Su Schrieffer Heeger Research Articles

Quantum geometric tensor and the topological characterization of the extended Su–Schrieffer–Heeger model

We investigate the quantum metric and topological Euler number in a cyclically modulated Su–Schrieffer–Heeger (SSH) model with long-range hopping terms. By computing the quantum geometry tensor, we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons, and we obtain the phase diagram of the model marked by the first Chern number. Furthermore, we also obtain the topological Euler number of the energy band based on the Gauss–Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. However, some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric, which leads to ill-defined non-integer topological Euler numbers. Nevertheless, the non-integer “Euler number” provides valuable insights and an upper bound for the absolute values of the Chern numbers.

A scheme for realizing nonreciprocal interlayer coupling in bilayer topological systems

Nonreciprocal interlayer coupling is difficult to practically implement in bilayer non-Hermitian topological photonic systems. In this work, we identify a similarity transformation between the Hamiltonians of systems with nonreciprocal interlayer coupling and on-site gain/loss. The similarity transformation is widely applicable, and we show its application in one- and two-dimensional bilayer topological systems as examples. The bilayer non-Hermitian system with nonreciprocal interlayer coupling, whose topological number can be defined using the gauge-smoothed Wilson loop, is topologically equivalent to the bilayer system with on-site gain/loss. We also show that the topological number of bilayer non-Hermitian C6v-typed domain-induced topological interface states can be defined in the same way as in the case of the bilayer non-Hermitian Su–Schrieffer–Heeger model. Our results show the relations between two microscopic provenances of the non-Hermiticity and provide a universal and convenient scheme for constructing and studying nonreciprocal interlayer coupling in bilayer non-Hermitian topological systems. This scheme is useful for observation of non-Hermitian skin effect in three-dimensional systems.Graphical

The reciprocating and bipolar non-Hermitian skin effect engineered by spin–orbit coupling

We theoretically study a one-dimensional non-Hermitian Su–Schrieffer–Heeger model with an imaginary gauge field and spin–orbit coupling. We find that, under open boundary conditions, the dispersions possess the reciprocating real–complex–real transitions with increasing the strength of spin–orbit coupling. Correspondingly, the bulk energy eigenstates exhibit a reciprocating non-Hermitian skin effect. This mechanism can be characterized by the generalized Brillouin zone moduli, which approaches zero or infinity at the transition points. We further demonstrate the non-zero winding number inside the loops of generalized Brillouin zone when the strength of intra-cell spin–orbit coupling is larger than the inter-cell one, which results in the bipolar non-Hermitian skin effect. As the intra-cell and inter-cell spin–orbit coupling strength becomes comparable, the bipolar non-Hermitian skin effect degenerates to the conventional non-Hermitian skin effect. Our work may pave the way for the non-Hermitian optoelectronic devices utilizing the reciprocating non-Hermitian skin effect and the bipolar non-Hermitian skin effect by engineering the spin–orbit coupling.

Edge state behavior in a Su–Schrieffer–Heeger like model with periodically modulated hopping

Su–Schrieffer–Heeger (SSH) model is one of the simplest models to show topological end/edge states and the existence of Majorana fermions. Here we consider a SSH like model both in one and two dimensions where a nearest neighbor hopping features spatially periodic modulations. In the 1D chain, we witness appearance of new in-gap end states apart from a pair of Majorana zero modes (MZMs) when the hopping periodicity go beyond two lattice spacings. The pair of MZMs, that appear in the topological regime, characterize the end modes each existing in either end of the chain. These, however, crossover to both-end end modes for small hopping modulation strength in a finite chain. Contrarily in a 2D SSH model with symmetric hopping that we consider, both non-zero and zero energy topological states appear in a finite square lattice even with a simple staggered hopping, though the zero energy modes disappear in a ribbon configuration. Apart from edge modes, the 2D system also features corner modes as well as modes with satellite peaks distributed non-randomly within the lattice. In both the dimensions, an increase in the periodicity of hopping modulation causes the zero energy Majorana modes to become available for either sign of the modulation. But interestingly with different periodicity for hopping modulations in the two directions, the zero energy modes in a 2D model become rarer and does not appear for all strength and sign of the modulation.

Planar and tunable quantum state transfer in a splicing Y-junction Su–Schrieffer–Heeger chain

We present a feasible scheme to implement a planar and tunable quantum state transfer (QST) via topologically protected zero-energy mode in a splicing Y-junction Su–Schrieffer–Heeger (SSH) chain. The introduction of the elaborate nearest-neighbor (NN) hopping enables one to generate a topological interface at the central site of the Y-junction. By modulating the NN hopping adiabatically in the chain, the quantum state initially prepared at the central site can be simultaneously transferred to the three endpoints of the Y-junction with the equal/unequal probabilities. The planar distribution of QST is expected to realize a quantum router, whose function is to make the quantum information on the central site (input port) appear equally/unequally at the three endpoints (output ports) with different directions. Moreover, the numerical simulations demonstrate that the scheme possesses the robustness on the fluctuations of the NN hopping and the on-site potential in the system. Furthermore, we show that the number of the output ports with different directions can be flexibly increased in an extended X-junction SSH chain, and the experimental feasibility for implementing special QST in a superconducting qubit-resonator system is briefly discussed. Our work extends the space distribution of QST from linear distribution to planar distribution and promotes the construction of large-scale quantum networks.

Cross-dimensional valley excitons from Förster coupling in arbitrarily twisted stacks of monolayer semiconductors

In stacks of transition metal dichalcogenide monolayers with arbitrary twisting angles, we explore a new class of bright excitons arising from the pronounced Förster coupling, whose dimensionality is tuned by its in-plane momentum. The low energy sector at small momenta is two-dimensional, featuring a Mexican Hat dispersion, while the high energy sector at larger momenta becomes three-dimensional (3D) with sizable group velocity both in-plane and out-of-plane. By choices of the spacer thickness, versatile surface or interface exciton modes localized at designated layers can emerge out of the cross-dimensional bulk dispersion for a topological origin, which can be mapped to the Su–Schrieffer–Heeger soliton. Moreover, step-edges in spacers can be exploited for engineering lateral interfaces to enable interlayer communication of the topological interface exciton. Combined with the polarization selection rule inherited from the monolayer building block, these exotic exciton properties open up new opportunities for multilayer design towards 3D integration of valley exciton optoelectronics.

Giant atom induced zero modes and localization in the nonreciprocal Su–Schrieffer–Heeger chain

A notable feature of non-Hermitian systems with skin effects is the sensitivity of their spectra and eigenstates to the boundary conditions. In the literature, three types of boundary conditions–periodic boundary condition, open boundary condition (OBC) and a defect in the system as a boundary, are explored. In this work we introduce the other type of boundary condition provided by a giant atom. The giant atom couples to a nonreciprocal Su–Schrieffer–Heeger (SSH) chain at two points and plays the role of defects. We study the spectrum and localization of eigenstates of the system and find that the giant atom can induce asymmetric zero modes. A remarkable feature is that bulk states might localize at the left or the right chain-atom coupling sites in weak localization regimes. This bipolar localization leads to Bloch-like states, even though translational invariance is broken. Moreover, we find that the localization is obviously weaker than the case with two small atoms or OBCs even in strong coupling regimes. These intriguing results indicate that nonlocal coupling of the giant atom to a nonreciprocal SSH chain weakens the localization of the eigenstates. We also show that the Lyapunov exponent in the long-time dynamics in real space can act as a witness of the localized bulk states.

Observation of Boundary‐Obstructed Topological Insulators in 2D Asymmetric Su–Schrieffer–Heeger Topological Electric Circuits

Topological insulators that support unique symmetry‐protected boundary states gain popularity in the field of electric circuit systems. Boundary‐obstructed topological (BOT) insulators, as a new class of topological insulators, do not utilize the closure of the bulk band during the topological phase transition, which displays as an obstruction of boundary states. In this work, the experimental realization of BOT insulators based on electric circuit systems are reported and it is further demonstrated experimentally that the corresponding hollow structure consisting of only edges and corners can offer corner states in the edge gap. In this research, a highly controllable platform is offered for investigating unique topological phenomena and the way is paved for observing boundary‐obstructed topology by electric circuit systems.

A randomized measurement toolbox for an interacting Rydberg-atom quantum simulator

We present a toolbox to probe quantum many-body states implemented on Rydberg-atoms quantum hardware via randomized measurements. We illustrate the efficacy of this measurement toolbox in the context of probing entanglement, via the estimation of the purity, and of verifying a ground-state preparation using measurements of the Hamiltonian variance. To achieve this goal, we develop and discuss in detail a protocol to realize independent, local unitary rotations. We benchmark the protocol by investigating the ground state of the one-dimensional Su–Schrieffer–Heeger model, recently realized on a chain of Rydberg atom, and the state resulting after a sudden quench in a staggered XY chain. We probe the robustness of our toolbox by taking into account experimental imperfections, such as pulse fluctuations and measurement errors.

Study of the energy spectrum of a few one-dimensional and quasi one-dimensional lattices through tight binding and density functional theory

A graphene nanoribbon (GNR) is a strip of carbon atoms having sp2 hybridization. It has wide application in nanoelectronics and opto-electronics. Usual fields of application are found in field effect transistors, interconnects, logic gates, sensors, energy storage, and photovoltaics. A single unit graphene nanoribbon is a long strip of graphene rings. Such a GNR structure may be seen as two one-dimensional carbon chains that are suitable connected with bonds. We have done tight binding calculations and density functional theory simulation of carbon chains. We study the single bond and double bond one-dimensional carbon chain and the alternate bond (t1-t2), also called a bond order system in one dimension and quasi one-dimensional chain. We find evidence for the emergence of multiple gaps in the energy spectrum of these systems. We have mapped the alternate bond system to the Su–Schrieffer–Heeger model (with a small modification) in one dimension and quasi one dimension. This is the first time such a mapping has been attempted and a comprehensive theoretical and computational study of these chains has been performed.

Quantum state transfer on 1-dimensional and 2-dimensional topological Heisenberg spin models

The geometrical significance of winding number is used to study the topological characteristics of a single-excitation Heisenberg XY spin model represented by the Su–Schrieffer–Heeger (SSH) model. Due to the presence of topological edge states, the quantum state may be conveyed from the first site to the last site with a high degree of fidelity. Also examined is a Heisenberg XY model on the kagome lattice. In the topological phase of the kagome lattice, the three corner states are not affected by the Dzyaloshinskii–Moriya (DM) interactions. For the kagome lattice with and without the DM interactions, the state of a spin on one corner site can be transferred to the other two corner sites in various ways during the topological phase. The quantum state transfer (QST) for the XY model on the kagome lattice without the DM interactions is symmetrical and can only produce a three-body entangled state. In contrast, the QST for the XY model on the kagome lattice with the DM interactions is in an asymmetrical form, similar to one-way propagation, and it can produce both product states and three-body entangled states.

Subsonic and supersonic polaron dynamics in polyacetylene

The paper considers the dynamics of a polaron in an electric field on a polyacetylene lattice. The one-electron non-adiabatic approximation based on the Su–Schrieffer–Heeger model is used. In an electric field, a polaron gains one of two saturated velocities, — subsonic and supersonic. Velocity bifurcation occurs in a very narrow range of electric field strengths. In an electric field, the polaron’s internal dynamic and electronic structure changes in a complex way. The supersonic polaron transforms the energy received from the electric field into monochromatic lattice oscillations. The generation of these oscillations is due to the internal polaron dynamics. The effective polaron mass is determined and m* ≈ 1.3. The negative effective mass was also identified. An analysis of the wave function and its phases provides useful information about the polaron evolution. The polaron dynamics is also studied depending on the mode of switching on and off the electric field and in multiple electric fields.

Topological degeneracy breaking in synthetic frequency lattice by Floquet engineering

Abstract Synthetic frequency dimensions provide important opportunities to investigate novel topological phenomena. Previously, many theoretical proposals have been studied and relevant experiments have been performed very recently. However, all these works consider models in the weak modulation regime, where static lattice models are constructed. Here we explore a Floquet Su–Schrieffer–Heeger (SSH) model with time-dependent hoppings in the frequency dimension by dynamically modulating ring resonators ultrastrongly, and find that the topological states, originally degenerate in conventional SSH lattices, are separated in eigenenergies. There exists a series of edge states from band folding at the 0 and π energy bandgaps, which exhibit complex multi-frequency oscillations due to the inclusion of counter-rotating terms with higher order oscillation frequencies. Such a system with stronger modulations can widen the bandgap and therefore it provides an effective way to localize pulses in synthetic frequency dimensions. Our work shows a photonic platform with the synthetic dimension in exploring exotic Floquet topological phenomena and shows potential applications in optical storage and communications.

Optimizing mixing in the Rudner–Levitov lattice

Here we discuss the optimization of mixing in finite linear and circular Rudner–Levitov lattices (Su–Schrieffer–Heeger lattices with a dissipative sublattice). We show that the presence of exceptional points in the systems’ spectra can lead to drastically different scaling of the mixing time with the number of lattice nodes, varying from quadratic to logarithmic. When operating in the region between maximal and minimal exceptional points, it is always possible to restore logarithmic scaling by choosing the initial state of the chain. Moreover, for the same localized initial state and values of parameters, a longer lattice might mix much faster than the shorter one. Also we demonstrate that an asymmetric circular Rudner–Levitov lattice can preserve logarithmic scaling of the mixing time for an arbitrarily large number of lattice nodes.

Footprint of a topological phase transition on the density of states

For a generalized Su–Schrieffer–Heeger model, the energy zero is always critical and hyperbolic in the sense that all reduced transfer matrices commute and have their spectrum off the unit circle. Disorder-driven topological phase transitions in this model are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that away from such a transition the density of states vanishes at zero energy with an explicitly computable Hölder exponent, while it has a characteristic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for martingales of suitably constructed comparison processes.

Symmetry‐Induced Selective Excitation of Topological States in Su–Schrieffer–Heeger Waveguide Arrays

The investigation of topological state transition in carefully designed photonic lattices is of high interest for fundamental research, as well as for applied studies such as manipulating light flow in on‐chip photonic systems. Herein, the topological phase transition between symmetric topological zero modes (TZM) and antisymmetric TZMs in Su–Schrieffer–Heeger mirror symmetric waveguides is reported. The transition of TZMs is realized by adjusting the coupling ratio between neighboring waveguide pairs, which is enabled by selective modulation of the refractive index in the waveguide gaps. Bidirectional topological transitions between symmetric and antisymmetric TZMs can be achieved with proposed switching strategy. Selective excitation of topological edge mode is demonstrated owing to the symmetry characteristics of the TZMs. The flexible manipulation of topological states is promising for on‐chip light flow control and may spark further investigations on symmetric/antisymmetric TZM transitions in other photonic topological frameworks.

Macroscopic Zeno Effect in a Su–Schrieffer–Heeger Photonic Topological Insulator

AbstractThe quantum Zeno effect refers to the slowdown of the decay of a quantum system due to frequent measurements. It has been extended beyond quantum systems, manifesting itself in such phenomena as the suppression of output beam decay by sufficiently strong absorption introduced in guiding optical systems. In this case, the phenomenon is termed as macroscopic Zeno effect. Here the observation of the macroscopic Zeno effect in a topological photonic system is reported. The phenomenon is based on the suppression of decay for only a subspace of edge modes that can propagate in the system and does not rely on the existence of exceptional points. By introducing controlled losses in one of the arms of a topological insulator comprising two closely positioned Su–Schrieffer–Heeger arrays, the macroscopic Zeno effect is demonstrated, which manifests itself in an increase of the transparency of the system with respect to the topological modes created at the interface between two arrays. The phenomenon remains robust against disorder in the non‐Hermitian topological regime. In contrast, coupling a topological array with a non‐topological one results in a monotonic decrease in output power with increasing absorption.

Phase-controlled topological plasmons in 1D graphene nanoribbon array

In this Letter, we report on the phase-controlled topological plasmons in 1D graphene nanoribbons (GNRs) based on a Su−Schrieffer−Heeger (SSH) model variant. By considering the dipole–dipole mode interactions, we first study the normal SSH model by an effective Hamiltonian and calculate the Zak phase as a topological invariant, finding that it is nontrivial (trivial) when the coupling distance is bigger (smaller) than half the period. Then, we reveal that the edge modes with fields highly localized at only one side exist in the model with nontrivial topology and shows the robustness of strong field confinement and extreme frequency stability against in-plane and out-of-plane disorders. Finally, we introduce the offset SSH model variant by vertically offsetting one of the GNR in SSH unit, which allows us to greatly engineer both the width of topological gap and the number of topological windows. The underlying physics are uncovered by defining a parameter called phase difference, which reveals that the topological edge modes appear (disappear) generally near the positions where the inter-unit coupling strength is bigger (smaller) than the intra-unit coupling strength, and, more notably, the phase difference is around even (odd) multiple numbers of π, which is much different from the normal SSH model where the topological phase is simply affected by the resonator distance. In addition to opening up a possibility to explore the fundamental physics of topologically protected graphene plasmons, this work also offers potential applications of these concepts to design graphene-based plasmon devices with immunity to structural imperfections.

Two-Dimensional Thermal Regulation Based on Non-Hermitian Skin Effect

The non-Hermitian skin effect has been applied in multiple fields. However, there are relatively few models in the field of thermal diffusion that utilize the non-Hermitian skin effect for achieving thermal regulation. Here, we propose two non-Hermitian Su–Schrieffer–Heeger (SSH) models for thermal regulation: one capable of achieving edge states, and the other capable of achieving corner states within the thermal field. By analyzing the energy band structures and the generalized Brillouin zone, we predict the appearance of the non-Hermitian skin effect in these two models. Furthermore, we analyze the time-dependent evolution results and assess the robustness of the models. The results indicate that the localized thermal effects of the models align with our predictions. In a word, this work presents two models based on the non-Hermitian skin effect for regulating the thermal field, injecting vitality into the design of non-Hermitian thermal diffusion systems.

Improvement on the topological localized interface enabled by chiral symmetry

Zero-energy topological states, which are protected by chiral symmetry against certain perturbations topologically, localize at interfaces between trivial and non-trivial phases in the Su–Schrieffer–Heeger (SSH) chain model. Here, we propose and demonstrate a method to manipulate chiral symmetry itself to improve the localized interfaces and enlarge the mode volume of topological states in the SSH model, thus optimizing the lasing performance of localized interfaces. As multiple defects corresponding to off-diagonal perturbations in an eigenmatrix are introduced, the topological state expands and extends to extra defects at the topological interface without breaking chiral symmetry. We apply the proposed method in electrical pumping semiconductor laser arrays to verify our theoretical prediction and optimize the output characteristics of the devices. The measured results of the proposed multi-defect SSH laser array show that the output power has been increased by 27%, and the series resistance and far-field divergence have been reduced by half compared to the traditional SSH laser array, establishing a high-performance light source for integrated silicon photonics, infrared light detection and ranging, and so on. Our work demonstrates that the proposed method is capable of improving topological localized interfaces and redistributing zero-energy topological states. Furthermore, our method can be applied to other platforms and inspire optimizations of more devices in broader areas.