Topologically-protected Edge Research Articles

Tunable topological edge and corner states in an all-dielectric photonic crystal.

Topological photonics has become a new and fascinating area in recent years, which enables electromagnetic waves to propagate with negligible backscattering and excellent robustness even when encountering sharp corners or defects. But the flexible tunability of edge and corner states is challenging once the topological photonic crystals (PhCs) have been fabricated. In this paper, we propose a new all-dielectric PhC with C3 symmetry constructed by hexagonal array of petal-like aperture embedded in silicon background. The proposed configuration has much wider energy gap than its triangular counterpart, and hence is suitable for wideband and high-capacity applications. When the apertures are filled with liquid crystals (LCs), the topologically-protected edge and corner states can be regulated through changing the refractive index of the LCs under different bias voltages. Moreover, the robustness of topological protection of edge and corner states is further demonstrated. This is the first demonstration of LC based tunable valley higher-order photonic topological insulator. The tunability of the proposed topological PhCs may be beneficial for development of tunable optical waveguides, reconfigurable topological microcavities, and other intelligent topological optical/terahertz devices.

Characterization of topological phase of superlattices in superconducting circuits

The recent experimental observation of topological magnon insulator states in a superconducting circuit chain marks a breakthrough for topological physics with qubits, in which a dimerized qubit chain has been realized. Here, we extend such a dimer lattice to superlattice with arbitrary number of qubits in each unit cell in superconducting circuits, which exhibits rich topological properties. Specifically, by considering a quadrimeric superlattice, we show that the topological invariant (winding number) can be effectively characterized by the dynamics of the single-excitation quantum state through time-dependent quantities. Moreover, we explore the appearance and detection of the topological protected edge states in such a multiband qubit system. Finally, we also demonstrate the stable Bloch-like-oscillation of multiple interface states induced by the interference of them. Our proposal can be readily realized in experiment and may pave the way towards the investigation of topological quantum phases and topologically protected quantum information processing.

Interaction of two quantum dots mediated by edge modes of coupled-cavity arrays.

Topological photonics is a hot topic in recent years. We combine it with the quantum optics and explore the dynamics of two quantum dots (QDs) separated by the finite coupled-cavity arrays (CCAs). The finite CCAs possessing the alternating hopping strengths will lead to the existence of the topological protected edge modes, also called zero energy modes, when the boundaries leave the weak hopping at two ends. Due to the two edge modes, i.e., symmetric and antisymmetric, with nearly degenerate frequencies, the dynamics of two QDs coupled to the cavities at both ends exhibit complicated behaviors. When the CCAs are composed of a large number of cavities, there are two kinds of phenomena: if the coupling between QDs and cavity is weak, two edge modes will cancel each other out and isolate two QDs deeply; if the coupling between QDs and cavities is large compared with hopping strength, the edge mode disappears and two QDs can be connected through extend modes. Importantly, when the CCAs are formed by a small number of cavities, energy can be transferred to each other between two QDs through the edge modes. Such energy transfer is topologically protected, and the period is long and easily controlled. We also investigate the effects of topologically protected quantum entangled states on such system and find that the quantum entanglement can be well kept or generated for appropriate choices of system parameters and initial states. The investigations enrich the manifestation of topological physics and are helpful to apply the topological protection to quantum computation and quantum communication.

Dynamics of transmission in disordered topological insulators

The discovery of topological insulators has opened new prospects for robust signal transport for electronic, phononic, and photonic devices. Though transport of topological protected edge states is robust to disorder, large fluctuations and lengthened average delay time are observed. Here, we consider a quasi-1d system following the Haldane model and generalize the idea of eigenchannel time delay to the topological system. Eigenchannel time delay indicates the excited density of states for the configuration and relates to the intensity integral inside the system. Taking advantage of this property, we point out a practical way to extract the central frequency and linewidth of localized modes excited in the topological system. This work links the fluctuation of time delay to the strength of disorder and discusses the scaling of time delay.

Broadband topological valley-projected edge-states transport in composite structure phononic crystal* *Project supported by the National Natural Science Foundation of China (Grant No. 61775050) and the Fundamental Research Funds for the Central Universities, China (Grant No. PA2019GDZC0098).

The topological valley transport, realized in phononic crystals, has aroused tremendous interest in these years. Many previous researches have further promoted the development of this transport phenomenon. Crucially, the bandwidth of the valley-projected edge mode has been an essential research topic. As is well known, the broadband will improve the adaptability of the acoustic edge-states, which will be more conducive to the transmission of information. Therefore, in this paper, we present a composite structure, composed of the atoms with different shapes forming a hexagonal lattice, which can achieve larger bandwidth than a single structure. Meanwhile, the results demonstrate that the topological protected edge states are also observed in our structure. Furthermore, the backscattering suppressions from associated valley-protected edge states under certain perturbations have also been investigated and demonstrated. Our work can provide a new idea for designing acoustic devices based on valley degree of freedom.

Multi-branch valley-chiral edge states of antisymmetric plate wave in phononic crystal plates with double-sided symmetric pillars

We build up phononic crystals (PCs) by attaching triangular pillars symmetrically to the double sides of a thin plate in the triangular lattice. The degeneracy of multiple Dirac cones is simultaneously lifted for the same PC plate, due to the mirror-inversion symmetry breaking caused by the rotation of triangular pillars; opposite Berry curves and valley Chern number are got at the lower and upper branches of each valley band. The multi-branch topological protected edge states are then created on the interface between PC plates with opposite topological phase. Interestingly enough, those edge states present various transverse symmetries and distinct robustness against sample imperfection such as cavity defect, rotation angle disorder, or sharp bending. Our results present the rich behaviors of multiple branches of topological edges.

Topological protection in non-Hermitian Haldane honeycomb lattices

Topological phenomena in non-Hermitian systems have recently become a subject of great interest in the photonics and condensed-matter communities. In particular, the possibility of observing topologically-protected edge states in non-Hermitian lattices has sparked an intensive search for systems where this kind of states are sustained. Here, we present the first study on the emergence of topological edge states in two-dimensional Haldane lattices exhibiting balanced gain and loss. In line with recent studies on other Chern insulator models, we show that edge states can be observed in the so-called broken $\mathcal{P}\mathcal{T}$-symmetric phase, that is, when the spectrum of the gain-loss-balanced system's Hamiltonian is not entirely real. More importantly, we find that such topologically protected edge states emerge irrespective of the lattice boundaries, namely zigzag, bearded or armchair.

Controllable photonic and phononic topological state transfers in a small optomechanical lattice

We propose a scheme to achieve the photonic and the phononic state transfers via the topological protected edge channel based on a one-dimensional small optomechanical lattice. We find that the optomechanical lattice can be mapped into a Su-Schrieffer-Heeger model after eliminating the counter rotating wave terms. By dint of the edge channel of the Su-Schrieffer-Heeger model, we show that the quantum state transfer between the photonic left and the right edge states can be achieved with a high fidelity. Especially, our scheme can also achieve another phononic state transfer based on the same channel via controlling the next-nearest-neighboring interactions between the cavity fields; this is different from the previous investigations achieving only one kind of quantum state transfer. Our scheme provides a novel, to the best of our knowledge, path to switch two different kinds of quantum state transfers in a controllable way.

A bipartite Kronig–Penney model with Dirac-delta potential scatterers

Here we present a simple extension to the age-old Kronig–Penney model, which is made to be bipartite by varying either the scatterer separations or the potential heights. In doing so, chiral (sublattice) symmetry can be introduced. When such a symmetry is present, topological chiral symmetry protected edge states are seen to exist in correspondence with the standard quantised Zak phase bulk invariant. This quantisation behaviour may also be observed within a ‘gauge’-invariant on-diagonal matrix element of a unit eigenvalue equation. The solution proceeds through the conventional scattering formalism used to study the Kronig–Penney model, which does not require further tight-binding approximations or mapping into a Su–Schrieffer–Heeger model. The cases in which chiral symmetry is absent are then seen to not host topologically protected edge states, as verified by zero bulk invariants.

Photonic Topological States in a Two-Dimensional Gyrotropic Photonic Crystal

Time-reversal symmetry (TRS) of electrons is associated with an anti-unitary operator with T 2 = − 1 , which induces Kramers degeneracy and plays an important role in realizing the quantum spin Hall effect (QSHE). By contrast, TRS of photons is described by T b 2 = 1 . We point out that due to this difference, TRS is not the necessary condition for the construction of the photonic analogue of the QSHE. Instead, by constructing an artificial pseudo TRS T p with T p 2 = − 1 in a photonic system, one can realize the photonic Kramers degeneracy and a pair of topological protected edge states, a photonic analogue of the QSHE. Specifically, by retrieving the optical parameters of materials with the pseudo TRS, we propose a photonic topological insulator (PTI) utilizing a pair of double-degenerate transverse electric (TE) and transverse magnetic (TM) polarizations to mimic the spin up and down states of the electron. We demonstrate that the unidirectional polarization-dependent transportation of TE and TM edge states can be realized in this system based on computer simulations. For all possible symmetry types, we check the robustness of these topological states by using a complete set of impurities, including three Pauli matrices and one complex conjugate operator. The results show that the PTI is protected by the pseudo TRS T p . In general, an arbitrary pair of optical polarizations on the Bloch sphere can be utilized to construct photonic pseudospin states and the PTI. Our findings confirm the physical meaning of the pseudo TRS and may provide guidance for future PTI designs.

Two-dimensional photonic crystal with ring degeneracy and its topological protected edge states

We propose a two-dimensional photonic crystal that possesses a degenerate ring in the momentum space. The photonic crystal is composed of the parallel-plate metal filled with a periodically arranged square array of metallic cylinders. Opening an air gap breaks the z-inversion symmetry, leading to the modes coupling (bi-anisotropy response) of TE and TM waves. This induced electric-magneto coupling, a similar role of the spin-orbit interaction in the condensed matters, results in a complete topological band gap around the degenerate frequency. The bulk bands below the band gap take non-zero Z2 topological invariant characterized by the evolution of the Berry phase. As a consequence, the interface of two photonic crystals with opposite bi-anisotropy supports topological protected edge states that exhibit one-way propagation and are highly resistant to disorders. Our work might be very useful for the design of topological photonic crystals and may serve as a platform for studying pseudo-spin photonics.

Two classes of singularities and novel topology in a specially designed synthetic photonic crystals.

Zak phase and topological protected edge state are usually studied in one-dimensional (1D) photonic systems with spatial inversion symmetry (SIS). In this work, we find that specific classes of 1D structure without SIS can be mapped to a system with SIS and also exhibit novel topology, which manifest as phase cut lines (PCLs) in our specially designed synthetic photonic crystals (SPCs). Zak phase defined in SIS is extended to depict the topology of PCLs after redefinition, and a topological protected edge state is also achieved in our 1D structure without SIS. In our SPCs, the relationship between Chern numbers in two dimensions (2D) and the extended Zak phases of 1D PCLs is given, which are bound by the first type singularities. Higher Chern numbers and multi-chiral edge states are achieved utilizing the concept of synthetic dimensions. The effective Hamiltonian is given, based on which we find that the band edges of each PCL play a role analogous to the valley pseudospin, and our SPC is actually a new type of valley photonic crystal that is usually studied in graphene-like honeycomb lattice. The chiral valley edge transport is also demonstrated. In higher dimensions, the shift of the first type singularity in expanded parameter space will lead to the Weyl point topological transition, which we proposed in our previous work. In this paper, we find a second type of singularity that manifests as a singular surface in our expanded parameter space. The shift of the singular surface will lead to the nodal line topological transition. We find the states on the singular surface possess extremely high robustness against certain randomness, based on which a topological wave filter is constructed.

Nontrivial topology and localization in the double exchange model with possible applications to perovskite manganites

The double exchange model describing the coupling between conduction electrons and localized magnetic moments is relevant for a large family of physical systems including manganites. Here we reveal that the one dimensional double exchange model with an incommensurate magnetic elliptical spiral is a topological insulator with a Chern number $2\mathbb{Z}$ in the two dimensional space with one physical dimension and one ancillary dimension spanned by the Goldstone mode of the spiral. Moreover, the electronic states can be localized for a strong local exchange coupling. The topological protected edge states are responsible for the pumping of electron charge, and give rise to multiferroic response. Our work uncovers hitherto undiscovered nontrivial topology and Anderson localization in the double exchange model with possible applications to perovskite manganites.

Exploring photonic topological insulator states in a circuit-QED lattice

We propose a simple protocol to explore the topological properties of photonic integer quantum Hall states in a one-dimensional circiut-QED lattice. By periodically modulating the on-site photonic energies in such a lattice, we demonstrate that this one-dimensional lattice model can be mapped into a two-dimensional integer quantum Hall insulator model. Based on the lattice-based cavity input–output theory, we show that both the photonic topological protected edge states and topological invariants can be clearly measured from the final steady state of the resonator lattice after taking into account cavity dissipation. Interestingly, we also find that the measurement signals associated with the above topological features are quite unambitious even in five coupled dissipative resonators. Our work opens up a new prospect of exploring topological states with a small-size dissipative quantum artificial lattice, which is quite attractive to the current quantum optics community.

Topological magnon bound states in periodically modulated Heisenberg XXZ chains

Strongly interacting topological states in multi-particle quantum systems pose great challenges to both theory and experiment. Recently, bound states of elementary spin waves (magnons) in quantum magnets have been experimentally observed in quantum Heisenberg chains comprising ultracold Bose atoms in optical lattices. Here, we explore a strongly interacting topological state called topological magnon bound-state in the quantum Heisenberg chain under cotranslational symmetry. We find that the cotranslational symmetry is the key to the definition of a topological invariant for multi-particle quantum states, which enables us to characterize the topological features of multi-magnon excitations. We calculate energy spectra, density distributions, correlations and Chern numbers of the two-magnon bound-states and show the existence of topological protected edge bound-states. Our study not only opens a new prospect to pursue topological magnon bound-states, but also gives insights into the characterization and understanding of strongly interacting topological states.

Deterministic interface states in photonic crystal with graphene-allotrope-like complex unit cells

Topological insulators have aroused much research interest in condensed matter physics in recent years. Topological protected edge states can propagate unidirectionally and backscattering free along the boundaries of the topological insulators' which will be important for future electronic devices for its immunity to defects. Topology is dependent only on the symmetry of lattice of the system rather than its specific wave form. Thus, based on the analogy between electronics and photons, photonic topological insulator has also been demonstrated both theoretically and experimentally. Graphene, composed of a monolayer of carbon atoms in honeycomb lattice, exhibits unusual properties due to its intriguing band diagram. Many types of graphene allotropes have been proposed theoretically. However, due to fabrication difficulties, most of graphene allotropes are unavailable. Here, we propose to study two dimensional (2D) photonic crystal (PC) with complex lattices, similar to that of graphene allotrope. The complex PC structure provides more degrees of freedom in manipulating its symmetry.Interface states can also exist in the interface region between two PCs, if they have different topological properties. Without any surface decoration, deterministic interface states can be created when bulk photonic band inversion can be induced and are demonstrated theoretically and experimentally in 2D PCs with square lattice. By controlling the parameters of PCs, their bulk photonic band properties are engineered and topological phase transition occurs. By inverting the bulk photonic band properties, interface states exist in the common band gaps for two PC systems in the gapped region. Similarly, we proceed to complex honeycomb lattice of PCs. By lowering its original C6v symmetry to C3v, C3, C2v and even C2 symmetry, the degeneracies of valley Dirac dispersion at the corners of Brillouin zone are lifted. Photonic band inversion occurs in all four symmetries and the deterministic interface states are numerically realized in the interface region between two PCs. Unidirectional propagation of interface state immune to backscattering along the interface channels is demonstrated if a source with proper optical vortex index is utilized. Due to its easy fabrication, PC is a perfect platform to explore the topological properties of complex lattice and these acquired topological optical states can be of benefit to the control the propagation of light in the photonic waveguide.

Transport and localization in a topological phononic lattice with correlated disorder

Recently proposed classical analogs of topological insulators in phononic lattices have the advantage of much more accessible experimental realization as compared to conventional materials. Drawn to their potential practical structural applications, we investigate how disorder, which is generically non-negligible in macroscopic realization, can attenuate the topologically protected edge (TPE) modes that constitute robust transmitting channels at zero disorder. We simulate the transmission of phonon modes in a quasi-one-dimensional classical lattice waveguide with mass disorder, and show that the TPE mode transmission remains highly robust ($\Xi\sim1$) in the presence of uncorrelated disorder, but diminishes when disorder is spatially correlated. This reduction in transmittance is attributed to the Anderson localization of states within the mass disorder domains. By contrast, non-TPE channels exhibit qualitatively different behavior, with spatial correlation in the mass disorder leading to significant transmittance reduction (enhancement) at low (high) frequencies. Our results demonstrate how topologically protected edge modes drastically modify the effect of spatial correlation on mode localization.

The Z2 classification of dimensional reduced Hopf insulators

Hopf insulators are characterized by a topological invariant called the Hopf index which classifies maps from three-sphere to two-sphere, instead of a Chern number or a Chern parity. In contrast to a topological insulator, the Hopf insulator is not protected by any kind of symmetry. By dimensional reduction, we argue that there exists a new type of index for 2D Hamiltonian with a vanishing Chern number. A specific model Hamiltonian with this nontrivial index is constructed. We also numerically calculate the topological protected edge modes of this dimensional reduced Hopf insulator and show that they are consistent with the classification.