Zak Phase Research Articles

Topological protection in a strongly nonlinear interface lattice

Mechanical topological insulators are well understood for linear and weakly nonlinear systems, however traditional analysis methods break down for strongly nonlinear systems since linear methods can not be applied in that case. We study one such system in the form of a one-dimensional mechanical analog of the Su-Schrieffer-Heeger interface model with strong nonlinearity of the cubic form. The frequency-energy dependence of the nonlinear bulk modes and topologically insulated mode is explored using Numerical continuation of the system's nonlinear normal modes (NNMs), and the linear stability of the NNMs are investigated using Floquet Multipliers (FMs) and Krein signature analysis. We find that the nonlinear topological lattice supports a family of topologically insulated NNMs that are parameterized by the total energy of the system and are stable within a range of frequencies. Next, it is shown that empirical calculations of the geometric Zak Phase can define an energy threshold to predict the excitability of the nonlinear topological mode, and that this threshold coincides with the energy that the topological NNM intersects the linear bulk-spectrum. These predictions are validated with numerical simulations of the nonlinear topological system. These results are also tested for parametric perturbations which preserve and break chirality in the system. Thus, we provide a new method for analyzing and predicting the existence of topologically insulated modes in a strongly nonlinear lattice based on the physical observable of band topology.

Topology of a parity–time symmetric non-Hermitian rhombic lattice

We investigate the topological properties of a trimerized parity–time () symmetric non-Hermitian rhombic lattice. Although the system is -symmetric, the topology is not inherited from the Hermitian lattice; in contrast, the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices. The bulk–boundary correspondence is valid and the Bloch bulk captures the band topology. Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach. In addition, the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice. Our findings reveal the topological properties of the -symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.

Topological edge states controlled by next-nearest-neighbor coupling and Peierls phase in a P T-symmetric trimerized lattice.

We study the topological features in a trimerized lattice of parity-time symmetry with comparable nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings as well as a Peierls phase. Eigen energies of four edge states in two bandgaps, of topological origin verified by the quantized total Zak phase, are surprisingly independent of the NNN coupling and the Peierls phase. Topological regions with respect to the intercell NN coupling, as the intracell NN coupling is fixed, can be extended with reinforced localization strengths for one pair of edge states but reduced with weakened localization strengths for the other pair of edge states, by increasing the NNN coupling. The partial overlapping between extended and reduced topological regions promises then a two-step phase transition of 'zero - two - four' edge states, viable to be periodically modulated by the Peierls phase.

Nonlinear Topological Effects in Optical Coupled Hexagonal Lattice.

Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson–hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.

Specialty topological fiber using periodic lattice geometries

Topological photonics provides a new platform to control and manipulate the flow of light in photonic devices that exhibit topological phase transitions. One way to introduce such transition of geometric phases in photonic waveguides is to modulate the structure of the photonic crystals, array waveguides, or the metamaterials in such a way that it eventually changes the bulk topology. Such topological waveguides localize the light at the boundary of the two distinct bulk systems with different topological invariant. Based on this fundamental physics, we demonstrate a robust and controlled propagation of light in a class of specialty optical fiber which exhibits the topological phase transition between a nontrivial one-dimensional (1D) periodic cylinder and its topologically trivial counterpart. The key idea is to explore the possibility of topological phase transition of 1D photonic bands over the momentum space by tuning the unit-cell dimension and its individual thicknesses. We have systematically computed the trivial and the nontrivial topological invariants, the Zak phase, of designed cylindrical 1D lattices with inversion symmetry over the first Brillouin zone. To establish the presence and the robustness of the localized edge mode, we have studied the reflectance of the geometry in the presence of disorder. Also, the simulated field profile and the back-reflectionless propagation of the edge mode through the fiber have been shown. Besides conventional fibers, such topological optical fibers are definitely a waveguide geometry for robust delivery of light and thus would be very significant to open up a new paradigm in guided wave photonics.

Spin-momentum locked interface modes based on transverse resonance and Zak phase in finite thickness dielectric slabs

The field of topological photonics has made great strides in the past decade with many new designs based on bandgap and band inversion structures that provide robust, unidirectional, and reflection-free propagation of energy. The topological invariant or Chern number of a metamaterial guarantees the existence of topologically protected edge modes. However, its mathematical application to real systems is not always straightforward and can be greatly simplified by reducing the dimensional complexity of the problem by the calculation of a Zak phase that determines the topological phase in just one dimension. This work explores two methods of creating interface modes with finite height dielectric slabs: (1) transverse resonance through variable edge truncation of a photonic crystal (PhC) or cell sliding and (2) a uni-axial topological phase by means of scaling the internal features of a unit cell. The proposed metamaterial devices use the same C4v symmetric unit cell structure on both sides of the interface and are finite in all three dimensions, allowing for easy fabrication, excitation, and implementation in real-world applications. The all-dielectric design also enables an easy transition to and from conventional PhC waveguides and lends itself well to operation in frequencies spanning across the microwave and optical spectrum without concerns of additional metallic losses in the THz region.

Straight-angled corner state in acoustic second-order topological insulator

Higher-order topological insulators, providing a horizon beyond the conventional bulk-interface correspondence, have zero-dimensional topological corner states at different corners, such as a right-angled corner. Here, we numerically demonstrate a hierarchy of band topology in a two-dimensional square-lattice sonic crystal. In our system, the acoustic bulk bands mimic the quantum spin Hall effect, while gapped interface states emerge in the bulk gap due to symmetry reduction and carry quantized Zak phases, which result in zero-dimensional topological corner states in the interface gap. By direct acoustic measurement, we observe that zero-dimensional topological corner states are confined at not only the right-angled corner, but also the straight-angled corner. Our results present this type of topological shape-dependent corner state, which possibly possesses potential application in controlling and localizing acoustic waves.

Experimentally Detecting Quantized Zak Phases without Chiral Symmetry in Photonic Lattices.

Symmetries play a major role in identifying topological phases of matter and in establishing a direct connection between protected edge states and topological bulk invariants via the bulk-boundary correspondence. One-dimensional lattices are deemed to be protected by chiral symmetry, exhibiting quantized Zak phases and protected edge states, but not for all cases. Here, we experimentally realize an extended Su-Schrieffer-Heeger model with broken chiral symmetry by engineering one-dimensional zigzag photonic lattices, where the long-range hopping breaks chiral symmetry but ensures the existence of inversion symmetry. By the averaged mean displacement method, we detect topological invariants directly in the bulk through the continuous-time quantum walk of photons. Our results demonstrate that inversion symmetry protects the quantized Zak phase but edge states can disappear in the topological nontrivial phase, thus breaking the conventional bulk-boundary correspondence. Our photonic lattice provides a useful platform to study the interplay among topological phases, symmetries, and the bulk-boundary correspondence.

Four dimensional second-order topological insulator based on a synthetic plasmonic metasurface.

It is possible to explore higher dimensional topological properties in lower dimensional structures by introducing additional synthetic dimensions. In this Letter, we construct a four-dimensional (4D) second-order topological insulator using gradient nanoparticle arrays arranged in a periodic lattice. The nanoparticle array has spatially varying inter-particle distance along x and y directions, which can be regarded as two synthetic dimensions. Different from higher-order topological insulators in classical wave systems, the higher-order topological states in this 4D system are protected by a pair of first Chern numbers in two-dimensional (2D) subspaces instead of by the quantized 2D Zak phases. It is shown that there exist (4-1)- and (4-2)-dimensional boundary states for both transverse and longitudinal collective resonant modes, which provides new, to the best of our knowledge, mechanisms for light confinement and control in such a plasmonic superlattice.

Multihole models for deterministically placed acceptor arrays in silicon

We compute the electronic structure of acceptor clusters in silicon by using three methods to include electron correlations: the full configuration interaction, the Heitler-London approximation, and the unrestricted Hartree-Fock method. We show both the HL approach and the UHF method are good approximations to the ground state of the full CI calculation for a pair of acceptors and for finite linear chains. The total energies for finite linear chains show the formation of a 4-fold degenerate ground state when there is a weak bond at the end of the chain, which is shown to be a manifold of topological edge states. We identify a change in the angular momentum composition of the ground state at a critical pattern of bond lengths and show it is related to a crossing in the Fock matrix eigenvalues. We also test the symmetry of the UHF solution and compare it to the full CI; the symmetry is broken under almost all the arrangements by the formation of a magnetic state in UHF, and we find further broken symmetries for some particular arrangements related to crossings or potential crossings between the Fock-matrix eigenvalues. We also compute the charge distributions across the acceptors obtained from the eigenvectors of the Fock matrix: with weak bonds at the chain ends, two holes are localized at either end of the chain while the others have a nearly uniform distribution over the middle; this implies the existence of the non-trivial edge states. We apply the UHF method to treat an infinite linear chain with periodic boundary conditions. We find the band structures in the UHF approximation and compute the Zak phases for the occupied Fock-matrix eigenvalues; we find they do not correctly predict the topological edge states in this interacting system. We find direct study of the quantum numbers characterising the edge states provides a better insight into their topological nature.

Local resonator stimulated polarization transition in metamaterials and the formation of topological interface states

Developing the acoustic-elastic analogy of a topological insulator has attracted extensive research attention in recent years. Designs developed in the literature strongly rely on reforming the host structure of a system to achieve topology transition. In this article, an innovative topological metamaterial is presented. Unlike topological phononic crystals, the proposed topological metamaterial has a uniform host structure without any phononic crystal features. The band inversion and polarization transition are achieved by manipulating the design of local resonators. It has been proven that by changing the coupling spring constant of local resonators, the mode topologies at the bound states associated with the band gaps can be converted. Three polarization transition points have been found from the bound evolution analysis, and all of them possess the capability of stimulating topological interface states in the corresponding band gaps. The Zak phase calculations have further ascertained the prediction about the topological interface states. The band structure and transmittance spectrum of a supercell lattice of the proposed topological metamaterial have been examined. The topological interface states, as well as the energy localization effects, have been successfully observed in all the three band gaps. The proposed metamaterial brings the convenience of creating topological interface states by carefully manipulating the local resonators only. The methodology presented in this work is thus of practical significance in transforming existing structures into topological systems without revising them.

Zak phases of chiral photonic crystals designed via transformation optics

We study a class of inhomogeneous chiral photonic crystals with topologically nontrivial bands and edge modes using the approach of non-coordinate transformation optics. By transforming the chiral photonic crystals into achiral binary photonic crystals, the closed-form solutions to the photonic bands of the chiral photonic crystals are obtained. We show that the topological edge modes of these chiral photonic crystals can be accurately predicted by the Zak phases developed from the topological band theory for achiral binary photonic crystals.

Identifying topological signature of 1D photonic lattice by Zak phase analysis and towards robust amplification of edge state

Light propagating through a periodic photonic lattice supporting Bloch modes is leaky in nature. Recently, it has been established that these 1D photonic lattices can provide robust lossless guidance of light by appropriate structural modulation following special models. However, these special models compel the structure to be quasi-periodic, which makes their fabrication less feasible. Thus, we propose a fabrication-feasible 1D periodic lattice having topological features, which is established by Zak phase analysis. We demonstrate the existence of an edge state in this specialty photonic lattice as well as its robustness against lattice dislocations. Through numerical simulation, we have also illustrated that these states can be amplified without any distortion by introducing homogeneous gain to the lattice substrate. This is a strikingly different phenomenon from the conventionally known fact that with the introduction of gain to periodic lattices, the propagating state of light spreads in the transverse direction due to enhanced complex coupling. Therefore, in this report, we present a study of the interplay between topology and non-Hermiticity from the standpoint of device application. Our findings suffice to propose the utility of these systems as on-chip optical interconnects and have the potential to be used as topological amplifiers within the regime of optical communication.

Nonlinear control of photonic higher-order topological bound states in the continuum

Higher-order topological insulators (HOTIs) are recently discovered topological phases, possessing symmetry-protected corner states with fractional charges. An unexpected connection between these states and the seemingly unrelated phenomenon of bound states in the continuum (BICs) was recently unveiled. When nonlinearity is added to the HOTI system, a number of fundamentally important questions arise. For example, how does nonlinearity couple higher-order topological BICs with the rest of the system, including continuum states? In fact, thus far BICs in nonlinear HOTIs have remained unexplored. Here we unveil the interplay of nonlinearity, higher-order topology, and BICs in a photonic platform. We observe topological corner states that are also BICs in a laser-written second-order topological lattice and further demonstrate their nonlinear coupling with edge (but not bulk) modes under the proper action of both self-focusing and defocusing nonlinearities. Theoretically, we calculate the eigenvalue spectrum and analog of the Zak phase in the nonlinear regime, illustrating that a topological BIC can be actively tuned by nonlinearity in such a photonic HOTI. Our studies are applicable to other nonlinear HOTI systems, with promising applications in emerging topology-driven devices.

Tunable terahertz topological edge and corner states in designer surface plasmon crystals

In this work, we study topological edge and corner states in two-dimensional (2D) Su-Schrieffer-Heeger lattices from designer surface plasmon crystals (DSPCs), where the vertical confinement of the designer surface plasmons enables signal detection without the need of additional covers for the sample. In particular, the formation of higher-order topological insulator can be determined by the two-dimensional Zak phase, and the zero-dimensional subwavelength corner states are found in the designed DSPCs at the terahertz (THz) frequency band together with the edge states. Moreover, the corner state frequency can be tuned by modifying the defect strength, i.e., the location or diameter of the corner pillars. This work may provide a new approach for confining THz waves in DSPCs, which is promising for the development of THz topological photonic integrated devices with high compactness, robustness and tunability.

Conjugated topological interface-states in coupled ring resonators

The optical properties of topological photonics have attracted much interest recently because its potential applications for robust unidirectional transmission that are immune to scattering at disorder. However, researches on topological series coupled ring resonators (T-SCRR) have been much less discussed. The existence of topological interface-states (TIS) in the T-SCRR is described for the first time in this article. An approach has been developed to achieve this goal via the band structure of dielectric binary ring resonators and the Zak phase of each bandgap. It is found that an ultra-high-Q with complete transmission is obtained by the conjugated topological series coupled ring resonators due to the excitation of conjugated topological interface-states, which is different from those in conventional TIS. Furthermore, the problem of transmission decreases resulting from high-Q increases in the traditional photonic system is significantly improved by this approach. These findings could pave a novel path for developing advanced high-Q filters, optical sensors, switches, resonators, communications and quantum information processors.

Construction of Optical Topological Cavities Using Photonic Crystals

A novel design of the Fabry–Pérot optical cavity is proposed, utilizing both the topological interface state structures and photonic bandgap materials with a controllable reflection phase. A one-to-one correspondence between the traditional Fabry–Pérot cavity and optical topological cavity is found, while the tunable reflection phase of the photonic crystal mirrors provides an extra degree of freedom on cavity mode selection. The relationship between the Zak phase and photonic bandgap provides theoretical guidance to the manipulation of the reflection phase of photonic crystals. The dispersions of interface states with different topology origins are explored. Linear interfacial dispersion emerging in photonic crystals with the valley–spin Hall effect leads to an extra n = 0 cavity mode compared to the Zak phase–induced deterministic interface states with quadratic dispersion. The frequency of the n = 0 cavity mode is not affected by the cavity length, whose quality factor can also be tuned by the thickness of the photonic crystal mirrors. With the recent help of topology photonics in the tuning reflection phase and dispersion relationship, we hope our results can provide more intriguing ideas to construct topological optical devices.

Topological plasmonic waveguides in triharmonic metal gratings

We study topological surface-plasmon-polaritons at optical frequencies in tri-harmonic diffraction gratings formed at a metal–dielectric interface. The latter are shown to well approximate a bipartite Kronig–Penney model. Topologically protected localised modes are then predicted to occur at the edges of the grating and at defects formed by the combination of two mirror antisymmetric corrugations, whose bulk invariant is a step-wise varying Zak phase in both cases. An interesting special case wherein the defect state is in-fact forbidden is also observed that reveals the fragility of such states despite their topological nature.

Probing the topology of the two-photon bands via time-dependent quantum walks.

Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the topological phase still remains challenging, especially in the entangled many-body case. Here, we propose an approach to independently probe single- and two-photon topological invariants from the time evolution of the two-photon state in a one-dimensional array of qubits. Extending the bulk-boundary correspondence to the two-photon scenario, we show that an appropriate choice of the initial state enables the retrieval of the topological invariant for the different types of the two-photon states in the interacting Su-Schrieffer-Heeger model. Our analysis of the Zak phase reveals additional facets of topological protection in the case of collapse of bound photon pairs.

Topological Graphene Plasmons in a Plasmonic Realization of the Su–Schrieffer–Heeger Model

Graphene hybrids, made of thin insulators, graphene, and metals can support propagating acoustic plasmons (AGPs). The metal screening modifies the dispersion relation of usual graphene plasmons leading to slowly propagating plasmons, with record confinement of electromagnetic radiation. Here, we show that a graphene monolayer, covered by a thin dielectric material and an array of metallic nanorods can be used as a robust platform to emulate the Su-Schrieffer-Heeger model. We calculate the Zak's phase of the different plasmonic bands to characterise their topology. The system shows bulk-edge correspondence: strongly localized interface states are generated in the domain walls separating arrays in different topological phases. We find signatures of the nontrivial phase which can directly be probed by far-field mid-IR radiation, hence allowing a direct experimental confirmation of graphene topological plasmons. The robust field enhancement, highly localized nature of the interface states, and their gate-tuned frequencies expand the capabilities of AGP-based devices.