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

: The exploration of topologically nontrivial states in condensed ma tt er systems, along with their novel transport properties, has garnered signi ﬁ cant research interest. This review aims to provide a comprehensive overview of representative topological phases, starting from the initial proposal of the quantum Hall insulator. We begin with a concise introduction, followed by a detailed examination of ﬁ rst-order topological quantum phases, including gapped and gapless systems, encompassing relevant materials and associated phenomena in experiment. Subsequently, we delve into the realm of exotic higher-order topological quantum phases, examining both theoretical propositions and experimental ﬁ ndings. Moreover, we discuss the mechanisms underlying the emergence of higher-order topology, as well as the challenges involved in experimentally verifying materials exhibiting such properties. Finally, we outline future research directions. This review not only systematically surveys various types of topological quantum states, spanning from ﬁ rst-order to higher-order, but also proposes potential approaches for realizing higher-order topological phases, thereby o ﬀ ering guidance for the detection of related quantum phenomena in experiments.


Introduction
Topological materials have garnered substantial attention in the field of condensed matter physics over the past decade [1][2][3][4]. This is due to their remarkable properties in boundary behavior, such as unidirectional and non-dissipative energy transport, which is protected against backscattering. These unique characteristics have significant applications in spintronics and quantum information sciences. Moreover, in recent years, the study of topological materials has expanded significantly into diverse fields, including optics [5], acoustics [6][7][8][9][10][11], and mechanics [12,13]. This expansion has opened up new opportunities for controlling and designing the propagation of classical waves.
Topology is a mathematical concept that captures the invariant properties of geometric objects under continuous transformations. Objects belonging to the same topological class can be continuously deformed into one another. This shared characteristic is expressed through topological invariants. Initially, topological properties were applied to describe the distribution states of materials in real space. Afterwards, the concept of topology was extended to momentum space. In this context, if the quantum mechanical wave function of a system can undergo an adiabatic deformation to transform into another wave function, they are considered topologically equivalent. For systems consisting of multiple particles, the simple combination of atomic wave functions yields a topologically trivial result. Therefore, if the wave function of a condensed matter system can be adiabatically transformed into the atomic limit, it is deemed the atomic insulator which is topologically trivial; otherwise, it is regarded as topologically nontrivial. Numerous reviews on topological materials have been published to date, but most of them primarily concentrate on early-stage first-order topological phases within the realms of electronic or electromagnetic fields [1][2][3][4]. However, these reviews overlook the emerging field of higher-order topological phases. In light of this, the objective of this article is to present a comprehensive examination of topological phases, including both firstorder and higher-order phases. The discussion will encompass realistic materials, underlying mechanisms, challenges, and future research directions. It is important to acknowledge that the content of this article is constrained by the authors' interests, expertise, and the article's length, resulting in the regrettable omission of many outstanding works.

First-Order Topological Quantum Phase
The quantum Hall system stands out as a pioneering example of a topologically nontrivial condensed matter system, marking the inception of research on topological quantum states. Building upon the quantum Hall effect, subsequent discoveries such as the quantum spin Hall effect, quantum anomalous Hall effect, topological insulators, and topological semimetals have unveiled a plethora of novel physical phenomena.

Quantum Hall Effect
In the 1980s, von Klitzing et al. [14] conducted a groundbreaking experiment where they measured the boundary conductivity of two-dimensional electronic materials subjected to an external magnetic field. Their discovery revealed that the conductivity σ was an integer multiple of a fundamental quantity, . Moreover, they observed a remarkable step-like change in conductance as the magnetic field strength increased, demonstrating a precise quantization unaffected by parameters such as boundary defects. This phenomenon, known as the "integer quantum Hall effect" [15], earned von Klitzing the Nobel Prize in Physics in 1985.
To elucidate the underlying physical mechanism of the integer quantum Hall effect, researchers have approached its explanation from various perspectives [16][17][18][19]. It is evident that a profound connection exists between the integer quantum Hall effect and the topology. Notably, Thouless et al. [17] provided a theoretical insight into the relationship between conductivity and the bulk state of the system. They derived a mathematical formula for calculating conductivity: 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 nH 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.
It is important to note that the aforementioned topological numbers solely rely on the properties of the bulk states, while the quantum Hall effect manifests as a boundary characteristic of a material. This unique feature is a defining aspect of topological materials, where the bulk properties dictate the conductive behavior at the boundary-a phenomenon known as the bulk-edge correspondence [2,3]. At the interface between two insulators with different Chern numbers, topological boundary states emerge and their number is equal to the difference between the two Chern numbers (Figure 1a). These boundary states enable the unimpeded propagation of electronic waves without interference from defects or impurities. Consequently, a dispersion curve appears within the band gap ( Figure 1b). This topological property can be understood in terms of the breaking of time reversal symmetry by the magnetic field. Electrons can only transport in a single direction and are immune to backward scattering, even in the presence of defects or impurities, and the direction of the topological state reverses only when the direction of the magnetic field is reversed.

Quantum Spin Hall Effect
The quantum Hall effect is a result of the presence of topologically protected boundary states under an external magnetic field. However, an intriguing question arises: do these non-dissipative boundary states also exist in the absence of an external magnetic field? In 1988, Haldane made a significant breakthrough by recognizing that the key to realizing the quantum Hall effect lies in the breaking of time reversal symmetry. He developed a tight-binding model based on a two-dimensional hexagonal lattice, where each unit cell features a properly positioned magnetic flux [18,21]. Through this model, he derived the corresponding energy band and wave function, leading to the prediction that the quantum Hall conductivity can also be obtained in the absence of a magnetic field. This phenomenon is known as the quantum anomalous Hall effect and is associated with a time-reversal-symmetry breaking topological insulator, also known as the Chern insulator [18,21].
Subsequently, the theoretical conjecture of the quantum Hall effect was further extended to systems with time reversal invariance, wherein the presence of both clockwise and anticlockwise boundary states is expected. Notably, these systems require the inclusion of heavy elements with a significant spin-orbit interaction. The spin-orbit coupling (SOC) plays a crucial role by linking the current direction of the boundary state to the spin direction of the electron. In other words, the spin orientation and orbital motion are locked, giving rise to the phenomenon known as the quantum spin Hall effect.
In a two-dimensional insulator with strong SOC, the SOC acts as an effective spindependent magnetic field, giving rise to stable conductive edge states at the material's boundary. Notably, these edge states exhibit a spin-dependent behavior where spin-up electrons travel in one direction while spin-down electrons travel in the opposite direction. This spin-dependent motion leads to the formation of two dispersion curves within the band gap, as depicted in Figure 2. Consequently, there is a net spin transport occurring in the forward and backward directions along the upper and lower boundaries, respectively, equivalent to segregating electrons in the quantum Hall state based on their spins. This remarkable behavior is a characteristic of the quantum spin Hall effect. As long as the time reversal invariance of the system remains unaltered by the absence of a magnetic field or magnetic impurity scatterings, the influence of SOC will continue to dominate the behavior of the topological edge states. A notable example in this context is the extension of the Haldane model to realistic materials. Kane and Mele [22] employed the tight-binding Hamiltonian for the hexagonal lattice to investigate the electronic properties of graphene: where + σ i c and σ j c are the creation and annihilation operators acting on lattice sites i and j, respectively. The terms <ij> and <<ij>> represent the summation over nearest neighbors and next nearest neighbors, respectively. The first term accounts for the overlapping of nearest neighbors with an overlapping integral represented by t. The second term incorporates the intrinsic SOC with a strength of λSO. The third term introduces Rashba SOC, characterized by a strength of λR, 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 λν, with representing the onsite energy difference of the A (B) sublattice. Moreover, and dij 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 . 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.
Furthermore, theoretical analyses revealed that the presence of SOC in graphene induced a small gap in the energy spectrum, resulting in an insulating bulk state and spindependent conductive states at the material's boundary [22]. Figure 3a displays the energy spectrum of the quantum spin Hall phase observed in zigzag-type graphene nanoribbons, whereas Figure 3b represents the energy spectrum of the trivial phase. In the quantum spin Hall effect, the edge states with spin-up and spin-down exhibit reversed currents due to time reversal symmetry. As a result, there is no net charge flow, resulting in a Hall conductance of zero. Consequently, the integration of Berry curvature over the Brillouin zone yields a value of zero, making it impossible to characterize the topological properties using the first Chern number. However, upon closer inspection of the two spin subsystems individually, their corresponding Chern numbers are found to be (1, −1), indicating a spin Chern number of 1. Building on this analysis, Kane and Mele [22] introduced a novel Z 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.
In order to determine the topological index Z2, Kane and Mele [22] introduced a matrix, denoted as m(k), with matrix elements that are given by the expression: where ) (k j u is the periodic part of the Bloch wave, and i and j range from 1 to N, representing the occupied energy band number. represents the time reversal operator, in which y σ is the Pauli matrix and K denotes the complex conjugate. It can be readily and m(k) is an antisymmetric matrix. In the case of an antisymmetric matrix, we can compute its Pfaffian (Pf) [22]: On the one hand, if the zeros of P(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: Kane and Mele's theoretical research brought significant innovation to the field. However, the practical realization of the quantum spin Hall effect faced challenges due to the weak SOC in carbon, resulting in a small band gap that is susceptible to thermal fluctuations. To overcome this obstacle, Bernevig et al. devised a groundbreaking approach to investigating topological insulators and the quantum spin Hall effect [23]. They identified HgTe as a material with strong SOC and proposed a semiconductor quantum well structure consisting of a thin layer of HgTe sandwiched between layers of CdTe. Theoretical predictions demonstrated that when the thickness of the HgTe layer exceeds a critical value, a robust quantum spin Hall effect emerges in this two-dimensional system. Subsequently, this effect was experimentally observed in HgTe quantum wells [24]. Notably, the presence of edge states offers a direct means to distinguish a quantum spin Hall insulator from a trivial insulator, as these states manifest as two one-dimensional conducting channels, each contributing a quantized conductivity of e 2 /h.
By the way, it is worth noting that the stability of quantum spin Hall insulators relies on the presence of time reversal symmetry. However, in real materials, this symmetry can be easily disrupted by perturbations, which raises concerns about the stability of quantum spin Hall states in real-world environments.

Quantum Anomalous Hall Effect
The discovery of the quantum spin Hall effect has significantly advanced the exploration of the quantum anomalous Hall effect. By introducing ferromagnetism and breaking the time reversal symmetry, one spin channel in the quantum spin Hall system is suppressed, thereby giving rise to the quantum anomalous Hall effect, as depicted in Figure  4. The quantum anomalous Hall effect offers an alternative avenue to harness the principles of the quantum Hall effect in the absence of an external magnetic field.
The initial breakthrough in the realization of the quantum anomalous Hall effect came from first-principles calculations, which suggested that by thinning and magnetically doping the three-dimensional topological insulator Bi2Se3, it could be possible to achieve this phenomenon [25]. Given the experimental availability of Bi2Se3 materials, the introduction of dopants such as Cr or Fe ions in the Bi2Se3 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 Bi2Se3 system the most promising candidate for realizing the quantum anomalous Hall effect.  Although theoretical predictions suggest the possibility of material realization, practical implementation of the quantum anomalous Hall effect faces challenges due to the presence of electron impurities caused by Se vacancy defects in Bi2Se3. 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 Cr0.15(Bi0.1Sb0.9)1.85Te3 [26]. The Hall conductance exhibits quantization in the region where the longitudinal conductance is zero, as illustrated in Figure 5. The quantum anomalous Hall effect, quantum Hall effect, and quantum spin Hall effect collectively form a comprehensive trio known as the quantum Hall family [27]. In these three quantized Hall effects, electrons exhibit non-dissipative motion along the material's boundary, while the interior remains insulating. However, there are distinct differences among them. In the quantum Hall effect, electrons of different spins move in a single direction. In the quantum spin Hall effect, electrons of different spins move in opposite directions. In the quantum anomalous Hall effect, only electrons with a spin opposite to the bulk magnetic moment move in a specific direction. The locking mechanism between the spin and the moving direction, as well as the number of edge channels is intrinsic to the material itself.
In the quantum anomalous Hall insulator, the interplay between spontaneous magnetic moment and SOC gives rise to a topologically nontrivial electronic structure, resulting in the emergence of the quantum Hall effect in the absence of an external magnetic field. This unique two-dimensional insulator, also referred to as a Chen insulator, distinguishes itself from ordinary insulators and topological insulators. Its non-zero topological index is determined by the distribution of the Berry curvature of the energy bands in the Brillouin zone [28], similar to the quantum Hall insulator. Additionally, due to the involvement of only one spin channel in conductivity, the non-dissipative conductivity of the quantum anomalous Hall system is less prone to interference compared to the quantum spin Hall system. This characteristic opens up potential applications in low-loss electronic and spintronic devices.

Three-Dimensional Topological Insulator
Starting from the two-dimensional quantum spin Hall insulator, researchers have explored the generalization of this concept to three-dimensional topological insulators with conductive surface states. The theoretical predictions have been successfully confirmed in practical experiments [29][30][31]. The surface states of three-dimensional topological insulators exhibit similar characteristics to the Dirac electronic structures observed in graphene, displaying a relativistic linear energy-momentum relationship [29][30][31]. However, unlike graphene, three-dimensional topological insulators possess only one Dirac cone without spin degeneracy. To achieve a three-dimensional topological insulator, a strong SOC is required to significantly influence the electronic structure. Therefore, semiconductors containing heavy elements and small band gaps are promising candidates, as the SOC effect is prominent in heavy elements. It is crucial for the band gap to be smaller than the energy scale of SOC to ensure that the transition to a topological insulator occurs.
The first discovered three-dimensional topological insulator is the BixSb1−x alloy [29][30][31]. Subsequently, the Bi2Se3 family, including Bi2Te3 and Sb2Te3, with space group [32][33][34][35]. Compared to the BixSb1−x alloy, the Bi2Se3 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 Bi2Se3 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 Bi2Se3, Bi2Te3, and Sb2Te3. However, Sb2Se3 is considered a trivial insulator due to the weak SOC provided by Sb atoms.
Here, the presence of strong SOC once again plays an important role in the correlation of all the spins of electrons. As illustrated in Figure 6a, at the Fermi level there is only a single spin state along each momentum on the surface, and the spin direction rotates as the momentum moves around the Fermi plane. When disorders or impurities are introduced onto the surface, scattering can occur among the surface states. However, the remarkable feature of topological insulators is that the bulk topological property ensures the persistence of the metallic surface states even in the presence of scattering. In other words, the existence of metallic surface states and their robustness to disorder are fundamental characteristics of three-dimensional topological insulators. Experimental evidence of the Dirac cone for the (111) surface state of Bi2Se3 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. According to the energy band theory of solids, materials can be classified into insulators (including semiconductors) and metals based on their conductivity properties. However, categorization can also be achieved based on the topology of electronic states. Topological insulators, in particular, represent a novel quantum state of matter that is fundamentally distinct from ordinary insulators and metals. In a topological insulator, the bulk exhibits an insulating state with an energy gap, while the surface possesses metallic states without a gap. Unlike typical surface states that arise from unsaturated bonds or surface reconstructions, these metallic surface states are not determined by the specific surface structure but rather by the unique topological characteristics of the bulk wave functions. Importantly, these conductive surface states remain stable based on the premise that certain symmetries (such as time reversal symmetry) are present, and they are resistant to the effects of impurities and disorders. Remarkably, electrons with different spins in a topological insulator move in opposite directions, offering the potential for transmitting information through spin rather than charge.
The three-dimensional strong topological insulator materials Bi2Se3, Bi2Te3, and Sb2Te3 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].

Dirac Semimetal
In 2004, Novoselov et al. achieved a significant breakthrough by successfully obtaining monolayer graphene through mechanical exfoliation [62]. Graphene, as a prominent example of a Dirac semimetal, exhibits Dirac fermions at the K and K' points in its Brillouin zone, leading to the observation of the quantum Hall effect in experimental studies [63]. More recently, graphene has regained attention due to its exceptional physical properties, including superlattice structures and superconductivity in magic angle graphene [64,65].
Na3Bi 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, Na3Bi's instability and sensitivity to air have posed challenges for further investigations. Another prominent three-dimensional Dirac semimetal, Cd3As2 [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 Cd3As2, 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 Cd3As2 nanosheets [77,78] provided a significant breakthrough, expanding the study of the quantum Hall effect beyond two-dimensional systems.

Weyl Semimetal
When either time reversal or spatial inversion symmetry is broken, Dirac semimetals undergo a transition to Weyl semimetals characterized by twofold-degenerate crossings.

Nodal Line Semimetal
In 2011, Burkov et al. [99] introduced the concept of nodal line semimetals, emphasizing the additional protection provided by crystal symmetry. Unlike independent points in reciprocal space, band-crossings in nodal line semimetals manifest as continuous lines or rings on the Fermi surface. These crossings can take the form of Dirac points or Weyl points, thereby classifying nodal line semimetals into Dirac nodal line semimetals (e.g., ZrSiS family [100][101][102][103][104]) or Weyl nodal line semimetals (e.g., PbTaSe2 [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.

Higher-Order Topological Quantum Phase
The discovery of the quantum Hall insulator marked the beginning of the quest for topologically nontrivial phases in quantum materials. Since then, there has been a growing desire to uncover additional quantum phases exhibiting intriguing topological behaviors and associated fundamental physical properties. As the understanding of topological states advanced, it was discovered that crystal spatial symmetry can provide protection to topological states, giving rise to topological crystalline insulators.

Topological Crystalline Insulator
In 2009, Kitaev utilized K-theory to derive the topological classification of systems with three internal symmetries: time reversal symmetry, particle-hole symmetry, and chiral symmetry, across various dimensions [107]. Subsequently, Stone, Chiu, and other researchers organized this classification into ten distinct topological classes [108,109], commonly known as the "ten-fold way", as depicted in Figure 10. However, the topological periodic table solely takes into account the influence of three non-spatial symmetries on the system's topological classification. In the context of condensed matter physics, this limited consideration of non-spatial symmetries is inadequate for materials with different crystal symmetries. Consequently, the assessment of how crystal space symmetry impacts the system's topological classification emerges as an urgent issue requiring attention. In 2011, Fu introduced the concept of topological crystalline insulators [110]. In systems without considering spin, he explored topological crystalline insulators protected by fourfold rotational symmetry C 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.
SnTe is the first topological crystalline insulator material to be theoretically predicted and experimentally verified as protected by mirror symmetry [111]. It exhibits mirror symmetry in the (001) and (110) directions and features an even number of Dirac conelike surface states without energy gaps on high-symmetry crystal surfaces, such as {001}, {110}, and {111}. Additionally, members of the same family, such as PbTe and PbSe, can also become topological crystalline insulators when subjected to band inversion under the influence of pressure, strain, or doping. Subsequently, ARPES experiments quickly confirmed the existence of Dirac cone-like surface states protected by mirror symmetry in SnTe and Pb1− Sn Se [112,113].
KHgSb is the second topological crystalline insulator material that was theoretically predicted and experimentally verified to be protected by glide symmetry [114,115]. Its surface exhibits an hourglass-shaped two-dimensional topological surface state protected by non-symmorphic symmetry. For this three-dimensional system with both glide symmetry and time reversal symmetry, a new Z 2 topological invariant can be defined, in addition to the time-reversal Z 2 invariant, to characterize the system's topology [116][117][118].
In 2019, Fang and Fu made a significant discovery of an anomalous type of topology in two-dimensional systems possessing both time reversal symmetry and C 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 Ba3Cd2As4, Ba3Zn2As4, and Ba3Cd2Sb4 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].
In recent years, physicists have advanced the theory of Topological Quantum Chemistry by exploring the connection between band representations and symmetry indicators [121]. This theory enables more precise topological classifications of materials within all 230 space groups, consequently leading to the discovery of a wide range of topological crystalline insulators protected by diverse crystal symmetries [122,123].
Later, researchers discovered that for certain insulators a fully opened energy gap appears on the (d−1)-dimensional surface, while (d−2)-dimensional topological boundary states exist. This type of topological insulator is called a higher-order topological insulator. It is important to note that higher-order topological insulators also rely on the safeguarding of crystal symmetry and can be seen as an extension of topological crystalline insulators. In the following discussion, we will delve into these types of exotic higherorder topological phases: higher-order topological insulators and higher-order topological semimetals.

Higher-Order Topological Insulator
Topological insulators are known for their remarkable feature of topologically protected boundary states. According to the bulk-edge correspondence [5,124], an m-dimensional topological insulator exhibits (m−1)-dimensional gapless boundary states. However, in 2017, researchers [125,126] introduced a new class of topological insulators that break the bulk-edge correspondence, known as higher-order topological insulators. These unique insulators possess lower-dimensional boundary states. Specifically, an m-dimensional n-order topological insulator features m−1, m−2, (m−n+1)-dimensional gapped boundary states, and (m−n)-dimensional gapless boundary states. For instance, in a threedimensional system, a second-order topological insulator displays hinge states, while a third-order topological insulator exhibits corner states. In a two-dimensional system, a second-order topological insulator will showcase zero-dimensional corner states, as illustrated in Figure 11. To date, the existence of higher-order topological insulators has been experimentally confirmed in various systems, including mechanical systems [127], microwave systems [128], electrical circuits [129], acoustic systems [130,131], photonic systems [132], and elastic systems [133]. In the following sections, we will introduce three notable models of higher-order topological insulators.

Quantized Quadrupole Topological Insulator
In 2017, Benalcazar, Bernevig, and Hughes introduced the concept of the quantum multipole moment insulator [134,135]. In classical physics, the primitive dipole, quadrupole, and octupole moments of a continuum volume charge density ρ(r) are defined as , respectively [136]. In the modern theory of polarization in crystals, the dipole moment pi corresponds to the Berry phase of the bulk electronic states: where BZ indicates the Brillouin zone and the Berry connection A is characterized by com- , where n u k represents the Bloch function of band n, and the index n is limited to the occupied energy bands. The dipole moment pi 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 qij or oijk as: respectively, where repeated indices are summed over. Here, However, it is important to note that the values of these moments are not arbitrary, but rather constrained to specific quantized values. Crystal symmetries impose restrictions on these moments [135], requiring that a moment that is odd under a particular symmetry either vanishes or takes on a nontrivial quantized value permitted by the lattice structure. For instance, in a two-band inversion-symmetric insulator at half-filling where there is only one electron per unit cell, the electron center of charge must either coincide with the atomic center (p 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.
Furthermore, it is important to consider that in two-dimensional topological insulators, one-dimensional topological edge states emerge within the bulk energy gap due to the reduction in dimensionality. From a physical perspective, these dimension-reductioninduced topological edge states can be understood through an alternative approach. The quantization of the bulk dipole moment, resulting from specific symmetries, leads to the accumulation of additional charge on the edges, giving rise to the formation of topological edge states [135]. Within this framework, all early stage two-dimensional topological insulators are classified as first-order topological insulators. Expanding upon the theory of dipole moments in crystalline insulators to encompass multipole moments, a new category of insulators, known as quantized electric quadrupole (octupole) insulators or higher-order topological insulators, has been defined. In these systems, all lower-order moments vanish, but the quadrupole (octupole) moment exhibits a non-zero quantized value. Consequently, for electric quadrupole and octupole insulators, there is no longer any additional charge accumulation on the edges or surfaces. In particular, the realization of multipole moment topological insulators can be achieved by incorporating positive and negative couplings into a tight-binding model. For instance, in the case of a two-dimensional quadrupole topological insulator, its tight-binding Hamiltonian can be described as follows: Subsequently, several research teams have successfully realized quadrupole moment higher-order topological insulators in various experimental systems. These include microwave circuits [128], phononic systems [129], and photonic systems [137], among others. For instance, in the case of photonic systems, Mittal et al. demonstrated the existence of a quantized quadrupole topological phase on the silicon photon platform, which exhibited zero-dimensional corner states [137]. In their experiment, depicted in Figure 15a, a two-dimensional lattice model composed of nano-photonic silicon ring resonators was used. The unit cell consisted of four ring resonators interconnected by resonators (shown in red and green). By adjusting the coupling strength and symbols via the manipulation of the band gap sizes of the lattice resonant ring and connecting resonant ring, a quantized bulk quadrupole moment was achieved, resulting in quantized dipole moments along the edge and quantized charges at the corners. When the coupling strength between unit cells exceeded that within the unit cells, the system exhibited nontrivial polarization and supported localized corner states within the band gap, as depicted in Figure 15b. Conversely, when the coupling strength was reversed, the system transitioned into a trivial insulator. Notably, these corner states were found to be more robust compared to those observed in topological insulators with zero quadrupole moment, as they exhibited greater resilience to disorder and avoided coupling into the bulk, as shown in Figure 15c. It is worth mentioning that other systems, such as gyromagnetic materials [138] and dielectric structures [139], also hold potential for realizing quadrupole insulators. 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.

Two-Dimensional Su-Schrieffer-Heeger (SSH) Model
The two-dimensional SSH model [140] provides another approach to realizing higher-order topological insulators. In contrast to the previous quadrupole higher-order topological insulator model that requires positive and negative couplings, the two-dimensional SSH model is simpler in its construction. The emergence of corner states in this higher-order SSH model can be attributed to dipole polarization, which describes the displacement of the average position of the Wannier center relative to the unit cell center [130]. The nontrivial two-dimensional SSH phase can be characterized by the two-dimensional Zak phase [141] and the bulk polarization can be defined as follows [142]: where S is the area of the first Brillouin zone and is the Berry connection. The indices m and n signify the energy bands. Taking the photonic system as an example, Xie et al. initially simulated the presence of local corner states and one-dimensional boundary states in a two-dimensional square lattice [143]. Subsequently, two separate research groups experimentally observed zero-dimensional corner states in the microwave frequency band using near-field scanning techniques. These observations were obtained in a square lattice consisting of dielectric cylinders [144,145]. By controlling the coupling strength both inside and outside the unit cells through adjusting the distances between the medium, the system exhibited a nontrivial phase with a polarization of (1/2, 1/2) when the inter-unit cell coupling strength exceeded the intra-unit cell coupling strength. Conversely, the system transitioned into a trivial phase with a polarization of (0, 0) when the inter-unit cell coupling strength was smaller.
In addition to square lattices, higher-order topological insulators can also be realized in Kagome lattices [142,146]. Figure 16a illustrates a second-order topological insulator configuration in a Kagome lattice [147]. When the inter-unit cell coupling exceeds the intra-unit cell coupling, the bulk polarization becomes (1/3, 1/3), indicating a displacement of the Wannier center from the unit cell center and establishing a nontrivial system characterized by edge states and corner states at the boundary. It is worth noting that in photonic crystals, a new type of higher-order topological phase emerges when considering far-field interactions between non-nearest-neighbor unit cells, which differs from the effects of tight-binding interactions. Experimental measurements of the density of states reveal two types of corner states, as shown in Figure 16b: one arising from bulk polarization (Figure 16c) and the other resulting from far-field interactions (Figure 16d). Furthermore, El Hassan et al. successfully demonstrated local corner states in the visible band using a Kagome lattice constructed with coupled waveguides [148].  [147]. Copyright 2020 Nature Publishing Group.

Three-Dimensional Higher-Order Topological Insulator
In 2017, Schindler et al. achieved a breakthrough in the field by introducing mass terms that disrupt the original symmetry in first-order topological insulators, leading to the discovery of a three-dimensional higher-order topological insulator [126]. The introduction of these additional mass terms creates effective mass domain walls at the corners or edges of the system, giving rise to the emergence of corner and hinge states [126].
The Hamiltonian of the original two-dimensional topological insulator is expressed as: where i = x, y. By introducing an additional mass term, denoted as ( ) y x mass k k H cos cos 2 − Δ = 0 σ τ y , which breaks both time reversal symmetry (T) and fourfold rotational symmetry (C4) while preserving the C4T symmetry, a higher-order topological insulator with chiral onedimensional 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. So far, extensive calculations have identified a range of materials that are predicted to exhibit the higher-order topological insulating phase. These include three-dimensional materials such as bismuth [150], transition metal disulfides XTe2 (X = Mo, W) [151], and shaft insulators like Bi2-xSmxSe3 [152] and EuIn2As [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].
Higher-order topological insulators offer a promising research platform for exploring topologically protected local states with numerous potential applications. For instance, in the field of photonics, the utilization of topological corner modes has shown promise in high-quality and low-threshold lasers [162,163], thereby enabling the development of compact laser devices. Additionally, the concept of surface wave photonic crystals with wide band gaps [164] holds potential for advancements in integrated photonics. Furthermore, in the realm of non-Hermitian systems, the coupling of cavities opens up possibilities for realizing higher-order topological insulators that differ from those found in condensed matter systems [165,166].

Higher-Order Topological Semimetal
Naturally, the concept of "higher-order" also applies to topological semimetals. In contrast to early stage first-order topological semimetals, higher-order topological phases exhibit lower-dimensional topological boundary states [142,[167][168][169][170][171]. For instance, threedimensional higher-order topological semimetals can possess hinge Fermi arcs or zerodimensional corner states, while two-dimensional higher-order topological semimetals may exhibit zero-dimensional corner states [167]. As an example, we consider a threedimensional higher-order semimetal called a quadrupole semimetal, formed by stacking two-dimensional quantum quadrupole moment insulators. The bulk energy band of this system features nodes, while its surface hosts one-dimensional gapless hinge Fermi arcs that connect the projections of the bulk nodes, as illustrated in Figure 18a.
The parameter γ 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.
In recent years, it has been discovered that some nodal line semimetals may also exhibit higher-order topology [174][175][176]. In 2020, Wang et al. [176] proposed a novel secondorder nodal line semimetal phase with spacetime inversion symmetry (PT) protection. This phase features a pair of nodal rings and is constructed by stacking two-dimensional second-order Chern insulators with off-diagonal corner states along the k 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.
Firstly, the Z 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]:  Figure 19. This switch implies the presence of nodal lines between the two planes. Moreover, within a specific kz region (−kR, kR) bounded by two nodal rings, if each plane in the region exhibits a nontrivial υR = 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. Secondly, each nodal ring is associated with a Z 2 -valued topological invariant ω, which corresponds to the quantized π Berry phase: where 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. Overall, the two charges, υ 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.
In recent years, significant advancements have been made in the experimental realization of higher-order nodal line semimetals. For instance, it has been proposed that materials such as XTe2 (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.
The exploration of topological materials has expanded beyond first-order topological states to higher-order topological states, leading to significant advancements in both theoretical understanding and experimental realization. However, the study of higher-order semimetals is still in its early stages, and despite the numerous intriguing phenomena proposed in theory, the discovery of suitable real materials remains limited and challenging.

Summary and Discussion
Our work provides a comprehensive overview of the research progress in topological phases. In the introduction, we introduce the concept of topology in condensed matter physics. In Section 2, we discuss various early stage first-order topological phases in detail, starting with the quantum Hall insulator as a pioneering example, followed by the quantum spin Hall insulator, quantum anomalous Hall insulator, and topological semimetal. Moving on to Section 3, we provide an exhaustive overview of two intriguing higher-order topological phases.
At present, the hotspot of research on higher-order topological insulators lies in the exploration of new types of topological materials to enrich the family of higher-order topological phases. However, progress in related experimental studies has been slow due to several reasons. Firstly, a relatively limited number of materials have been predicted. While two-dimensional higher-order topological electronic states have been confirmed in certain models such as the square lattice [134,135], Kagome lattice, and honeycomb lattice [130][131][132][133], these models have stringent requirements on materials, often limited to a few carbon-based materials, while others are mostly artificial systems. Moreover, there have been few reports on theoretical models related to three-dimensional higher-order topological insulators. Secondly, the growth of electronic materials and the determination of their topological properties still pose challenges in experimental settings. Lastly, current research on higher-order topological insulators mainly focuses on electronic materials, with a lack of exploration in the realm of phononic materials. These unfavorable factors impede further progress in the fundamental theory and experimental investigation of higher-order topological insulators.
Currently, there are only four proposed mechanisms in theory for the formation of higher-order topological insulators in electronic systems. These mechanisms are as follows: (1) the breathing lattice mechanism. This mechanism involves modifying hopping terms in high-symmetry lattices to induce a "shrinking and swelling" pattern. The reduction in symmetry leads to the lifting of degeneracy of the Dirac bands, resulting in nontrivial higher-order topological energy gaps [142,181]. However, this mechanism has mainly been demonstrated experimentally in artificial acoustic, optical, and elastic systems. (2) The double-band inversion mechanism. In this mechanism, two sets of bands with opposite parities are inverted to induce higher-order topological energy gaps [182]. Experimental confirmation of this mechanism has been observed in only a few three-dimensional electronic materials, such as Bi [150], Td-WTe2 [183], and Bi4Br4 [184]. (3) The inplane 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. -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.
In recent years, there has been significant theoretical research on topological states, focusing on the manipulation of electrons and phonons both domestically and internationally. Notable contributions include the following: (1) Noguchi et al. proposed the use of van der Waals stacking to achieve a transition from a trivial phase to a higher-order topological phase. Different stacking arrangements were shown to lead to different topological states in Bi4Br4 [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 inplane 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.
In addition, while topological physics has been extensively studied in various electronic and phononic systems, the SOC mechanism that is well-known in electronic systems to open first-order topological energy gaps is not applicable to phononic systems due to the absence of SOC for phonons. As a result, there have been no reports of the coexistence of topological electrons and phonons in the same material for a long time. However, a recent breakthrough has been made by the author's team, who demonstrated the coexistence of higher-order topological electrons and higher-order topological phonons in graphdiyne with a Kekulé lattice [158], providing the first evidence of higherorder topological phononic states in real materials. This novel topological phase, which combines both electronic and phononic degrees of freedom, offers a promising platform for investigating the interplay between higher-order topological electronic and phononic states. It is worth noting that when localized higher-order topological electronic states and phononic states overlap in space, strong electron-phonon coupling can occur. This coupling is of particular interest in the context of Bardeen-Cooper-Schrieffer (BCS) superconductivity theory, as strong electron-phonon coupling is a key factor in the emergence of superconductivity. Therefore, it is conceivable that higher-order topological insulators exhibiting the coexistence of electronic and phononic states may hold potential for higherorder topological superconductivity. Nevertheless, due to the early stage of research in this field, our understanding of these materials and their formation mechanisms remains limited.
Based on the above discussions, we believe that further investigations can be conducted in the following areas: (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.