Mohapatra  _et al._ (2019).

##  VII Summary and Outlook

To summarize, we developed a multi-band model Hamiltonian and performed
density functional calculations to study the electronic and magnetic structure
of SrIrO3 and examined the role of onsite Coulomb repulsion and spin-orbit
coupling in the cubic and distorted structural framework. Furthermore, we used
SrIrO3 as a prototype to examine the electronic structure of low-spin $d^{5}$
perovskites in general by building phase diagram and also smoothly varied the
rotation and tilting of the IrO6 octahedra to bring a third dimension into it.
Our study reveals that eight quantum phases, namely, paramagnetic metal,
paramagnetic Dirac semimetal, ferromagnetic metal, ferromagnetic insulator,
G-type antiferromagnetic metal, G-type antiferromagnetic insulator, canted
antiferromagnetic metal, and canted antiferromagnetic insulator as shown in
Figs. 5 and 7 of the main text. The mechanism driving such phases are
explained in details. Further, we find that each of them form a soft-boundary
to allow a continuous phase transition from one phase to the other by varying
the interaction strengths. The phase diagram is validated by mapping the
ground state of the reported low-spin $d^{5}$ perovskites CaIrO3, BaIrO3,
SrRhO3, LaRuO3 and YRuO3 in the phase diagrams.

By scanning the periodic table, we see that low-spin $d^{5}$ transition metal
oxide perovskites can be designed by exploring the following group
combinations: I-X, II-IX, III-VIII (KPdO3, RbPdO3, MgRhO3, ScRuO3, ScOsO3 and
YOsO3, etc.). Theoretically, thermodynamical stability of such systems can be
examined and experimental synthesis can be attempted with the advent of state
of the art synthesis techniques such as atom by atom deposition methods and
high pressure methods. Furthermore, the two formula unit double perovskite
transition metal oxides with $d^{5}$ state can be thought of as sister members
where similar competing interactions govern the system. In this way, the
phases proposed in the phase diagram can be achieved. The interaction
strengths can be varied under external stimuli such as pressure and strain, as
well as through changing the chemical composition, design of heterostructures,
etc. to induce quantum phase transition in these systems. As a whole, we
believe that the present study will trigger experimental and theoretical
studies to envisage novel quantum phases and applications.

##  VIII Acknowledgement

The authors would like to thank HPCE, IIT Madras for providing the
computational facility. This work is funded by the Department of Science and
Technology, India, through grant No. CRG/2020/004330.

##  Appendix A TRANSFORMATION OF HOPPING MATRICES UNDER ROTATION AND TILTING
OF OCTAHEDRA

In this appendix we briefly explain the transformation of TB matrices for
hopping between different sublattices under rotation and tilting of octahedra.
Distortion can be described in the form of octahedral rotation and tilting
which varies from one Ir-site to the another as shown in Fig. 1(a). Here, $+$
or $-$ signs on each Ir-site indicates the clockwise or counterclockwise
rotation and tilting of octahedra. The transformation of hopping matrices is
determined by a site dependent $5$ $\times$ $5$ rotation matrix (Eq. A4)
Mohapatra  _et al._ (2018); Tinkham (1964). Denoting unrotated and rotated
basis in the order (xy,yz,xz,x2 \- y2,3z2 \- r2) for A sublattice as
$\ket{\alpha}$ and $\ket{\alpha^{\prime}}$, for B sublattice as $\ket{\beta}$
and $\ket{\beta^{\prime}}$, so that $\ket{\alpha^{\prime}}$ = R $\ket{\alpha}$
and $\ket{\beta^{\prime}}$ = R′ $\ket{\beta}$, where R and R′ are
corresponding rotation matrices for A and B sublattices, the hopping integral
in the rotated basis is then given by,

$\tilde{H}_{\alpha^{\prime},\beta^{\prime}}=\bra{\alpha^{\prime}}H\ket{\beta^{\prime}}\\\
=\bra{\alpha}R^{T}HR^{\prime}\ket{\beta}$ (15)

Hence, the Hamiltonian in the rotated basis is given by,

$\tilde{H}=R^{T}HR^{\prime}$ (16)

The total rotation matrix for each site for Ir site can be described as the
multiplication of two individual rotation and tilt matrices. The product is
given by,

$R(\theta_{\mathrm{r}},\theta_{\mathrm{t}})=R(\theta_{\mathrm{r}})R(\theta_{\mathrm{t}})$
(17)

where R($\theta_{\mathrm{r}}$) denotes the pure rotation about the z axis
whereas R($\theta_{\mathrm{t}}$) denotes the rotation about the crystal axis
$a$. For pure rotation about z axis, the Euler angles
($\alpha$,$\beta$,$\gamma$) = ($\theta_{\mathrm{r}}$,0,0) whereas for tilting
($\alpha$,$\beta$,$\gamma$) = (-45∘,$\theta_{\mathrm{t}}$,45∘), respectively.
Using these Euler angles, R($\theta_{\mathrm{r}}$) and
R($\theta_{\mathrm{t}}$) can be obtained using R given by

$R=\frac{1}{2}\begin{pmatrix}\beta{{}^{\prime}_{+}}^{2}\cos(2\alpha+2\gamma)/2&\beta{{}^{\prime}_{+}}\sin\beta\cos(2\alpha+\gamma)&\beta{{}^{\prime}_{+}}\sin\beta\sin(2\alpha+\gamma)&\beta{{}^{\prime}_{+}}^{2}\sin(2\alpha+2\gamma)/2&\sqrt{3}\sin^{2}\beta\\\\[5.0pt]
-\beta{{}^{\prime}_{-}}^{2}\cos(2\alpha-2\gamma)/2&+\beta{{}^{\prime}_{-}}\sin\beta\cos(2\alpha-\gamma)&-\beta{{}^{\prime}_{-}}\sin\beta\sin(2\alpha-\gamma)&+\beta{{}^{\prime}_{-}}^{2}\sin(2\alpha-2\gamma)/2&\sin{2\alpha}\\\\[20.00003pt]
-\beta{{}^{\prime}_{+}}\sin\beta\cos(\alpha+2\gamma)&\beta{{}^{\prime}_{+}}\beta^{\prime\prime}_{-}\cos(\alpha+\gamma)&\beta{{}^{\prime}_{+}}\beta^{\prime\prime}_{-}\sin(\alpha+\gamma)&-\beta{{}^{\prime}_{+}}\sin\beta\sin(\alpha+2\gamma)&\sqrt{3}\sin\alpha\\\
-\beta{{}^{\prime}_{-}}\sin\beta\cos(\alpha-2\gamma)&+\beta{{}^{\prime}_{-}}\beta^{\prime\prime}_{+}\cos(\alpha-\gamma)&-\beta{{}^{\prime}_{-}}\beta^{\prime\prime}_{+}\sin(\alpha-\gamma)&+\beta{{}^{\prime}_{-}}\sin\beta\sin(\alpha-2\gamma)&\sin{2\beta}\\\\[20.00003pt]
\beta{{}^{\prime}_{+}}\sin\beta\sin(\alpha+2\gamma)&-\beta{{}^{\prime}_{+}}\beta^{\prime\prime}_{-}\sin(\alpha+\gamma)&\beta{{}^{\prime}_{+}}\beta^{\prime\prime}_{-}\cos(\alpha+\gamma)&-\beta{{}^{\prime}_{+}}\sin\beta\cos(\alpha+2\gamma)&\sqrt{3}\cos\alpha\\\
+\beta{{}^{\prime}_{-}}\sin\beta\sin(\alpha-2\gamma)&-\beta{{}^{\prime}_{-}}\beta^{\prime\prime}_{+}\sin(\alpha-\gamma)&-\beta{{}^{\prime}_{-}}\beta^{\prime\prime}_{+}\cos(\alpha-\gamma)&+\beta{{}^{\prime}_{-}}\sin\beta\cos(\alpha-2\gamma)&\sin{2\beta}\\\\[20.00003pt]
-\beta{{}^{\prime}_{+}}^{2}\sin(2\alpha+2\gamma)/2&-\beta{{}^{\prime}_{+}}\sin\beta\sin(2\alpha+\gamma)&\beta{{}^{\prime}_{+}}\sin\beta\cos(2\alpha+\gamma)&\beta{{}^{\prime}_{+}}^{2}\cos(2\alpha+2\gamma)/2&\sqrt{3}\sin^{2}\beta\\\\[5.0pt]
+\beta{{}^{\prime}_{-}}^{2}\sin(2\alpha-2\gamma)/2&-\beta{{}^{\prime}_{-}}\sin\beta\sin(2\alpha-\gamma)&-\beta{{}^{\prime}_{-}}\sin\beta\cos(2\alpha-\gamma)&\beta{{}^{\prime}_{-}}^{2}\cos(2\alpha-2\gamma)/2&\cos{2\alpha}\\\\[20.00003pt]
-\sqrt{3}\sin^{2}\beta\sin 2\gamma&\sqrt{3}\sin 2\beta\sin\gamma&-\sqrt{3}\sin
2\beta\cos\gamma&\sqrt{3}\sin^{2}\beta\cos
2\gamma&2-3\sin^{2}\beta\par\end{pmatrix}$ (18)

where $\beta_{\pm}^{\prime}$ = 1 $\pm$ $\cos\beta$ and
$\beta_{\pm}^{\prime\prime}$ = 2 $\cos\beta$ $\pm$ 1

In the rotated basis, the TB hopping matrices between A to B sublattice can be
obtained by using the following equation,

$\tilde{H}_{\mathrm{AB}}=R^{T}(-\theta_{\mathrm{r}},-\theta_{\mathrm{t}})H_{\mathrm{AB}}R(\theta_{\mathrm{r}},-\theta_{\mathrm{t}})$
(19)

where $H_{\mathrm{AB}}$ is the Hamiltonian is the unrotated basis. In compact
form $H_{\mathrm{AB}}$ is expressed as,

$H_{\mathrm{AB}}=\begin{pmatrix}H_{\mathrm{AB}}^{\uparrow\uparrow}&H_{\mathrm{AB}}^{\uparrow\downarrow}\\\\[5.69046pt]
H_{\mathrm{AB}}^{\downarrow\uparrow}&H_{\mathrm{AB}}^{\downarrow\downarrow}\end{pmatrix}$
(20)

with H${}^{\uparrow\uparrow}_{\mathrm{AB}}$
=H${}^{\downarrow\downarrow}_{\mathrm{AB}}$ and
H${}^{\uparrow\downarrow}_{\mathrm{AB}}$ =
H${}^{\downarrow\uparrow}_{\mathrm{AB}}$ = 0. The submatrix
H${}_{\mathrm{AB}}^{\uparrow\uparrow}$ in the expanded form is given by,

$H_{\mathrm{AB}}^{\uparrow\uparrow}=\begin{pmatrix}k_{1}F_{1}&0&0&0&0\\\\[5.69046pt]
0&k_{2}F_{1}&k_{3}F_{2}&0&0\\\\[5.69046pt]
0&k_{3}F_{2}&k_{2}F_{1}&0&0\\\\[5.69046pt]
0&0&0&k_{4}F_{1}&k_{5}F_{2}\\\\[5.69046pt]
0&0&0&k_{5}F_{2}&k_{6}F_{1}\\\\[5.69046pt] \end{pmatrix}$ (21)

where,

$\displaystyle F_{1}$ $\displaystyle=\cos(k_{x}/2)\cos(k_{y}/2)$
$\displaystyle F_{2}$ $\displaystyle=\sin(k_{x}/2)\sin(k_{y}/2)$
$\displaystyle k_{1}$ $\displaystyle=4t_{1}$ $\displaystyle k_{2}$
$\displaystyle=2t_{1}$ $\displaystyle k_{3}$ $\displaystyle=-2t_{1}$
$\displaystyle k_{4}$ $\displaystyle=3t_{4}$ $\displaystyle k_{5}$
$\displaystyle=\sqrt{3}t_{4}$ $\displaystyle k_{6}$ $\displaystyle=t_{4}$ (22)

Similarly, Hopping between A to A sublattice is given by,

$\tilde{H}_{\mathrm{AA}}=R^{T}(-\theta_{\mathrm{r}},-\theta_{\mathrm{t}})H_{\mathrm{AA}}R(-\theta_{\mathrm{r}},-\theta_{\mathrm{t}})$
(23)

where,

$H_{\mathrm{AA}}^{\uparrow\uparrow}=\begin{pmatrix}k_{7}F_{3}&0&0&0&k_{8}F_{4}\\\\[5.69046pt]
0&k_{9}F_{6}&0&0&0\\\\[5.69046pt] 0&0&k_{9}F_{5}&0&0\\\\[5.69046pt]
0&0&0&k_{10}F_{3}&0\\\\[5.69046pt] k_{8}F_{4}&0&0&0&k_{11}F_{3}\end{pmatrix}$
(24)

and the corresponding dispersions and hopping strength are given as,

$\displaystyle F_{3}$ $\displaystyle=\cos{k_{x}}+\cos{k_{y}}$ $\displaystyle
F_{4}$ $\displaystyle=\cos{k_{x}}-\cos{k_{y}}$ $\displaystyle F_{5}$
$\displaystyle=\cos{k_{x}}$ $\displaystyle F_{6}$ $\displaystyle=\cos{k_{y}}$
$\displaystyle k_{7}$ $\displaystyle=\frac{3}{2}t_{2}$ $\displaystyle k_{8}$
$\displaystyle=-\frac{\sqrt{3}}{2}t_{7}$ $\displaystyle k_{9}$
$\displaystyle=2t_{3}$ $\displaystyle k_{10}$ $\displaystyle=2t_{6}$
$\displaystyle k_{11}$ $\displaystyle=\frac{1}{2}t_{5}$ (25)

Similarly, the transformation and corresponding sublattice hopping matrices
for interlayer coupling are given by,

$\displaystyle\tilde{H}_{\mathrm{AC}}$
$\displaystyle=R^{T}(-\theta_{\mathrm{r}},-\theta_{\mathrm{t}})H_{\mathrm{AC}}R(-\theta_{\mathrm{r}},\theta_{\mathrm{t}})$
(26) $\displaystyle\tilde{H}_{\mathrm{AD}}$
$\displaystyle=R^{T}(-\theta_{\mathrm{r}},-\theta_{\mathrm{t}})H_{\mathrm{AD}}R(\theta_{\mathrm{r}},\theta_{\mathrm{t}})$
(27)

$\displaystyle H_{\mathrm{AC}}^{\uparrow\uparrow}$
$\displaystyle=\begin{pmatrix}0&0&0&0&0\\\\[5.69046pt]
0&k_{12}F_{7}&0&0&0\\\\[5.69046pt] 0&0&k_{12}F_{7}&0&0\\\\[5.69046pt]
0&0&0&0&0\\\\[5.69046pt] 0&0&0&0&k_{13}F_{7}\end{pmatrix}$ (28)

$\displaystyle H_{\mathrm{AD}}^{\uparrow\uparrow}$
$\displaystyle=\begin{pmatrix}k_{14}F_{8}&k_{15}F_{11}&k_{15}F_{10}&0&0\\\\[5.69046pt]
k_{15}F_{11}&k_{16}F_{8}&k_{17}F_{9}&k_{18}F_{10}&k_{19}F_{11}\\\\[5.69046pt]
k_{15}F_{10}&k_{17}F_{9}&k_{16}F_{8}&k_{18}F_{11}&k_{19}F_{10}\\\\[5.69046pt]
0&k_{18}F_{10}&k_{18}F_{11}&k_{20}F_{8}&k_{21}F_{9}\\\\[5.69046pt]
0&k_{19}F_{11}&k_{19}F_{10}&k_{21}F_{9}&k_{22}F_{8}\end{pmatrix}$ (29)

and the corresponding dispersion relations and hopping strength are given as,

$\displaystyle F_{7}$ $\displaystyle=\cos(k_{z}/2)$ $\displaystyle F_{8}$
$\displaystyle=\cos(k_{x}/2)\cos(k_{y}/2)\cos(k_{z}/2)$ $\displaystyle F_{9}$
$\displaystyle=\sin(k_{x}/2)\sin(k_{y}/2)\cos(k_{z}/2)$ $\displaystyle F_{10}$
$\displaystyle=\sin(k_{x}/2)\sin(k_{z}/2)\cos(k_{y}/2)$ $\displaystyle F_{11}$
$\displaystyle=\sin(k_{y}/2)\sin(k_{z}/2)\cos(k_{x}/2)$ $\displaystyle k_{12}$
$\displaystyle=2t_{1}$ $\displaystyle k_{13}$ $\displaystyle=2t_{4}$
$\displaystyle k_{14}$ $\displaystyle=4t_{3}$ $\displaystyle k_{15}$
$\displaystyle=-2\sqrt{2}t_{3}$ $\displaystyle k_{16}$
$\displaystyle=(3t_{2}+2t_{3})$ $\displaystyle k_{17}$
$\displaystyle=-(3t_{2}-2t_{3})$ $\displaystyle k_{18}$
$\displaystyle=-\frac{3}{\sqrt{2}}t_{7}$ $\displaystyle k_{19}$
$\displaystyle=-\sqrt{\frac{3}{2}}t_{7}$ $\displaystyle k_{20}$
$\displaystyle=(\frac{3}{2}t_{5}+2t_{6})$ $\displaystyle k_{21}$
$\displaystyle=(-\sqrt{3}t_{5}+2t_{6})$ $\displaystyle k_{22}$
$\displaystyle=(\frac{t_{5}}{2}+6t_{6})$ (30)

##  Appendix B GEOMETRICAL DESIGN FOR INDUCING ROTATION AND TILTING

Figure 9: Schematic illustration of geometrical design for inducing rotation
in the $ab$ plane. Here, a (= $\sqrt{2}a_{\mathrm{0}}$) and a′ are the lattice
parameters of the undistorted and distorted structures, respectively.

The geometrical design for inducing rotation in the $ab$ plane is shown in
Fig. 9. This geometrical design differs from the equilibrium orthorhombic
structure. For the former, the Ir-O bond lengths are uniform while the Ir-O-Ir
bond-angles vary to facilitate the rotation and tilting. However, for the
latter both bond-lengths and bond angles are anisotropic to minimize the
energy. Therefore, this geometrical design can explicitly examine the effect
of rotation and tilting on spin ordering. To do a quantitative measure of it,
we have specifically considered the case of $bc$ plane tilting only.

For analyzing the effect of structural distortions, a
$\sqrt{2}a_{\mathrm{0}}\times\sqrt{2}a_{\mathrm{0}}\times 2a_{\mathrm{0}}$
supercell is designed. Further distortion is induced in the supercell geometry
by displacing the oxygen atoms in the $ab$ (for rotation) as well in the $bc$
(for tilting) plane keeping the Ir-O bond length fixed. The geometrical design
for inducing rotation is shown in Fig. 9. For facilitating
$\theta_{\mathrm{r}}$, oxygen atoms are displaced from their mean position
O(x,y) to O′(x′, y′). In this process the lattice parameter changes from $a$
to a′ to maintain the constant bond length. By using similar geometry the
tilting is designed in the $bc$ plane. The lattice parameter for distorted
structure and coordinates of displaced oxygen atoms are related by the
following equations.

$\displaystyle x^{\prime}$
$\displaystyle=\frac{a^{\prime}\cos{(\phi+45^{\circ})}}{2\sqrt{2}\cos{\phi}}$
(31) $\displaystyle y^{\prime}$
$\displaystyle=\frac{a^{\prime}\sin{(\phi+45^{\circ})}}{2\sqrt{2}\cos{\phi}}$
(32) $\displaystyle a^{\prime}$ $\displaystyle=a\cos{\phi}$ (33)

Figure 10: Planar view of $ab$ and $bc$ planes for (a,b) real orthorhombic
structure, and (c,d) for the designed supercell geometry.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Pesin and Balents (2010) D. Pesin and L. Balents, Nat. Phys. 6, 376 (2010).
  * Bergholtz and Liu (2013) E. J. Bergholtz and Z. Liu, Int. J. Mod. Phys. B 27, 1330017 (2013).
  * Baskaran  _et al._ (2007) G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. 98, 247201 (2007).
  * Takagi  _et al._ (2019) H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nat. Rev. Phys. 1, 264 (2019).
  * Okamoto  _et al._ (2007) Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Phys. Rev. Lett. 99, 137207 (2007).
  * Yan and Felser (2017) B. Yan and C. Felser, Annu. Rev. of Condens. Matter Phys. 8, 337 (2017).
  * Vafek and Vishwanath (2014) O. Vafek and A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014).
  * Wan  _et al._ (2012) X. Wan, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. Lett. 108, 146601 (2012).
  * Wan  _et al._ (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
  * Witczak-Krempa  _et al._ (2013) W. Witczak-Krempa, A. Go, and Y. B. Kim, Phys. Rev. B 87, 155101 (2013).
  * Topp  _et al._ (2018) G. E. Topp, N. Tancogne-Dejean, A. F. Kemper, A. Rubio, and M. A. Sentef, Nat. Commun. 9, 4452 (2018).
  * Rau  _et al._ (2014) J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett. 112, 077204 (2014).
  * Chaloupka  _et al._ (2013) J. c. v. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 110, 097204 (2013).
  * Reuther  _et al._ (2014) J. Reuther, R. Thomale, and S. Rachel, Phys. Rev. B 90, 100405 (2014).
  * Takahashi  _et al._ (2019) S. K. Takahashi, J. Wang, A. Arsenault, T. Imai, M. Abramchuk, F. Tafti, and P. M. Singer, Phys. Rev. X 9, 031047 (2019).
  * Kenney  _et al._ (2019) E. M. Kenney, C. U. Segre, W. Lafargue-Dit-Hauret, O. I. Lebedev, M. Abramchuk, A. Berlie, S. P. Cottrell, G. Simutis, F. Bahrami, N. E. Mordvinova, G. Fabbris, J. L. McChesney, D. Haskel, X. Rocquefelte, M. J. Graf, and F. Tafti, Phys. Rev. B 100, 094418 (2019).
  * Nauman  _et al._ (2017) M. Nauman, Y. Hong, T. Hussain, M. S. Seo, S. Y. Park, N. Lee, Y. J. Choi, W. Kang, and Y. Jo, Phys. Rev. B 96, 155102 (2017).
  * Kim  _et al._ (2008) B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008).
  * Watanabe  _et al._ (2013) H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett. 110, 027002 (2013).
  * Yan  _et al._ (2015) Y. J. Yan, M. Q. Ren, H. C. Xu, B. P. Xie, R. Tao, H. Y. Choi, N. Lee, Y. J. Choi, T. Zhang, and D. L. Feng, Phys. Rev. X 5, 041018 (2015).
  * Fujiyama  _et al._ (2012) S. Fujiyama, K. Ohashi, H. Ohsumi, K. Sugimoto, T. Takayama, T. Komesu, M. Takata, T. Arima, and H. Takagi, Phys. Rev. B 86, 174414 (2012).
  * Chen  _et al._ (2015) Y. Chen, Y.-M. Lu, and H.-Y. Kee, Nat. Commun. 6, 6593 (2015).
  * Nie  _et al._ (2015) Y. F. Nie, P. D. C. King, C. H. Kim, M. Uchida, H. I. Wei, B. D. Faeth, J. P. Ruf, J. P. C. Ruff, L. Xie, X. Pan, C. J. Fennie, D. G. Schlom, and K. M. Shen, Phys. Rev. Lett. 114, 016401 (2015).
  * Fujioka  _et al._ (2017) J. Fujioka, T. Okawa, A. Yamamoto, and Y. Tokura, Phys. Rev. B 95, 121102 (2017).
  * Fujioka  _et al._ (2019) J. Fujioka, R. Yamada, M. Kawamura, S. Sakai, M. Hirayama, R. Arita, T. Okawa, D. Hashizume, M. Hoshino, and Y. Tokura, Nat. Commun. 10, 362 (2019).
  * Fujioka  _et al._ (2021) J. Fujioka, R. Yamada, T. Okawa, and Y. Tokura, Phys. Rev. B 103, L041109 (2021).
  * Carter  _et al._ (2012) J.-M. Carter, V. V. Shankar, M. A. Zeb, and H.-Y. Kee, Phys. Rev. B 85, 115105 (2012).
  * Cheng  _et al._ (2013) J.-G. Cheng, T. Ishii, H. Kojitani, K. Matsubayashi, A. Matsuo, X. Li, Y. Shirako, J.-S. Zhou, J. B. Goodenough, C. Q. Jin, M. Akaogi, and Y. Uwatoko, Phys. Rev. B 88, 205114 (2013).
  * Xia  _et al._ (2017) H. Xia, J. Dai, Y. Xu, Y. Yin, X. Wang, Z. Liu, M. Liu, M. A. McGuire, X. Li, Z. Li, C. Jin, Y. Yang, J. Zhou, and Y. Long, Phys. Rev. Materials 1, 024406 (2017).
  * Mathi Jaya  _et al._ (1991) S. Mathi Jaya, R. Jagadish, R. S. Rao, and R. Asokamani, Phys. Rev. B 43, 13274 (1991).
  * Singh (2003) D. J. Singh, Phys. Rev. B 67, 054507 (2003).
  * Ji  _et al._ (2020) K. Ji, A. Paul, E. Solana-Madruga, A. M. Arevalo-Lopez, U. V. Waghmare, and J. P. Attfield, Phys. Rev. Materials 4, 091402 (2020).
  * Zeb and Kee (2012) M. A. Zeb and H.-Y. Kee, Phys. Rev. B 86, 085149 (2012).
  * Torrance  _et al._ (1992) J. B. Torrance, P. Lacorre, A. I. Nazzal, E. J. Ansaldo, and C. Niedermayer, Phys. Rev. B 45, 8209 (1992).
  * García-Muñoz  _et al._ (1992) J. L. García-Muñoz, J. Rodríguez-Carvajal, P. Lacorre, and J. B. Torrance, Phys. Rev. B 46, 4414 (1992).
  * Miyasaka  _et al._ (2003) S. Miyasaka, Y. Okimoto, M. Iwama, and Y. Tokura, Phys. Rev. B 68, 100406 (2003).
  * Zhou  _et al._ (2005) J.-S. Zhou, J.-Q. Yan, and J. B. Goodenough, Phys. Rev. B 71, 220103 (2005).
  * Zhou  _et al._ (2010) J.-S. Zhou, J. A. Alonso, V. Pomjakushin, J. B. Goodenough, Y. Ren, J.-Q. Yan, and J.-G. Cheng, Phys. Rev. B 81, 214115 (2010).
  * Vogt  _et al._ (2003) T. Vogt, J. A. Hriljac, N. C. Hyatt, and P. Woodward, Phys. Rev. B 67, 140401 (2003).
  * Kim  _et al._ (2017) B. Kim, P. Liu, and C. Franchini, Phys. Rev. B 95, 115111 (2017).
  * Liu  _et al._ (2015) P. Liu, S. Khmelevskyi, B. Kim, M. Marsman, D. Li, X.-Q. Chen, D. D. Sarma, G. Kresse, and C. Franchini, Phys. Rev. B 92, 054428 (2015).
  * Zhao  _et al._ (2008) J. G. Zhao, L. X. Yang, Y. Yu, F. Y. Li, R. C. Yu, Z. Fang, L. C. Chen, and C. Q. Jin, J. App. Phys. 103, 103706 (2008).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Dudarev  _et al._ (1998) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
  * Longo  _et al._ (1971) J. Longo, J. Kafalas, and R. Arnott, J. Solid State Chem. 3, 174 (1971).
  * Modic  _et al._ (2014) K. A. Modic, T. E. Smidt, I. Kimchi, N. P. Breznay, A. Biffin, S. Choi, R. D. Johnson, R. Coldea, P. Watkins-Curry, G. T. McCandless, J. Y. Chan, F. Gandara, Z. Islam, A. Vishwanath, A. Shekhter, R. D. McDonald, and J. G. Analytis, Nature Communications 5, 4203 (2014).
  * Boseggia  _et al._ (2013) S. Boseggia, R. Springell, H. C. Walker, H. M. Rønnow, C. Rüegg, H. Okabe, M. Isobe, R. S. Perry, S. P. Collins, and D. F. McMorrow, Phys. Rev. Lett. 110, 117207 (2013).
  * Chaloupka and Khaliullin (2016) J. Chaloupka and G. Khaliullin, Phys. Rev. B 94, 064435 (2016).
  * Biffin  _et al._ (2014) A. Biffin, R. D. Johnson, I. Kimchi, R. Morris, A. Bombardi, J. G. Analytis, A. Vishwanath, and R. Coldea, Phys. Rev. Lett. 113, 197201 (2014).
  * Slater and Koster (1954) J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).
  * Luo (2013) Q. Luo, Numerical Study of Hubbard Model for Iron-based Superconductors, PhD diss.,University of Tennessee (2013).
  * Kanamori (1963) J. Kanamori, Progress of Theoretical Physics 30, 275 (1963).
  * Mohapatra  _et al._ (2019) S. Mohapatra, S. Aditya, R. Mukherjee, and A. Singh, Phys. Rev. B 100, 140409 (2019).
  * Kobayashi  _et al._ (1994) H. Kobayashi, M. Nagata, R. Kanno, and Y. Kawamoto, Materials Research Bulletin 29, 1271 (1994).
  * Hogan  _et al._ (2016) T. Hogan, L. Bjaalie, L. Zhao, C. Belvin, X. Wang, C. G. Van de Walle, D. Hsieh, and S. D. Wilson, Phys. Rev. B 93, 134110 (2016).
  * Meyers  _et al._ (2019) D. Meyers, Y. Cao, G. Fabbris, N. J. Robinson, L. Hao, C. Frederick, N. Traynor, J. Yang, J. Lin, M. H. Upton, D. Casa, J.-W. Kim, T. Gog, E. Karapetrova, Y. Choi, D. Haskel, P. J. Ryan, L. Horak, X. Liu, J. Liu, and M. P. M. Dean, Scientific Reports 9, 4263 (2019).
  * Mohapatra  _et al._ (2018) S. Mohapatra, C. Bhandari, S. Satpathy, and A. Singh, Phys. Rev. B 97, 155154 (2018).
  * Tinkham (1964) M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill, New York (1964).