# Research Progress of Topological Quantum Materials: From First-Order to Higher-Order

^{*}

## Abstract

**:**

## 1. Introduction

## 2. First-Order Topological Quantum Phase

#### 2.1. Quantum Hall Effect

**k**for the i-th energy band, and ${\overline{u}}^{i}(\mathit{k},\mathit{r})$ denotes the complex conjugate of the Bloch state. The index i spans all energy bands below the Fermi energy level. It has been mathematically proven that n

_{H}is an exact integer known as the TKNN number [17], serving as a topological invariant characterizing the quantum Hall effect. Equation (1) is closely associated with the first Chern class in topology [20], introduced by Shiing-Shen Chern. Consequently, the TKNN numbers are often referred to as Chern numbers [19]. An insulator with a Chern number of zero is classified as a trivial insulator, while a non-zero Chern number designates a Chern insulator. The integral term in Equation (1) corresponds to the Berry curvature, a quantity that describes the intrinsic property of Bloch states within the first Brillouin zone.

#### 2.2. Quantum Spin Hall Effect

_{SO}. The third term introduces Rashba SOC, characterized by a strength of λ

_{R}, which breaks the mirror symmetry about the z-axis. The fourth term introduces a staggered sublattice potential that breaks the twofold rotational symmetry within the plane. The strength of the potential on the A (B) sublattice is denoted by λ

_{ν}, with ${\xi}_{i}=+1(-1)$ representing the onsite energy difference of the A (B) sublattice. Moreover, ${\nu}_{ij}=(2/\sqrt{3}){({\mathit{d}}_{il}\times {\mathit{d}}_{lj})}_{z}=\pm 1$ and

**d**

_{ij}represents the lattice vector pointing from lattice site j to i, and

**s**is the spin Pauli operator. σ and σ’ represent the spin index, and $\overline{\sigma}=-\sigma $. This is the Kane-Mele model capturing the quantum phase transition between a quantum spin Hall insulator (a first-order topological insulator) and an ordinary insulator in graphene.

_{2}topological index to describe this spin edge state, distinguishing it from trivial insulating states. Therefore, all two-dimensional insulators that preserve time reversal symmetry can be categorized into two classes based on their Z

_{2}index: trivial insulators with Z

_{2}= 0 and topological insulators with Z

_{2}= 1.

_{2}, Kane and Mele [22] introduced a matrix, denoted as m(

**k**), with matrix elements that are given by the expression:

**k**) is an antisymmetric matrix. In the case of an antisymmetric matrix, we can compute its Pfaffian (Pf) [22]:

**k**) within the Brillouin zone are discrete, the topological index Z

_{2}is determined by the parity of the number of zeros in half of the Brillouin zone, denoted as B

^{+}. On the other hand, if the zeros of P(

**k**) are continuous, Z

_{2}is determined by the parity of half the number of sign changes of P(

**k**) along the boundary of B

^{+}. These two cases can be expressed in a unified manner as follows:

^{2}/h.

#### 2.3. Quantum Anomalous Hall Effect

_{2}Se

_{3}, it could be possible to achieve this phenomenon [25]. Given the experimental availability of Bi

_{2}Se

_{3}materials, the introduction of dopants such as Cr or Fe ions in the Bi

_{2}Se

_{3}thin films was considered. Through the ferromagnetic exchange mechanism between these dopants, the thin films could attain a stable ferromagnetic insulating state with well-defined band gaps and quantized edge states. This makes the doped Bi

_{2}Se

_{3}system the most promising candidate for realizing the quantum anomalous Hall effect.

_{2}Se

_{3}. These impurities prevent the strict requirements for achieving the quantum anomalous Hall effect from being met. After years of hard working, Xue’s group observed the quantum anomalous Hall effect experimentally for the first time without the need for an external magnetic field in the magnetically doped thin film of Cr

_{0.15}(Bi

_{0.1}Sb

_{0.9})

_{1.85}Te

_{3}[26]. The Hall conductance exhibits quantization in the region where the longitudinal conductance is zero, as illustrated in Figure 5.

#### 2.4. Three-Dimensional Topological Insulator

_{x}Sb

_{1−x}alloy [29,30,31]. Subsequently, the Bi

_{2}Se

_{3}family, including Bi

_{2}Te

_{3}and Sb

_{2}Te

_{3}, with space group ${D}_{3d}^{5}$ ($R\overline{3}m$) was also discovered [32,33,34,35]. Compared to the Bi

_{x}Sb

_{1−x}alloy, the Bi

_{2}Se

_{3}samples [32,33] are relatively easier to prepare and exhibit topological insulator behavior even at high temperatures, with a band gap larger than 0.1 eV. Notably, the band gap of Bi

_{2}Se

_{3}is approximately 0.3 eV (equivalent to 3600 K), which greatly exceeds the room temperature energy scale, making it a potential candidate for low-loss spintronic devices at room temperature. First-principles calculations [36] have revealed that the introduction of SOC results in a band inversion at the Γ point, predicting the topological insulator properties of Bi

_{2}Se

_{3}, Bi

_{2}Te

_{3}, and Sb

_{2}Te

_{3}. However, Sb

_{2}Se

_{3}is considered a trivial insulator due to the weak SOC provided by Sb atoms.

_{2}Se

_{3}is shown in Figure 6b. It is worth noting that the surfaces of three-dimensional topological insulators do not exhibit complete resistance to backscattering, unlike their two-dimensional counterparts. Even in the absence of magnetic impurities, electrons with the same helical chirality in edge states can still undergo scattering. However, their topological nature guarantees that the metallic behavior does not easily disappear in these systems.

_{2}Se

_{3}, Bi

_{2}Te

_{3}, and Sb

_{2}Te

_{3}are highly desirable for research and experimental investigations due to their ease of growth and handling. These materials serve as ideal platforms for studying and verifying various properties of surface states, including ideal transport without backscattering [37], weak anti-localization effect [38], and more. Furthermore, magnetically doped films of these materials have been utilized to experimentally demonstrate the quantum anomalous Hall effect [39].

#### 2.5. Topological Semimetal

#### 2.5.1. Dirac Semimetal

_{3}Bi is a notable example of a three-dimensional topological Dirac semimetal, initially predicted by theory [41] and subsequently confirmed by experimental studies [66,67,68,69]. It possesses time reversal and threefold rotational symmetry, with two Dirac points along the Γ–A direction. Experimental observations, using techniques such as ARPES [66,67], have revealed the presence of Dirac cones and Fermi arcs connecting the bulk Dirac points, as depicted in Figure 7. Furthermore, transport measurements have demonstrated intriguing phenomena, including negative magnetoresistance attributed to chiral anomalies [68] and quantum oscillation [69]. However, Na

_{3}Bi’s instability and sensitivity to air have posed challenges for further investigations. Another prominent three-dimensional Dirac semimetal, Cd

_{3}As

_{2}[70], is protected by time reversal and fourfold rotational symmetry, with Dirac points located along the Γ–A direction. This has been confirmed by ARPES experiments [71,72,73]. In addition, other intriguing phenomena have also been observed in Cd

_{3}As

_{2}, such as quantum oscillation [74] and negative magnetoresistance arising from chiral anomalies [75,76]. Notably, in 2017, the observation of a three-dimensional quantum Hall effect in Cd

_{3}As

_{2}nanosheets [77,78] provided a significant breakthrough, expanding the study of the quantum Hall effect beyond two-dimensional systems.

#### 2.5.2. Weyl Semimetal

_{2}[87], MoTe

_{2}[88,89,90,91,92], MoP

_{2}[93], and TaIrTe

_{4}[94]. Fermi arcs and chiral anomalies are considered essential characteristics of Weyl semimetals, and their existence can be verified through ARPES and transport experiments.

_{2}, on the other hand, has garnered significant attention as the first predicted type-II Weyl semimetal. ARPES experiments have successfully captured the Fermi arc in WTe

_{2}[95], while transport studies have demonstrated unsaturated giant magnetoresistance and anisotropic negative magnetoresistance effects [96]. Subsequent experiments have further unveiled intriguing phenomena, including quantum oscillation [97] and the emergence of superconducting states under high pressure [98]. These remarkable characteristics establish WTe

_{2}as an excellent platform for exploring diverse quantum phenomena.

#### 2.5.3. Nodal Line Semimetal

_{2}[105]). Notably, nodal line semimetals exhibit distinctive nontrivial topological surface states known as drumhead-type states, contrasting the Fermi arc surface states observed in Dirac and Weyl semimetals. Furthermore, nodal line semimetals serve as a bridge for the transformation between different topological states. Figure 9 illustrates the relationship between the symmetries and electronic structures of different topological states. By introducing SOC and breaking crystal symmetries, various topological semimetals and topological insulators can be realized based on a spinless Dirac nodal line semimetal [106]. For instance, a spinless Dirac nodal line semimetal can be transformed into a Dirac semimetal by combining time reversal, inversion symmetry, and n-fold rotational symmetry. However, it is important to note that the realization of Dirac semimetals and other topological states is not limited to the method shown in Figure 9.

## 3. Higher-Order Topological Quantum Phase

#### 3.1. Topological Crystalline Insulator

_{4}and time reversal symmetry using a tight-binding model. He introduced a new Z

_{2}invariant based on the C

_{4}rotational invariants and discovered gapless topological surface states protected by nontrivial Z

_{2}invariants on the (001) surface while maintaining C

_{4}symmetry. Further research on topological crystalline insulators revealed that among 230 space groups, there generally exist seven topological invariants related to crystal symmetries, including translation, mirror reflection, glide reflection, rotation, screw rotation, spatial inversion, and S

_{4}symmetry.

_{1−x;}Sn

_{x;}Se [112,113].

_{2}topological invariant can be defined, in addition to the time-reversal Z

_{2}invariant, to characterize the system’s topology [116,117,118].

_{n}rotational symmetry (n = 2, 4, and 6). Typically, any two-dimensional lattice with time reversal and C

_{n}symmetries should feature 2n Dirac cones. However, in this peculiar state, only n massless Dirac fermions are protected by time reversal and C

_{n}[119]. Subsequently, it was discovered that Ba

_{3}Cd

_{2}As

_{4}, Ba

_{3}Zn

_{2}As

_{4}, and Ba

_{3}Cd

_{2}Sb

_{4}represent a class of topological crystalline insulators exhibiting a C

_{2}rotational anomaly. These compounds showcase two surface Dirac cones on both upper and lower surfaces perpendicular to the rotation axis. Additionally, (d−2)-dimensional helical hinge states exist along the edges parallel to the rotation axis, connecting the anomalous Dirac cones on the upper and lower surfaces [120].

#### 3.2. Higher-Order Topological Insulator

#### 3.2.1. Quantized Quadrupole Topological Insulator

**r**) are defined as ${p}_{i}={\displaystyle \int {d}^{3}\mathit{r}\rho (\mathit{r}){r}_{i}}$, ${q}_{ij}={\displaystyle \int {d}^{3}\mathit{r}\rho (\mathit{r}){r}_{i}}{r}_{j}$ and ${o}_{ijk}={\displaystyle \int {d}^{3}\mathit{r}\rho (\mathit{r}){r}_{i}}{r}_{j}{r}_{k}$, respectively [136]. In the modern theory of polarization in crystals, the dipole moment p

_{i}corresponds to the Berry phase of the bulk electronic states:

**A**is characterized by components ${\left[{A}_{i}(\mathit{k})\right]}^{mn}=-i\left.\u2329{u}_{k}^{m}\right|{\partial}_{{k}_{i}}\left|{u}_{k}^{n}\right.\u232a$, where $\left|{u}_{k}^{n}\right.\u232a$ represents the Bloch function of band n, and the index n is limited to the occupied energy bands. The dipole moment p

_{i}is essentially associated with the presence of surface charge. Benalcazar et al. [134,135] derived the electromagnetic properties of a two-dimensional insulator with a square shape and a three-dimensional insulator with a cubic shape, having only a nonvanishing q

_{ij}or o

_{ijk}as:

_{i}= 0) or be positioned halfway between centers (p

_{i}= ±e/2). Any other position of the electron would violate the inversion symmetry, as depicted in Figure 13. As a consequence, the correlation between the multipole moments in Equation (7) leads to the quantization of q

_{ij}and o

_{ijk}as well.

#### 3.2.2. Two-Dimensional Su–Schrieffer–Heeger (SSH) Model

#### 3.2.3. Three-Dimensional Higher-Order Topological Insulator

_{4}) while preserving the C

_{4}T symmetry, a higher-order topological insulator with chiral one-dimensional gapless hinge states can be achieved. This is illustrated in Figure 17. Additionally, Langbehn et al. [149] proposed a three-dimensional second-order topological insulator with broken time reversal symmetry.

_{2}(X = Mo, W) [151], and shaft insulators like Bi

_{2−x}Sm

_{x}Se

_{3}[152] and EuIn

_{2}As [153]. Additionally, two-dimensional materials like twisted bilayer graphene [154,155], graphdiyne [156,157,158], graphyne [159], monolayer FeSe [160], and covalent organic frameworks [161] have also been considered. Notably, in 2022, our research group made a significant discovery regarding the unique coexistence of topological electron and phonon behavior in graphdiyne [138].

#### 3.3. Higher-Order Topological Semimetal

_{z}in Equation (8) [173]:

_{x,y}represents the intra-unit cell coupling along the x and y directions. In Figure 18b, the k

_{z}-dependent energy spectrum of the system with open boundaries in both the x and y directions is shown. It reveals a fourfold-degenerate zero-energy flat band that terminates at the nodes. This confirms the presence of hinge Fermi arc states connecting the projection of bulk Dirac points. Therefore, the system exhibits second-order topologically nontrivial boundary states characteristic of a higher-order topological semimetal.

_{z}direction. Unlike previously known topological semimetal phases that possess protected states only at fixed-order boundaries, this new topological phase exhibits topological states both at the surface and at the hinge regions, as depicted in Figure 19. This unique behavior arises from the bulk of the material being protected by two nontrivial topological charges, namely the real Chern number υ

_{R}(also called the second Stiefel-Whitney number [177,178]) and the one-dimensional winding number ω. The definitions of these charges are provided as follows.

_{2}-valued real Chern number υ

_{R}is defined as a horizontal two-dimensional plane parallel to the concerned nodal ring. It can be determined by extracting the parity eigenvalues at time reversal invariant momentum points on this plane and then using the following equation [179,180]:

_{R}is nontrivial (υ

_{R}= 1) for one plane and trivial (υ

_{R}= 0) for another plane, it indicates a switch in topology along k

_{z}, as depicted in Figure 19. This switch implies the presence of nodal lines between the two planes. Moreover, within a specific k

_{z}region (−k

_{R}, k

_{R}) bounded by two nodal rings, if each plane in the region exhibits a nontrivial υ

_{R}= 1, i.e., a two-dimensional real Chern insulator, they necessarily possess protected zero modes at a pair of PT-connected corners. Collectively, the corner zero modes from all these nontrivial planes form two hinge Fermi arcs within the three-dimensional system.

_{2}-valued topological invariant ω, which corresponds to the quantized π Berry phase:

**A**represents the Berry connection for the occupied bands and C denotes a closed path encircling the ring (as depicted by the blue cycle with an arrow in Figure 19). Physically, a nontrivial ω (ω = 1) gives rise to a pair of drumhead surface bands confined within the projected nodal rings. This is in contrast to early stage nodal line semimetals, which typically exhibit a single drumhead surface band.

_{R}and ω, give rise to distinct topological boundary modes at different boundaries. The real Chern number υ

_{R}manifests as a pair of hinge Fermi arcs that connect the projected bulk nodal lines on PT-connected hinges, serving as a fingerprint of the second-order topology. On the other hand, the topological charge ω guarantees the existence of double drumhead surface bands.

_{2}(X = Mo, W) [151] and three-dimensional ABC-stacked graphene [157,174,177] have the potential to exhibit higher-order nodal line semimetal behavior. However, the experimental realization of higher-order Weyl or Dirac semimetals has, thus far, been limited to acoustic and photonic crystals [168,170,171], with their realization in condensed matter systems still remaining elusive.

## 4. Summary and Discussion

_{2}[183], and Bi

_{4}Br

_{4}[184]. (3) The in-plane Zeeman field mechanism. By applying an in-plane magnetic field to a topological insulator, time reversal or crystal symmetries are broken, inducing the helical Dirac boundary states or surface states to open the higher-order topological energy gaps [153,185,186,187,188]. While some magnetic higher-order topological insulators are expected to exhibit this mechanism, experimental confirmation is still lacking. (4) The structure bending mechanism. This mechanism involves bending the planar structure of a two-dimensional topological crystalline insulator to break crystal symmetry and induce the opening of higher-order topological energy gaps. β-Sb monolayers [189] are proposed as materials where this mechanism can be observed, but experimental verification is still needed. Additionally, the research on the formation mechanisms of higher-order topological insulators in phononic systems is still largely unexplored. Therefore, it is crucial to uncover more formation mechanisms of higher-order topological insulators and identify additional material systems to accelerate progress in experimental detection.

_{4}Br

_{4}[184]. (2) Ezawa suggested that stacking two-dimensional weak topological insulators with varying interlayer coupling strengths can realize both strong and weak three-dimensional higher-order topological insulator phases [190]. (3) Hughes et al. constructed a three-dimensional higher-order topological semimetal by stacking two-dimensional quadrupole square lattices [172]. (4) Liu et al. proposed that applying an electric field can break the mirror symmetry along the (001) crystal direction in SnTe thin films, resulting in a gap in the boundary states [191]. (5) Chen et al. suggested that applying in-plane strain can amplify the influence of twist angles on the phononic structure of twisted multilayer graphene, providing an effective means of manipulating the phononic structure [192]. (6) Jiang et al. proposed that applying biaxial strain to monolayer hexagonal boron nitride can effectively adjust the position of the topological phonon band gap [193]. These theoretical studies demonstrate that factors such as stress, electric field, and stacking can effectively control the electronic and phononic structures of topological materials. Consequently, this opens up new avenues for further research on higher-order topological phases. By exploring the possibilities of topological phase transitions through the application of stress, external fields, interlayer coupling, and other external factors, it becomes possible to manipulate the original electronic and phononic structures of the systems.

- (1)
- Studying the models of two-dimensional higher-order topological states and exploring suitable material systems. This entails conducting in-depth theoretical analyses and systematic explorations of the underlying physical mechanisms based on lattice models. It is crucial to recognize that the topological properties of electronic and phononic material systems are influenced by distinct factors. In the case of electronic systems, these factors encompass the lattice structure, atomic orbital types, and SOC. Whereas, for phononic systems, the relevant factors include the lattice structure and atomic vibration modes.
- (2)
- Investigating the control of two-dimensional higher-order topological states through external manipulation. Building upon existing models and discovered real materials for two-dimensional higher-order topological states, studies should be conducted of the influence of various external factors (such as stress, electric field, magnetic field, and stacking) on the electronic and phononic structures. Future works should also study potential higher-order topological phase transitions, analyze the underlying physical mechanisms, and derive applicable rules and guidelines.
- (3)
- Exploring the novel category of higher-order topological phases that involve the coexistence of electrons and phonons. Studies can utilize the breathing lattice mechanism to explore this new type of higher-order topological phase where electrons and phonons coexist, as well as uncovering the underlying formation mechanism, thereby establishing a platform for investigating the interplay between electronic and phononic higher-order topological states. Additionally, studies should be conducted of the potential for higher-order topological superconductivity within this context.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The integer quantum Hall edge state in two-dimensional electronic systems under a strong magnetic field. (

**a**) Spatial image. (

**b**) Energy band image.

**Figure 2.**The quantum spin Hall edge states in two-dimensional electronic systems. (

**a**) Spatial image. (

**b**) Energy band image.

**Figure 3.**Calculation results for a zigzag-type graphene nanoribbon. (

**a**) The energy spectrum of the quantum spin Hall phase. The edge states appearing in the bulk band gap cross at ka = π. (

**b**) The energy spectrum of the trivial insulating phase. Reprinted with permission from [22]. Copyright 2005 American Physical Society.

**Figure 4.**The quantum anomalous Hall edge states in magnetically doped thin films. (

**a**) Spatial image. (

**b**) Energy band image.

**Figure 5.**Experimental measurement results of the quantum anomalous Hall effect. The magnetic field dependence of (

**a**) Hall resistivity and (

**b**) longitudinal resistivity at different gate voltages at 30 mK. Reprinted with permission from [26]. Copyright 2013 American Association for the Advancement of Science.

**Figure 6.**(

**a**) The helical Dirac cone surface states of three-dimensional topological insulators. The green and blue arrows indicate the distribution of electronic spins. (

**b**) The electronic structure of the topological insulator Bi

_{2}Se

_{3}. The top image shows the band dispersion of the (111) surface measured via angle-resolved photoelectron spectroscopy (ARPES). The bottom image shows the three-dimensional Brillouin zone and the two-dimensional projected Brillouin region for the surface. Reprinted with permission from [33]. Copyright 2009 Nature Publishing Group.

**Figure 7.**(

**a**) Crystal structure of Na

_{3}Bi. Reprinted with permission from [66]. Copyright 2014 American Association for the Advancement of Science. (

**b**) Fermi surface map of the Na

_{3}Bi sample at photon energy 55 eV. DP1 and DP2 denote the two bulk Dirac points. Reprinted with permission from [67]. Copyright 2015 American Association for the Advancement of Science.

**Figure 8.**Electronic structure of the (001) surface state in TaAs. (

**a**) Curvature intensity plots of ARPES data. (

**b**) Fermi surface of the TaAs (001) surface, showing the locations of projected Weyl points. Reprinted with permission from [44]. Copyright 2015 American Physical Society.

**Figure 10.**Periodic table of topological insulators and superconductors. δ: = d−D, where d is the space dimension and D+1 is the codimension of defects; the leftmost column (A, AIII, …, CI) denotes the ten symmetry classes of fermionic Hamiltonians, which are characterized by the presence or absence of time reversal (T), particle-hole (C), and chiral (S) symmetries of different types denoted by ±1. Reprinted with permission from [109]. Copyright 2016 American Physical Society.

**Figure 12.**Quadrupole and octupole moments. (

**a**) Bulk quadrupole moment q

_{xy}and its boundary polarizations p

_{i}and corner charges Q. (

**b**) Bulk octupole moment o

_{xyz}and its surface quadrupoles q

_{ij}, hinge polarizations p

_{i}, and corner charges Q. Reprinted with permission from [134]. Copyright 2017 American Association for the Advancement of Science.

**Figure 13.**Two one-dimensional lattices with one atomic site (blue dots) and one electron (red circles) per unit cell. The electric dipole moment p

_{i}is ±e/2 in (

**a**) and 0 in (

**b**) due to the difference in the electron center of charge.

**Figure 14.**(

**a**) The tight-binding model with a quantized quadrupole moment q

_{xy}. γ and λ represent two hopping strengths and the dashed lines represent hopping terms with negative signs. (

**b**) The energy spectrum as a function of γ/λ for a system with open boundary conditions in the x and y directions and the electron charge density distribution in the nontrivial phase. Reprinted with permission from [134]. Copyright 2019 Nature Publishing Group.

**Figure 15.**(

**a**) Schematic of the photonic quadrupole topological system composed of ring resonators. (

**b**) In the case of a non-zero quantized quadrupole moment, the measured spatial intensity profile shows the localized corner modes. (

**c**) In the case of a zero quantized quadrupole moment, the measured spatial intensity profile shows the corner modes coupled into the bulk modes. Reprinted with permission from [137]. Copyright 2019 Nature Publishing Group.

**Figure 16.**(

**a**) Schematic diagram of the photonic crystal structure with a Kagome lattice. (

**b**) Experimentally measured density of states. (

**c**) Type I corner state. (

**d**) Type II corner state. Reprinted with permission from [147]. Copyright 2020 Nature Publishing Group.

**Figure 17.**(

**a**) Hinge states of the three-dimensional second-order topological insulator with broken time reversal symmetry. The red and blue arrows indicate chiral hinge currents with inverse directions. (

**b**) The energy spectrum of the three-dimensional chiral higher-order topological insulator, where red lines are the spectrum of hinge states. Reprinted with permission from [126]. Copyright 2018 American Association for the Advancement of Science.

**Figure 18.**(

**a**) Schematic diagram of the hinge Fermi arc of the three-dimensional quadrupole moment semimetal. Reprinted with permission from [172]. Copyright 2018 American Physical Society. (

**b**) The k

_{z}-dependent energy spectrum of the system with open boundaries in both the x and y directions. Reprinted with permission from [173]. Copyright 2020 American Physical Society.

**Figure 19.**Phase schematic diagram of the second-order nodal line semimetal. Reprinted with permission from [174]. Copyright 2022 American Physical Society.

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**MDPI and ACS Style**

Liu, B.; Zhang, W.
Research Progress of Topological Quantum Materials: From First-Order to Higher-Order. *Symmetry* **2023**, *15*, 1651.
https://doi.org/10.3390/sym15091651

**AMA Style**

Liu B, Zhang W.
Research Progress of Topological Quantum Materials: From First-Order to Higher-Order. *Symmetry*. 2023; 15(9):1651.
https://doi.org/10.3390/sym15091651

**Chicago/Turabian Style**

Liu, Bing, and Wenjun Zhang.
2023. "Research Progress of Topological Quantum Materials: From First-Order to Higher-Order" *Symmetry* 15, no. 9: 1651.
https://doi.org/10.3390/sym15091651