Gapped topological kink states and topological corner states in honeycomb lattice

Higher-order topological phases in crystalline and non-crystalline systems: a review

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

Atomically Thin Current Pathways in Graphene through Kekulé-O Engineering

We demonstrate that the current flow in graphene can be guided on atomically thin current pathways by the engineering of Kekulé-O distortions. A grain boundary in these distortions separates the system into topologically distinct regions and induces a ballistic domain-wall state. The state is independent of the orientation of the grain boundary with respect to the graphene sublattice and permits guiding the current on arbitrary paths. As the state is gapped, the current flow can be switched by electrostatic gates. Our findings are explained by a generalization of the Jackiw-Rebbi model, where the electrons behave in one region of the system as Fermions with an effective complex mass, making the device not only promising for technological applications but also a test-ground for concepts from high-energy physics. An atomic model supported by DFT calculations demonstrates that the system can be realized by decorating graphene with Ti atoms.

Intrinsic Second-Order Topological Insulator in Two-Dimensional Covalent Organic Frameworks

As an intriguing topological phase, higher-order topological insulators have attracted tremendous attention, but the candidate materials are limited in artificial and inorganic systems. In this work, we propose a universal approach to search for two-dimensional (2D) second-order topological insulators (SOTIs) in covalent organic frameworks (COFs) with C₃ symmetric cores. The underlying mechanism is illustrated through tight-binding calculations in a star lattice, showing the 2D SOTI in an overlooked energy window between two Kagome-bands with four types of nontrivial band structures. The emergence of the unique topological edge and corner states can be understood from the Su-Schrieffer-Heeger model. Furthermore, using the frontier orbital of the monomer building block as an indicator, the 2D SOTI is directly confirmed in three realistic COFs by first-principles calculations. Our results not only extend the concept of organic topological insulators from first-order to second-order but also demonstrate the universal existence of intrinsic higher-order topology in 2D COFs.

Observation of Emergent Dirac Physics at the Surfaces of Acoustic Higher-Order Topological Insulators

Using 3D sonic crystals as acoustic higher-order topological insulators (HOTIs), 2D surface states described by spin-1 Dirac equations at the interfaces between the two sonic crystals with distinct topology but the same crystalline symmetry are discovered. It is found that the Dirac mass can be tuned by the geometry of the two sonic crystals. The sign reversal of the Dirac mass reveals a surface topological transition where the surface states exhibit zero refractive index behavior. When the surface states are gapped, 1D hinge states emerge due to the topology of the gapped surface states. The zero refractive index behavior and the emergent topological hinge states are confirmed experimentally. This study reveals a multidimensional Wannier orbital control that leads to extraordinary properties of surface states and unveils an interesting topological mechanism for the control of surface waves.

Higher-order topological states in two-dimensional Stampfli-Triangle photonic crystals

In this Letter, the higher-order topological state (HOTS) and its mechanism in two-dimensional Stampfli-Triangle (2D S-T) photonic crystals (PhCs) is explored. The topological corner states (TCSs) in 2D S-T PhCs are based on two physical mechanisms: one is caused by the photonic quantum spin Hall effect (PQSHE), and the other is caused by the topological interface state. While the former leads to the spin-direction locked effect which can change the distribution of the TCSs, the latter is conducive to the emergence of multiband TCSs in the same structure due to the characteristics of plentiful photonic bandgap (PBG) and broadband in 2D S-T PhCs. These findings allow new, to the best of our knowledge, insight into the HOTS, and are significant to the future design of photonic microcavities, high-quality factor lasers, and other related integrated multiband photonic devices.

Observation and control of pseudospin switching in a finite-width topological photonic crystal

Finite-size effect plays a significant role in topology photonics not to mention in reality all experimental setups are in finite-size. A photonic bandgap is opened in the topological edge state dispersion if a topological photonic crystal with finite width is considered, and the bandgap size relies on the finite-size effect. Pseudospin-preserving and pseudospin-flipping processes can be realized when a selectively switch of the pseudospin of edge states are customized by our designs. Our microwave experiments also successfully demonstrate pseudospin switch-on and -off behaviors in a finite-width photonic crystal. By combining photonic crystals with finite widths, a multi-tunneling proposal of topological photonic crystals can also be achieved. Our study of the finite-size effect will provide new approaches and thoughts to improve the development of topological photonic devices in the future.

Second harmonic generation enhancement and directional emission from topological corner state based on the quantum spin Hall effect

Topological corner state has attracted much research interests since it does not obey the conventional bulk-edge correspondence and enables tightly confined light within small volumes. In this work, we demonstrate an enhanced second harmonic generation (SHG) from a topological corner state and its directional emission. To this end, we design an all-dielectric topological photonic crystal based on optical quantum spin Hall effect. In this framework, pseudospin states of photons, topological phase, and topological corner state are subsequently constructed by engineering the structures. It is shown that a high Q-factor of 3.66×10¹¹ can be obtained at the corner state, showing strong confinement of light at the corner. Consequently, SHG is significantly boosted and manifests directional out-of-plane emission. More importantly, the enhanced SHG has robustness against a broad class of defects. These demonstrated properties offer practical advantages for integrated optical circuits.

Topological kink states in graphene

Due to the unique band structure, graphene exhibits a number of exotic electronic properties that have not been observed in other materials. Among them, it has been demonstrated that there exist the one-dimensional valley-polarized topological kink states localized in the vicinity of the domain wall of graphene systems, where a bulk energy gap opens due to the inversion symmetry breaking. Notably, the valley-momentum locking nature makes the topological kink states attractive to the property manipulation in valleytronics. This paper systematically reviews both the theoretical research and experimental progress on topological kink states in monolayer graphene, bilayer graphene and graphene-like classical wave systems. Besides, various applications of topological kink states, including the valley filter, current partition, current manipulation, Majorana zero modes and etc, are also introduced.

Realistic flat-band model based on degenerate p-orbitals in two-dimensional ionic materials

Though several theoretical models have been proposed to design electronic flat-bands, the definite experimental realization in two-dimensional atomic crystal is still lacking. Here we propose a novel and realistic flat-band model based on threefold degenerate p-orbitals in two-dimensional ionic materials. Our theoretical analysis and first-principles calculations show that the proposed flat-band can be realized in 1T layered materials of alkali-metal chalogenides and metal-carbon group compounds. Some of the former are theoretically predicted to be stable as layered materials (e.g., K₂S), and some of the latter have been experimentally fabricated in previous works (e.g., Gd₂CCl₂). More interestingly, the flat-band is partially filled in the heterostructure of a K₂S monolayer and graphene layers. The spin polarized nearly flat-band can be realized in the ferromagnetic state of a Gd₂CCl₂ monolayer, which has been fabricated in experiments. Our theoretical model together with the material predictions provide a realistic platform for the study of flat-bands and related exotic quantum phases.

Floquet Second-Order Topological Phases in Momentum Space

Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers are shown to be measurable from the chiral dynamics of wave packets. Under open boundary conditions, multiple quartets Floquet corner modes with zero and π quasienergies emerge in the system and coexist with delocalized bulk states at the same quasienergies, forming second-order Floquet topological bound states in the continuum. The number of these corner modes is further counted by the bulk topological invariants according to the relation of bulk-corner correspondence. Our findings thus extend the study of HOTPs to momentum-space lattices and further uncover the richness of HOTPs and corner-localized bound states in continuum in Floquet systems.

Non-Abelian Braiding of Dirac Fermionic Modes Using Topological Corner States in Higher-Order Topological Insulator

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. Finally, the braiding operators for Dirac fermionic modes, especially their explicit matrix forms, are analytically derived and compared with the case of Majorana zero modes.