Quantum superposition demonstrated higher-order topological bound states in the continuum

Higher-Order Topological In-Bulk Corner State in Pure Diffusion Systems

Compared with conventional topological insulator that carries topological state at its boundaries, the higher-order topological insulator exhibits lower-dimensional gapless boundary states at its corners and hinges. Leveraging the form similarity between Schrödinger equation and diffusion equation, research on higher-order topological insulators has been extended from condensed matter physics to thermal diffusion. Unfortunately, all the corner states of thermal higher-order topological insulator reside within the band gap. Another kind of corner state, which is embedded in the bulk states, has not been realized in pure diffusion systems so far. Here, we construct higher-dimensional Su-Schrieffer-Heeger models based on sphere-rod structure to elucidate these corner states, which we term "in-bulk corner states." Because of the anti-Hermitian properties of diffusive Hamiltonian, we investigate the thermal behavior of these corner states through theoretical calculation, simulation, and experiment. Furthermore, we study the different thermal behaviors of in-bulk corner state and in-gap corner state. Our results would open a different gate for diffusive topological states and provide a distinct application for efficient heat dissipation.

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.

Valley edge states as bound states in the continuum

Bound states in the continuum (BICs) are spatially localized states with energy embedded in the continuum spectrum of extended states. The combination of BICs physics and nontrivial band topology theory givs rise to topological BICs, which are robust against disorders and meanwhile, the merit of conventional BICs is attracting wide attention recently. Here, we report valley edge states as topological BICs, which appear at the domain wall between two distinct valley topological phases. The robustness of such BICs is demonstrated. The simulations and experiments show great agreement. Our findings of valley related topological BICs shed light on both BICs and valley physics, and may foster innovative applications of topological acoustic devices.

Demonstration of bound states in the continuum in substrate integrated waveguides

Substrate integrated waveguides (SIWs) components play a crucial role in microwave devices fabricated by printed circuit board (PCB) technology. Bound states in the continuum (BICs) have high-quality factors that approach infinity. So far, there is little research on BICs in SIWs. Therefore, we studied a symmetry-protected BIC generated by the coupling between SIW and SIW resonators to fill this gap. Using the revised coupled mode theory (CMT), we explored the mechanism of resonance generation in this system. In addition, the effect of the geometrical parameters on the resonance is also investigated and higher Q3dB factors are obtained. The findings offer new insights into the design of BIC devices by traditional PCB technology, thus contributing to future applications in the integrated circuits field.

Non-Abelian Topological Bound States in the Continuum

Bound states in the continuum (BICs), which are spatially localized states with energies lying in the continuum of extended modes, have been widely investigated in both quantum and classical systems. Recently, the combination of topological band theory with BICs has led to the creation of topological BICs that exhibit extraordinary robustness against disorder. However, the previously proposed topological BICs are only limited in systems with Abelian gauge fields. Whether non-Abelian gauge fields can induce topological BICs and how to experimentally explore these phenomena remains unresolved. Here, we report the theoretical and experimental realization of non-Abelian topological BICs, which are generated by the interplay between two inseparable pseudospins and can coexist in each pseudospin subspace. This unique characteristic necessitates non-Abelian couplings that lack any Abelian counterparts. Furthermore, the non-Abelian couplings can also offer a new avenue for constructing topological subspace-induced BICs at bulk dislocations. Those exotic phenomena are observed by non-Abelian topolectrical circuits. Our results establish the connection between topological BICs and non-Abelian gauge fields, and serve as the catalyst for future investigations on non-Abelian topological BICs across different platforms.

Dynamical Detection of Topological Spectral Density

Local density of states (LDOS) is emerging as powerful means of exploring classical-wave topological phases. However, the current LDOS detection method remains rare and merely works for static situations. Here, we introduce a generic dynamical method to detect both the static and Floquet LDOS, based on an elegant connection between dynamics of chiral density and local spectral densities. Moreover, we find that the Floquet LDOS allows to measure out Floquet quasienergy spectra and identify topological π modes. As an example, we demonstrate that both the static and Floquet higher-order topological phase can be universally identified via LDOS detection, regardless of whether the topological corner modes are in energy gaps, bands, or continuous energy spectra without band gaps. Our study opens a new avenue utilizing dynamics to detect topological spectral densities and provides a universal approach of identifying static and Floquet topological phases.

Fractal photonic anomalous Floquet topological insulators to generate multiple quantum chiral edge states

Anomalous Floquet topological insulators with vanishing Chern numbers but supporting chiral edge modes are attracting more and more attention. Since the existing anomalous Floquet topological insulators usually support only one kind of chiral edge mode even at a large lattice size, they are unscalable and unapplicable for multistate topological quantum systems. Recently, fractal topological insulators with self-similarity have been explored to support more nontrivial modes. Here, we demonstrate the first experimental realization of fractal photonic anomalous Floquet topological insulators based on dual Sierpinski carpet consisting of directional couplers using the femtosecond laser direct writing. The fabricated lattices support much more kinds of chiral edge states with fewer waveguides and enable perfect hopping of quantum states with near unit transfer efficiency. Instead of zero-dimensional bound modes for quantum state transport in previous laser direct-written topological insulators, we generate multiple propagating single-photon chiral edge states in the fractal lattice and observe high-visibility quantum interferences. These suggest the successful realization of highly indistinguishable single-photon chiral edge states, which can be applied in various quantum operations. This work provides the potential for enhancing the multi-fold manipulation of quantum states, enlarging the encodable quantum information capacity in a single lattice via high-dimensional encoding and many other fractal applications.

Photonic topological subspace-induced bound states in the continuum

Bound states in the continuum (BICs) are intriguing localized states that possess eigenvalues embedded within the continuum of extended states. Recently, a combination of topological band theory and BIC physics has given rise to a novel form of topological matter known as topological BICs. In this work, we experimentally demonstrate the photonic topological subspace-induced BICs. By using femtosecond-laser writing, we experimentally establish a photonic nontrivial three-leg ladder lattice, thereby directly observe the localized propagation of two kinds of topological edge states which exist at different boundaries. Interestingly, such edge states appear in the continuum of the bulk modes, and the topological properties are inherited from its independent subspace Hamiltonian which contains a celebrated Su-Schrieffer-Heeger lattice. This work not only presents a novel, to the best of our knowledge, platform for investigating topological physics in optics, but also unveils exciting prospects for future exploration of other remarkable BICs.

Two Individual Super-Bound State Modes within Band Gap with Ultra-High Q Factor for Potential Sensing Applications in the Terahertz Wave Band

Bound states in the continuum (BICs) garnered significant research interest in the field of sensors due to their exceptionally high-quality factors. However, the wide-band continuum in BICs are noise to the bound states, and it is difficult to control and filter. Therefore, we constructed a top-bottom symmetric cavity containing three high permittivity rectangular columns. The cavity supports a symmetry-protected (SP) superbound state (SBS) mode and an accidental (AC) SBS mode within the bandgap. With a period size of 5 × 15, the bandgap effectively filters out the continuum, allowing only the bound states to exist. This configuration enabled us to achieve a high signal-to-noise ratio and a wide free-spectral-range. The AC SBS and the SP SBS can be converted into quasi-SBS by adjusting different parameters. Consequently, the cavity can function as a single-band sensor or a dual-band sensor. The achieved bulk sensitivity was 38 µm/RIU in terahertz wave band, and a record-high FOM reached 2.8 × 108 RIU-1. The effect of fabrication error on the performance for sensor application was also discussed, showing that the application was feasible. Moreover, for experimental realization, a 3D schematic was presented. These achievements pave the way for compact, high-sensitivity biosensing, multi-wavelength sensing, and other promising applications.

Quasi-bound states in the continuum induced by supercell coupling

In this paper, we propose what we believe to be a novel coupling mechanism for generating quasi-bound states in the continuum (quasi-BIC) in symmetrical metasurface structures. We demonstrate for the first time in theoretical predictions that supercell coupling can induce quasi-BIC(s). We utilize the coupled mode theory (CMT) to analyze the physical mechanism for the generation of quasi-bound states in such symmetrical structures, which result from our investigation of the coupling between sub-cells that are separated from supercells. We verify our theory by using both full-wave simulations and experiments.

Acoustic Higher-Order Weyl Semimetal with Bound Hinge States in the Continuum

Higher-order topological phases have raised widespread interest in recent years with the occurrence of the topological boundary states of dimension two or more less than that of the system bulk. The higher-order topological states have been verified in gapped phases, in a wide variety of systems, such as photonic and acoustic systems, and recently also observed in gapless semimetal phase, such as Weyl and Dirac phases, in systems alike. The higher-order topology is signaled by the hinge states emerging in the common band gaps of the bulk states and the surface states. In this Letter, we report our first prediction and observation of a new type of hinge states, the bound hinge states in the continuum (BHICs) bulk band, in a higher-order Weyl semimetal implemented in phononic crystal. In contrast to the hinge state in gap, which is characterized by the bulk polarization, the BHIC is identified by the nontrivial surface polarization. The finding of the topological BHICs broadens our insight to the topological states, and may stimulate similar researches in other systems such as electronic, photonic, and cold atoms systems. Our Letter may pave the way toward high-Q acoustic devices in application.

Universal Mirror-Stacking Approach for Constructing Topological Bound States in the Continuum

Bound states in the continuum (BICs) are counterintuitive localized states with eigenvalues embedded in the continuum of extended states. Recently, nontrivial band topology is exploited to enrich the BIC physics, resulting in topological BICs (TBICs) with extraordinary robustness against perturbations or disorders. Here, we propose a simple but universal mirror-stacking approach to turn nontrivial bound states of any topological monolayer model into TBICs. Physically, the mirror-stacked bilayer Hamiltonian can be decoupled into two independent subspaces of opposite mirror parities, each of which directly inherits the energy spectrum information and band topology of the original monolayer. By tuning the interlayer couplings, the topological bound state of one subspace can move into and out of the continuum of the other subspace continuously without hybridization. As representative examples, we construct one-dimensional first-order and two-dimensional higher-order TBICs, and demonstrate them unambiguously by acoustic experiments. Our findings will expand the research implications of both topological materials and BICs.

Organic Higher-Order Topological Insulators: Heterotriangulene-Based Covalent Organic Frameworks

The ability to design and control the chemical characteristics of covalent organic frameworks (COFs) offers a new avenue for the development of functional materials, especially with respect to topological properties. Based on density functional theory calculations, by varying the core units through the choice of bridging groups [O, C═O, CH₂, or C(CH₃)₂] and the linker units [acetylene, diacetylene, or benzene], we have designed heterotriangulene-based COFs that are predicted to be two-dimensional higher-order topological insulators (TIs). The higher-order TI characteristics of these COFs are identified via their topological invariants and the presence of in-gap topological corner modes and gapped edge states. The frontier molecular orbital energies of the building moieties play an important role in determining the size of the higher-order TI gap, which we find to be highly dependent on linker units. We also examined the deposition of the COFs on a boron nitride substrate to assess the feasibility of experimental observation of a higher-order TI phase in the organic layer. This work thus provides new insights into heterotriangulene-based COFs and guidance for the exploration of purely organic topological materials.

Progress in Topological Mechanics

Topological mechanics is rapidly emerging as an attractive field of research where mechanical waveguides can be designed and controlled via topological methods. With the development of topological phases of matter, recent advances have shown that topological states have been realized in the elastic media exploiting analogue quantum Hall effect, analogue quantum spin Hall effect, analogue quantum valley Hall effect, higher-order topological physics, topological pump, topological lattice defects and so on. This review aims to introduce the experimental and theoretical achievements with defect-immune protected elastic waves in mechanical systems based on the abovementioned methods, respectively. From these discussions, we predict the possible perspective of topological mechanics.