We present a two-dimensional (2D) lattice model that exhibits nontrivial topological phase in the absence of Berry curvature. Instead, connection provides model, whose integration over momentum space, so-called 2D Zak phase, yields fractional wave polarization each direction. These polarizations manifest themselves as degenerated edge states with opposite parities model.

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 structural imperfections. However, development nanoscale standing-wave cavities in topological photonics is rather slow, despite its importance when building densely-integrated integrated circuits. In this Letter, we report crystal nanocavity based on corner state, supported at 90-degrees-angled...

Abstract The quantum spin Hall effect lays the foundation for topologically protected manipulation of waves, but is restricted to one-dimensional-lower boundaries systems and hence limits diversity integration topological photonic devices. Recently, conventional bulk-boundary correspondence band topology has been extended higher-order cases that enable explorations states with codimensions larger than one such as hinge corner states. Here, we demonstrate a in two-dimensional crystal. Owing...

A topological electric quadrupole is a recently proposed concept that extends the theory of polarization crystals to higher orders. Such phase supports states localized on both edges and corners. In this work, we show in honeycomb lattice, helical edge pseudospin-polarized corner appear by making use pseudospin degree freedom related point group symmetry. Furthermore, argue general condition for emergence (pseudo)spinful existence either mirror or time-reversal Our results offer way...

Topological photonic crystals are designed based on the concept of Zak's phase rather than topological invariants such as Chern number and spin number, which rely existence a nonvanishing Berry curvature. Our (PCs) made pure dielectrics sit square lattice obeying ${C}_{4v}$ point-group symmetry. Two varieties PCs considered: one closely resembles electronic two-dimensional Su-Schrieffer-Heeger model, other continues an extension this analogy. In both cases, transitions induced by adjusting...

Topological defects in solid-state materials are crystallographic imperfections that local perturbations cannot remove. Owing to their nontrivial real-space topology, topological such as dislocations and disclinations could trap anomalous states associated with momentum-space topology. The topology of can be characterized by the Burgers vector $\mathbf{B}$, which is usually a fixed fraction integer lattice constant materials. Here we show dielectric photonic crystal---an artificial...

Typical higher-order topological systems require the fine-tuning of hopping textures and external fields, which considerably hinders their practical realization. Based on a simple picture that corners are "edges" edges, we determine in already-thoroughly-studied monolayer graphene, corner states appear without introducing any additional effects. Unlike quadrupole insulators, owing to degenerate Dirac points emergence depends angle edge geometries. We provide useful expression for indication...

We study topological states of honeycomb photonic crystals in the absence inversion symmetry using plane wave expansion and finite element methods. The breaking lattice leads to contrasting valley indices, i.e., valley-dependent Chern numbers momentum space. find that corner appear for 60 ° degree corners, but absent other which can be understood as sign flip number at corner. Our results provide an experimentally feasible platform exploring higher-order topology systems.

The Su-Schrieffer-Heeger (SSH) model is fundamental in topological insulators and relevant to understanding higher-order phases. This study explores the relationship between $n$-dimensional SSH its $(n\ensuremath{-}1)$-dimensional counterpart, identifying a hierarchical structure Hamiltonian that allows us solve an arbitrary analytically. By generalizing bulk-edge correspondence principle dimensions fashion using vectored Zak phase, we reveal type of insulator called insulators. In this...

Structural properties of the ship-transport network China (STNC) are studied in light recent investigations complex networks. STNC is composed a set routes and ports located along sea or river. Network including degree distribution, correlations, clustering, shortest path length, centrality, betweenness different definitions topology. It found that geographical constraint plays an important role topology STNC. We also study traffic flow based on weighted representation, demonstrate weight...

We present a two-dimensional (2D) lattice model that exhibits nontrivial topological phase in the absence of Berry curvature. Instead, connection provides model, whose integration over momentum space, so-called 2D Zak phase, yields fractional wave polarization each direction. These polarizations manifest themselves as degenerated edge states with opposite parities model.

The integration of topological concepts into electronic energy band theory has been a transformative development in condensed matter physics. Since then, this paradigm broadened its reach, extending to variety physical systems, including open ones. In study, we employ analogs the generalized $n$-dimensional Su-Schrieffer-Heeger model, cornerstone understanding insulators and higher-order states, unveil dimensional hierarchy states within thermal diffusive networks. Unlike their counterparts,...

In crystalline systems, higher-order topology, characterized by topological states of codimension greater than one, usually arises from the mismatch between Wannier centers and atomic sites, leading to filling anomalies. However, this phenomenon is less understood in aperiodic such as quasicrystals, where bulk are absent. study, we examine a modification Fibonacci chains squares derived typical model, two-dimensional Su-Schrieffer-Heeger investigate their properties. We discover that...

Focusing on the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model, we propose an additive rule between real-space topological invariant s of disclinations (related to Burgers vector B ) and reciprocal-space p bulk wave functions (the vectored Zak phase). The disclination-induced bound states in 2D SSH model appear only if ( + /2 π is nonzero modulo lattice constant. These disclination-bound are robust against perturbations respecting C 4 point group symmetry other within amplitude...

Breaking Hermiticity in topological systems gives rise to intriguing phenomena, such as the exceptional topology and non-Hermitian skin effect. In this work, we study a crystalline insulator sitting on Kekul\'e texture-modulated honeycomb lattice with balanced gain loss. We find that gaplessness of edge states system is insensitive geometries under moderate strength loss, unlike cases Hermitian insulators depend crucially. focus two types loss configurations, which are $PT$ symmetric...

Topological band theory has emerged as a powerful framework to classify and understand the electronic properties of materials. semimetals, which have protected crossings near Fermi level include Dirac Weyl points, lines, or surfaces, generally remain uncommon. Hypervalent compounds exhibit tunable highly degenerate nonbonding states driving crossings, so they could provide an effective platform explore topological semimetallic phases. Here, we identify topology structure hypervalent hydrides...

Tournaments of the game Go can be represented as a directed network in which vertices are players and link is pointing from winner to loser for each game. In this article, we present some interesting results Asian players, composed 756 9473 tournaments. It found that topological structure displays small-world property significant rich-club phenomenon where high-degree nodes tightly interconnected. addition, consider weighted version network, find weights obey power-law distributions, while...

Abstract We numerically study the energy band structures and corresponding wavefunctions of carbon nanotubes under circularly polarized irradiation perpendicular to tube axis on basis Floquet–Bloch theory. focus two typical frequencies, ħ Ω ≪ γ ∼ γ, where ≈ 3 eV is hopping graphene. Circularly found open gaps for metallic zigzag near Fermi shift degenerate points armchair in spectra away from K K′ points. Furthermore, high-frequency localizes either side nanotubes; particular, localized have...

Quantum coherence is important in quantum mechanics, and its essence from superposition principle. We study the of any two pure states that their arbitrary superposition, obtain relationship between them. In case have support on orthogonal subspaces, simple, is, difference state average them smaller than 1. other cases, we different a little more complicated relationships. Furthermore, also lower bound superpositions.

Band topology and related spin (or pseudo-spin) physics of photons provide us with a new dimension for manipulating light, which is potentially useful information communication data storage. Especially, the quantum Hall effect where electromagnetic waves propagate along surfaces samples strong spin-momentum locking, paves way achieving topologically protected photonic transport. Recently, conventional bulk-edge correspondence band has been extended to higher-order cases that enables...

In this paper, we present a unified framework of multiple attractors including multistability, multiperiodicity and multichaos. Multichaos, which means that the chaotic solution system lies in different disjoint invariant sets with respect to initial values, is very interesting important dynamical behavior, but it never addressed before best our knowledge. By constructing logistic map, show chaos can exist according value parameter p. end, by derived compact set Lorenz system, are...

We demonstrate a photonic crystal nanocavity based on topological corner state. The design exploits the hierarchical bulk-edge-corner correspondence for two-dimensional crystal, opening deterministic route to introduce nanocavities in photonics platforms. © 2019 Author(s)

Abstract Topological materials are renowned for their ability to harbor robust states localized at surfaces, edges, and corners. Accompanying the formation of these topological states, fractional charges appear on peripheral unit cells. Recently, bound disclinations crystallographic defects induced by nontrivial bulk invariants have been theoretically predicted experimentally confirmed. This is so-called bulk-disclination correspondence. Here, besides correspondence, we demonstrate an...