The concept of topological phases has been generalized to higher-order topological insulators and superconductors with novel boundary states on corners or hinges. Meanwhile, recent experimental advances in controlling dissipation (such as gain and loss) open new possibilities in studying non-Hermitian topological phases. Here, we show that higher-order topological corner states can emerge by simply introducing staggered on-site gain/loss to a Hermitian system in trivial phases. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal nested-Wilson-loop and edge-polarization theory, which can be applied to a wide class of non-Hermitian systems with higher-order topological orders. The theory gives rise to topological invariants characterizing the non-Hermitian topological multipole moments (i.e., corner states) that are protected by reflection or chiral symmetry. Such gain/loss induced higher-order topological corner states can be experimentally realized using photons in coupled cavities or cold atoms in optical lattices.
The bulk-edge correspondence characterizes topological insulators and superconductors. We generalize this concept to the bulk-corner correspondence and the edge-corner correspondence in two dimensions. In the bulk-corner (edge-corner) correspondence, the topological number is defined for the bulk (edge), while the topological phase is evidenced by the emergence of zero-energy corner states. It is shown that the boundary-obstructed topological phases recently proposed are the edge-corner-correspondence type, while the higher-order topological phases are classified into the bulk-corner-correspon…
Photonic crystals have provided a controllable platform to examine excitingly new topological states in open systems. In this work, we reveal photonic topological corner states in a photonic graphene with mirror-symmetrically patterned gain and loss. Such a nontrivial Wannier-type higher-order topological phase is achieved through solely tuning on-site gain/loss strengths, which leads to annihilation of the two valley Dirac cones at a time-reversal-symmetric point, as the gain and loss change the effective tunneling between adjacent sites. We find that the symmetry-protected photonic corner mo…
We analyze the robustness of corner modes in topological photonic crystals, taking a $C_6$-symmetric breathing honeycomb photonic crystal as an example. First, we employ topological quantum chemistry and Wilson loop calculations to demonstrate that the topological properties of the bulk crystal stem from an obstructed atomic limit phase. We then characterize the topological corner modes emerging within the gapped edge modes employing a semi-analytical model, determining the appropriate real space topological invariants. For the first time, we provide a detailed account of the effect of long-ra…
We numerically and experimentally study corner states in a continuous elastic plate with em-bedded bolts in a hexagonal pattern. While preserving C6 crystalline symmetry, the system can transition from a topologically trivial to a non-trivial configuration. We create interfacial corners of 60° and 120° by adjoining trivial and non-trivial topological configurations. Due to the rich interaction between the bolts and the continuous elastic plate, we find a variety of corner states with and without topological origin. Notably, some of the corner states are highly localized and tunable. By taking …
We numerically demonstrate that the topological corner states residing in the corners of higher-order topological insulator possess non-Abelian braiding properties. Such topological corner states are Dirac fermionic modes other than Majorana zero-modes. We claim that Dirac fermionic modes protected by nontrivial topology also support non-Abelian braiding. An analytical description on such non-Abelian braiding is conducted based on the vortex-induced Dirac-type fermionic modes. The braiding operator for Dirac fermionic modes is also analytically derivated and compared with the Majorana zero-mod…
In two-dimensional higher-order topological insulators, the corner states are separated by a non-negligible distance. The crystalline symmetries protect the robustness of their corner states with long-range entanglement, which are robust against time-reversal breaking perturbations. Here, we demonstrate the possibility of direct control of the topological corner states by introducing the spin degree of freedom in a rhombus-shaped Kekulé nanostructure with local magnetization and local electric potential. By applying a local magnetization on one corner, the other corner can also be strongly spi…
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…
Based on the tight-binding model calculations and photonic experimental visualization on graphene, we report the domain-wall-induced gapped topological kink states and topological corner states. In graphene, domain walls with gapless topological kink states could be induced either by sublattice symmetry breaking or by lattice deformation. We find that the coexistence of these two mechanisms will induce domain walls with gapped topological kink states. Significantly, the intersection of these two types of domain wall gives rise to topological corner state localized at the crossing point. Throug…
Topological states of light represent counterintuitive optical modes localized at boundaries of finite-size optical structures that originate from the properties of the bulk. Being defined by bulk properties, such boundary states are insensitive to certain types of perturbations, thus naturally enhancing robustness of photonic circuitries. Conventionally, the N-dimensional bulk modes correspond to (N-1)-dimensional boundary states. The higher-order bulk-boundary correspondence relates N-dimensional bulk to boundary states with dimensionality reduced by more than 1. A special interest lies in m…
Topological phonics has emerged as a novel approach to engineer the flow of light and provides unprecedented means for developing diverse photonic elements, including robust optical waveguides immune to structural imperfections. However, the development of nanoscale standing-wave cavities in topological photonics is rather slow, despite its importance when building densely-integrated photonic integrated circuits. In this Letter, we report a photonic crystal nanocavity based on a topological corner state, supported at a 90-degrees-angled rim of a two dimensional photonic crystal. A combination …
Electric circuits are known to realize topological quadrupole insulators. We explore electric circuits made of capacitors and inductors forming the breathing Kagome and pyrochlore lattices. They are known to possess three phases (trivial insulator, higher-order topological insulator and metallic phases) in the tight-binding model. The topological phase is characterized by the emergence of zero-energy corner states. A topological phase transition is induced by tuning continuously the capacitance, which is possible by using variable capacitors. It is found that the two-point impedance yields hug…
We discuss some bulk-surfaces gapped Hamiltonians on a lattice with corners and propose a periodic table for topological invariants related to corner states aimed at studies of higher-order topological insulators. Our table is based on four things: (1) the definition of topological invariants, (2) a proof of their relation with corner states (3) computations of K-groups and (4) a construction of explicit examples.
Recently there is a surge of interests in the so-called topologically protected corner states in 2D and 3D systems. Such systems are considered as high order topological insulators. Wannier centers are used as topological invariants to characterize bulk systems. The existence of corner states is considered as a reflection of the topological non-triviality of bulk energy bands. We demonstrate that the Wannier centers are not topological invariants by showing they depend on the choices of unit cells. The same bulk system can be considered as both topological and trivial with two equally possible…
A higher-order topological insulator (HOTI) in two dimensions is an insulator without metallic edge states but with robust zero-dimensional topological boundary modes localized at its corners. Yet, these corner modes do not carry a clear signature of their topology as they lack the anomalous nature of helical or chiral boundary states. Here, we demonstrate using immunity tests that the corner modes found in the breathing kagome lattice represent a prime example of a mistaken identity. Contrary to previous theoretical and experimental claims, we show that these corner modes are inherently fragi…
We investigate topological lasers in combination of nonlinear, non-Hermitian and topological lattice systems based on a quench dynamics starting from one site. We consider explicitly the topological laser in the Su-Schrieffer-Heeger (SSH) model with two topological edge states and the second-order topological laser in the breathing Kagome lattice with three topological corner states. Once we stimulate any one site, after a delay, all sites belonging to the topological edge or corner states begin to emit stable laser light depending on the density of states, although no wave propagation is obse…
We consider Toeplitz operators defined on a concave corner-shaped subset of the square lattice. We obtain a necessary and sufficient condition for these operators to be Fredholm. We further construct a Fredholm concave corner Toeplitz operator of index one. By using this, a relation between Fredholm indices of quarter-plane and concave corner Toeplitz operators is clarified. As an application, topological invariants and corner states for some bulk-edges gapped Hamiltonians on two-dimensional (2-D) class AIII and 3-D class A systems with concave corners are studied. Explicit examples clarify th…
We consider a system consisting of two tunnel-coupled two-dimensional topological insulators proximitized by a top and bottom superconductor with a phase difference of $π$ between them. We show that this system exhibits a time-reversal invariant second-order topological superconducting phase characterized by the presence of a Kramers pair of Majorana corner states at all four corners of a rectangular sample. We furthermore investigate the effect of a weak time-reversal symmetry breaking perturbation and show that an in-plane Zeeman field leads to an even richer phase diagram exhibiting two non…
Topologically gapless edge states, characterized by topological invariants and Berry's phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons and phonons. Recently, a new family of topological phases, dictated by the bulk polarization, has been observed, leading to the discovery of the higher-order topological insulators (HOTIs). So far, the HOTIs are only demonstrated in discrete mechanical and electromagnetic systems and electrical circuits with the quantized quadrupole polarization. Here, we realize the higher-order t…
Higher-order topological insulators in two dimensions have states that localize at their corners, called corner states. In this paper, we reveal characteristics of the penetration depth of their corner states by using the Benalcazar-Bernevig-Hughes model. First, we show that when we change the energy of the corner states toward the end of the edge gap by adding an on-site potential to the corner site, the penetration depth along the edge diverges toward infinity while the penetration depth into the bulk remaining finite. We analytically derive the corner-state wavefunction in a form of ellipti…
Recently discovered photonic higher-order topological insulators enable unprecedented flexibility in the robust localization of light in structures of different dimensionality. While the potential of the two-dimensional systems is currently under active investigation, only a few studies explore the physics of the three-dimensional higher-order topological insulators. Here we propose a three-dimensional structure with cubic symmetry exhibiting vanishing bulk polarization but nonzero corner charge and hosting a zero-dimensional corner state mediated by the long-range interactions. We trace the e…
We theoretically study two-dimensional exciton-polariton lattices and predict that non-Hermitian topological corner modes can be formed under non-resonant pumping. As a generalization of the non-Hermitian skin effect, all eigenstates are localized at the two corners in our model. This is also a higher dimensional topology compared to other proposals in exciton-polariton systems and we find that it allows propagating signals in the bulk of the system to travel around defects, which is not possible in one-dimensional topological lattices or two-dimensional lattices with Hermitian edge states. Fu…
Higher-order topological phase in 2-dimensional (2D) systems is characterized by in-gap corner states, which are hard to detect and utilize. We numerically investigate transport properties of topological corner states in 2D honeycomb lattice, where the second-order topological phase is induced by an in-plane Zeeman field in the conventional Kane-Mele model. Through engineering multihollow structures with appropriate boundaries in honeycomb lattice, multiple corner states emerge, which greatly increases the probability to observe them. A typical two-probe setup is built to study the transport f…
The recent discovery of higher-order topological insulators (HOTIs) has significantly extended our understanding of topological phases of matter. Here, we predict that second-order corner states can emerge in the dipolar-coupled dynamics of topological spin textures in two-dimensional artificial crystals. Taking a breathing honeycomb lattice of magnetic vortices as an example, we derive the full phase diagram of collective vortex gyrations and identify three types of corner states that have not been discovered before. We show that the topological "zero-energy" corner modes are protected by a g…
Recently realized higher order topological insulators have taken a surge of interest among the theoretical and experimental condensed matter community. The two-dimensional second order topological insulators give rise to zero-dimensional localized corner modes that reside within the band gap of the system along with edge modes that inhabit a band edge next to bulk modes. Thanks to the topological nature, information can be trapped at the corners of these systems, which will be unhampered even in the presence of disorder. Being localized at the corners, the exchange of information among the cor…
A two-dimensional second-order topological superconductor exhibits a finite gap in both bulk and edges, with the nontrivial topology manifesting itself through Majorana zero modes localized at the corners, i.e., Majorana corner states. We investigate a time-reversal-invariant topological superconductor in two dimension and demonstrate that an in-plane magnetic field could transform it into a second-order topological superconductor. A detailed analysis reveals that the magnetic field gives rise to mass terms which take distinct values among the edges, and Majorana corner states naturally emerge…
Recently, a new type of second-order topological insulator has been theoretically proposed by introducing an in-plane Zeeman field into the Kane-Mele model in the two-dimensional honeycomb lattice. A pair of topological corner states arise at the corners with obtuse angles of an isolated diamond-shaped flake. To probe the corner states, we study their transport properties by attaching two leads to the system. Dressed by incoming electrons, the dynamic corner state is very different from its static counterpart. Resonant tunneling through the dressed corner state can occur by tuning the in-plane…
The higher-order topological insulators (HOTIs), with such as the topological corner states, emerge as a thriving topic in the field of topological physics. But few connections have been found for the HOTIs with the well explored first-order topological insulators described by the Z_2 index. However, most recently, a proposal asserts that a significant bridge can be established between the HOTIs and the Z_2 topological insulators. When subject to an in-plane Zeeman field, the corner states, the signature of the HOTIs, can be induced in a Z_2 topological insulator. Such Zeeman field can be prod…
Topological photonics provides a new paradigm in studying cavity quantum electrodynamics with robustness to disorder. In this work, we demonstrate the coupling between single quantum dots and the second-order topological corner state. Based on the second-order topological corner state, a topological photonic crystal cavity is designed and fabricated into GaAs slabs with quantum dots embedded. The coexistence of corner state and edge state with high quality factor close to 2000 is observed. The enhancement of photoluminescence intensity and emission rate are both observed when the quantum dot i…
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…
Parafermions are a natural generalization of Majorana fermions. We consider a breathing Kagome lattice with complex hoppings by imposing $\mathbb{Z}_{3}$ clock symmetry in the complex energy plane. It is a non-Hermitian generalization of the second-order topological insulator characterized by the emergence of topological corner states. We demonstrate that the topological corner states are parafermions in the present $\mathbb{Z}_{3}$ clock-symmetric model. It is also shown that the model is realized in electric circuits properly designed, where the parafermion corner states are observed by impe…
Higher-order topological insulators have triggered great interests because of exhibitions of non-trivial bulk topology on lower-dimensional boundaries like corners and hinges. While such interesting phases have been investigated in a plethora of systems by tuning staggered tunneling strength or manipulating existing topological phases, here we show that a higher-order topological phase can be driven solely by mirror-symmetric onsite potentials. We first introduce a simple chain model in one dimension that mimics the Su-Schrieffer-Heeger-like model. However, due to the lack of internal symmetri…
Square-root topological insulators are recently-proposed intriguing topological insulators, where the topologically nontrivial nature of Bloch wave functions is inherited from the square of the Hamiltonian. In this paper, we propose that higher-order topological insulators can also have their square-root descendants, which we term square-root higher-order topological insulators. There, emergence of in-gap corner states is inherited from the squared Hamiltonian which hosts higher-order topology. As an example of such systems, we investigate the tight-binding model on a decorated honeycomb latti…
We demonstrate that the presence of a localized state at the corner of an insulating domain is not always a predictor of a certain non-trivial higher-order topological invariant, even though they appear to co-exist in the same Hamiltonian parameter space. Our analysis of $C^n$-symmetric crystalline insulators and their multi-layer stacks reveals that topological corner states are not necessarily correlated with other well-established higher-order boundary observables, such as fractional corner charge or filling anomaly. In a $C^3$-symmetric breathing Kagome lattice, for example, we show that t…
We theoretically demonstrate that the second-order topological insulator with robust corner states can be realized in two-dimensional $\mathbb{Z}_2$ topological insulators by applying an in-plane Zeeman field. Zeeman field breaks the time-reversal symmetry and thus destroys the $\mathbb{Z}_2$ topological phase. Nevertheless, it respects some crystalline symmetries and thus can protect the higher-order topological phase. By taking the Kane-Mele model as a concrete example, we find that spin-helical edge states along zigzag boundaries are gapped out by Zeeman field whereas in-gap corner state at…
Quantized electric quadrupole insulators have recently been proposed as novel quantum states of matter in two spatial dimensions. Gapped otherwise, they can feature zero-dimensional topological corner mid-gap states protected by the bulk spectral gap, reflection symmetries and a spectral symmetry. Here we introduce a topolectrical circuit design for realizing such corner modes experimentally and report measurements in which the modes appear as topological boundary resonances in the corner impedance profile of the circuit. Whereas the quantized bulk quadrupole moment of an electronic crystal do…
The quest for new realizations of higher-order topological system has garnered much recent attention. In this work, we propose a paradigmatic brick lattice model where corner modes requires protection by nonsymmorphic symmetry in addition to two commuting mirror symmetries. Unlike the well-known square corner mode lattice, it has an odd number of occupied bands, which necessitates a different definition for the $\mathbb Z_2\times \mathbb Z_2$ topological invariant. By studying both the quadrupolar polarization and effective edge model, our study culminates in a phase diagram containing two dis…
Higher-order topological insulators, which support lower-dimensional topological boundary states than the first-order topological insulators, have been intensely investigated in the integer dimensional systems. Here, we provide a new paradigm by presenting experimentally a higher-order topological phase in a fractal-dimensional system. Through applying the Benalcazar, Bernevig, and Hughes model into a Sierpinski carpet fractal lattice, we uncover a squeezed higher-order phase diagram featuring the abundant corner states, which consist of zero-dimensional outer corner states and 1.89-dimensiona…
We theoretically study emergent edge and corner states in monolayer blue phosphorus (blue phosphorene) using the first-principles calculation and tight-binding model. We show that the existence of the Wannier orbitals at every bond center yields edge states both in zigzag and armchair nanoribbons. The properties of the edge states can be well described by a simple effective Hamiltonian for uncoupled edge orbitals, where the structural relaxation near the boundary significantly affects the edge band structure. For corner states, we examine two types of corner structures consisting of zigzag and…
Topological corner states are exotic topological boundary states that are bounded to zero-dimensional geometry even the dimension of systems is large than one. As an elegant physical correspondence, their numbers are dictated by the bulk topological invariants. So far, all previous realizations of HOTIs are hallmarked by $\mathbb{Z}_2$ topological invariants and therefore have only one corner state at each corner. Here we report an experimental demonstration of $\mathbb{Z}$-class HOTI phases in electric circuits, hosting $N$ corner modes at each single corner structure. By measuring the impeda…
Recently, higher-order topological phases that do not obey the usual bulk-edge correspondence principle have been introduced in electronic insulators and brought into classical systems, featuring with in-gap corner/hinge states. So far, second-order topological insulators have been realized in mechanical metamaterials, microwave circuit, topolectrical circuit and acoustic metamaterials. Here, using near-field scanning measurements, we show the direct observation of corner states in second-order topological photonic crystal (PC) slabs consisting of periodic dielectric rods on a perfect electric…
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 show that monolayer graphene intrinsically hosts higher-order topological corner states, in which electrons are localized topologically at atomic sizes. The emergence of the topological corner states in graphene is due to a nontrivial product of the Zak phases for two independent directions, which can be handily calculated graphically by using the bulk wavefunctions. We give an explicit expression that indicates the existence of topological corner states for various geometric edges and corner angles. We also demonstrate the nontrivial localization nature of the topological corner states in …
We propose the concept of 2D non-Abelian topological insulator which can explain the energy distributions of the edge states and corner states in systems with parity-time symmetry. From the viewpoint of non-Abelian band topology, we establish the constraints on the 2D Zak phase and polarization. We demonstrate that the corner states in some 2D systems can be explained as the boundary mode of the 1D edge states arising from the multi-band non-Abelian topology of the system. We also propose the use of off-diagonal Berry phase as complementary information to assist the prediction of edge states i…
