Chern dartboard insulator: sub-Brillouin zone topology and skyrmion multipoles

Chern insulators are classic examples of non-interacting systems with nontrivial bulk topology ¹, in which the quantized Hall conductivity observed in transport experiments is related to the first Chern number defined in the Brillouin zone (BZ) ². The associated quantum anomalous Hall effect has also been observed  ^{3, 4, 5, 6, 7} along with robust gapless edge states ^{4, 5, 8, 9}. In this work, we show that the nontrivial topology can appear locally in the BZ, even when the global topology is trivial. The idea relies on the concept of sub-Brillouin zone (sBZ) topology where the topological invariant is defined in a fraction of the BZ. We introduce a family of delicate topological systems ^{10, 11}, termed as the Chern dartboard insulators (CDIs), which exhibit the nontrivial sBZ topology. The $n$-th order Chern dartboard insulators, CDI_n for short, have quantized first Chern numbers inside $1/2n$ of the BZ. These reduced Chern numbers are protected by $n$ mirror symmetries (Fig. 1a). These systems cannot be captured by the theories of tenfold way ^{12, 13}, symmetric indicators ^{14, 15, 16} or topological quantum chemistry ^{17, 18}. In addition, all the CDIs exhibit multicellular and even noncompact topology ^{10, 11, 19}, i.e., the Wannier functions cannot be entirely localized to $\delta$-functions due to the reduced Chern numbers. Similar to the returning Thouless pump (RTP) insulators, all the CDIs can be captured by the quantized Berry phases along the high-symmetry lines (HSLs), and the topology can be trivialized by adding trivial atomic bands into either the occupied or unoccupied space.

Interestingly, for the two-band CDIs, skyrmion multipoles appear as a manifestation of the sBZ topology in the BZs. Contrary to Chern insulators that exhibit meron-antimeron pairs, the BZs of CDIs consist of HSLs which pin the pseudospins in the same direction. The (anti)skyrmions live in the sBZs bounded by the HSLs, thereby exhibiting the multipole structures (Fig. 2). The total number of (anti)skyrmions in the sBZ indeed corresponds to the reduced Chern number.

Here we observe that with the sBZ topology, all the CDIs host gapless edge states, even with Möbius fermions and midgap corner states in certain cases. However, contrary to Chern insulators, by including weak disorders (compared to the bulk gap) that obey mirror symmetries, one can in general gap out the edge states. In this sense, the edge (corner) states are as robust as the two-dimensional (2D) weak TIs or the multipole-moment insulators protected by crystalline symmetries ^{20, 21, 22, 23}. Nevertheless, under the sharp boundary condition, the gapless edge states can be protected at certain high-symmetry edges.

Theory. We aim at finding the possible topology protected by $n$ mirror symmetries $\mathcal{M}_{1},\mathcal{M}_{2},...,\mathcal{M}_{n}$. These symmetries divide the BZ into the irreducible BZs, which become the fundamental domain to define the topology. A possible realization is to consider the systems with the same mirror symmetry representation $\mathcal{M}_{1},...,\mathcal{M}_{n}=\sigma_{z}\otimes I$,

$\displaystyle\mathcal{M}_{i}H(\bm{k})\mathcal{M}_{i}^{-1}=H(R_{i}\bm{k}),$

(1)

where $\sigma_{z}$ and $I$ are the Pauli and identity matrix, and $R_{i}$ represent the mirror reflections in the $\bm{k}$ space. Here, the basis orbitals are chosen such that the mirror symmetry representation is diagonal. Therefore, the projection matrix onto the occupied space at the HSLs is in a block diagonal form. Notice that the system also has $C_{n}$ symmetry with trivial representation $C_{n}=I$.

Next, we consider the models in which all the occupied states at the HSLs have the same mirror representations. The blocks in the projection matrix are thus composed of zero and identity matrices, which correspond to the unoccupied and occupied space, respectively. Up to a $\bm{k}$-dependent $U(1)$ phase, the HSLs are mapped to a point in the Hilbert space, and each sBZ enclosed by the HSLs is topologically equivalent to a compact manifold. The first Chern number is thus well-defined in the irreducible BZ,

$$\mathcal{C}_{n}=-\frac{1}{2\pi}\int_{\text{irBZ}}d^{2}k\Tr F_{xy},$$

(2)

where $F_{xy}$ is the non-Abelian Berry curvature, and $\mathcal{C}_{n}$ is the reduced Chern number of the $n$-th order CDIs.

The simplest CDI₁ is protected by one mirror symmetry and has the quantized reduced Chern number inside half of the BZ. There are two types of CDI₁: Type I has the opposite mirror representations at the two different HSLs. An example is given by the two-band tight-binding Hamiltonian with the mirror symmetry $\mathcal{M}_{y}=\sigma_{z}$,

	$\displaystyle H^{I}_{1}(\bm{k})$		$\displaystyle=\cos k_{x}\sin k_{y}\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}$		(3)
			$\displaystyle+(m+\cos k_{y})\sigma_{z},$

where $m$ is a tunable parameter. In this model, the two HSLs sit at $k_{y}=0,\pi$. The basis orbitals consist of an $s$ orbital and a $p_{y}$ orbital. At $-1<m<1$, this model has a quantized reduced Chern number $\mathcal{C}_{1}=1$ inside the upper half BZ, and has a flat-band limit at $m=0$. Type I CDI₁ has a quantized bulk polarization if the total number of the occupied bands is odd ²⁰.

On the other hand, type II CDI₁ has the same mirror representations at the two different HSLs. A flat-band model is given by

	$\displaystyle H^{II}_{1}(\bm{k})$		$\displaystyle=\frac{1}{2}(1+\cos k_{x})\sin 2k_{y}\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}$		(4)
			$\displaystyle+\frac{1}{2}[(1+\cos k_{x})(\cos 2k_{y}-1)+2]\sigma_{z}.$

Interestingly, the bulk polarization $P_{x}=\int^{2\pi}_{0}\Tr A_{x}dk_{x}$ shows the RTP behavior ^{10, 11} in both cases. The RTP invariant is given by the difference of polarizations along the HSLs $\Delta P_{x}=P_{x}(k_{y}=\pi)-P_{x}(k_{y}=0)$. Notably, this invariant can be related to the quantized reduced Chern number through the Stokes theorem.

The sBZ topology can be visualized by mapping (3) to the Bloch sphere with $k_{x}\rightarrow\phi,k_{y}\rightarrow\theta$ ^{24, 25}. This mapping is only meaningful when $k_{y}\in[0,\pi]$. The upper half BZ is mapped to the entire sphere through the Mercator projection, thereby hosting the reduced Chern number $\mathcal{C}_{1}=1$ (Figs. 1b and 2a). Meanwhile, the lower half BZ $k_{y}\in[-\pi,0]$ is also mapped to the entire sphere, but now with a negative sign $\mathcal{C}_{1}=-1$ due to mirror symmetry. The quantization of the reduced Chern number inside the half BZ can be understood as follows: under the mirror symmetry $\mathcal{M}_{y}=\sigma_{z}$, the $k_{y}=0$ and $\pi$ HSLs are mapped to the north and south poles, respectively, as they have opposite representations. Therefore, the half BZ has a topology equivalent to $S^{2}$, and the reduced Chern number $\mathcal{C}_{1}$ is well-defined (Fig. 1b). In contrast to Type I CDI₁, Type II CDI₁ has well-defined skyrmions inside the half BZ, as shown in Figure 2b. The sBZ topology is more complicated as both the HSLs are mapped to the north pole, and it can be directly related to the existence of skyrmions.

Higher-order CDIs, on the other hand, are quite different from the CDI₁s as they cannot be captured by the RTP invariant. In particular, the symmetry representations of the valence bands (or the conduction bands) at all the HSLs are exactly the same as the symmetry representation of one of the basis orbitals. Figures 2c, 2d and 2e plot the peudospin textures of the two-band higher-order CDIs. All these cases have blue-centered skyrmions or antiskyrmions inside the irreducible BZs, which lead to the nontrivial reduced Chern number $\mathcal{C}_{n}=1$. Finally, we note that there are two types of CDI₃s with different configurations of the irreducible BZs. Here we plot Type I CDI₃ that corresponds to the configuration in Fig. 1a.

By inspecting the symmetry representations at the HSLs alone, one cannot detect the CDIs. However, the CDIs still have different band representations from the ones with $\delta$-like Wannier functions, since the homotopic inequivalence occurs from the quantized reduced Chern number. Moreover, the $n=2,4,6$ CDIs are noncompact atomic insulators, where the orthonormal Wannier functions cannot be strictly local and compact ¹⁹. Note that these models are quite different from the cases studied in Ref. ¹⁹, where the noncompactness arises from  the obstruction of the lattices that leads to obstructed atomic insulators. Here, the CDIs are not obstructed and the noncompactness arises due to the multicellularity.

Gapless edge states. Similar to the Chern insulators, the appearance of gapless edge states in CDIs can be explained by the domain walls, with the exception of Type I CDI₁s. The domain walls arise because of the band inversions inside the irreducible BZs. For Type I CDI₁s, there are no isolated band-inversion points and the gapless edge states only exist in the directions perpendicular to the HSLs. For other CDIs, there must exist isolated band-inversion points inside the irreducible BZ, owing to the quantized reduced Chern number $\mathcal{C}_{n}$. Inside the irreducible BZ, the topology shares similar behavior with regular Chern insulators. If one tries to close the gap by creating massive Dirac cones at the band-inversion points, the minimal low-energy Dirac Hamiltonian with mirror symmetry representation $\sigma_{z}\otimes I$ can be expressed as

$\displaystyle H(\bm{k},r)=k_{x}\Gamma_{x}+k_{y}\Gamma_{y}+m(r)\sigma_{z}\otimes I,$

(5)

where the three matrices $\Gamma_{x},\Gamma_{y},\sigma_{z}\otimes I$ anticommute with each other, and $m(r)$ is a position-dependent mass term. Since $m(r)<0$ inside the bulk, the gapless edge states appear as the domain walls between the bulk and the vacuum in which $m(r)>0$ ²⁶. In the two-band models, the band-inversion points are exactly associated with the skyrmion centers in Fig. 2.

It is worth emphasizing that there is a fundamental difference between the Chern insulators and the CDIs. In a Chern insulator with $\mathcal{C}=1$, we have only one band-inversion point inside the BZ. Thus, the chiral gapless edge states appear due to the domain walls between the bulk and the vacuum. However, for CDIs with $\mathcal{C}_{n}=1$, we have $2n$ band-inversion points with opposite signs of reduced Chern number inside the BZ under mirror symmetries. It follows that the edges host $n$ gapless edge states with positive chirality and $n$ gapless edge states with negative chirality near $E=0$, regardless of the edge terminations. The bulk-boundary correspondence for all the CDIs, except for Type I CDI₁, can be written as,

$\displaystyle N^{+}_{e}=N^{-}_{e}=n\mathcal{C}_{n},$

(6)

where $N^{\pm}_{e}$ is the minimal number of gapless edge states with positive (negative) chirality near $E=0$. Here, we emphasize again that this picture only works in the low-energy region. One can expect that the gapless edge states with positive and negative chiralities may rejoin at high energy such that they are disconnected from the bulk bands. In this sense, the gapless edge states in the CDIs are as robust as the delicate TIs. In Fig. 3, we plot the nanoribbon band structures for all the CDIs. It is easy to observe that the bulk-boundary correspondence Eq. (6) is satisfied. Finally, the edge states in Fig. 3c, Fig. 3d and Fig. 3f are doubly degenerate because of mirror symmetry that relates the two boundaries.

Midgap Corner States. By cutting the edges in the (1,1) and (1,-1) directions of Type I CDI₁, we obtain two protected midgap corner states with quantized $1/2$ charges. Notice that the unit cell is preserved, and the upper and lower corners have two atoms instead of one atom. Contrary to the 2D weak TIs where there are plenties of midgap edge states, Type I CDI₁ host two protected midgap corner states at the upper and lower corners (Fig. 4). This is similar to the midgap corner states in the model with a quantized polarization ²⁸, where the corner states appear at the corners that correspond to the direction of polarization. Therefore, Type I CDI₁ also serves as the simplest two-band models with protected midgap corner states.

Möbius Fermions. Interestingly, the edge states parallel to the $x$-axis in Type II CDI₁ show even more striking phenomenon. We expect that there are two chiral gapless edge states with opposite chirality near $E=0$, regardless of the edge terminations due to the band inversions inside the irreducible BZ. In the flat-band limit, the edge Hamiltonian describes the phase transition point of the four-band SSH model, and the edge states form the Möbius fermions ²⁹ (Fig. 3b). The energy of the two edge states are $E_{1}(k_{x})=\sin(k_{x}/2)$ and $E_{2}(k_{x})=-\sin(k_{x}/2)$. Clearly, they show the Möbius twist: the energies are only periodic in $k_{x}\in[0,4\pi]$, and a single edge Dirac cone appears at $k_{x}=0$. Since the edge Dirac cone is protected by the quantized reduced Chern number, the Dirac cone cannot disappear although the position can shift from $k_{x}=0$. The $e^{ik_{x}/2}$ dependence is also clearly shown in the edge states (see Supplementary Note 2 for details). In addition to Type II CDI₁, the Möbius fermions also appear in both types of CDI₃s (Fig. 3f). Finally, we note that the Möbius fermions can only appear with odd numbers of Dirac cones in the CDIs.

We introduce a novel concept of sBZ topology that manifests itself in different classes of CDIs. CDIs host gapless edge states in general and can even develop nontrivial Möbius fermions or midgap corner states in certain cases. Although here we only consider specific examples of the sBZ topology, one can easily generalize the same argument to systems in higher dimensions or with different symmetries/constraints. The sBZ topology thus opens a fertile area for new topological systems with nontrivial surface responses. The different physical properties from the global topology make the realization and classification of the sBZ topology a new frontier of research in topological materials. Thanks to the recent advances in detecting local Berry curvature in various systems ^{30, 31}, and in realizing topological phases in nanophotonic silicon ring resonators ³², the realization and observation of CDIs and their exotic edge states is expected in the near future.

Conventions and definitions. With translational symmetry, the second quantized tight-binding Hamiltonian can be written into the Bloch form,

$\displaystyle\hat{H}=\sum_{\bm{k}}c^{\dagger}_{i,\bm{k}}\big{[}H(\bm{k})\big{]}_{ij}c_{j,\bm{k}},$

(7)

where

$\displaystyle c_{j,\bm{k}}=\frac{1}{\sqrt{N_{t}}}\sum_{\bm{R}}e^{i\bm{k}\cdot\bm{R}}c_{j,\bm{R}}$

(8)

is the electron annihilation operator. Here, $j=1,...,2N$ labels the basis orbitals and spins, $\bm{R}$ labels the unit cell position, and $N_{t}$ is the total number of the unit cells. We use the convention that the Bloch Hamiltonian $H(\bm{k})$ is periodic under a translation of a reciprocal lattice vector $\bm{G}$:

$\displaystyle H(\bm{k})=H(\bm{k}+\bm{G}).$

(9)

The intra-cell eigenstates are defined by:

$\displaystyle H(\bm{k})|u^{l}(\bm{k})\rangle=E^{l}(\bm{k})|u^{l}(\bm{k})\rangle,$

(10)

where $l=1,...,2N$ are the band indices and $E^{l}(\bm{k})$ is the eigenenergy. Note that one can generically choose a smooth and periodic gauge for the eigenstates of CDIs since the total Chern number is zero. Considering the half-filling band insulators, the intra-cell states can be decomposed into the valence states $|u^{l}_{v}\rangle$ and the conduction states $|u^{l}_{c}\rangle$, where $l=1,...,N$. The Bloch state is given by $|\psi^{l}(\bm{k})\rangle=e^{i\bm{k}\cdot\bm{R}}|u^{l}(\bm{k})\rangle$.

The non-Abelian Berry connection for the valence bands is defined as:

$\displaystyle\bm{A}_{lm}(\bm{k})=i\langle u^{l}_{v}(\bm{k})|\bm{\nabla}|u^{m}_{v}(\bm{k})\rangle,$

(11)

and the non-Abelian Berry curvature in two dimensions is:

$\displaystyle F_{xy,lm}=\partial_{x}A_{y,lm}-\partial_{y}A_{x,lm}-i\big{[}A_{x},A_{y}\big{]}_{lm}.$

(12)

Skyrmion number. The topology of two-band CDIs can be visualized using the psudospin textures. We first expand the Bloch Hamiltonian into the Pauli matrices:

$\displaystyle H(\bm{k})=\sum_{i}d_{i}(\bm{k})\sigma_{i},$

(13)

where $\sigma_{i}$ with $i=x,y,z$ are the Pauli matrices. The reduced Chern number can be defined as the degree of the map from the irreducible BZ to $S^{2}$:

$\displaystyle\mathcal{C}_{n}=-\frac{1}{4\pi}\int_{\text{irBZ}}d^{2}k\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial k_{x}}\times\frac{\partial\bm{n}}{\partial k_{y}}\right)$

(14)

where $\bm{n}(\bm{k})=\bm{d}(\bm{k})/|\bm{d}(\bm{k})|$ are the unit vectors that define the space of $S^{2}$. The reduced Chern number measures how many times the irreducible BZ wraps around $S^{2}$.

The peudospin textures of CDIs are plotted in Fig. 2. The first Chern number can be calculated as the sum of the indices around either $\bm{n}_{0}=(0,0,1)$ or $\bm{n}_{0}=(0,0,-1)$ inside the BZ, with a sign difference ³³. Note that this choice is just for convenience as one can easily do an arbitrary unitary transformation. The reduced Chern number is

$\displaystyle\mathcal{C}_{n}=\sum_{i}S_{i},$

(15)

where $S_{i}$ is the index around the south pole $\bm{n}_{0}=(0,0,-1)$ with the blue center in the plot, and $i$ denotes the different skyrmions inside the BZ. Since the irreducible BZ is a two-dimensional manifold, the indices can be calculated simply as the winding numbers of the vectors around the south pole. Notice that the Poincaré–Hopf theorem constrains the total indices including those around the north pole $\bm{n}_{0}=(0,0,1)$ are summed to zero. This is consistent with Eq. (14) since a nonzero Chern number implies the north pole and the south pole must be wrapped around nontrivially.

Tight-binding models. Here we list the two-band spinless tight-binding models used to construct the figures in the main text for CDIs. The $n=1,2,4$ CDIs are built in the simple square lattice with primitive lattice vectors $\bm{a}_{1}=(1,0)$ and $\bm{a}_{2}=(0,1)$ in the unit of a lattice constant. The CDI₃ is built in the triangular lattice with primitive lattice vectors $\bm{a}_{1}=(1,0)$ and $\bm{a}_{2}=(1/2,\sqrt{3}/2)$ in the unit of a lattice constant. The basis orbitals are all placed on the atoms. The numerical results of the midgap corner states and the nanoribbon band structures are performed using the P_YTHTB package ³⁴.

Type I CDI₁:

	$\displaystyle H^{I}_{1}(\bm{k})$		$\displaystyle=\cos k_{x}\sin k_{y}\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}$		(16)
			$\displaystyle+(m+\cos k_{y})\sigma_{z},$

where $m$ is a parameter. The basis orbital consists of a $s$ orbital and a $p_{y}$ orbital. The Hamiltonian has the mirror symmetry $\mathcal{M}_{y}=\sigma_{z}$. For $-1<m<1$, this model has a quantized reduced Chern number $\mathcal{C}_{1}=1$ inside the upper half BZ, see Fig. 1. The system has a flat-band limit when $m=0$.

Type II CDI₁:

	$\displaystyle H^{II}_{1}(\bm{k})$		$\displaystyle=\cos k_{x}\sin 2k_{y}\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}$		(17)
			$\displaystyle+(m+\cos 2k_{y}-\cos k_{x})\sigma_{z},$

where $m$ is a parameter. The basis orbital consists of a $s$ orbital and a $p_{y}$ orbital. The Hamiltonian has the mirror symmetry $\mathcal{M}_{y}=\sigma_{z}$. For $0<m<2$, this model has a quantized reduced Chern number $\mathcal{C}_{1}=1$ inside the upper half BZ, see Fig. 1. The system has a flat-band limit in the following form:

	$\displaystyle H^{IIc}_{1}(\bm{k})$		$\displaystyle=\frac{1}{2}(1+\cos k_{x})\sin 2k_{y}\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}$		(18)
			$\displaystyle+\frac{1}{2}[(1+\cos k_{x})(\cos 2k_{y}-1)+2]\sigma_{z}.$

CDI₂:

	$\displaystyle H_{2}(\bm{k})=$		$\displaystyle-\sin k_{x}\sin 2k_{y}\sigma_{x}+\sin 2k_{x}\sin k_{y}\sigma_{y}$		(19)
			$\displaystyle+(m+\cos 2k_{x}+\cos 2k_{y})\sigma_{z}.$

where $m$ is a parameter. The basis orbital consists of a $s$ orbital and a $d_{xy}$ orbital. The Hamiltonian has two mirror symmetries $\mathcal{M}_{x}=\mathcal{M}_{y}=\sigma_{z}$. For $0<m<2$, this model has a quantized reduced Chern number $\mathcal{C}_{2}=1$ inside the upper right quarter of the BZ, see Fig. 1. We set $m=1.0$ for all the figures.

CDI₄:

	$\displaystyle H_{4}(\bm{k})=($		$\displaystyle-\sin k_{x}\sin 4k_{y}+\sin 4k_{x}\sin k_{y})\sigma_{x}$
	$\displaystyle+($		$\displaystyle\sin 2k_{x}\sin 4k_{y}-\sin 4k_{x}\sin 2k_{y})\sigma_{y}$
	$\displaystyle+\big{[}$		$\displaystyle m+\cos 2k_{x}+\cos 2k_{y}+\cos 4k_{x}$		(20)
			$\displaystyle+\cos 4k_{y}+4\cos k_{x}\cos k_{y}\big{]}\sigma_{z},$

where $m$ is a parameter. The basis orbital consists of a $s$ orbital and a second orbital that is odd under four mirror symmetries. The Hamiltonian has four mirror symmetries $\mathcal{M}_{x}=\mathcal{M}_{y}=\mathcal{M}_{x+y}=\mathcal{M}_{x-y}=\sigma_{z}$. For $2<m<4$, this model has a quantized reduced Chern number $\mathcal{C}_{4}=1$ inside the irreducible BZ, see Fig. 1. We set $m=3.0$ for all the figures.

Type I CDI₃:

	$\displaystyle H_{3}(\bm{k})=\big{[}$		$\displaystyle\sin\frac{5}{2}k_{x}\sin\frac{\sqrt{3}}{2}k_{y}-\sin 2k_{x}\sin\sqrt{3}k_{y}$
			$\displaystyle+\sin\frac{1}{2}k_{x}\sin\frac{3\sqrt{3}}{2}k_{y}\big{]}\sigma_{x}$
	$\displaystyle+\big{[}$		$\displaystyle-\cos\frac{5}{2}k_{x}\sin\frac{\sqrt{3}}{2}k_{y}-\cos 2k_{x}\sin\sqrt{3}k_{y}$
			$\displaystyle+\cos\frac{1}{2}k_{x}\sin\frac{3\sqrt{3}}{2}k_{y}\big{]}\sigma_{y}$
	$\displaystyle+\bigg{[}$		$\displaystyle m+\sum_{a=1}^{6}\big{(}e^{i\bm{t}_{2}(a)\cdot\bm{k}}+\frac{1}{2}e^{2i\bm{t}_{2}(a)\cdot\bm{k}}\big{)}\bigg{]}\sigma_{z},$		(21)

where $m$ is a parameter and $\bm{t}_{2}(a)=\sqrt{3}[\cos(\pi a/3-\pi/6),\sin(\pi a/3-\pi/6)]^{T}$. The basis orbitals consist of a $s$ orbital and a $f_{y(3x^{2}-y^{2})}$ orbital. The Hamiltonian has three mirror symmetries $\mathcal{M}_{y}=C_{3}\mathcal{M}_{y}C_{3}^{-1}=C_{3}^{2}\mathcal{M}_{y}C_{3}^{-2}=\sigma_{z}$. The $\sigma_{z}$ term contains the hoppings with range $\bm{a}_{1}+\bm{a}_{2}$, $2(\bm{a}_{1}+\bm{a}_{2})$, and also the ones generated by all the $C_{6}$ rotations. The $\sigma_{x}$ and $\sigma_{y}$ terms contain the hoppings with range $\bm{a}_{1}+2\bm{a}_{2}$ and the ones generated by three mirror symmetries. For $0<m<4.5$, this model has a quantized reduced Chern number $\mathcal{C}_{3}=1$ inside the irreducible BZ, see Fig. 1. We set $m=2.0$ for all the figures.

The data for all the figures are all available by requesting.

The authors thank Yi-Chun Hung for helpful discussions. Y.C.C. and Y.J.K. were partially supported by the Ministry of Science and Technology (MOST) of Taiwan under grants No. 108-2112-M-002-020-MY3, 110-2112-M-002-034-MY3, 111-2119-M-007-009, and by the National Taiwan University under Grant No. NTU-CC-111L894601. Y.P.L. received the fellowship support from the Emergent Phenomena in Quantum Systems (EPiQS) program of the Gordon and Betty Moore Foundation.

Y.C.C conceived the ideas and performed the theoretical and numerical analysis. Y.P.L provided the idea to look at the pseudospin textures and Möbius fermions. Y.J.K. supervised the project. Y.C.C, Y.P.L and Y.J.K. wrote the manuscript.

The authors declare no competing interests.