Macroscopic two-dimensional sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be controlled simultaneously. The bulk band gap forms an acoustic analog of topological crystalline insulators with edge states which are gapped due to symmetry reduction on the edges. In the presence of mirror symmetry, the band topology of the edge states can be characterized by the Zak phase, illustrating the band topology in a hierarchy of dimensions, which is at the heart of higher-order topology. Moreover, the edge band gap can be closed without closing the bulk band gap, revealing an independent topological transition on the edges. The rich topological transitions in both bulk and edges can be well-described by the symmetry eigenvalues at the high-symmetry points in the bulk and surface Brillouin zones. We further analyze the higher-order topology in the shrunken sonic crystals where slightly different physics but richer corner and edge phenomena are revealed. In these systems, the rich, multidimensional topological transitions can be exploited for topological transfer among zero-, one- and two- dimensional acoustic modes by controlling the geometry.
Topological insulators with unique gapless edge states have revolutionized the understanding of electronic properties in solid materials. These gapless edge states are dictated by the topological invariants associated with the quantization of generalized Berry phases of the bulk energy bands through the bulk-edge correspondence, a paradigm that can also be extended to acoustic and photonic systems. Recently, high-order topological insulators (HOTIs) are proposed and observed, where the bulk topological invariants result in gapped edge states and in-gap corner or hinge states, going beyond the …
Topological phases of matter have been extensively studied for their intriguing bulk and edge properties. Recently, higher-order topological insulators with boundary states that are two or more dimensions lower than the bulk states, have been proposed and investigated as novel states of matter. Previous implementations of higher-order topological insulators were based on two-dimensional (2D) systems in which 1D gapped edge states and 0D localized corner states were observed. Here we theoretically design and experimentally realize a 3D higher-order topological insulator in a sonic crystal with …
We discuss the change of the corner charge for noninteracting two dimensional insulators with inversion symmetry undergoing adiabatic evolution. We show that the change of the corner charge is accounted for by the adiabatic current flowing along the edges of the system. The study of systems with quasi-1D geometry is necessary to derive the analytical expression for the adiabatic current. This fact suggests that the change of the corner charge is neither a purely bulk nor edge effect, but rather a mixed one. The derived adiabatic current was examined and shows good agreement with the numerical …
We study topological states of honeycomb photonic crystals in absence of inversion symmetry using plane wave expansion and finite element methods. The breaking of inversion symmetry in honeycomb lattice leads to contrasting topological valley indices, i.e., the valley-dependent Chern numbers in momentum space. We find that the topological corner states appear for 60$^\circ$ degree corners, but absent for other corners, which can be understood as the sign flip of valley Chern number at the corner. Our results provide an experimentally feasible platform for exploring valley-dependent higher-orde…
We propose theoretically a reconfigurable two-dimensional (2D) hexagonal sonic crystal with higher-order topology protected by the six-fold, $C_6$, rotation symmetry. The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit-cell. In the nontrivial phase, the sonic crystal realizes the topological spin Hall effect in a higher-order fashion: (i) The edge states emerging in the bulk band gap exhibits partial spin-momentum locking and are gapped due to the reduced spatial symmetry at the edges. (ii) The gapped edge states, on the other hand, stab…
Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe$_2$ with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin p…
SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s-wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetry-protected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introduci…
Topologically protected gapless edge states are phases of quantum matter which behave as massless Dirac fermions, immunizing against disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with topological corner states have been theoretically predicted in electric systems, and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems, electrical circuits, optical and sonic crystals, and elastic phononic plates. Here, we demonstrate a pseudospin-valley-coupled phononic TI, which simultaneously exhibits gapped edge states and to…
We realize an elastic second-order topological insulator hosting both one-dimensional gapped edge states and zero-dimensional in-gap corner modes in the double-sided pillared phononic crystal plates with square lattice. Changing the width of two neighbor pillars breaks the inversion symmetry and induces the band inversion to emulate the quantum spin Hall effect where the gapless edge states are obtained. Further breaking the space-symmetry at interface, the gapless edge states are gapped and inducing the edge topological transitions and then giving rise to the zero-dimensional in-gap corner mo…
We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of "higher-order" topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We show that the surface states of higher-order topological insulators and superconductors can be thought of as globally irremovable topological defects and provide a complete classification of these inversion-protected phases in any spatial dimension for…
We study corner states on a flat band in the square lattice. To this end, we introduce a two dimensional model including Su-Schrieffer-Heeger type bond alternation responsible for corner states as well as next-nearest neighbor hoppings yielding flat bands. The key symmetry of the model for corner states is space-time inversion ($\cal PT$) symmetry, which guarantees quantized Berry phases. This implies that edge states as well as corner states would show up if boundaries are introduced to the system. We also argue that an infinitesimal $\cal PT$ symmetry-breaking perturbation could drive flat b…
We have performed angle-resolved photoemission spectroscopy (ARPES) of LaSb and CeSb, a candidate of topological insulator. Using soft-x-ray photons, we have accurately determined the three-dimensional bulk band structure and revealed that the band inversion at the Brillouin-zone corner - a prerequisite for realizing topological-insulator phase - is absent in both LaSb and CeSb. Moreover, unlike the ARPES data obtained with soft-x-ray photons, those with vacuum ultraviolet (VUV) photons were found to suffer significant $k_z$ broadening. These results suggest that LaSb and CeSb are topologicall…
Topological phases have spurred unprecedented abilities for sound, light and matter engineering and recent progress has shown how waves not only confine at the interfaces between topologically distinct insulators, but in the form of zero-dimensional non-propagating states bound to defects or corners. Majorana-like bound states have recently been observed in man-made Kekulé textured lattices. We show here how the acoustic version of the associated Jackiw-Rossi vortex embodies a Thouless pumping process, in which the spectral flow of corner states adiabatically merge with the said Majorana-like …
Although single-layer transition-metal dichalcogenides with novel valley functionalities are promising candidate to realize valleytronic devices, the essential understanding of valley depolarization mechanisms is still incomplete. Based on pump-probe experiments performed for MoSe2 and WSe2 monolayers and corroborating analysis from density functional calculations, we demonstrate that coherent phonons at the K-point of the Brillouin zone can effectively mediate the valley transfer of electron carriers. In the MoSe2 monolayer case, we identify this mode as the flexural acoustic ZA(K) mode, whic…
We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for the TTIs can be realized on any bipartite lattice where the inversion exchanges sublattices. The protecting symmetries lead to the classifying space of Hamiltonians being unitary symmetric matrices, and therefore Takagi's factorization can be performed. Particularly, the global Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D, there is a …
Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming t…
The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. …
Graphene/h-BN has emerged as a model van der Waals heterostructure, and the band structure engineering by the superlattice potential has led to various novel quantum phenomena including the self-similar Hofstadter butterfly states. Although newly generated second generation Dirac cones (SDCs) are believed to be crucial for understanding such intriguing phenomena, so far fundamental knowledge of SDCs in such heterostructure, e.g. locations and dispersion of SDCs, the effect of inversion symmetry breaking on the gap opening, still remains highly debated due to the lack of direct experimental res…
The search for exotic new topological states of matter in widely accessible materials, for which the manufacturing process is mastered, is one of the major challenges of the current topological physics. Here we predict higher order topological insulator state in quantum wells based on the most common semiconducting materials. By successively deriving the bulk and boundary Hamiltonians, we theoretically prove the existence of topological corner states due to cubic symmetry in quantum wells with double band inversion. We show that the appearance of corner states does not depend solely on the cry…
Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801…
Generally, the topological corner state in two-dimensional second-order topological insulator (2D SOTI) is equivalent to the well-known domain wall state, originated from the mass-inversion between two adjacent edges with phase shift of pi. In this work, go beyond this conventional physical picture, we report a fractional mass-kink induced 2D SOTI in monolayer FeSe with canted checkerboard antiferromagnetic (AFM) order by analytic model and first-principles calculations. The canted spin associated in-plane Zeeman field can gap out the quantum spin Hall edge state of FeSe, forming a fractional …
Higher-order topology realizes topologically robust corner modes as a manifestation of nontriviality. We theoretically propose non-Hermitian skin effects which stem from second-order topology of chiral-symmetric Hermitian systems. It is found that the skin modes are localized at the corners. We demonstrate two types of second-order topological skin effects by two-dimensional intrinsic and extrinsic second-order topology. The intrinsic second-order topological skin effect is characterized topologically by bulk inversion symmetry as well as chiral symmetry. Meanwhile, the extrinsic second-order …
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；College of Physics, Hebei Normal University（河北师范大学物理学院）；Hefei National Laboratory（合肥国家实验室）
We establish an analytic low-energy theory describing higher-order topological insulator (HOTI) phases in quasicrystalline systems. We apply this to a model consisting of two stacked Haldane models with oppositely propagating edge modes, analogous to the Kane-Mele model, and with a $30^\circ$ twist. We show that the resulting localized modes at corners, characteristic of a HOTI, are not associated with conventional mass inversions but are instead associated with what we dub "fractional mass kinks". By generalizing the low-energy theory, we establish a classification for arbitrary $ n $-fold ro…
We propose second-order topological insulators (SOTIs) whose lattice structure has the hexagonal symmetry $C_{6}$ in three and two dimensions. We start with a three-dimensional weak topological insulator constructed on the stacked triangular lattice, which has only side topological surface states. We then introduce an additional mass term which gaps out the side surface states but preserves the hinge states. The resultant system is a three-dimensional SOTI. The bulk topological quantum number is shown to be the $\mathbb{Z}_{3}$ index protected by the inversion time-reversal symmetry $IT$ and t…
Higher-order topological insulators have been introduced in the precursory Benalcazar-Bernevig-Hughes quadrupole model, but no electronic compound has been proposed to be a quadrupole topological insulator (QTI) yet. In this work, we predict that Ta$_2M_3$Te$_5$ ($M=$ Pd, Ni) monolayers can be 2D QTIs with second-order topology due to the double-band inversion. A time-reversal-invariant system with two mirror reflections (M$_x$ and M$_y$) can be classified by Stiefel-Whitney numbers ($w_1, w_2$) due to the combined symmetry $TC_{2z}$. Using the Wilson loop method, we compute $w_1=0$ and $w_2=1…
Higher-order topological insulators exhibit multidimensional topological physics and unique application values due to their ability of integrating stable boundary states at multiple dimensions in a single chip. However, for signal-processing applications in high-frequency mechanical systems, the current realizations of higher-order topological mechanical materials are still limited to large-scale systems for kilohertz or lower frequencies. Here, we report the experimental observation of a on-chip micromechanical metamaterial as higher-order topological insulator for high-frequency mechanical w…
Higher Order Topological Insulators (HOTI) are $d$-spatial dimensional systems featuring topologically protected gap-less states at their $(d-n)$-dimensional boundaries. With the help of \textit{ab-initio} calculations and tight binding models along with symmetry considerations we show that monolayer buckled honeycomb structures of group-V elements (Sb,As), which have already been synthesized, belong in this category and have a charge fractionalization of $\frac{e}{2}$ at the corner states as well as weak topological edge states, all protected by their properties under the inversion operation …
In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials -- monolayers of centrosymmetric $β$-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric $γ$-phase MoTe$_2$ and WTe$_2$ have been identified as type-II Weyl semimetals. However, ARPES and STM probes of these TMDs have revealed huge, "arc-like" surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this letter, we use first-principles calculati…
Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here we identify the analog of quadrupole order in Maxwell's equations for a photonic crystal (PhC) and identify quadrupole topological photonic crystals formed through a band inversion process. Unlike prior studies relying on threaded…
Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in $d-2$ or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of…
We perform both lattice dynamics analysis and molecular dynamics simulations to demonstrate the existence of topologically protected phonon modes in a two-dimensional, monolayer hexagonal boron nitride sheet. The topological phonon modes are found to be localized at an in-plane interface that divides the system into two regions of distinct valley Chern numbers. The dispersion of this topological phonon mode crosses over the frequency gap [1123, 1278] cm^{-1}, which is opened through analogy with the quantum valley Hall effect by breaking inversion symmetry of the boron and nitride atoms in the…
The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of b…
Based on first-principles calculations and tight-binding model analysis, we propose monolayer graphdiyne as a candidate material for a two-dimensional higher-order topological insulator protected by inversion symmetry. Despite the absence of chiral symmetry, the higher-order topology of monolayer graphdiyne is manifested in the filling anomaly and charge accumulation at two corners. Although its low energy band structure can be properly described by the tight-binding Hamiltonian constructed by using only the $p_z$ orbital of each atom, the corresponding bulk band topology is trivial. The nontr…
Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a three-dimensional crystal. A second-order topological phase can be induced by the presence of a bulk crystalline symmetry. Building on Shiozaki and Sato's complete classification of bulk crystalline phases with an order-two crystalline symmetry [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)], such as mirror reflection, twofold rotation, or inversion symmetry, we classify all co…
We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion $I_{ST}$ symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an $I_{ST}$-symmetric Wannier representation. Moreove…
We propose magnetic second-order topological insulators (SOTIs). First, we study a three-dimensional model. It is pointed out that the previously proposed topological hinge insulator has actually surface states along the [001] direction in addition to hinge states. We gap out these surface states by introducing magnetization, obtaining a SOTI only with hinge states. The bulk topological number is the $Z_2$ index protected by the combined symmetry of the four-fold rotation and the inversion symmetry. We next study two dimensional magnetic SOTIs, where the corner states are robust also in the pr…
We theoretically study the lattice distortion induced first and second order topological phase transition in rectangular FeSe$_{x}$Te$_{1-x}$ monolayer. When compressing the lattice constant in one direction, our first principles calculation shows that the FeSe$_x$Te$_{1-x}$ undergoes a band inversion at $Γ$ point in a wide dopping range, say $x\in(0.0,0.7)$, which ensures coexistence of the topological band state and the high-T$_{\rm c}$ superconductivity. This unidirectional pressure also leads to the C$_4$ symmetry breaking which is necessary for the monolayer FeSe$_x$Te$_{1-x}$ to support …
We propose two mechanisms to realize the second order topological insulator (SOTI) state in spinless hexagonal lattices, viz., chemical modification and anti-Kekulé/Kekulé distortion of hexagonal lattice. Correspondingly, we construct two models and demonstrate the nontrivial band topology of the SOTI state characterized by the second Stiefel-Whitney class $w_2$ in the presence of inversion symmetry ($\textit{P}$) and time-reversal symmetry ($\textit{T}$). Based on the two mechanisms and using first-principles calculations and symmetry analysis, we predict three categories of real light elemen…
Topological insulator (TI) is an exciting discovery because of its robustness against disorder and interactions. Recently, higher-order TIs have been attracting increasing attention, because they host 1D topologically-protected hinge states in 3D or 0D corner states in 2D. A significantly critical issue is whether the higher-order TIs also survive interactions, but it is still unexplored. We study the effects of weak Coulomb interaction on a 3D second-order TI, with the help of a renormalization group calculation. We find that the 3D higher-order TIs are always unstable, suffering from two typ…
Hexagonal two-dimensional materials with broken inversion symmetry (as BN or transition metal dichalcodenides) are known to sustain chiral phonons with finite angular momentum, adding a further useful degree of freedom to the extraordinary entangled (electrical, optical, magnetic and mechanical) properties of these compounds. However, because of lattice symmetry constraints, such chiral modes are constrained to the corners of the Brillouin zone, allowing little freedom for manipulating the chiral features. In this work, we show how the application of uniaxial strain leads to the existence of n…
Motivated by the triumph and limitation of graphene for electronic applications, atomically thin layers of group VI transition metal dichalcogenides are attracting extensive interest as a class of graphene-like semiconductors with a desired band-gap in the visible frequency range. The monolayers feature a valence band spin splitting with opposite sign in the two valleys located at corners of 1st Brillouin zone. This spin-valley coupling, particularly pronounced in tungsten dichalcogenides, can benefit potential spintronics and valleytronics with the important consequences of spin-valley interp…
Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotation symmetry of order three, four or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney …
In insulating crystals, it was previously shown that defects with two fewer dimensions than the bulk can bind topological electronic states. We here further extend the classification of topological defect states by demonstrating that the corners of crystalline defects with integer Burgers vectors can bind 0D higher-order end (HEND) states with anomalous charge and spin. We demonstrate that HEND states are intrinsic topological consequences of the bulk electronic structure and introduce new bulk topological invariants that are predictive of HEND dislocation states in solid-state materials. We d…
We explore topological properties of a modulated Haldane model (MHM) in which the strength of the nearest-neighbor and next-nearest-neighbor terms is made unequal and the three-fold rotational symmetry $\mathcal{C}_3$ is broken by introducing a trimerization term ($|t_{1w(2w)}|< t_{1s(2s)}$) in the Hamiltonian. Using the parameter $η={t_{1w}}/{t_{1s}}= t_{2w}/t_{2s}$, we show that the MHM supports a transition from the quantum anomalous Hall insulator (QAHI) to a HOTI phase at $η=\pm 0.5$. The MHM also hosts a zero-energy corner mode on a nano-disk that can transition to a trivial insulator wi…
We theoretically study spin and charge currents through finite-size two-dimensional $s$-wave superconductor/uniform ferromagnet/s-wave superconductor (SFS) junctions with intrinsic spin-orbit interactions (ISOIs) using a quasiclassical approach. Considering experimentally realistic parameters, we demonstrate that the combination of spontaneously broken time-reversal symmetry and lack of inversion symmetry can result in spontaneously accumulated spin currents at the edges of finite-size two-dimensional magnetic SF hybrids. Due to the spontaneous edge spin accumulation, the corners of the F wire…
We investigate a topological switch between second-order topological insulators (SOTIs) and topological crystalline insulators (TCIs). Both the SOTI and the TCI are protected by the mirror and inversion symmetries, for which we define the bulk topological numbers of the same type. We take examples of square nanodisks on the square lattice and hexagonal nanodisks on the triangular lattice. When inplane magnetic field is introduced parallel to one of the helical edges, the system becomes a TCI. The conductance along the edge is 1 in the unit of the conductance quantum $e^2/h$. As the inplane fie…
