Unidirectional Edge States Research Articles

Pseudospin modes of surface acoustic wave and topologically protected sound transmission in phononic crystal

The manipulation of surface acoustic wave (SAW) in phononic crystal plays an important role in the applications of SAW. The introduction of topological acoustic theory has opened a new field for SAW in phononic crystals. Here we construct pseudospin modes of SAW and topological phase transition along the surface of phononic crystal. The local SAW propagation is realized by air cylindrical holes in honeycomb lattice arranged on rigid substrate, and the Dirac cone is formed at the <i>K</i> point of the first Brillouin zone. Furthermore, using the band-folding theory, double Dirac cones can be formed at the center <i>Г<sub>s</sub></i> point in the Brillouin zone of compound cell that contains six adjacent cylindrical air holes. The double Dirac cone can be broken to form two degenerated states and complete band gap by only shrinking or expanding the spacing of adjacent holes in the compound cell. It is found that the direction of energy is in a clockwise or counterclockwise direction, thus the pseudospin modes of SAW are constructed. The shrinkage-to-expansion of the compound cell leads to band inversion, and the system changes from trivial state to nontrivial state, accompanied by the phase transition. According to the bulk-boundary correspondence, the unidirectional acoustic edge states can be found at the interface between trivial system and nontrivial system. Then we can construct a topologically protected waveguide to realize the unidirectional transmission of surface waves without backscattering. This work provides a new possibility for manipulating the SAW propagating on the surface of phononic crystals and may be useful for making the acoustic functional devices based on SAW.

Inhibition of tunneling and edge state control in polariton topological insulators

We address the inhibition of tunneling in polariton condensates confined in a potential landscape created by a honeycomb array of microcavity pillars in the presence of spin-orbit coupling and Zeeman splitting in the external magnetic field. The coupling rate between the microcavity pillars can be strongly impacted even by weak out-of-phase temporal modulations of the depths of the corresponding potential wells. When such a modulation is implemented in truncated honeycomb arrays that realize a polariton topological insulator, which supports unidirectional edge states in the presence of spin-orbit coupling and Zeeman splitting, it allows controlling the velocity of the states. The origin of the phenomenon is the dynamical modulation with a proper frequency, which notably changes the dispersion of the system and the group velocity of edge states. We show that such a control is possible for modulation frequencies close to resonances for the inhibition of tunneling in a two-well configuration. Edge states considerably slow down, and even stop completely, when the modulation frequency approaches a resonant value, while above such a frequency, splitting of the edge states into wavepackets moving with different velocities occurs.

Realization of a magneto-optical near-zero index medium by an unpaired Dirac point

We realize an unpaired Dirac cone at the center of the first Brillouin zone, using a gyromagnetic photonic crystal with broken square sub-lattice symmetry and broken time reversal symmetry. The behavior of the Dirac modes can be described by a gyromagnetic effective medium model with near-zero refractive index, and Voigt parameter near unity. When two domains are subjected to opposite magnetic biases, there exist unidirectional edge states along the domain wall. This establishes a link between topological edge states and the surface waves of homogenous magneto-optical media.

Asymmetric Cherenkov emission in a topological plasmonic waveguide

Here, we investigate the Cherenkov emission by an array of moving electric charges in the vicinity of a topologically nontrivial gyrotropic material. It is shown that the nonreciprocal material response may result in a robustly asymmetric Cherenkov emission, such that the spectrum of the emitted radiation and the stopping power depend strongly on the sign of the particle velocity. It is demonstrated that the main emission channels are determined by the unidirectional edge states supported by the topological material. We consider as examples both magnetized plasmas and Weyl semi-metals. The latter may exhibit a spontaneous nonreciprocal response without a biasing magnetic field.

Resonant Edge‐State Switching in Polariton Topological Insulators

AbstractTopological insulators are unique devices supporting unidirectional edge states at their interfaces. Due to topological protection, such edge states persist in the presence of disorder and do not experience backscattering upon interaction with defects. Despite the topological protection and the fact that such states at the opposite edges of an insulator carry opposite currents, a physical mechanism exists allowing topological excitations propagating at opposite edges to be resonantly coupled. Such a mechanism uses weak periodic temporal modulations of the system parameters and does not affect the internal symmetry and topology of the system. This mechanism is illustrated in truncated honeycomb arrays of microcavity pillars, where topological insulation is possible for polaritons under the combined action of spin–orbit coupling and Zeeman splitting in the external magnetic field. The temporal modulation of the potential leads to a periodic switching between topological states with the same Bloch momentum, but located at the opposite edges. The switching rate is found to increase for narrower ribbon structures and for larger modulation depth, though it is changing nonmonotonically with the Bloch momentum of the input edge state. These results provide a promising realization of a coupling device based on topologically protected states.

Rashba sandwiches with topological superconducting phases

We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topological phases in the same setup. Our model relies only on conventional non-topological ingredients. The bilayer setup consists of an s-wave superconductor sandwiched between two two-dimensional electron gas layers with strong Rashba spin-orbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev superconducting pairing is dominant. In addition, an in-plane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in non-centrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Electrically tunable robust edge states in graphene-based topological photonic crystal slabs

Topological photonic crystals are optical structures supporting topologically protected unidirectional edge states that exhibit robustness against defects. Here, we propose a graphene-based all-dielectric photonic crystal slab structure that supports two-dimensionally confined topological edge states. These topological edge states can be confined in the out-of-plane direction by two parallel graphene sheets. In the structure, the excitation frequency range of topological edge states can be dynamically and continuously tuned by varying bias voltage across the two parallel graphene sheets. Utilizing this kind of architecture, we construct Z-shaped channels to realize topological edge transmission with diffrerent frequencies. The proposal provides a new degree of freedom to dynamically control topological edge states and potential applications for robust integrated photonic devices and optical communication systems.

Nonreciprocal lasing in topological cavities of arbitrary geometries.

Resonant cavities are essential building blocks governing many wave-based phenomena, but their geometry and reciprocity fundamentally limit the integration of optical devices. We report, at telecommunication wavelengths, geometry-independent and integrated nonreciprocal topological cavities that couple stimulated emission from one-way photonic edge states to a selected waveguide output with an isolation ratio in excess of 10 decibels. Nonreciprocity originates from unidirectional edge states at the boundary between photonic structures with distinct topological invariants. Our experimental demonstration of lasing from topological cavities provides the opportunity to develop complex topological circuitry of arbitrary geometries for the integrated and robust generation and transport of photons in classical and quantum regimes.

Unidirectional edge states in topological honeycomb-lattice membrane photonic crystals.

Photonic analogs of electronic systems with topologically non-trivial behavior such as unidirectional scatter-free propagation has tremendous potential for transforming photonic systems. Like in electronics topological behavior can be observed in photonics for systems either preserving time-reversal (TR) symmetry or explicitly breaking it. TR symmetry breaking requires magneto-optic photonics crystals (PC) or generation of synthetic gauge fields. For on-chip photonics that operate at optical frequencies both are quite challenging because of poor magneto-optic response of materials or substantial nanofabrication challenges in generating synthetic gauge fields. A recent work by Ma, et al. [Phys. Rev. Lett.114, 223901 (2015)] based on preserving pseudo TR symmetry offers a promising design scheme for observing unidirectional edge states in a modified honeycomb photonic crystal (PC) lattice of circular rods that offers encouraging alternatives. Here we propose through bandstructure calculations the inverse system of modified honeycomb PC of circular holes in a dielectric membrane which is more attractive from fabrication standpoint for on-chip applications. We observe trivial and non-trivial bandgaps as well as unidirectional edge states of opposite helicity propagating in opposite directions at the interface of a trivial and non-trivial PC structures. Around 1550nm operating wavelength ~55nm of bandwidth is possible for practicable values of design parameters (lattice constant, hole radii, membrane thickness, scaling factor etc.) and robust to reasonable variations in those parameters.

Majorana flat bands, chiral Majorana edge states, and unidirectional Majorana edge states in noncentrosymmetric superconductors

We study two-dimensional noncentrosymmetric nodal superconductors under Zeeman field and clarify the field angle dependence of topological properties. It has been shown that the nodal excitation acquires an excitation gap due to the Zeeman field perpendicular to anti-symmetric spin-orbit coupling, and then gapful topological superconductivity is realized. We show that the system undergoes gapful-gapless transition against tilting of the field. The gapless phase remains to show a finite band gap and unusual Majorana edge states in between the bulk bands. The Majorana edge states naturally propagate in a same direction between oppositely-oriented edges. We elucidate relations of such unidirectional Majorana edge states with chiral edge states in the gapful topological superconducting phase and previously studied Majorana flat bands at zero Zeeman field. A compact formula of topological invariants characterizing the edge states is given. The gapful-gapless topological phase transition and associated evolution of Majorana states are demonstrated in a model for $D+p$-wave superconductivity. Experimental realization in recently fabricated cuprate heterostructures and heavy fermion thin films is discussed.

Bulk-edge correspondence for topological photonic continua

Here, building on our previous work [Phys. Rev. B, 92, 125153, (2015)], it is shown that the propagation of unidirectional gapless edge states at an interface of two topologically distinct electromagnetic continua with a well-behaved asymptotic electromagnetic response is rigorously predicted by the bulk-edge correspondence principle. We work out detailed examples demonstrating that when the spatial-cut off of the nonreciprocal part of the material response is considered self-consistently in the solution of the relevant electromagnetic problem, the number of unidirectional gapless edge modes is identical to the difference of the Chern numbers of the bulk materials. Furthermore, it is shown how the role of the spatial cut-off can be imitated in realistic systems using a tiny air gap with a specific thickness. This theory provides a practical roadmap for the application of topological concepts to photonic platforms formed by nonreciprocal electromagnetic continua.

Modulational instability and solitary waves in polariton topological insulators

Optical microcavities supporting exciton–polariton quasi-particles offer one of the most powerful platforms for the investigation of the rapidly developing area of topological photonics in general, and of photonic topological insulators in particular. Energy bands of the microcavity polariton graphene are readily controlled by a magnetic field and influenced by the spin-orbit (SO) coupling effects, a combination leading to the formation of linear unidirectional edge states in polariton topological insulators as very recently predicted. In this work we depart from the linear limit of non-interacting polaritons and predict instabilities of the nonlinear topological edge states resulting in the formation of the localized topological quasi-solitons, which are exceptionally robust and immune to backscattering wave packets propagating along the graphene lattice edge. Our results provide a background for experimental studies of nonlinear polariton topological insulators and can influence other subareas of photonics and condensed matter physics, where nonlinearities and SO effects are often important and utilized for applications.

Manipulating Unidirectional Edge States Via Magnetic Plasmonic Gradient Metasurfaces

We show that a strongly enhanced coupling of spatially propagating electromagnetic waves to self-guiding unidirectional edge states (UESs) can be achieved by engineering a magnetic plasmonic gradient metasurface (GMS) made of an array of ferrite rods. The conversion efficiency of the incident photons into self-guiding UESs exhibits a transition from zero on an ordinary periodic surface to nearly 80 % on a surface incorporating a GMS. The underlying physics lies in that the magnetic plasmonic GMS enables a direct excitation of the edge states due to the band-folding or momentum compensation effect, which are in turn transformed into the self-guiding UESs on the ordinary periodic surface. The excitation of the UESs can also be revealed by considering the partial wave scattering amplitudes of the constituent rods on the surface, which manifests a change from a standing wave in the region subject to an external illumination to a self-guiding wave propagating and confined on the surface, a signature of UESs. The magnetic plasmonic GMS can also be used to implement the unidirectional phase control of the UES and the nonreciprocal Goos-Hanchen shift as a consequence of the time-reversal-symmetry breaking nature of the system and the strong coupling of the incident wave. In addition, the unidirectional features are shown to be flexibly controlled by either tailoring the gradient or tuning the external magnetic field, adding considerably to the performance of the magnetic plasmonic GMS systems.

Zero-frequency and extremely slow elastic edge waves in mechanical granular graphene

We report here the theoretical description and analysis of edge elastic waves in a semi-infinite mechanical granular graphene structure. The granular graphene, composed of spherical beads arranged in a single-layer honeycomb structure, is studied for two types of edge configurations: zigzag and armchair. Due to the existence of the rotational degrees of freedom of the grains, rotation-associated shear, bending and torsional couplings between the neighbor beads are activated. The dispersion curves of the edge waves are theoretically derived and numerically analyzed for various configurations of bead couplings, as well as the existence of edge states when the torsional or/and bending rigidities are weak/vanishing. Quasi-flat edge mode dispersion curves with near zero frequency are observed for both the zigzag and armchair edges. These quasi-flat dispersion curves, supporting the propagation of waves with extremely slow group velocity, tend to be perfect zero-frequency modes for zero torsional rigidity or vanish for zero bending rigidity, indicating that weak bending and torsional interbead interactions are critical in the transformation of zero-frequency modes into extremely slow propagating modes. These results on edge waves in mechanical granular graphene structures with rotational degrees of freedom are the necessary preliminary step for the design of granular meta-graphenes with artificial symmetry breaking for inducing topologically protected unidirectional edge states.

Topological Photonic Quasicrystals: Fractal Topological Spectrum and Protected Transport

We show that it is possible to have a topological phase in two-dimensional quasicrystals without any magnetic field applied, but instead introducing an artificial gauge field via dynamic modulation. This topological quasicrystal exhibits scatter-free unidirectional edge states that are extended along the system's perimeter, contrary to the states of an ordinary quasicrystal system, which are characterized by power-law decay. We find that the spectrum of this Floquet topological quasicrystal exhibits a rich fractal (self-similar) structure of topological "minigaps," manifesting an entirely new phenomenon: fractal topological systems. These topological minigaps form only when the system size is sufficiently large because their gapless edge states penetrate deep into the bulk. Hence, the topological structure emerges as a function of the system size, contrary to periodic systems where the topological phase can be completely characterized by the unit cell. We demonstrate the existence of this topological phase both by using a topological index (Bott index) and by studying the unidirectional transport of the gapless edge states and its robustness in the presence of defects. Our specific model is a Penrose lattice of helical optical waveguides - a photonic Floquet quasicrystal; however, we expect this new topological quasicrystal phase to be universal.

The emergence of Dirac points in photonic crystals with mirror symmetry.

We show that Dirac points can emerge in photonic crystals possessing mirror symmetry when band gap closes. The mechanism of generating Dirac points is discussed in a two-dimensional photonic square lattice, in which four Dirac points split out naturally after the touching of two bands with different parity. The emergence of such nodal points, characterized by vortex structure in momentum space, is attributed to the unavoidable band crossing protected by mirror symmetry. The Dirac nodes can be unbuckled through breaking the mirror symmetry and a photonic analog of Chern insulator can be achieved through time reversal symmetry breaking. Breaking time reversal symmetry can lead to unidirectional helical edge states and breaking mirror symmetry can reduce the band gap to amplify the finite size effect, providing ways to engineer helical edge states.

Linear and nonlinear traveling edge waves in optical honeycomb lattices

Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudomagnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudofields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schr\odinger equation along the edge. Nonlinear states termed edge solitons are predicted analytically and are found numerically.