SSH Model Research Articles

Engineering negative coupling and corner modes in a three-dimensional acoustic topological network

Higher-order topological insulators have aroused massive attention due to their fancy topological properties. Artificial metamaterials, with their exquisite fabrications and measurements, offer an ideal approach to investigate topological properties in classical systems. Therein lower-dimensional corner states obtained in the three-dimensional (3D) Su-Schrieffer-Heeger (SSH) model and the octupole insulator have been extensively explored in acoustic systems. However, the existing acoustic negative couplings in the quantized multipole topological phases are mostly formed by shifting the connecting waveguides between every two resonators, which support a restricted dipole resonance mode. Here we establish subwavelength acoustic third-order topological insulators based on the theoretical frame of acoustic transmission lines and provide an innovative way for freely engineering acoustic negative couplings that work in arbitrary resonance modes. Furthermore, topological invariant calculations and numerical simulations in analogous acoustic networks validate the occurrence of topological corner states in both the 3D SSH model and octupole insulator. We foresee that the proposed methodology will broaden future avenues for studying atypical topological quantum effects with negative couplings in the classical macroscale context.

Winding number and Zak phase in multi-band SSH models

We use multi-band SSH models to demonstrate a prescription for calculating the correct Zak phase and winding number of multi-band systems. We verify our prescription by comparing the resultant winding number of a four-band SSH model with the edge states of a semi-infinite chain which we find by solving the equation of motion. We then also carry out an extensive comparison with the numerical results obtained by solving the matrix eigenvalue problem of finite chains with various number of sites. As a double check of our prescription, we also confirm the bulk edge correspondence in a six-band SSH model. Similar to the usual SSH model, the winding numbers associated to the left and right boundaries in a finite chain may be different if space inversion symmetry is violated in the system. We believe the prescription we propose here may also be applied to other 1D multi-band systems.

Asymptotically exact photonic approximations of chiral symmetric topological tight-binding models

Topological photonic edge states, protected by chiral symmetry, are attractive for guiding wave energy as they can allow for more robust guiding and greater control of light than alternatives; however, for photonics, chiral symmetry is often broken by long-range interactions. We look to overcome this difficulty by exploiting the topology of networks, consisting of voids and narrow connecting channels, formed by the spaces between closely spaced perfect conductors. In the limit of low frequencies and narrow channels, these void–channel systems have a direct mapping to analogous discrete mass–spring systems in an asymptotically rigorous manner and therefore only have short-range interactions. We demonstrate that topological tight-binding models that are protected by chiral symmetries, such as the SSH model and square-root semimetals, are reproduced for these void–channel networks with appropriate boundary conditions. We anticipate, moving forward, that this paper provides a basis from which to explore continuum photonic topological systems, in an asymptotically exact manner, through the lens of a simplified tight-binding model.

Topological States in Two-Dimensional Su-Schrieffer-Heeger Models

We study the topological properties of the generalized two-dimensional (2D) Su-Schrieffer-Heeger (SSH) models. We show that a pair of Dirac points appear in the Brillouin zone (BZ), consisting a semimetallic phase. Interestingly, the locations of these Dirac points are not pinned to any high-symmetry points of the BZ but tunable by model parameters. Moreover, the merging of two Dirac points undergoes a novel topological phase transition, which leads to either a weak topological insulator or a nodal-line metallic phase. We demonstrate these properties by constructing two specific models, which we referred as type-I and type-II 2D SSH models. The feasible experimental platforms to realize our models are also discussed.

Dynamical relaxation of correlators in periodically driven integrable quantum systems

We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where $n$ is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale ${n}_{c}$ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.

Critical phase boundary and finite-size fluctuations in the Su-Schrieffer-Heeger model with random intercell couplings

A dimerized fermion chain, described by Su-Schrieffer-Heeger (SSH) model, is a well-known example of 1D system with a non-trivial band topology. An interplay of disorder and topological ordering in the SSH model is of a great interest owing to experimental advancements in synthesized quantum simulators. In this work, we investigate a special sort of a disorder when inter-cell hopping amplitudes are random. Using a definition for $\mathbb{Z}_2$-topological invariant $\nu\in \{ 0; 1\}$ in terms of a non-Hermitian part of the total Hamiltonian, we calculate $\langle\nu\rangle$ averaged by random realizations. This allows to find (i) an analytical form of the critical surface that separates phases of distinct topological orders and (ii) finite size fluctuations of $\nu$ for arbitrary disorder strength. Numerical simulations of the edge modes formation and gap suppression at the transition are provided for finite-size system. In the end, we discuss a band-touching condition derived within the averaged Green function method for a thermodynamic limit.

Topological phases of non-Hermitian SSH model with spin-orbit coupling

Inspired by the traditional bulk-boundary correspondence that can be broken by the non-Hermitian skin effect, we study the topological properties of the non- Hermitian Su-Schrieffer-Heeger (SSH) model with spin-orbit coupling (SOC). Considering the relationship of the non-Hermitian skin effect and the non-Hermitian Aharonov-Bohm (AB) effect, the topological counterpart is constructed. It finds that the system can be studied in the two subspaces. The analytical forms of the topological invariants are obtained to distinguish different topological phases. The numerical solutions verify the above analysis. The method is expected to apply the system with the non-Hermitian skin effect.

Generalized Su-Schrieffer-Heeger Model in One Dimensional Optomechanical Arrays

We propose an implementation of a generalized Su-Schrieffer-Heeger (SSH) model based on optomechanical arrays. The topological properties of the generalized SSH model depend on the effective optomechanical interactions which can be controlled by strong driving fields. Three phases including one trivial and two distinct topological phases are found in the generalized SSH model. The phase transition can be observed by turning the strengths and phases of the effective optomechanical interactions via adjusting the driving fields. Moreover, four types of edge states can be created in generalized SSH model of an open chain under single-particle excitation, and the dynamical behaviors of the excitation in the open chain are related to the topological properties under the periodic boundary condition. We show that the edge states can be pumped adiabatically along the optomechanical arrays by periodically modulating the amplitude and frequency of the driving fields, and the state pumping is robust against small disorders. The generalized SSH model based on the optomechanical arrays provides us a controllable platform to engineer topological phases for photons and phonons, which may have potential applications in controlling the transport of photons and phonons.

Topological properties of non-isotropic two-dimensional SSH model

<sec>The one-dimensional (1D) Su-Schrieffer-Heeger (SSH) chain is a model that has been widely studied in the field of topological physics. The two-dimensional (2D) SSH model is a 2D extension of the 1D SSH chain and has many unique physical properties. It is a higher-order topological insulator (HOTI), in which corner states with bound states in the continuum (BIC) properties will arise between the second energy band and the third energy band. There are two different topological phases in the isotropic 2D SSH model, and a topological phase transition will happen when the intracell coupling strength is equal to the intercell coupling strength.</sec><sec>In this paper, we first break the isotropy of the isotropic 2D SSH model, defining the ratio of the <i>x</i>-directional coupling strength to the <i>y</i>-directional coupling strength as <i>α</i> and the ratio of the intercell coupling strength to the intracell coupling strength as <i>β</i>, which represent the strength of the topological property and anisotropy respectively. We use <i>α</i> and <i>β</i> to calibrate all possible models, classify them as three different types of phases, and draw their phase diagrams.Then we argue when the energy gap between the second energy band and the third energy band emerges over the entire Brillouin zone.</sec><sec>Meanwhile, we use a method to calculate the spatial distribution of polarization when the model is half-filled, and it is shown that there is 1/2 polarization localized at the edges in the direction with larger intracell coupling, but no edge polarization in the other direction. The edge polarization excites the edge dipole moment, giving rise to a topological edge state in the energy gap. At the same time, when the model has an entire open boundary, the dipole moment directs the charge to accumulate on the corners, which can be observed from the local charge density distribution. This type of fractional charge is a filling anomaly and formed spontaneously by the lattice to maintain electrical neutrality and rotational symmetry simultaneously. This fractional charge induces the aforementioned corner state. And by its nature of filling anomaly, this corner state is better localized and robust. It will not couple with the bulk state as long as the rotational symmetry or chirality of the model is not broken.</sec><sec>Finally, we construct an acoustic resonant cavity model: a rectangular shaped resonant cavity is used to simulate individual lattice points and the coupling strength between the lattice points is controlled by varying the diameter of the conduit between the resonant cavities. According to the Comsol calculation results, we can see that the topological properties of the anisotropic two-dimensional SSH model are well simulated by this model.</sec>

Non-Hermitian skin effect in a domain-wall system

The non-Hermitian skin effect is one of the most striking features in non-Hermitian physics. It reveals a novel phenomenon in a non-Hermitian system that the bulk wave function and energy spectrum are sensitively dependent on the boundary conditions. The concept of generalized Brillouin zones has been proposed to characterize bulk wave functions in such systems . Based on generalized Brillouin zones, non-Bloch topological invariants can reconstruct the non-Hermitian bulk-edge correspondence. Previous discussion of the non-Hermitian skin effect mainly focused on open boundary conditions, and the calculation of generalized Brillouin zones needs to be reconsidered under domain-wall boundary conditions. The paper introduces the related researches of the non-Hermitian skin effect in domain-wall systems, including the general form of the generalized Brillouin zone equation in a one-dimensional single-band model, non-Bloch topological invariants in non-Hermitian SSH (Su-Schieffer-Heeger) model, and the experimental realization of the non-Hermitian skin effect in one-dimensional quantum walk system.

Topological states in the super-SSH model

The topological edge state distributes along the edge of a topological insulator which has advantages in prohibiting radiation and reflection in the evolution dynamics because of the topological protection property. The Su-Schrieffer-Heeger (SSH) model provides the simplest lattice configuration that supports topological edge states. Here, we investigate the properties of an extended SSH model – super-SSH model – with three sites in a unit cell for one-dimensional case and nine sites in a unit cell for two-dimensional case. Theoretical analysis and numerical simulation demonstrate that topological edge states and topological defect states are supported in the super-SSH model. This work extends the form of SSH model and may serve as a novel platform for developing photonic techniques based on topological phase transition.

The topological counterparts of non-Hermitian SSH models

Inspired by the relevance between the asymmetric coupling amplitude and the imaginary gauge field, we construct the counterpart of the non-Hermitian SSH model. The idea is the nonzero imaginary magnetic flux vanishing when the boundary condition changes from periodic to open. The zero imaginary magnetic flux of the counterpart leads to the eliminating of the non-Hermitian skin effect and the non-Hermitian Aharonov–Bohm effect which ensures the recovery of the conventional bulk-boundary correspondence from the non-Bloch bulk-boundary correspondence. We explain how some the non-Hermitian models can be transformed to the non-Hermitian SSH models and how the non-reciprocal hopping in the non-Hermitian SSH models can be transformed from one term to the other terms by the similarity transformations. We elaborate why the effective imaginary magnetic flux disappears due to the interplay of the non-reciprocal hoppings in the partner of the non-Hermitian SSH model. As the results, we obtain the topological invariants of the non-Hermitian SSH model in analytical form defined in conventional Brillouin zone. The non-Hermitian SSH model in domain configuration on a chain is discussed with this method. The technique gives an alternative way to study the topological properties of non-Hermitian systems.

Qubit-photon bound states in topological waveguides with long-range hoppings

Quantum emitters interacting with photonic band-gap materials lead to the appearance of qubit-photon bound states that mediate decoherence-free, tunable emitter-emitter interactions. Recently, it has been shown that when these band-gaps have a topological origin, like in the photonic SSH model, these qubit-photon bound states feature chiral shapes and certain robustness to disorder. In this work, we consider a more general situation where the emitters interact with an extended SSH photonic model with longer range hoppings that displays a richer phase diagram than its nearest-neighbour counterpart, e.g., phases with larger winding numbers. In particular, we first study the features of the qubit-photon bound states when the emitters couple to the bulk modes in the different phases, discern its connection with the topological invariant, and show how to further tune their shape through the use of giant atoms, i.e., non-local couplings. Then, we consider the coupling of emitters to the edge modes appearing in the different topological phases. Here, we show that giant atoms dynamics can distinguish between all different topological phases, as compared to the case with local couplings. Finally, we provide a possible experimental implementation of the model based on periodic modulations of circuit QED systems. Our work enriches the understanding of the interplay between topological photonics and quantum optics.

Weyl nodal-ring semimetallic behavior and topological superconductivity in crystalline forms of Su-Schrieffer-Heeger chains

We consider a three-dimensional model of coupled Su-Schrieffer-Heeger (SSH) chains. The analytically soluble model discussed here reliably reproduces the features of the band structure of crystalline polyacetylene as obtained from density-functional theory. We show that when a certain interchain coupling is sufficiently increased, the system develops a ring of Weyl nodes. We argue that such an increase could be achieved experimentally by intercalation or extreme pressure. With the addition of a simple intraorbital pairing term we find that the system supports an exotic superconducting state with drumhead surface states and annular Majorana states localized on the surface. In addition to suggesting a real material realization of a nodal ring semimetal and possibly topological superconductivity, our results provide a different perspective on the SSH model, demonstrating that a simple extension of this broadly impacting model can once again provide fundamental insights on the topological behavior of condensed matter systems.

Controlling optical beam thermalization via band-gap engineering

We establish dispersion engineering rules that allow us to control the thermalization process and the thermal state of an initial beam propagating in a multimode nonlinear photonic circuit. To this end, we have implemented a kinetic equation (KE) approach in systems whose Bloch dispersion relation exhibits bands and gaps. When the ratio between the gap-width to the band-width is larger than a critical value, the KE has stationary solutions which differ from the standard Rayleigh-Jeans (RJ) distribution. The theory also predicts the relaxation times above which such non-conventional thermal states occur. We have tested the validity of our results for the prototype SSH model whose connectivity between the composite elements allows to control the band-gap structure. These spectral engineering rules can be extended to more complex photonic networks that lack periodicity but their spectra consist of groups of modes that are separated by spectral gaps.

Tight-binding model in optical waveguides: Design principle and transferability for simulation of complex photonics networks

Integrated optical waveguides have been widely explored in the context of quantum simulation for various physical models based on paraxial diffraction of light, which are described by an equivalent Schr\"odinger equation. The physics in these systems can be formulated using the tight-binding models, in which the coupling between the waveguides can be tuned independently in a wide range, providing an excellent platform for simulation of various phenomena. In this work, we build a tight-binding model with parameters transported directly from two coupled waveguides, which are controlled by dielectric constant change, site distance, and geometries. This design principle can greatly save the simulation and experimental cost in real implementation. As compared with results from the exact simulation based on Maxwell equations, our numerical results demonstrate that the physics of the one- or two-dimensional large lattice systems could be well described by our tight-binding model, exhibiting the excellent transferability of the parameters in two coupled waveguides. In addition, some applications are further discussed: we show how to realize the topological Su-Schrieffer-Heeger (SSH) model, kinked SSH model, and study their associated topological phases and edge modes. Some two-dimensional models based on our tight-binding models are also discussed. Lastly, more intriguing applications, such as nonlinearity or disorder-induced effect and generation of the gauge potential are also briefly discussed. Our work intuitively provides an effective reference route for designing models for experiments and demonstrates the practicality of using the tight-binding approximation to solve complicated models. The design principle demonstrated in this work paves the foundation for the application of optical waveguides based on more complicated models, and will be readily verified experimentally in the near future.

Topological edge modes in one-dimensional photonic crystals containing metal

Topological phases of matter has been developing rapidly in recent decades due to their unique topological edge states. In this paper, we analyze the surface modes of a one-dimensional periodic metal-vacuum multilayer structure and find that it can be seen as the optical analogy of the Su-Schrieffer-Heeger (SSH) model. There are two symmetric and antisymmetric edge modes, whose fields are mainly concentrated on the two outermost interfaces. By introducing off-diagonal perturbation to some layers, we find that the edge modes are topologically protected, that is, have good robustness. However, the surface modes in the one-dimensional structure are not completely consistent with the SSH model, especially in the number of extended modes and the existence conditions of edge modes. As some extended modes are missing in our model, it can improve the coupling efficiency between atom and edge modes. Our results not only provide a new platform for the study of robust topological edge modes, but also have potential applications in information transmission, power transfer, and so on.

Connection between the winding number and the Chern number

Bulk-edge correspondence is one of the most distinct properties of topological insulators. In particular, the 1D winding number ν has a one-to-one correspondence to the number of edge states in a chain of topological insulators with boundaries. By properly choosing the unit cells, we carry out numerical calculation to show explicitly in the extended SSH model that the winding numbers corresponding to the left and right unit cells may be used to predict the numbers of edge states on the two boundaries in a finite chain. Moreover, by drawing analogy between the SSH model and QWZ model, we show that the extended SSH model may be generalized to the extended QWZ model. By integrating the “magnetic field” over the momentum strip 0≤p2≤π,0≤p12π in the Brillouin zone, we show a identity relating the 2D Chern number and the difference between the 1D winding numbers at p2=0 and p2=π.

Dynamics of entanglement after exceptional quantum quench

We investigate a quantum quench from a critical to an exceptional point. The initial state, prepared in the ground state of a critical hermitian system, is time evolved with a non-hermitian SSH model, tuned to its exceptional point. The single particle density matrix exhibits supersonic modes and multiple light cones, characteristic to non-hermitian time evolution. These propagate with integer multiples of the original Fermi velocity. In the long time limit, the fermionic Green's function decays spatially as $1/x^2$, in sharp contrast to the usual $1/x$ decay of non-interacting fermions. The entanglement entropy is understood as if all these supersonic modes arise from independent quasiparticles (though they do not), traveling with the corresponding supersonic light cone velocity. The entropy production rate decreases with time and develops plateaus during the time evolution, signaling the distinct velocities in the propagation of non-local quantum correlations. At late times, the entanglement entropy saturates to a finite value, satisfying a volume law.

Observation of topological bound states in a double Su-Schrieffer-Heeger chain composed of split ring resonators

The observation of abundant topological properties in optical platforms has attracted people's attention due to their great advantages in fundamental topological research and practical applications. Here, we use a double Su-Schrieffer-Heeger (SSH) chain structure composed of split ring resonators (SRRs) to mimic a Kitaev model. By properly tuning the orientation angles of the slits of the SRRs, we can ingeniously design the complex coupling distribution. We experimentally measure the topological phase transition of the double SSH model by adjusting the distance between the two chains and observe the topological bound states in the topological nontrivial phase. Importantly, inserting a trivial structure into the topological chain, we show the coupling of two topological bound states. By increasing the length of the inserted trivial chain, we observe that two topological bound states gradually approach each other and degenerate in the critical state. In addition, we theoretically study the controllable topological bound states and exceptional point physics formed by topological bound states. Our findings not only clearly show the establishing process of the photonic bound states in a double SSH chain, but also may facilitate some applications such as sensors, energy transfer, and switch with topological protection.5 MoreReceived 2 September 2020Revised 20 January 2021Accepted 25 January 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.013122Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMetamaterialsTopological effects in photonic systemsTopological phases of matterAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics