Mid-IR Electromagnetic Topological Edge Modes: Design, Fabrication, and Polarization States Confirmed by High-Angle-Resolution Reflection Spectroscopy

We successfully fabricated silicon-on-insulator (SOI) photonic crystal (PhC) slabs in which electromagnetic topological band gaps and edge modes were materialized for symmetric transverse-electric-like modes. Because the structure of our specimens can be regarded as symmetric about the horizontal middle plane of the PhC slab, their symmetric and antisymmetric eigenmodes were rigorously separated, so we could achieve genuine photonic band gaps for the former. We fabricated those specimens by electron beam lithography of the top Si layer of SOI wafers and successive plasma-enhanced chemical vapor deposition of a ${\mathrm{SiO}}_{2}$ capping layer. We confirmed the complete common band gap of topologically trivial and nontrivial PhCs and the topological edge modes on the boundary between them by angle-resolved reflection spectroscopy in the mid-infrared range. This is an observation of the common band gap and edge modes that materialized in symmetric PhC slabs without a membrane structure, that is, without the etching of a sacrificial layer under the PhC to increase the refractive-index contrast and the width of the common band gap.

In this paper, optically controllable and topologically protected plasmon transport is implemented via a topological nanohole plasmonic waveguide coupled to a standard edge mode of a graphene metasurface. By introducing nanoholes with different sizes in the unit cell, one breaks the spatial-inversion symmetry of a graphene metasurface in which the topological waveguide is constructed, leading to the emergence of topologically protected modes located in a nontrivial band-gap. Based on the strong Kerr effect and tunable optical properties of graphene, the coupling between the edge and topological interface modes can be efficiently controlled by optical means provided by an optical pump beam injected in a bulk mode. In particular, by tuning the power inserted in the bulk mode, one can control the difference between the wave-vectors of the topological and edge modes and consequently the optical power coupled in the topological mode. Our results show that when the pump power approaches a specific value, the edge and topological modes become phase-matched and the topological waveguide mode can be efficiently excited. Finally, we demonstrated that the optical coupling is strongly dependent on the group-velocity of the pump mode, a device feature that can be important in practical applications.

The topological characteristics, including invariant topological orders, band inversion, and the topological edge mode (TEM) in the photonic insulators, have been widely studied. Whether people can take advantage of intriguing topological modes in simple one-dimensional systems to implement some practical applications is an issue which people are increasingly concerned about. In this work, based on a photonic dimer chain composed of ultra-subwavelength resonators, we verify experimentally that the TEM in the effective second-order parity-time (PT) system is immune to the inner disorder perturbation, and can be used to realize the long-range wireless power transfer (WPT) with high transmission efficiency. To intuitively show the TEM can be used for WPT, a power signal source is used to excite the TEM. It can be clearly seen that two LED lamps with 0.5-W at both ends of the structure are lighted up with the aid of TEMs. In addition, in order to solve the special technical problems of standby power loss and frequency tracking, we further propose that a WPT system with effective third-order PT symmetry can be constructed by using one topological interface mode and two TEMs. Inspired by the long-range WPT with TEMs in this work, it is expected to use more complex topological structures to achieve energy transmission with more functions, such as the WPT devices whose direction can be selected flexibly in the quasiperiodic or trimer topological chains.

The well-known Su-Schrieffer-Heeger (SSH) model predicts that a chain of sites with alternating coupling constant exhibits two topological distinct phases, and at the truncated edge of the topological nontrivial phase there exists topologically protected edge modes. Such modes are named zero-energy modes as their eigenvalues are located exactly at the midgaps of the corresponding bandstructures. The previous publications have reported a variety of photonic realizations of the SSH model, however, all of these studies have been restricted in the systems of time-reversal-symmetry (TRS), and thus the important question how the breaking of TRS affects the topological edge modes has not been explored. In this work, to the best of our knowledge, we study for the first time the topological zero-energy modes in the systems where the TRS is broken. The system used here is semiconductor microcavities supporting exciton-polariton quasi-particle, in which the interplay between the spin-orbit coupling stemming from the TE-TM energy splitting and the Zeeman effect causes the TRS to break. We first study the topological edge modes occurring at the edge of one-dimensional microcavity array that has alternative coupling strengths between adjacent microcavity, and, by rigorously solving the Schrdinger-like equations (see Eq.(1) or Eq.(2) in the main text), we find that the eigen-energies of topological zero-energy modes are no longer pinned at the midgap position:rather, with the increasing of the spin-orbit coupling, they gradually shift from the original midgap position, with the spin-down edge modes moving toward the lower band while the spin-up edge modes moving towards the upper band. Interestingly enough, the mode profiles of these edge modes remain almost unchanged even they are approaching the bulk transmission bands, which is in sharp contrast to the conventional defect modes that have an origin of bifurcation from the Bloch mode of the upper or lower bands. We also study the edge modes in the two-dimensional microcavity square array, and find that the topological zero modes acquire mobility along the truncated edge due to the coupling from the adjacent arrays. Importantly, owing to the breaking of the TRS, a pair of counterpropagating edge modes, of which one has a momentum k and the other has -k, is no longer of energy degeneracy; as a result the scattering between the forward-and backward-propagating modes is greatly suppressed. Thus, we propose the concept of the one-dimensional topological zero-energy modes that are propagating along the two-dimensional lattice edge, with extremely weak backscattering even on the collisions of the topological zero-energy modes with structural defects or disorder.

We study the dynamic long range interaction induced topological photonic bands and edge modes in a one-dimensional (1D) array of strongly dispersive gyromagnetic resonant cylinders. We propose a 1D topological model for such a dispersive gyromagnetic system and demonstrate that the dynamic long-range interaction can lead to localized topological edge modes, while the quasistatic interaction alone does not. Different from the conventional Su-Schrieffer-Heeger model that has only nearest-neighbor interactions, we find that the normal modes of the system coupled strongly to the photon mode of the background medium and the dynamic effects create a different band gap. Our results indicate the importance of the dynamic long-range interactions on the band structures and topological edge modes in strongly dispersive gyromagnetic systems.

Non-Hermitian systems have attracted much attention during the past few years, both theoretically and experimentally. The existence of non-Hermiticity can induce multiple exotic phenomena that cannot be observed in Hermitian systems. In this work, we introduce a new non-Hermitian system called the non-Hermitian mosaic dimerized lattice. Unlike the regular nonreciprocal lattices where asymmetric hoppings are imposed on every hopping term, here in the mosaic dimerized lattices the staggered asymmetric hoppings are only added to the nearest-neighboring hopping terms with equally spaced sites. By investigating the energy spectra, the non-Hermitian skin effect (NHSE), and the topological phases in such lattice models, we find that the period of the mosaic asymmetric hopping can influence the system’s properties significantly. For a system with real system parameters, we find that as the strength of asymmetric hopping increases, the energy spectra of the system under open boundary conditions will undergo a real-imaginary or real-complex transition. As to the NHSE, we find that when the period is odd, there appears no NHSE in the system and the spectra under open boundary conditions (OBCs) and periodic boundary conditions (PBCs) are the same (except for the topological edge modes under OBCs). If the period of the mosaic asymmetric hopping is even, the NHSE will emerge and the spectra under different boundary conditions exhibit distinctive structures. The PBC spectra form loop structures, indicating the existence of point gaps that are absent in the spectra under OBCs. The point gap in the PBC spectrum is shown to be the topological origin of the NHSE under OBCs, which also explains the NHSE in our mosaic dimerized lattices. To distinguish whether the bulk states of the system under OBCs are shifted to the left or right end of the one-dimensional lattice due to the NHSE, we define a new variable called the directional inverse participation ratio (dIPR). The positive dIPR indicates that the state is localized at the right end while the negative dIPR corresponds to the states localized at the left end of the one-dimensional lattice. We further study the topological zero-energy edge modes and characterize them by calculating the Berry phases based on the generalized Bloch Hamiltonian method. In addition, we also find that the topological edge modes with nonzero but constant energy can exist in the system. Our work provides a new non-Hermitian lattice model and unveils the exotic effect of mosaic asymmetric hopping on the properties of non-Hermitian systems.

Topological phonon polaritons (TPhPs) are promising optical modes relevant in long-range radiative heat transfer, information processing and infrared sensing, whose topological protection is expected to enable their robust existence and transport. In this work we show that TPhPs can be supported in one-dimensional (1D) bichromatic silicon carbide nanoparticle (NP) chains, and demonstrate that they can considerably enhance radiative heat transfer for an array much longer than the wavelength of radiation. By introducing incommensurate or commensurate modulations on the interparticle distances, the NP chain can be regarded as an extension of the off-diagonal Aubry-Andr\'e-Harper (AAH) model. By calculating the eigenstate spectra with respect to the modulation phase that creates a synthetic dimension, we demonstrate that under this type of modulation the chain supports nontrivial topological modes localized over the boundaries, since the present system inherits the topological property of two-dimensional integer quantum Hall systems. In this circumstance the gap-labeling theorem and corresponding Chern number can be used to characterize the features of band gaps and topological edge modes. Based on many-body radiative heat transfer theory for a set of dipoles, we theoretically show the presence of topological gaps and midgap TPhPs can substantially enhance radiative heat transfer for an array much longer than the wavelength of radiation. We show how the modulation phase that acts as the synthetic dimension can tailor the radiative heat transfer rate by inducing or annihilating topological modes. We also discuss the role of dissipation in the enhancement of radiative heat transfer. These findings therefore provide a fascinating route for tailoring near-field radiative heat transfer based on the concept of topological physics.

We investigate the topological edge modes of surface plasmon polaritons (SPPs) in a non-Hermitian system composed of graphene pair arrays with alternating gain and loss. The topological edge modes emerge when two topologically distinct graphene arrays are connected. The gain and loss present in the system provide additional ways to control the propagation loss and field distributions of the topological edge modes. Moreover, the existence of the topological edge modes is related to the broken parity-time (PT) symmetry. We show the beam diffraction can be steered by tuning the chemical potential of graphene. Thanks to the strong confinement of SPPs, the topological edge modes can be squeezed into a lateral width of ~λ/70. We also show such modes can be realized in lossy graphene waveguides without gain. The study provides a promising approach to realizing robust light transport and optical switches on a deep-subwavelength scale.

We analyze the local density of electromagnetic states (LDOS) in finite 1D topological gratings to unveil the unique nature of topological edge modes. At the frequency corresponding to the topological edge mode, the LDOS intensity decays exponentially from the topological phase transition interface toward both sides. Our results show that the topological edge state’s LDOS peak exceeds that of a defect mode by over 1.5× and that of a band edge mode by nearly 100×, with a correspondingly narrower FWHM, benefiting from intrinsic topological protection and strong spatial confinement. We investigate how structural symmetry breaking and polarization effects modify the LDOS distribution. In addition, the coupling between topological edge modes of different orders and dipoles with various orientations is analyzed through LDOS calculations. By adjusting the local perturbation strength of the nearest and next-nearest neighboring unit cells at the topological interface, the LDOS intensity at the resonance frequencies of different-order topological edge modes can be modulated, thereby enabling selection of the dominant cavity mode. Consequently, LDOS calculations offer valuable insights for optimizing light–matter interaction and reducing laser thresholds. This work provides practical guidelines for designing topological lasers with improved efficiency and narrow spectral linewidth.

Many magnetic materials are predicted to exhibit bosonic topological edge modes in their excitation spectra, because of the nontrivial topology of their magnon, triplon, or other quasiparticle band structures. However, there is a discrepancy between theory prediction and experimental observation, which suggests some underlying mechanism that intrinsically suppresses the expected experimental signatures, like the thermal Hall current. Many-body interactions that are not accounted for in the noninteracting quasiparticle picture are most often identified as the reason for the absence of the topological edge modes. Here we report persistent bosonic edge modes at the boundaries of a ladder quantum paramagnet with gapped triplon excitations in the presence of the full many-body interaction. We use tensor network methods to resolve topological edge modes in the time-dependent spin-spin correlations and the dynamical structure factor, which is directly accessible experimentally. We further show that signatures of these edge modes survive even when the noninteracting quasiparticle theory breaks down; we discuss the topological phase diagram of the model, demonstrate the fractionalization of its low-lying excitations, and propose potential material candidates.

We analyze chiral topological edge modes in a non-Hermitian variant of the 2D Dirac equation. Such modes appear at interfaces between media with different "masses" and/or signs of the "non-Hermitian charge." The existence of these edge modes is intimately related to exceptional points of the bulk Hamiltonians, i.e., degeneracies in the bulk spectra of the media. We find that the topological edge modes can be divided into three families ("Hermitian-like," "non-Hermitian," and "mixed"); these are characterized by two winding numbers, describing two distinct kinds of half-integer charges carried by the exceptional points. We show that all the above types of topological edge modes can be realized in honeycomb lattices of ring resonators with asymmetric or gain-loss couplings.

Symmetry-protected topological phases1-4 cannot be described by any local order parameter and are beyond the conventional symmetry-breaking model5. They are characterized by topological boundary modes that remain stable under symmetry respecting perturbations1-4,6-8. In clean, gapped systems without disorder, the stability of these edge modes is restricted to the zero-temperature manifold; at finite temperatures, interactions with mobile thermal excitations lead to their decay9-11. Here we report the observation of a distinct type of topological edge mode12-14, which is protected by emergent symmetries and persists across the entire spectrum, in an array of 100 programmable superconducting qubits. Through digital quantum simulation of a one-dimensional disorder-free stabilizer Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles for a wide range of initial states. We show that the interaction between these edge modes and bulk excitations can be suppressed by dimerizing the stabilizer strength, leading to an emergent U(1) × U(1) symmetry in the prethermal regime of the system. Furthermore, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence, despite the system being disorder-free and at finite temperature. Our results establish a viable digital simulation approach15-18 to experimentally study topological matter at finite temperature and demonstrate a potential route to construct long-lived, robust boundary qubits in disorder-free systems.

We study the edge modes of a finite LC-resonator circuit that consists of alternatingly arranged inductors, separated by identical capacitors, each of which is grounded. The circuit has the configuration of the Su–Schrieffer–Heeger (SSH) model, and one-to-one correspondence between these two models can be established. Interestingly, when the corresponding SSH model is in the topological nontrivial regime, only one topological edge mode appears in the finite LC-resonator circuit, and the other is absent. Through the analysis of the circuit Hamiltonian, we find that only by reducing the boundary capacitance can the missing topological edge mode occur, but the reappeared topological mode is different from the standard one. When the boundary capacitance is lower than a critical value, a Tamm mode related to the local resonance also appears. When the finite circuit system is in the trivial regime, the replacement of the boundary capacitor could result in only the Tamm mode due to the restriction of the bulk-edge correspondence. The present study of the influence of the boundary configuration on the topological edge modes will deepen our understanding of topolectrical circuits.

Finite topologically non-trivial systems are often characterised by the presence of bound states at their physical edges. These topological edge modes can be distinguished from usual Shockley waves energetically, as their energies remain finite and in-gap. On a clean 1D or reducible 2D model, in either the commensurate or semi-infinite case, the edge modes can be obtained analytically, as shown in [PRL 71, 3697 (1993)] and [PRA 89, 023619 (2014)]. We put forward a method for obtaining the spectrum and wave functions of topological edge modes for arbitrary finite lattices, including the incommensurate case. A small number of parameters are easily determined numerically, with the form of the eigenstates remaining fully analytical. We also obtain the bulk modes in the finite system analytically and their eigenenergies, which lie within the infinite-size limit continuum. Our method is general and can be easily applied to obtain the properties of non-topological models and/or extended to include impurities. As an example, we consider the case of an impurity located next to one edge of a 1D system, equivalent to a softened boundary in a separable 2D model. We show that a localised impurity can have a drastic effect on the edge modes of the system. Using the periodic Harper and Hofstadter models to illustrate our method, we find that, on increasing the impurity strength, edge states can enter or exit the continuum, and a trivial Shockley state bound to the impurity may appear. The fate of the topological edge modes in the presence of impurities can be addressed by quenching the impurity strength. We find that at certain critical impurity strengths, the transition probability for a particle initially prepared in an edge mode to decay into the bulk exhibits discontinuities that mark the entry and exit points of edge modes from and into the bulk spectrum.

Topological phases have recently witnessed a rapid progress in non-Hermitian systems. Here we study a one-dimensional non-Hermitian Aubry-Andr\'e-Harper model with imaginary periodic or quasiperiodic modulations. We demonstrate that the non-Hermitian off-diagonal AAH models can host zero-energy modes at the edges. In contrast to the Hermitian case, the zero-energy mode can be localized only at one edge. Such a topological phase corresponds to the existence of a quarter winding number defined by eigenenergy in momentum space. We further find the coexistence of a zero-energy mode located only at one edge and topological nonzero energy edge modes characterized by a generalized Bott index. In the incommensurate case, a topological non-Hermitian quasicrystal is predicted where all bulk states and two topological edge states are localized at one edge. Such topological edge modes are protected by the generalized Bott index. Finally, we propose an experimental scheme to realize these non-Hermitian models in electric circuits. Our findings add a new direction for exploring topological properties in Aubry-Andr\'e-Harper models.