Su Schrieffer Heeger Research Articles

Deterministic interface modes in two-dimensional acoustic systems

The interface state in two-dimensional (2D) sonic crystals (SCs) was obtained based on trying or cutting approach, which greatly limits its practical applications. In this paper, we theoretically demonstrate that one category of interface states can deterministically exist at the boundary of two square-lattice SCs due to the geometric phase transitions of bulk bands. First, we derive a tight-binding formalism for acoustic waves and introduce it into the 2D case. Furthermore, the extended 2D Zak phase is employed to characterize the topological phase transitions of bulk bands. Moreover, the topological interface states can be deterministically found in the nontrivial bandgap. Finally, two kinds of SCs with the [Formula: see text] symmetry closely resembling the 2D Su–Schrieffer–Heeger (SSH) model are proposed to realize the deterministic interface states. We find that tuning the strength of intermolecular coupling by contacting or expanding the scatterers can effectively induce the bulk band inversion between the trivial and nontrivial crystals. The presence of acoustic interface states for both cases is further demonstrated. These deterministic interface states in 2D acoustic systems will be a great candidate for future waveguide applications.

Tunable Topological Beam Splitter in Superconducting Circuit Lattice

In the usual Su–Schrieffer–Heeger (SSH) model with an even number of lattice sites, the topological pumping between left and right edge states cannot be easily realized since the edge states occupy two-end sites simultaneously. Here we propose a scheme to investigate the topological edge pumping in an even-sized periodically modulated SSH model mapped by a one dimensional superconducting transmission line resonators array. We find that the photon initially prepared in the first resonator can be finally observed at the two-end resonators with a certain proportion. The final photon splitting at the two-end resonators indicates that the present superconducting circuit is expected to realize the topological beam splitter. Further, we demonstrate that the splitting proportion between the two-end resonators can be arbitrarily tuned from 1 to 0, implying the potential feasibility of implementing the tunable topological beam splitter. Meanwhile, we also show that the tunable topological beam splitter is immune to the mild disorder added into the system due to the topology protection of the zero energy modes, and find that the tunable topological beam splitter is much more robust to the global on-site disorder compared with the nearest neighbor disorder. Our work greatly extends the practical application of topological matter in quantum information processing and opens up a new way towards the engineering of topological quantum optical device.

Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices

We report on the experimental demonstration of two-photon quantum walks at the edge of a photonic Su–Schrieffer–Heeger lattice and compare them to those observed when launching photons at the edge of a homogeneous lattice. Whereas at the topological edge, one of the photons primarily remains close to the edge, both photons penetrate freely from the trivial edge into the bulk. This behavior manifests also in the average inter-particle distance, which is significantly larger at the topological edge. Hence, for a given propagation length, the entangled two-photon state launched at the topological edge extends over a wider domain of the lattice.

Generalized Bloch band theory for non-Hermitian bulk–boundary correspondence

Abstract Bulk–boundary correspondence is the cornerstone of topological physics. In some non-Hermitian topological systems this fundamental relation is broken in the sense that the topological number calculated for the Bloch energy band under the periodic boundary condition fails to reproduce the boundary properties under the open boundary. To restore the bulk–boundary correspondence in such non-Hermitian systems a framework beyond the Bloch band theory is needed. We develop a non-Hermitian Bloch band theory based on a modified periodic boundary condition that allows a proper description of the bulk of a non-Hermitian topological insulator in a manner consistent with its boundary properties. Taking a non-Hermitian version of the Su–Schrieffer–Heeger model as an example, we demonstrate our scenario, in which the concept of bulk–boundary correspondence is naturally generalized to non-Hermitian topological systems.

Non-Bloch band theory and bulk–edge correspondence in non-Hermitian systems

Abstract In this paper, we review our non-Bloch band theory in 1D non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su–Schrieffer–Heeger model, the bulk–edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge states.

Dirac equation perspective on higher-order topological insulators

In this Tutorial, we pedagogically review recent developments in the field of non-interacting fermionic phases of matter, focusing on the low-energy description of higher-order topological insulators in terms of the Dirac equation. Our aim is to give a mostly self-contained treatment. After introducing the Dirac approximation of topological crystalline band structures, we use it to derive the anomalous end and corner states of first- and higher-order topological insulators in one and two spatial dimensions. In particular, we recast the classical derivation of domain wall bound states of the Su–Schrieffer–Heeger (SSH) chain in terms of crystalline symmetry. The edge of a two-dimensional higher-order topological insulator can then be viewed as a single crystalline symmetry-protected SSH chain, whose domain wall bound states become the corner states. We never explicitly solve for the full symmetric boundary of the two-dimensional system but instead argue by adiabatic continuity. Our approach captures all salient features of higher-order topology while remaining analytically tractable.

Non-Hermitian generalizations of extended Su–Schrieffer–Heeger models

Non-Hermitian generalizations of the Su–Schrieffer–Heeger (SSH) models with higher periods of the hopping coefficients, called the SSH3 and SSH4 models, are analyzed. The conventional construction of the winding number fails for the Hermitian SSH3 model, but the non-Hermitian generalization leads to a topological system due to a point gap on the complex plane. The non-Hermitian SSH3 model thus has a winding number and exhibits the non-Hermitian skin effect. Moreover, the SSH3 model has two types of localized states and a zero-energy state associated with special symmetries. The total Zak phase of the SSH3 model exhibits quantization, and its finite value indicates coexistence of the two types of localized states. Meanwhile, the SSH4 model resembles the SSH model, and its non-Hermitian generalization also exhibits the non-Hermitian skin effect. A careful analysis of the non-Hermitian SSH4 model with different boundary conditions shows the bulk-boundary correspondence is restored with the help of the generalized Brillouin zone or the real-space winding number. The physics of the non-Hermitian SSH3 and SSH4 models may be tested in various simulators.

Weakly nonlinear topological gap solitons in Su-Schrieffer-Heeger photonic lattices.

We study both theoretically and experimentally the effect of nonlinearity on topologically protected linear interface modes in a photonic Su-Schrieffer-Heeger (SSH) lattice. It is shown that under either focusing or defocusing nonlinearity, this linear topological mode of the SSH lattice turns into a family of topological gap solitons. These solitons are stable. However, they exhibit only a low amplitude and power and are thus weakly nonlinear, even when the bandgap of the SSH lattice is wide. As a consequence, if the initial beam has modest or high power, it will either delocalize, or evolve into a soliton not belonging to the family of topological gap solitons. These theoretical predictions are observed in our experiments with optically induced SSH-type photorefractive lattices.

Coherent Interactions in One-Dimensional Topological Photonic Systems and Their Applications in All-Optical Logic Operation.

Topological photonics has become an active subfield of photonics analogous to the electronic counterpart, and the bulk-edge correspondence leads to robust topologically protected interfacial states. However, a single-topological interface mode with fixed energy cannot be easily manipulated, hindering its applications in optical devices. Here, we study coupled-waveguide arrays mapped to a one-dimensional Su-Schrieffer-Heeger system with two coupled topological interfaces. This configuration greatly increases device versatility and tunability while keeping the confinement of coupled-interface modes inherited from the topological properties nearly intact. Theoretically predicted oscillations between coupled interfaces is experimentally observed. The spatial and energetic isolation of the coupled interface states from the bulk modes is experimentally observed and theoretically confirmed by calculating the degree of localization of the eigenstates, which is found to be comparable to a single-interface state. Finally, a proof-of-principle, all-optical logic circuit is fabricated based on coupled interfaces, demonstrating its potential in assembling on-chip topological optical devices.

State-Dependent Topological Invariants and Anomalous Bulk-Boundary Correspondence in Non-Hermitian Topological Systems with Generalized Inversion SymmetrySupported by the National Natural Science Foundation of China (Grant Nos. 11674026 and 11974053).

Breakdown of bulk-boundary correspondence in non-Hermitian (NH) topological systems with generalized inversion symmetries is a controversial issue. The non-Bloch topological invariants determine the existence of edge states, but fail to describe the number and distribution of defective edge states in non-Hermitian topological systems. The state-dependent topological invariants, instead of a global topological invariant, are developed to accurately characterize the bulk-boundary correspondence of the NH systems, which is very different from their Hermitian counterparts. At the same time, we obtain the accurate phase diagram of the one-dimensional non-Hermitian Su–Schrieffer–Heeger model with a generalized inversion symmetry from the state-dependent topological invariants. Therefore, these results will be helpful for understanding the exotic topological properties of various non-Hermitian systems.

Topological Phase Transition and Eigenstates Localization in a Generalized Non‐Hermitian Su–Schrieffer–Heeger Model

AbstractThe topological properties of a generalized non‐Hermitian Su–Schrieffer–Heeger model are investigated and it is demonstrated that the non‐Hermitian phase transition and the non‐Hermitian skin effect can be induced by intra‐cell asymmetric coupling under open boundary conditions. Through investigating and calculating the non‐Hermitian winding number with generalized Brillouin zone theory, it is found that the present non‐Hermitian system has an exact bulk‐boundary correspondence relationship. Meanwhile, the non‐Hermitian winding number is used to characterize the non‐Hermitian phase transition and determine the phase transition boundary, and it is found that the non‐Hermitian phase transition is not completely induced by the asymmetric coupling strength. By means of the mean inverse participation ratio, the factors that affect the eigenstates localization are shown and it is revealed that large system size or large asymmetric coupling strength can leave the system in the localized state. Additionally, it is found that for the asymmetric coupling strength and the system size, the eigenstates localization is much more sensitive to the asymmetric coupling strength.

Augmented Hückel molecular orbital model of π‐electron systems: from topology to metric. I. General theory

AbstractThe π‐electron molecule is planar (but the strict planarity requirement may be relaxed), with a set of electrons residing on the (valence) molecular orbitals of the π‐symmetry (πMOs). The quantum state of these electrons influences the geometry of the molecule, most notably the bond lengths. The Hückel molecular orbital (HMO) model provides the simplest (yet quite effective) description of the πMOs. Longuet‐Higgins and Salem, and their followers, made efforts to augment this model by making its empirical parameters dependent on the bond lengths; this approach was later often referred to as the SCβ HMO model. In the present paper, we introduce the AugHMO model that is a generalization of these earlier efforts. We show that the mathematical formulation of this model necessarily implies the principle of a bond‐order–bond‐length (BO‐BL) relationship that is universal for a given type of bond (C–C, C–N, etc.) in all π‐electron molecules. A plausible assumption that the BO‐BL relationship is a linear one leads to a family of workable computational methods, such as the Hückel–Su–Schrieffer–Heeger method and the Hückel–Longuet–Higgins–Salem method. Our AugHMO model is developed with the following goals in mind: (i) to offer a predictive tool for calculating the bond lengths in π‐electron molecules, capable of dealing with very large systems; (ii) to provide insights into the problem of coupling between the π‐electrons and the bond lengths; and (iii) to offer a simple educational model illustrating the principles of geometry optimization in quantum chemistry (involving the gradient and Hessian of the molecular energy).

Realization of multidimensional sound propagation in 3D acoustic higher-order topological insulator

Higher-order topological insulators (TIs) develop the conventional bulk-boundary correspondence theory and increase the interest in searching innovative topological materials. To realize a higher-order TI with a wide passband of one-dimensional (1D) and two-dimensional (2D) transportation modes, we design three-dimensional non-trivial and trivial sonic crystals whose combination mimics the Su–Schrieffer–Heeger model. The topological boundary states can be found at the interfaces, including the zero-dimensional corner state, 1D hinge state, and 2D surface state. The fabricated sample with the bent two-dimensional and one-dimensional acoustic channels exhibits the multidimensional sound propagation and verifies the mode transition among the complete bandgap, hinge mode, and surface mode. The bandwidth of the single-mode hinge state achieves a large relative bandwidth of 9.1% in which sound transports one-dimensionally without significant leak into the surfaces or the bulk. The higher-order topological states in the study pave the way for sound manipulation in multiple dimensions.

Molecular dynamic simulation of photogenerated soliton pairs in trans-polyacetylene

Abstract The formation and evolution of photogenerated soliton pairs in trans-polyacetylene is simulated by using a nonadiabatic dynamical method. Electron–lattice interaction as well as electron correlation effects are both included by employing the Su–Schrieffer–Heeger+Pariser–Parr–Pople model. We adopt the multiconfigurational time-dependent Hartree–Fock formulas which allows for the distinguishing between charged and neutral soliton pairs. It is found that charged soliton pairs are created in the initial stage of the evolution after photoexcitation. Subsequently, the formed charged soliton pairs transform into neutral ones during the dynamical evolutions. The conversion time from charged to neutral state depends on the strength of Coulomb interactions. However, neutral soliton pairs are suppressed in the non-interacting limit, which is in good agreement with the previous estimates based on the Su–Schrieffer–Heeger model. We hope these results can provide further insight into some of the basic issues of interest in photogenerated processes in conducting polymers.

Zitterbewegung of Dirac quasiparticles emerged in a Su-Schrieffer–Heeger lattice* *Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFA0301803 and 2016YFA0302800), the National Natural Science Foundation of China (Grant Nos. 11604103, 11704132, 11822403, and 91636218), the NSAF, China (Grant Nos. U1801661 and U1830111), the PCSIRT, China (Grant No. IRT1243), the Natural Science Foundation of Guangdong

We analytically and numerically investigate the dynamical properties of the tilted dispersion relativistic quasiparticles emerged in a cold atomic optical lattice system. By introducing the next nearest neighboring (NNN) hopping term into Su–Schrieffer–Heeger (SSH) model, the Dirac quasiparticles with tilted dispersion relation are realized. The results show that the tilted dispersion causes a drift in relativistic quasiparticles rather than affecting interference behavior between inner states. To be specific, the relativistic phenomena of the quasiparticles induced by the inner state interference (such as Zitterbewegung, Klein paradox, etc.) is completely unaffected by the tilted dispersion. In order to distinguish the drift induced by tilted dispersion and common initial velocity, we calculate the momentum distribution of the relativistic quasiparticles. We obtain the difference between the drift induced by initial velocity and tilted dispersion. The former affects the ZB, while the latter does not. By using this character, we propose a quench dynamics scheme to obtain a stable mono-spin state. The proposed cold atomic lattice system would provide a promising platform in exploring the intrinsic exotic physics of relativistic quasiparticles and the related systems.

Quantum Ising chain with boundary dephasing

Abstract We study the quantum Ising chain with boundary dephasing. By doubling the Hilbert space, the model is mapped to the Su–Schrieffer–Heeger model with imaginary chemical potential at the edges. We show analytically and numerically that the Liouvillian gap, i.e. the inverse relaxation time of the model, scales with the system size $ N $ as $ N^{-3} $.

Realization of hierarchical topological transitions and high-Q-response corner states in second-order topological photonic crystals

Topological photonic crystals (PCs) provide a convenient method of electromagnetic wave processing. Recently, higher-order photonic topological insulators have been investigated as an intriguing topological phase beyond the usual bulk–edge correspondence. Previous studies of corner states in higher-order topological insulators mainly focus on topological multipole systems realized by introducing a negative coupling between constituent units or based on the two-dimensional Su–Schrieffer–Heeger model. In this work, we study a second-order topological insulator of square lattice that can host one-dimensional edge states and zero-dimensional corner states by means of breaking the symmetry of the structure. A corner mode with a high quality factor over 22 000 is formed by combining two types of such PCs with different topological phases. The significant topological phase transitions in the bulk and at the edges can be realized by simply changing the diameter of the constituent cylinders. Our results provide a topological way to guide and trap electromagnetic waves.

Bound state in the continuum in topological inductor–capacitor circuit

Bound states in the continuum (BICs) have received increasing attention from researchers because of their potential applications in photonic crystal fibers, as well as in lasing, sensing, and surface acoustic wave devices. BICs have been experimentally observed in acoustic resonators, photonic crystal slabs, and optical waveguides. Herein, we constructed a topological inductor–capacitor (LC) circuit to observe BICs; the circuit consists of two identical LC Su–Schrieffer–Heeger chains coupled using a middle chain with capacitors. In addition, a BIC in a non-Hermitian system was also experimentally observed by adding different resistances to the original LC circuit. Our experimental and simulation results thus prove that an electrical circuit could be used as a platform to realize BIC. Furthermore, we believe that such a circuit can be extended to two- and three-dimensional models and higher frequencies, making it suitable for electrical device applications, such as antennae, filters, and radio frequency devices.

Topological and Nontopological Edge States Induced by Qubit‐Assisted Coupling Potentials

AbstractIn the usual Su–Schrieffer–Heeger (SSH) chain, the topology of the energy spectrum is divided into two categories in different parameter regions. Here, the topological and nontopological edge states induced by qubit‐assisted coupling potentials in circuit quantum electrodynamics (QED) lattice modeled as a SSH chain are studied. It is found that, when the coupling potential added on only one end of the system raises to a certain extent, the strong coupling potential will induce a new topologically nontrivial phase accompanied by the appearance of a nontopological edge state, and the novel phase transition leads to the inversion of odd–even effect directly. Furthermore, the topological phase transitions when two unbalanced coupling potentials are injected into both ends of the circuit QED lattice are studied, and it is found that the system exhibits three distinguishing phases with multiple flips of energy bands. These phases are significantly different from the previous phase induced via unilateral coupling potential due to the existence of a pair of nontopological edge states. The scheme provides a feasible and visible method to induce different topological and nontopological edge states through controlling the qubit‐assisted coupling potentials in circuit QED lattice both in experiment and theory.

Topological spintronics in a polyacetylene molecule device

Using the Su–Schrieffer–Heeger Hamiltonian and exploiting the Green’s function method in the framework of the Landauer–Büttiker formalism, the topological and spin dependent electron transport properties of a trans polyacetylene molecule are studied. It is found that molecules with the intracell single carbon–carbon bonding and the even number of monomers in their chains have two edge states and possess topological properties though their Hamiltonians do not respect the chiral symmetry. A perpendicular exchange magnetic field and two perpendicular and transverse electric fields are used to induce and manipulate the quantum spin dependent electron transport properties. The exchange field induces the spin polarization in different electron energy regions which are expanded by stronger exchange fields. Therefore this proposed device works as a perfect spin filter. The spin polarization can be manipulated by applying the perpendicular electric field and remains robust against the transverse electric field variations.