††thanks: Lu Liu and Yuntian Liu contributed equally to this work.††thanks: Lu
Liu and Yuntian Liu contributed equally to this work.

# Orbital-doublet-driven even spin Chern insulators

Lu Liu  Department of Physics and Shenzhen Institute for Quantum Science and
Engineering (SIQSE), Southern University of Science and Technology, Shenzhen
518055, China  Laboratory for Computational Physical Sciences (MOE), State Key
Laboratory of Surface Physics, and Department of Physics, Fudan University,
Shanghai 200433, China  Yuntian Liu  Department of Physics and Shenzhen
Institute for Quantum Science and Engineering (SIQSE), Southern University of
Science and Technology, Shenzhen 518055, China  Jiayu Li  Department of
Physics and Shenzhen Institute for Quantum Science and Engineering (SIQSE),
Southern University of Science and Technology, Shenzhen 518055, China  Hua Wu
Corresponding author<EMAIL_ADDRESS>Laboratory for Computational Physical
Sciences (MOE), State Key Laboratory of Surface Physics, and Department of
Physics, Fudan University, Shanghai 200433, China  Shanghai Qi Zhi Institute,
Shanghai 200232, China  Collaborative Innovation Center of Advanced
Microstructures, Nanjing 210093, China  Qihang Liu  Corresponding author.
<EMAIL_ADDRESS>Department of Physics and Shenzhen Institute for Quantum
Science and Engineering (SIQSE), Southern University of Science and
Technology, Shenzhen 518055, China  Guangdong Provincial Key Laboratory of
Computational Science and Material Design, Southern University of Science and
Technology, Shenzhen 518055, China

###### Abstract

Quantum spin Hall insulators hosting edge spin currents hold great potential
for low-power spintronic devices. In this work, we present a universal
approach to achieve a high and near-quantized spin Hall conductance plateau
within a sizable bulk gap. Using a nonmagnetic four-band model Hamiltonian, we
demonstrate that an even spin Chern (ESC) insulator can be accessed by tuning
the sign of spin-orbit coupling (SOC) within a crystal symmetry-enforced
orbital doublet. With the assistance of a high spin Chern number of $C_{S}=2$
and spin $U$(1) quasi-symmetry, this orbital-doublet-driven ESC phase is
endowed with the near-double-quantized spin Hall conductance. We identify 12
crystallographic point groups supporting such a sign-tunable SOC. Furthermore,
we apply our theory to realistic examples, and show the phase transition from
a trivial insulator governed by positive SOC in RuI3 monolayer to an ESC
insulator dominated by negative SOC in RuBr3 monolayer. This orbital-doublet-
driven ESC insulator, RuBr3, showcases nontrivial characteristics including
helical edge states, near-double-quantized spin Hall conductance, and robust
corner states. Our work provides new pathways in the pursuit of the long-
sought quantum spin Hall insulators.

## Introduction

Two-dimensional (2D) quantum spin Hall (QSH) insulators have garnered
significant interest for their promising applications in spintronics and
magnetoelectronics [1, 2, 3, 4]. They manifest topologically protected helical
edge states where the spin is locked to the momentum through spin-orbit
coupling (SOC) and time-reversal symmetry (TRS), providing dissipationless
spin transports ideal for low-power magnetic memory devices. The first
predictions of realistic QSH insulators identified graphene [5] and the HgTe
quantum well [6] as candidates, each characterized by a SOC-induced inverted
bulk gap along with a pair of helical edge states within this gap. This
topological phase is generally characterized by the topological invariant
$Z_{2}=1$, which also serves as the symmetry indicator for TRS-preserved
systems [7]. Over the years, this $Z_{2}=1$ topological phase has been
observed in several quantum wells [8, 9, 10] and pristine 2D materials such as
WTe2, bismuthene, Na3Bi, and germanene [11, 12, 13, 14, 15, 16].

Figure 1: Schematic for designing ESC insulators with $C_{S}=2$: by tuning the
sign of SOC within an orbital doublet from positive to negative, a phase
transition from $C_{S}=0$ to $C_{S}=2$ can be realized.

In addition to the $Z_{2}$ index, the spin Chern number $C_{S}$, also
established as a topological invariant, is directly related to the number of
pairs of helical edge states [17]. In particular, when the real-spin component
$S_{z}$ remains preserved, $C_{S}$ defines the quantized spin Hall conductance
(SHC) as $\sigma_{xy}^{S}=C_{S}\frac{e}{2\pi}$. These two invariants are
related by $Z_{2}=$ mod ($C_{S},2$). Therefore, QSH insulators with two pairs
of helical edge states in the $C_{S}=2$ regime are considered to be trivial
within conventional $Z_{2}$ classification. However, experiments have observed
near-double-quantized conductance in twisted bilayers WSe2 and MoTe2 [18, 19],
demonstrating that QSH effects can indeed manifest in even spin Chern (ESC)
insulators. Recently, we have emphasized the pivotal role of spin $U$(1)
quasi-symmetry for the near-quantization of SHC in TRS-preserved $Z_{2}=1$ or
such $Z_{2}=0$ systems, as well as TRS-broken cases [20]. Beyond theoretical
predictions of the ESC phase in monolayer $\alpha$-Sb/Bi [21, 22] and TRS-
broken Fe2BrMgP monolayer and TiTe bilayer [23], we have predicted near-
double-quantized SHC in twisted bilayer transition metal dichalcogenides and
monolayer RuBr3 [20] protected by spin $U$(1) quasi-symmetry. In this work, we
will present a general approach to realize an ESC phase with a symmetry-
protected near-double-quantized SHC within a large bulk gap, which would be an
ideal platform for observing QSH effects and further promote applications of
QSH insulators.

First, using a nonmagnetic four-band model Hamiltonian, we demonstrate that an
ESC phase with $C_{S}=2$ can be accessed by tuning the sign of SOC within a
crystal symmetry-enforced orbital doublet. Such an orbital-doublet-driven ESC
phase is endowed with two notable features: (i) a sizable bulk gap opened by
first-order spin-preserved SOC; (ii) a high near-quantized SHC approaching 2
(in unit of $e/2\pi$) protected by spin $U$(1) quasi-symmetry. Thereafter, we
enumerate 12 crystallographic point groups supporting the orbital doublets
with sign-alterable SOC effects. Furthermore, we present realistic examples to
demonstrate our theory. As shown below, a trivial insulator driven by positive
SOC transforms into a nontrivial ESC insulator induced by negative SOC, as
observed in the transition from monolayer RuI3 to RuBr3. In addition to the
topologically nontrivial features such as the near-double-quantized SHC and
two pairs of helical edge states, we further show robust in-gap corner states
that is associated with the slightly gapped edge states in RuBr3.

## Computational Details

Density functional theory calculations are performed using the full-potential
augmented plane wave plus the local orbital code (Wien2k) [24]. The optimized
lattice constants of RuI3 and RuBr3 monolayers are $a=b=$ 6.667 Å and 6.159 Å,
respectively. A vacuum slab of 15 Å is set along the $c$ axis for both
systems. The muffin-tin sphere radii are chosen to be 2.2 bohr for Ru atoms
and 2.4 bohr for both I and Br atoms. The cutoff energy of 14 Ry is set for
plane wave expansions of interstitial wave functions. We use the
11$\times$11$\times$1 $k$-mesh for integration over the Brillouin zone. SOC is
included by the second variational method with scalar relativistic wave
functions. Electron correlation of Ru $4d$ electrons is taken into account by
adopting a typical Hubbard $U$ of 2 eV and a Hund’s exchange of 0.5 eV [25].
The Wannier functions of Ru 4$d$, I 5$p$, and Br 4$p$ orbitals are constructed
using Wien2wannier [26] and WANNIER90 [27] without performing maximally
localized procedures. The topological edge states and SHC are calculated by
the iterative Green’s function and the Kubo formula [28], respectively, as
implemented in WannierTools package [29].

## Results

### I. Symmetry and model of even spin Chern phase

Figure 2: Band structures of the model Hamiltonian in Eq. (1) with the
parameters $A=B=0,C=D=0.3$: from (a) $E=0.1$ to (b) $E=-0.1$, the band
inversion marks a phase transition from a trivial insulator to an ESC
insulator with $C_{S}=2$.

To begin with, we will show that a nontrivial ESC phase can be realized within
a nonmagnetic four-band model Hamiltonian based on an orbital doublet that is
characterized by a 2D irreducible representation (irrep). We consider a
typical doublet formed by $p_{x}$ and $p_{y}$ orbitals as $p_{\pm}=(p_{x}\pm
ip_{y})/\sqrt{2}$, where the subscript $+$/$-$ denotes orbital angular
momentum $l_{z}=+1$/$-1$. To generate a 2D irrep furnished by the $p_{\pm}$
doublet, here we consider a $D_{6h}$ point group, of which the generators are
three-fold rotation symmetry $C_{3z}$ along the $z$ axis, two-fold rotation
symmetry $C_{2z}$/$C_{2y}$ along the $z$/$y$ axis, and space inversion
symmetry $I$. In the basis of
$\\{|p_{+},\uparrow\rangle,|p_{-},\uparrow\rangle,|p_{+},\downarrow\rangle,|p_{-},\downarrow\rangle\\}$,
the representation of symmetry operations is given by
$C_{3z}=e^{i\frac{\pi}{3}\sigma_{z}}\otimes e^{i\frac{2\pi}{3}\tau_{z}}$,
$C_{2z}=e^{i\frac{\pi}{2}\sigma_{z}}\otimes-\tau_{0}$,
$C_{2y}=e^{i\frac{\pi}{2}\sigma_{y}}\otimes-\tau_{x}$, $I=\mathbb{I}_{2\times
2}\otimes-\mathbb{I}_{2\times 2}$, and TRS $T=\mathcal{K}\cdot
i\sigma_{y}\otimes\tau_{x}$, where $\mathcal{K}$ is the complex conjugation
operator, $\mathbb{I}_{2\times 2}$ is a $2\times 2$ identity matrix, and
$\sigma_{x,y,z}$ and $\tau_{x,y,z}$ are Pauli matrices for spin and orbital
degrees of freedom, respectively. By imposing those symmetries, we derive the
generic form of the effective Hamiltonian as follows:

$\displaystyle H(\bm{k})=$
$\displaystyle\epsilon_{0}(\bm{k})\mathbb{I}_{4\times
4}+C[(k_{x}^{2}-k_{y}^{2})\sigma_{0}\otimes\tau_{x}-2k_{x}k_{y}\sigma_{0}\otimes\tau_{y}]$
(1)
$\displaystyle+D(k_{x}^{2}+k_{y}^{2})\sigma_{z}\otimes\tau_{z}+E\sigma_{z}\otimes\tau_{z}$

with $\epsilon_{0}(\bm{k})=A-B(k_{x}^{2}+k_{y}^{2})$. Note that the symmetry
preserves the term $E\sigma_{z}\otimes\tau_{z}$ which is contributed by the
first-order spin-preserved SOC. The resulting electronic structure consists of
two sets of doubly degenerate bands protected by $I$ and $T$ symmetry,
yielding an energy gap 2$E$. This bulk gap 2$E$, opened by first-order SOC,
can reach $\sim$100 meV to against thermal fluctuation and local disorder.
Furthermore, the change of the sign of $E$ from positive to negative marks a
phase transition accompanied by band inversion, as shown in Fig. 2. Note that
such band inversion does not change the $Z_{2}$ index of the system because
the wavefunctions of the lowest conduction band and the highest valence band
at $\Gamma$ share the same parity. However, we find that such a band inversion
signifies a topological phase transition from a trivial insulator to a
nontrivial ESC insulator characterized by $C_{S}=2$.

We note that such topological phase transition, driven by altering the sign of
the SOC within orbital doublets, can be achieved by orbital engineering.
Specifically, some $d$-orbital doublets, undergoing transformations identical
to the $p_{\pm}$ doublet but with an opposite $l_{z}$, can contribute negative
SOC in contrast to the positive one within $p_{\pm}$. We identify 12
crystallographic point groups that can support the sign-alterable SOC within
specific orbital doublets, as listed in Table 1. For instance, the orbital
doublet $d_{\pm 2}=|l_{z}=\pm 2\rangle$ is supported by
$(C_{3h},D_{3h},C_{6},C_{6v},C_{6h},D_{6},D_{6h})$ point groups. Under the
rotational symmetries that can distinguish the two states in an orbital
doublet, the $d_{-2}$ state transform as $p_{+}$, and $d_{+2}$ transforms as
$p_{-}$, e.g., the symmetry operation $C_{3z}$ introduces a phase factor
$e^{-i\frac{2\pi}{3}\tau_{z}}$ to $d_{\pm 2}$ but an opposite phase factor
$e^{i\frac{2\pi}{3}\tau_{z}}$ to $p_{\pm}$. Therefore, the $d_{\pm 2}$ doublet
will yield a negative splitting when SOC emerges, in contrast to the positive
SOC-splitting in $p_{\pm}$. Similarly, the $e_{\pm}^{\prime}$ doublet
supported by ($C_{3},C_{3v},D_{3},D_{3d},S_{6}$) point groups is formulated as
$e^{\prime}_{\pm}=\pm\cos\alpha|l_{z}=\pm 2\rangle-\sin\alpha|l_{z}=\mp
1\rangle$, where $\sin^{2}\alpha$ varies from 0 to 1/3 depending on local
$d$-orbital environments [30]. The $e_{\pm}^{\prime}$ transforms as $p_{\mp}$,
and thus also provides a negative SOC. Note that two other $d$-orbital
doublets listed in Table 1, specifically $d_{\pm}=|l_{z}=\pm 1\rangle$ and
$e^{\prime}_{g\pm}=\sin\alpha|l_{z}=\mp 2\rangle\mp\cos\alpha|l_{z}=\pm
1\rangle$, both yield the positive SOC just like that in $p_{\pm}$.

Table 1: Crystallographic point groups that permit orbital doublets with a sign-tunable SOC (both positive and negative). point groups | doublets | SOC-sign
---|---|---
$C_{3h},D_{3h},C_{6},C_{6v},C_{6h},D_{6},D_{6h}$ | $p_{\pm},d_{\pm}$ | +
| $d_{\pm 2}$ | $-$
$C_{3},C_{3v},D_{3},D_{3d},S_{6}$ | $p_{\pm},e_{g\pm}^{\prime}$ | +
| $e_{\pm}^{\prime}$ | $-$

We emphasize that among the 12 crystallographic point groups in Table 1, while
lowering symmetries from the highest symmetric point group $D_{6h}$ [Eq. (1)]
may introduce additional terms, the low-energy physics at the $\Gamma$ point
remains intact. For instance, in point group $D_{3d}$, the term
$F[(k_{x}^{2}-k_{y}^{2})\sigma_{x}\otimes\tau_{z}+2k_{x}k_{y}\sigma_{y}\otimes\tau_{z}]$
emerges [20], serving as spin-mixing perturbations. More notably, within the
eigenspace of the model Hamiltonian in Eq. (1), which is spanned by an orbital
doublet combined with electron spin, spin $U$(1) quasi-symmetry is present to
eliminate the first-order spin-mixing perturbation [20, 31]. Such a symmetry
plays a pivotal role for protecting QSH effects in realistic materials.
Consequently, despite a trivial $Z_{2}$ index, the ESC systems described by
our model can exhibit a near-double-quantized SHC plateau within a sizable
bulk gap. In addition, the edge state would open a small gap by spin-mixing
perturbation. These features are further confirmed by realistic 2D examples
presented in the following section.

### II. Realistic materials with tunable SOC

Figure 3: (a) Crystal structure of RuI3 and RuBr3 monolayers with Ru and I/Br
atoms represented by red and gray balls, respectively. In the bottom panel,
the 1$a$, 2$c$, and 3$f$ maximal Wyckoff positions within the $c=0$ plane are
denoted. (b) Band structures of RuI3 monolayer without SOC. The Fermi level is
set at the zero energy.

We take the RuI3 and RuBr3 monolayers as examples to demonstrate an ESC phase
that is accessible through tuning the sign of SOC. The three-dimensional form
of RuI3 has been crystallized in a rhombohedral structure with space group
$R\bar{3}$ [32], and its 2D counterpart is in the space group $P\bar{3}1m$,
providing the little point group $D_{3d}$ at the $\Gamma$ point, which is
included in Table 1. Recent studies have shown that, due to intricate SOC
effects combined with strong Ru-I hybridization, RuI3 exhibits paramagnetic
behavior and undergoes a metal-to-insulator transition from bulk to monolayer
[32, 33, 34, 35, 36]. Therefore, RuI3 monolayer would be a great platform for
investigating SOC effects on topological characteristics. Moreover, RuBr3
monolayer is also of interest for the variation of the relative importance of
the SOC at the Ru and ligand Br/I sites, and for the possibly new topological
properties.

We first present the band structures of RuI3 monolayer in the absence of SOC.
Fig. 3(b) illustrates that, without SOC, two isolated bands around the Fermi
level form crossings at $\Gamma$ and K points. This band degeneracy is
protected by the crystal symmetry and can be lifted by SOC. As shown in Figs.
4 and 5(a), a bulk gap is opened when SOC emerges, signifying the RuI3
monolayer as a band insulator [individual I and Ru SOC effect in Figs. 4(a)
and 4(c), and joint one in Fig. 5(a)]. To characterize the topological phase
of RuI3 monolayer, we calculate the $Z_{2}$ index by computing the parity
eigenvalues of valence bands at two time-reversal-invariant momenta [37],
namely $\Gamma$ and M. The same parity at $\Gamma$ and M yields a $Z_{2}=0$.
As a result, we find that RuI3 monolayer is a $Z_{2}$ trivial insulator.

It is worth to note that within the category of topologically trivial
insulators, there exists a special subgroup known as obstructed atomic
insulators (OAIs), as proposed based on topological quantum chemistry (TQC)
theory [38, 39, 40, 41, 42]. Within the TQC framework, for topologically
trivial insulators, the band representation (BR) of all occupied bands is a
sum of elementary band representations (EBRs) induced from atomic orbitals at
maximal Wyckoff positions. And OAI refers to the situation that some of those
Wyckoff positions are empty sites without atoms occupied. By calculating the
BR of valence bands, we find that the BR decompositions of RuI3 monolayer have
to include an EBR at empty Wyckoff position $1a$, i.e., the center of the
honeycomb lattice, see Fig. 3(a). Therefore, RuI3 monolayer falls into the
category of OAIs. This is also captured by the emergence of obstructed
metallic edge states, as shown in Fig. 5(b), which appears when one cuts the
edge containing the obstructed $1a$ site.

Figure 4: Band structures of RuI3 monolayer with (a) only I SOC active and (c)
only Ru SOC active. Combined with electron spin, the orbital doublets (b)
$p_{\pm}$ of I $5p$ states and (d) $e^{\prime}_{\pm}$ of Ru $4d$-$t_{2g}$
states form the co-irreps around the Fermi level at the $\Gamma$ point,
undergoing positive and negative SOC-splitting, respectively.

We now take a close look at SOC effects. As shown above, SOC-splitting is
responsible for the band gap of RuI3 monolayer. When we examine the individual
contributions of SOC from Ru and I elements, we find that the band splitting
in RuI3 is primarily driven by I SOC, as evidenced by the same co-irrep
feature of the lowest conduction bands and highest valence bands, i.e.,
($\Gamma_{5}^{-}\oplus\Gamma_{6}^{-}$)-over-$\Gamma_{4}^{-}$, for both Figs.
4(a) and 5(a). In contrast, when Ru SOC is considered independently, as shown
in Fig. 4(c), the band gap at $\Gamma$ point is inverted, yielding a negative
splitting $\Gamma_{4}^{-}$-over-($\Gamma_{5}^{-}\oplus\Gamma_{6}^{-}$). Such
SOC-sign-change behavior is well predicted as the case of $D_{3d}$ in Table 1.

Figure 5: (a) Band structures and (b) edge states and SHC of RuI3 monolayer
with SOC; (c) and (d) corresponding results of RuBr3 monolayer.

Despite the fact that either positive or negative SOC-splitting of the orbital
doublet does not change $Z_{2}$, the model Hamiltonian in Eq. (1) predicts
that the SOC-sign-change triggers a topological phase transition between the
trivial $C_{S}=0$ phase and the nontrivial $C_{S}=2$ phase. To provide a
realistic material candidate for the latter case, we naturally move to RuBr3,
taking into account the reduced SOC strength associated with Br $4p$ electrons
and their weaker hybridization with Ru $4d$ states as compared to I $5p$
electrons. As anticipated, our results show a band inversion from RuI3 to
RuBr3, as evidenced by the SOC-induced splitting at $\Gamma$ shifting from a
positive ($\Gamma_{5}^{-}\oplus\Gamma_{6}^{-}$)-over-$\Gamma_{4}^{-}$
configuration to a negative
$\Gamma_{4}^{-}$-over-($\Gamma_{5}^{-}\oplus\Gamma_{6}^{-}$) one, see Figs.
5(a) and 5(c).

### III. Nontrivial features in ESC insulator

Despite both RuI3 and RuBr3 belong to the $Z_{2}=0$ phase, their distinct
topological features are evident in the edge and SHC behaviors. In stark
contrast to RuI3, RuBr3 exhibits four metallic edge states and two Dirac-like
edge crossings, see Figs. 5(b) and 5(d). A closer examination of these edge
crossings reveals a small gap of 3 meV, which is opened by spin-mixing
perturbations. Moreover, unlike the absent SHC in RuI3, RuBr3 exhibits a SHC
plateau within a large bulk gap of 130 meV, and the SHC value of 1.93 closely
approaches the quantized value of 2. Note that these topologically nontrivial
features in RuBr3 are protected by a nonzero $C_{S}$ and spin $U$(1) quasi-
symmetry [20]. Thus, our findings highlight the orbital-doublet-driven ESC
insulators, as described by our nonmagnetic four-band model Hamiltonian, as an
ideal platform for realizing QSH effects.

Figure 6: Spatial distributions of the state at the Fermi level for (a)
hexagonal, (b) triangular, and (c) rhomboid-shaped nanodisks of RuBr3.

In addition, we note that in ESC insulators, the small edge gap induced by
spin-mixing perturbations may lead to in-gap corner states through the mass-
kink on the edges [43]. We construct nanodisks with hexagonal, triangular, and
rhomboid shapes for RuBr3 and plot the real-space distributions of the state
at the Fermi level, determined by valence electron counting. As shown in Fig.
6, we find that the in-gap states are well localized at the corners,
independent of the geometry. Our results are accordance with theoretical
predictions about robust corner states in TRS-preserved $C_{S}=2$ systems
[43]. As a result, manifold nontrivial characteristics embedded in the
orbital-doublet-driven ESC insulators, including helical edge states, high
near-quantized SHC, and robust in-gap corner modes, enrich their potential
applications spanning various fields.

## Summary

To summarize, we develop a nonmagnetic four-band model Hamiltonian based on a
crystal symmetry-enforced orbital doublet. We propose a generic approach to
realize a nontrivial ESC phase with $C_{S}=2$ by tuning the sign of SOC within
orbital doublets, which can be supported in 12 crystallographic point groups.
Realistic 2D examples, specifically the evolution from RuI3 monolayer to
RuBr3, demonstrate that a trivial $C_{S}=0$ insulator governed by positive SOC
transforms into a nontrivial $C_{S}=2$ insulator dominated by negative SOC.
Moreover, we show that such orbital-doublet-driven ESC insulators manifest
nontrivial features, including two pairs of helical edge states, high near-
quantized SHC, and robust in-gap corner modes. Our work presents a universal
strategy to design ESC insulators featuring a near-double-quantized SHC
plateau within a large bulk gap, offering new insights into the exploration of
QSH insulators.

## Acknowledgements

Q. L. acknowledges support by National Key R$\&$D Program of China under Grant
No. 2020YFA0308900, National Natural Science Foundation of China under Grant
No. 12274194, Guangdong Provincial Key Laboratory for Computational Science
and Material Design under Grant No. 2019B030301001, Shenzhen Science and
Technology Program under Grant No. RCJC20221008092722009, the Science,
Technology and Innovation Commission of Shenzhen Municipality under Grant No.
ZDSYS20190902092905285 and Center for Computational Science and Engineering of
Southern University of Science and Technology. H. W. acknowledges support by
the National Natural Science Foundation of China under Grants No. 12174062 and
No. 12241402.

## References

  * Kane and Mele [2005a] C. L. Kane and E. J. Mele, Quantum spin $\mathrm{H}$all effect in graphene, Phys. Rev. Lett. 95, 226801 (2005a).
  * Bernevig  _et al._ [2006a] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin $\mathrm{H}$all effect and topological phase transition in $\mathrm{H}$g$\mathrm{T}$e quantum wells, Science 314, 1757 (2006a).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Kane and Mele [2005b] C. L. Kane and E. J. Mele, Quantum spin $\mathrm{H}$all effect in graphene, Phys. Rev. Lett. 95, 226801 (2005b).
  * Bernevig  _et al._ [2006b] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin $\mathrm{H}$all effect and topological phase transition in $\mathrm{H}$g$\mathrm{T}$e quantum wells, Science 314, 1757 (2006b).
  * Kane and Mele [2005c] C. L. Kane and E. J. Mele, ${Z}_{2}$ topological order and the quantum spin $\mathrm{H}$all effect, Phys. Rev. Lett. 95, 146802 (2005c).
  * König  _et al._ [2007] M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum spin $\mathrm{H}$all insulator state in $\mathrm{H}$g$\mathrm{T}$e quantum wells, Science 318, 766 (2007).
  * Roth  _et al._ [2009] A. Roth, C. Brüne, H. Buhmann, L. W. Molenkamp, J. Maciejko, X.-L. Qi, and S.-C. Zhang, Nonlocal transport in the quantum spin $\mathrm{H}$all state, Science 325, 294 (2009).
  * Knez  _et al._ [2011] I. Knez, R.-R. Du, and G. Sullivan, Evidence for helical edge modes in inverted $\mathrm{InAs}/\mathrm{GaSb}$ quantum wells, Phys. Rev. Lett. 107, 136603 (2011).
  * Tang  _et al._ [2017] S. Tang, C. Zhang, D. Wong, Z. Pedramrazi, H.-Z. Tsai, C. Jia, B. Moritz, M. Claassen, H. Ryu, S. Kahn, J. Jiang, H. Yan, M. Hashimoto, D. Lu, R. G. Moore, C.-C. Hwang, C. Hwang, Z. Hussain, Y. Chen, M. M. Ugeda, Z. Liu, X. Xie, T. P. Devereaux, M. F. Crommie, S.-K. Mo, and Z.-X. Shen, Quantum spin $\mathrm{H}$all state in monolayer 1T′-$\mathrm{W}\mathrm{T}$e2, Nature Phys. 13, 683 (2017).
  * Fei  _et al._ [2017] Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, Edge conduction in monolayer $\mathrm{W}\mathrm{T}$e2, Nature Phys. 13, 677 (2017).
  * Wu  _et al._ [2018a] S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Observation of the quantum spin $\mathrm{H}$all effect up to 100 $\mathrm{K}$elvin in a monolayer crystal, Science 359, 76 (2018a).
  * Reis  _et al._ [2017] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schäfer, and R. Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin $\mathrm{H}$all material, Science 357, 287 (2017).
  * Collins  _et al._ [2018] J. L. Collins, A. Tadich, W. Wu, L. C. Gomes, J. N. B. Rodrigues, C. Liu, J. Hellerstedt, H. Ryu, S. Tang, S.-K. Mo, S. Adam, S. A. Yang, M. S. Fuhrer, and M. T. Edmonds, Electric-field-tuned topological phase transition in ultrathin Na3Bi, Nature 564, 390 (2018).
  * Bampoulis  _et al._ [2023] P. Bampoulis, C. Castenmiller, D. J. Klaassen, J. van Mil, Y. Liu, C.-C. Liu, Y. Yao, M. Ezawa, A. N. Rudenko, and H. J. W. Zandvliet, Quantum spin $\mathrm{H}$all states and topological phase transition in germanene, Phys. Rev. Lett. 130, 196401 (2023).
  * Sheng  _et al._ [2006] D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, Quantum spin-$\mathrm{H}$all effect and topologically invariant Chern numbers, Phys. Rev. Lett. 97, 036808 (2006).
  * Kang  _et al._ [2024a] K. Kang, Y. Qiu, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Observation of the double quantum spin Hall phase in moiré WSe2, arXiv: 2402.04196 (2024a).
  * Kang  _et al._ [2024b] K. Kang, B. Shen, Y. Qiu, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Observation of the fractional quantum spin Hall effect in moiré MoTe2, arXiv:2402.03294 (2024b).
  * Liu  _et al._ [2024] L. Liu, Y. Liu, J. Li, H. Wu, and Q. Liu, Quantum spin Hall effect protected by spin ${U}$(1) quasi-symmetry, arXiv:2402.13974 (2024).
  * Bai  _et al._ [2022] Y. Bai, L. Cai, N. Mao, R. Li, Y. Dai, B. Huang, and C. Niu, Doubled quantum spin $\mathrm{H}$all effect with high-spin Chern number in $\alpha$-antimonene and $\alpha$-bismuthene, Phys. Rev. B 105, 195142 (2022).
  * Wang  _et al._ [2022] B. Wang, X. Zhou, Y.-C. Lin, H. Lin, and A. Bansil, High spin-Chern-number insulator in $\alpha$-antimonene with a hidden topological phase, arXiv:2202.04162 (2022).
  * Xue  _et al._ [2023] Y. Xue, W. Xu, B. Zhao, J. Zhang, and Z. Yang, Antiferromagnetic quantum spin Hall insulators with high spin Chern numbers, Phys. Rev. B 108, 075138 (2023).
  * Blaha  _et al._ [2020] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, Wien2k: An apw+lo program for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020).
  * Anisimov  _et al._ [1993] V. I. Anisimov, I. V. Solovyev, M. A. Korotin, M. T. Czyżyk, and G. A. Sawatzky, Density-functional theory and NiO photoemission spectra, Phys. Rev. B 48, 16929 (1993).
  * Kuneš  _et al._ [2010] J. Kuneš, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held, Wien2wannier: From linearized augmented plane waves to maximally localized wannier functions, Comp. Phys. Commun. 181, 1888 (2010).
  * Mostofi  _et al._ [2008] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178, 685 (2008).
  * Yao and Fang [2005] Y. Yao and Z. Fang, Sign changes of intrinsic spin $\mathrm{H}$all effect in semiconductors and simple metals: First-principles calculations, Phys. Rev. Lett. 95, 156601 (2005).
  * Wu  _et al._ [2018b] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, WannierTools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018b).
  * Dai and Whangbo [2005] D. Dai and M.-H. Whangbo, Analysis of the uniaxial magnetic properties of high-spin d6 ions at trigonal prism and linear two-coordinate sites: Uniaxial magnetic properties of Ca3Co2O6 and Fe[C(SiMe3)3]2, Inorg. Chem. 44, 4407 (2005).
  * Li  _et al._ [2023] J. Li, A. Zhang, Y. Liu, and Q. Liu, Group theory on quasi-symmetry and protected near degeneracy, arXiv:2311.12738 (2023).
  * Ni  _et al._ [2022] D. Ni, X. Gui, K. M. Powderly, and R. J. Cava, Honeycomb-structure RuI3, a new quantum material related to $\alpha$-RuCl3, Adv. Mater. 34, 2106831 (2022).
  * Nawa  _et al._ [2021] K. Nawa, Y. Imai, Y. Yamaji, H. Fujihara, W. Yamada, R. Takahashi, T. Hiraoka, M. Hagihala, S. Torii, T. Aoyama, T. Ohashi, Y. Shimizu, H. Gotou, M. Itoh, K. Ohgushi, and T. J. Sato, Strongly electron-correlated semimetal RuI3 with a layered honeycomb structure, J. Phys. Soc. Jpn. 90, 123703 (2021).
  * Kaib  _et al._ [2022] D. A. S. Kaib, K. Riedl, A. Razpopov, Y. Li, S. Backes, I. Mazin, and R. Valentí, Electronic and magnetic properties of the RuX3 (X=Cl, Br, I) family: two siblings—and a cousin?, npj Quantum Mater. 7, 75 (2022).
  * Zhang  _et al._ [2022] Y. Zhang, L.-F. Lin, A. Moreo, and E. Dagotto, Theoretical study of the crystal and electronic properties of $\alpha$-RuI3, Phys. Rev. B 105, 085107 (2022).
  * Liu  _et al._ [2023a] L. Liu, K. Yang, G. Wang, D. Lu, Y. Ma, and H. Wu, Contrasting electronic states of RuI3 and RuCl3, Phys. Rev. B 107, 165134 (2023a).
  * Fu and Kane [2007] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76, 045302 (2007).
  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017).
  * Cano  _et al._ [2018] J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. G. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Building blocks of topological quantum chemistry: Elementary band representations, Phys. Rev. B 97, 035139 (2018).
  * Xu  _et al._ [2021] Y. Xu, L. Elcoro, G. Li, Z.-D. Song, N. Regnault, Q. Yang, Y. Sun, S. Parkin, C. Felser, and B. A. Bernevig, Three-dimensional real space invariants, obstructed atomic insulators and a new principle for active catalytic sites, arXiv:2111.02433 (2021).
  * Gao  _et al._ [2022] J. Gao, Y. Qian, H. Jia, Z. Guo, Z. Fang, M. Liu, H. Weng, and Z. Wang, Unconventional materials: the mismatch between electronic charge centers and atomic positions, Sci. Bull. 67, 598 (2022).
  * Liu  _et al._ [2023b] X.-R. Liu, H. Deng, Y. Liu, Z. Yin, C. Chen, Y.-P. Zhu, Y. Yang, Z. Jiang, Z. Liu, M. Ye, D. Shen, J.-X. Yin, K. Wang, Q. Liu, Y. Zhao, and C. Liu, Spectroscopic signature of obstructed surface states in SrIn2P2, Nat. Commun. 14, 2905 (2023b).
  * Hung  _et al._ [2023] Y.-C. Hung, B. Wang, A. Bansil, and H. Lin, Time-reversal soliton pairs in even spin-Chern-number higher-order topological insulators, arXiv:2303.04031 (2023).