Subwavelength arrays of plasmonic nanoparticles allow us to control the behaviour of light at the nanoscale. Here, we develop an eigenmode analysis, employing a coupled electromagnetic dipole formalism, which permits us to isolate the contribution to the far-field radiation of each array mode. Specifically, we calculate the far-field radiation patterns by bulk, edge and corner out-of-plane eigenmodes in a finite 2D Su-Schrieffer-Heeger (SSH) array of plasmonic nanoparticles with out-of-plane dipolar resonances. The breaking of symmetries in multipartite unit cells is exploited to tailor the optical properties and far-field radiation of the resonant modes. We prove that the antisymmetric modes are darker and have higher Q-factors than their symmetric counterparts. Also, the out-of-plane nature of the dipolar resonances imposes that all bulk $Γ$-modes are dark, while corner and edge states need extra in-plane symmetries to cancel the far-field radiation; radiation patterns in turn become more complex and concentrated along the array plane with increasing array size.
We investigate the direct thermal emission spectrum of a plasmonic Su-Schrieffer-Heeger chain of InSb nanoparticles as well as its far-field emission due to near-field scattering by a sharp tip as it is used in scattering type thermal microscopes. We discuss the measurability of the topological phase transition in such far-field and near-field experiments and highlight the signatures of the topological edge modes. We further discuss the impact of a substrate.
We study the emergence and disappearance of defect states in the complex Su-Schrieffer-Heeger (cSSH) model, a non-Hermitian one-dimensional lattice model containing gain and loss on alternating sites. Previous studies of this model have focused on the existence of a non-Hermitian defect state that is localized to the interface between two cSSH domains, and is continuable to the topologically protected defect state of the Hermitian Su-Schrieffer-Heeger (SSH) model. For large gain/loss magnitudes, we find that these defect states can disappear into the continuum, or undergo pairwise spontaneous …
A network model that can describe light propagation in one-dimensional ring-resonator arrays with a dimer structure is studied as a Su-Schrieffer-Heeger-type Floquet network. The model can be regarded as a Floquet system without periodic driving and exhibits quasienergy band structures of the ring propagation phase. Resulting band gaps support deterministic edge states depending on hopping S-matrices between adjacent rings. The number of edge states is one if the Zak phase is $π$. If the Zak phase is 0, the number is either zero or two. The criterion of the latter number is given analytically …
A non-Hermitian topological insulator with real spectrum is interesting in the theory of non-Hermitian extension of topological systems. We find an experimentally realizable example of a two dimensional non-Hermitian topological insulator with real spectrum. We consider two-dimensional Su-Schrieffer-Heeger (SSH) model with gain and loss. We introduce non-Hermitian polarization vector to explore topological phase and show that topological edge states in the band gap exist in the system.
We numerically determine the robustness of the lasing edge modes in a spin-torque oscillator array that realizes the non-Hermitian Su-Schrieffer-Heeger model. Previous studies found that the linearized dynamics can enter a topological regime in which the edge mode is driven into auto-oscillation, while the bulk dynamics are suppressed. Here we investigate the full non-linear and finite-temperature dynamics, whose understanding is essential for spin-torque oscillators-based applications. Our analysis shows that the lasing edge mode dynamics persist in the non-linear domain for a broad range of …
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 co…
We study the effect of photonic spin-orbit coupling (SOC) in micropillar lattices on the topological edge states of a one-dimensional chain with a zigzag geometry, corresponding to the Su-Schrieffer-Heeger model equipped with an additional internal degree of freedom. The system combines the strong hopping anisotropy of the $p$-type pillar modes with the large TE-TM splitting in Bragg microcavities. By resolving the photoluminescence emission in energy and polarization we probe the effects of the resulting SOC on the spatial and spectral properties of the edge modes. We find that the edge modes…
Topological physics strongly relies on prototypical lattice model with particular symmetries. We report here on a theoretical and experimental work on acoustic waveguides that is directly mapped to the one-dimensional Su-Schrieffer-Heeger chiral model. Starting from the continuous two dimensional wave equation we use a combination of monomadal approximation and the condition of equal length tube segments to arrive at the wanted discrete equations. It is shown that open or closed boundary conditions topological leads automatically to the existence of edge modes. We illustrate by graphical const…
We investigate the radiative heat transport along a Su-Schrieffer-Heeger chain of InSb nanoparticles. We show that in the topological non-trivial phase, the edge modes dominate the radiative heat transport despite their strong localization at the edges of the finite chain due to a long-range coupling of the first and last particle. We further discuss the scaling laws of the heat transfer with respect to the chain length for the longitudinal and transversal band- and edge modes and conclude that both type of modes obey the same scaling law.
We have theoretically investigated Su-Schrieffer-Heeger chains modelled as optical lattices (OL) loaded with exciton-polaritons. The chains have been subject to the resonant pumping of the edge site and shaken in either adiabatic or high-frequency regime. The topological state has been controlled by the relative phases of the lasers constructing the OL. The dynamic problem of the occupation of the lattice sites and eigenstates has been semi-classically solved. Finally, the analysis of the occupation numbers evolution has revealed that gapless, topologically trivial and non-trivial chain config…
Non-Hermiticity alters topology with the presence of non-Hermitian factors in topological systems. Most existing non-Hermitian topological systems derive their topological phases from Hermitian components, that is, the gain and loss of the system are considered simultaneously. In this work, we reveal two-dimensional non-Hermitian high-order topological insulator based on one-dimensional SSH chain, the nontrivial topology of which induced by total-loss. By introducing the loss of a specific configuration, we get a band gap with corner and edge states whose topology is characterized by the gappe…
The second hyperpolarizability $γ_N(-3ωω,ω,ω)$ of $N$ double-bond finite chain of trans-polyactylene is analyzed using the Su-Schrieffer-Heeger model to explain qualitative features of the size-dependence behavior of $γ_N$. Our study shows that $γ_N/N$ is {\it nonmonotonic} with $N$ and that the nonmonotonicity is caused by the dominant contribution of the intraband transition to $γ_N$ in polyenes. Several important physical effects are discussed to reduce quantitative discrepancies between experimental and our results
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 propose to realize Dirac states in an inclined two-dimensional Su-Schrieffer-Heeger model on a square lattice. We show that a pair of Dirac points protected by space-time inversion symmetry appear in the semimetal phase. The locations of these Dirac points are not pinned to any high-symmetry points of the Brillouin zone but are tunable through parameter modulations. Interestingly, the merging of two Dirac points undergoes a topological phase transition that leads to either a weak topological insulator or a nodal-line semimetal. We provide a systematic analysis of these topological phases fr…
We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a "conjugated-pseudo-Hermiticity" which we show is responsible for a quantized "complex" Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topologica…
By introducing the next-nearest-neighboring (NNN) intersite coupling, we investigate the eigenenergies of the $\cal PT$-symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials at the end sites. It is found that with the strengthening of NNN coupling, the particle-hole symmetry is destroyed. As a result, the bonding band is first narrowed and then undergoes the top-bottom reversal followed by the its width's increase, whereas the antibonding band is widened monotonously. In this process, the topological state extends into the topologically-trivial regio…
Su-Schrieffer-Heeger (SSH) model on two-dimensional square lattice exhibits a topological phase transition, which is related to the Zak phase determined by bulk band topology. The strong modulation of electron hopping causes nontrivial charge polarization even in the presence of inversion symmetry. The energy band structures and topological edge states have been calculated numerically in previous studies. Here, however, full energy spectrum and explicit form of wave functions for two-dimensional bulk and one-dimensional ribbon geometries of SSH model are analytically derived using wave mechani…
We study continuations of topological edge states in the Su-Schrieffer-Heeger model with on-site cubic (Kerr) nonlinearity, which is a 1D nonlinear photonic topological insulator (TI). Based on the topology of the underlying spatial dynamical system, we establish the existence of nonlinear edge states (edge solitons) for all positive energies in the topological band gap. We discover that these edge solitons are stable at any energy when the ratio between the weak and strong couplings is below a critical value. Above the critical coupling ratio, there are energy intervals where the edge soliton…
In the present work we discuss the propagation of excitons across a one-dimensional Su-Schrieffer-Heeger lattice, which possesses both harmonic oscillations and weak quartic anharmonicities. When quantizing these vibrational degrees of freedom we identify several phonon-conserving nonlinearities, each one with a different impact on the excitonic transport. Our analysis identifies a dominant nonlinear correction to the phonon hopping which leads to a strong enhancement of exciton conduction compared to a purely linear vibrational dynamics. Thus quartic lattice nonlinearities can be exploited to…
The emergence of non equilibrium topological phases in low dimensional systems offers an interesting route for material properties engineering. We analyze the dynamical modulation of two coupled one-dimensional chains, described by the Su-Schrieffer-Heeger model. We find that the interplay of driving interactions and interchain coupling leads to the emergence of non-equilibrium edge states with nontrivial topological properties. Using an effective Hamiltonian approach, we quantify the emergent topological phases via the winding number and show that oscillations in the mean pseudospin polarizat…
T-distributed stochastic neighborhood embedding (tSNE) is used as a tool to reveal the phase diagram of the Su-Schrieffer-Heeger model and some of its extended and non-Hermitian variants. Bloch vectors calculated at different points in the parameter space are mapped to a two-dimensional reduced space. The clusters in the reduced space are used to visualize different phase regions included in the input. The tSNE mapping is shown to be effective even in the challenging case of the non-Hermitian extended model where five different phases are present. An example of using wavefunction input, instea…
Coupling Majorana qubits with other qubits is an absolute essential in storing, manipulating and transferring informations for topological quantum computing. We theoretically propose a manner to coupling Majorana qubits with solitons, another kind of topological impurities, which was first studied in the spinless Su-Schrieffer-Heeger (SSH) model. We presents a NOT operation on the Majorana qubit with moving the complementary soliton through heterostructure adiabatically. Based on these two topological impurities, the operation is robust against local disorder. Furthermore, we find that the sol…
We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be…
Topological insulators are a new class of materials that have attracted significant attention in contemporary condensed matter physics. They are different from the regular insulators and they display novel quantum properties that also involve the idea of `topology', an area of mathematics. Some of the fundamental ideas behind the topological insulators, particularly in low-dimensional condensed matter systems such as poly-acetylene chains, can be understood using a simple one-dimensional toy model popularly known as the Su-Schrieffer-Heeger model or the SSH model. This model can also be used a…
Custodial symmetries are common in the Standard Model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory (memcircuit). In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spreads the state across the circuit. However, the resulting state is still protected against perturba…
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 enhanced by strong driving optical 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 external driving fields. Moreover, four types of edge states can be created in generalized …
We study the extended Su-Schrieffer-Heeger model with both the nearest-neighbor and next-nearest-neighbor hopping strengths being cyclically modulated and find the family of the model system exhibiting topologically nontrivial phases, which can be characterized by a nonzero Chern number defined in a two-dimensional space spanned by the momentum and modulation parameter. It is interesting that the model has a similar phase diagram as the well-known Haldane's model. We propose to use photonic crystal systems as the idea systems to fabricate our model systems and probe their topological propertie…
We explore theoretically the effect of inter and intra cell spin-orbit couplings on topological properties of a generalized Su-Schrieffer-Heeger model with multipartite lattice structure containing even number of sites per unit cell. We show that the spin-orbit couplings enrich topological phase diagrams so that nontrivial topological regions in the space of parameters extend significantly which is suitable for potential applications. We present an analytical formula for winding number in terms of sublattice number signaling that there are two nontrivial topological phases in the space of para…
We recently unveiled that antiferromagnetism (AFM) order can be dominantly induced by the bond Su-Schrieffer-Heeger (SSH) electron-phonon coupling (EPC), which is beyond the conventional wisdom that AFM order is usually driven by strong Coulomb interactions. Nevertheless, many aspects of the interplay between EPC and strong electronic interactions on AFM ordering remains unexplored. Here, we investigate the Su-Schrieffer-Heeger-Hubbard (SSHH) model with bond SSH phonons and onsite Hubbard interactions by large-scale quantum Monte-Carlo (QMC) simulations and obtain its ground-state phase diagra…
The topologically nontrivial edge states induce PT transition in Su-Schrieffer-Heeger (SSH) chain with one pair of gain and loss at boundaries. In this study, we investigated a pair of PT-symmetric defects located inside the SSH chain, in particular, the defects locations are at the chain centre. The PT symmetry breaking of the bound states leads to the PT transition, the PT-symmetric phases and the localized states were studied. In the broken PT-symmetric phase, all energy levels break simultaneously in topologically trivial phase; however, two edge states in topologically nontrivial phase ar…
We study the topological physics in nonlinear Schrödinger systems on lattices. We employ the quench dynamics to explore the phase diagram, where a pulse is given to a lattice point and we analyze its time evolution. There are two system parameters $λ$ and $ξ$, where $λ$ controls the hoppings between the neighboring links and $ξ$ controls the nonlinearity. The dynamics crucially depends on these system parameters. Based on analytical and numerical studies, we derive the phase diagram of the nonlinear Su-Schrieffer-Heeger (SSH) model in the ($λ,ξ$) plane. It consists of four phases. The topologi…
We investigate topological and disorder effects in non-Hermitian systems with chiral symmetry. The system under consideration consists in a finite Su-Schrieffer-Heeger chain to which two semi-infinite leads are attached. The system lacks the parity-time and time-reversal symmetries and is appropriate for the study of quantum transport properties. The complex energy spectrum is analyzed in terms of the chain-lead coupling and chiral disorder strength, and shows substantial differences between chains with even and odd number of sites. The mid-gap edge states acquire a finite lifetime and are bot…
In this work, we investigate some aspects of an acoustic analogue of the two-dimensional Su-Schrieffer-Heeger model. The system is composed of alternating cross-section tubes connected in a square network, which in the limit of narrow tubes is described by a discrete model coinciding with the two-dimensional Su-Schrieffer-Heeger model. This model is known to host topological edge waves, and we develop a scattering theory to analyze how these waves scatter on edge structure changes. We show that these edge waves undergo a perfect reflection when scattering on a corner, incidentally leading to a…
The high Chern number phases with the Chern number |C| > 1 are observed in this study of a periodically driven extended Su-Schrieffer-Heeger (E-SSH) model with a cyclic parameter. Besides the standard intra-dimer and the nearest-neighbor (NN) inter-dimer hopping of the SSH model, an additional next-nearest-neighbor (NNN) hopping is considered in the E-SSH model. The cyclic parameter, which plays the role of a synthetic dimension, is invoked as a modulation of the hopping strengths. A rigorous analysis of different phase diagrams has shown multiple Floquet topological phase transitions among th…
Topological electronics has extended its richness to non-electronic systems where phonons and magnons can play the role of electrons. In particular, topological phases of magnons can be enabled by the Dzyaloshinskii-Moriya interaction (DMI) which acts as an effective spin-orbit coupling. We show that besides DMI, an alternating arrangement of Heisenberg exchange interactions critically determines the magnon band topology, realizing a magnonic analog of the Su-Schrieffer-Heeger model. On a honeycomb ferromagnet with perpendicular anisotropy, we calculate the topological phase diagram, the chira…
We propose an anti-parity-time (anti-PT ) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model, where the large non-Hermiticity constructively creates nontrivial topology and greatly expands the topological phase. In the anti-PT -symmetric SSH model, the gain and loss are alternatively arranged in pairs under the inversion symmetry. The appearance of degenerate point at the center of the Brillouin zone determines the topological phase transition, while the exceptional points unaffect the band topology. The large non-Hermiticity leads to unbalanced wavefunction distribution in the broken an…
We show anomalous features of Majorana Bound State leakage in the situation where topological Rashba nanowire is dimerized according to the Su-Schrieffer-Heeger (SSH) scenario and an impurity is present at one of the ends of the system. We find that two topological branches: usual, indigenous to Rashba nanowire and dimerized one, existing as a result of SSH dimerization of nanowire, have different asymmetry of spin polarization that can be explained by opposite order of bands taking part in topological transitions. Additionally, introduction of an impurity to the dimerized nanowire influences …
The Su-Schrieffer-Heeger model describes fermions that hop on a one-dimensional chain with staggered hopping amplitude, where the unit cell contains two sites, or two sublattices. In this work we consider the Su-Schrieffer-Heeger model with superconducting pairing and show that the sublattice index acts as an additional quantum number in the classification of Cooper pairs, giving rise to inter- and intra-sublattice odd-frequency pair correlations in the bulk. Interestingly, this system behaves as a two band superconductor where the bulk odd-frequency correlations depend solely on the intrinsic…
We consider two interacting bosons in a dimerized Su-Schrieffer-Heeger (SSH) lattice. We identify a rich variety of two-body states. In particular, for open boundary conditions and moderate interactions, edge bound states (EBS) are present even for the dimerization that does not sustain single-particle edge states. Moreover, for large values of the interactions, we find a breaking of the standard bulk-boundary correspondence. Based on the mapping of two interacting particles in one dimension onto a single particle in two dimensions, we propose an experimentally realistic coupled optical fibers…
