SSH Model Research Articles

Topological flat band with higher winding number in a superradiance lattice

A five-level M-type scheme in atomic ensembles is proposed to generate a one-dimensional bipartite superradiance lattice in momentum space. By taking advantage of this tunable atomic system, we show that various types of Su-Schrieffer-Heeger (SSH) model, including the standard SSH and extended SSH model, can be realized. Interestingly, it is shown that through changing the Rabi frequencies and detunings in our proposed scheme, there is a topological phase transition from topological trivial regime with winding number being 0 to topological non-trivial regime with winding number being 2. Furthermore, a robust flat band with higher winding number (being 2) can be achieved in the above topological non-trivial regime, where the superradiance spectra can be utilized as a tool for experimental detection. Our proposal would provide a promising approach to explore new physics, such as fractional topological phases, in the flat bands with higher topological number.

Exponential decay of the critical points in a discrete model of polyacetylene

Abstract In this paper we consider stationary states of the SSH model for infinite polyacetylene chains that are homoclinic or heteroclinic connections between two-periodic dimerized states. We prove that such connections converge exponentially fast to the corresponding asymptotic periodic states.

Topological boundary states in engineered quantum-dot molecules on the InAs(111)A surface: Odd numbers of quantum dots

Atom manipulation by scanning tunneling microscopy was used to construct quantum dots on the InAs(111)A surface. Each dot comprised six ionized indium adatoms. The positively charged adatoms create a confining potential acting on surface-state electrons, leading to the emergence of a bound state associated with the dot. By lining up the dots into N-dot chains with alternating tunnel coupling between them, quantum-dot molecules were constructed that revealed electronic boundary states as predicted by the Su-Schrieffer-Heeger (SSH) model of one-dimensional topological phases. Dot chains with odd N were constructed such that they host a single end or domain-wall state, allowing one to probe the localization of the boundary state on a given sublattice by scanning tunneling spectroscopy. We found probability density also on the forbidden sublattice together with an asymmetric energy spectrum of the chain-confined states. This deviation from the SSH model arises because the dots are charged and create a variation in on-site potential along the chain—which does not remove the boundary states but shifts their energy away from the midgap position. Our results demonstrate that topological boundary states can be created in quantum-dot arrays engineered with atomic-scale precision. Published by the American Physical Society 2024

Realizations of Su-Schrieffer-Heeger (SSH) edge states in two-dimensional hydrocarbon systems

The Su-Schrieffer-Heeger (SSH) model of one-dimensional (1D) diatomic and four-atom chains, exhibit a topological phase transition characterized by the Zak phase. However, a challenge arises from the inherent difficulty of maintaining strong structural stability in real 1D nanostructures. Here, we show how to realize periodic 1D chains, reminiscent of the SSH model, in a two-dimensional (2D) system. These chains form a quasi-1D chain topological insulator (CTI) where the interchain coupling can be neglected. Based on first-principles calculations, we proposed that such CTIs can be realized in dumbbell (DB) C40H14 and DB C40H12 monolayers. The monolayers are CTIs, with a type of weak topological state, and the topological phase transition can be achieved by unit cell transformation or the application of 2D strain. Furthermore, increasing the number of DB C10H4 rings can enlarge the distance between the chains, corresponding to line defects within the monolayer, providing a possible strategy for experimental synthesis.

Impact of impurities on the topological boundaries and edge state localization in a staggered chain of atoms: SSH model and its topoelectrical circuit realization

We study the Su-Schrieffer-Hegger model, perhaps the simplest realization of a topological insulator, in the presence of an embedded impurity superlattice. We consider the impact of the said impurity by changing the hopping amplitudes between them and their nearest neighbors in the topological boundaries and the edge state localization in the chain of atoms. Within a tight-binding approach and through a topolectrical circuit simulation, we consider three different impurity-hopping amplitudes. We found a relaxation of the condition between hopping parameters for the topologically trivial and non-trivial phase boundary and a more profound edge state localization given by the impurity position within the supercell.

Topological edge states in reconfigurable multi-stable mechanical metamaterials

Multi-stable mechanical structures find cutting-edge applications across various domains due to their reconfigurability, which offers innovative possibilities for engineering and technology advancements. This study explores the emergence of topological states in a one-dimensional chain-like multi-stable mechanical metamaterial composed of bistable units through a combination of mechanical and optical experiments. Drawing inspiration from the SSH (Su-Schrieffer-Heeger) model in condensed matter physics, we leverage the unique mechanical properties of the reconfigurable ligament-oscillator metamaterial to engineer a system with coexisting topological phases. Based on the one-dimensional periodic bistable chain, there is an exponential decay diffusion of elastic energy from both end boundaries towards the interior of the body. Experimental characterizations demonstrate the existence of stable topological phases within the reconfigurable multi-stable mechanical metamaterial. The findings underscore the potential of reconfigurable mechanical metamaterials as versatile platforms for flexibly exploring and manipulating topological phenomena, with applications ranging from impact resistance to energy harvesting and information processing.

Observation of edge states derived from topological helix chains.

Introducing the concept of topology has revolutionized materials classification, leading to the discovery of topological insulators and Dirac-Weyl semimetals1-3. One of the most fundamental theories underpinning topological materials is the Su-Schrieffer-Heeger (SSH) model4,5, which was developed in 1979-decades before the recognition of topological insulators-to describe conducting polymers. Distinct from the vast majority of known topological insulators with two and three dimensions1-3, the SSH model predicts a one-dimensional analogue of topological insulators, which hosts topological bound states at the endpoints of a chain4-8. To establish this unique and pivotal state, it is crucial to identify the low-energy excitations stemming from bound states, but this has remained unknown in solids because of the absence of suitable platforms. Here we report unusual electronic states that support the emergent bound states in elemental tellurium, the single helix of which was recently proposed to realize an extended version of the SSH chain9,10. Using spin- and angle-resolved photoemission spectroscopy with a micro-focused beam, we have shown spin-polarized in-gap states confined to the edges of the (0001) surface. Our densityfunctional theory calculations indicate that these states are attributed to the interacting bound states originating from the one-dimensional array of SSH tellurium chains. Helices in solids offer a promising experimental platform for investigating exotic properties associated with the SSH chain and exploring topological phases through dimensionality control.

Theory of acoustic polarons in the two-dimensional SSH model applied to the layered superatomic semiconductor Re6Se8Cl2.

Layered superatomic semiconductors, whose building blocks are atomically precise molecular clusters, exhibit interesting electronic and vibrational properties. In recent work [Tulyagankhodjaev et al., Science 382, 438 (2023)], transient reflection microscopy revealed quasi-ballistic exciton dynamics in Re6Se8Cl2, which was attributed to the formation of polarons due to coupling with acoustic phonons. Here, we characterize the electronic, excitonic, and phononic properties with periodic density functional theory. We further parameterize a polaron Hamiltonian with nonlocal (Su-Schrieffer-Heeger) coupling to an acoustic phonon to study the polaron ground state binding energy and dispersion relation with variational wavefunctions. We calculate a polaron binding energy of about 10meV at room temperature, and the maximum group velocity of our polaron dispersion relation is 1.5 km/s, which is similar to the experimentally observed exciton transport velocity.

Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems

Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we apply a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models, but only when using the metric tensor containing both left and right eigenvectors. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. One remarkable example of this is the mathematical agreement between topological phase transition curves and lightlike paths in general relativity, suggesting a possibility of simulating space-time in non-Hermitian systems. We find that the metric in non-Hermitian phases degenerates in such a way that it effectively reduces the dimensionality of the quantum geometry by one. This implies that within linear response theory, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate. Published by the American Physical Society 2024

Topological phase transitions in the non-Hermitian SSH model with long-range hopping terms induced by gains and losses

The latest progress in the one-dimensional chain SSH model provide the basic understanding of non-Hermitian topological phases. By periodically introducing the gains and losses of imaginary potential ig,-ig,-ig,ig with anti-PT symmetry, we have investigated the topological property of non-Hermitian SSH model with long-range hopping terms. We find that the gains and losses of imaginary potential causes the model to exhibit non-Hermiticity, and the non-Hermiticity induces system to undergo topological phase transitions from topologically trivial phase with non-Hermitian winding number 0 to topologically nontrivial phase with non-Hermitian winding number 1 and −1. The introduction of gains and losses greatly broadens the topological non-trivial region and play a non-trivial role in the model. We fully characterize the topological phase transitions by the band gap closing-reopening, the non-Hermitian winding numbers v and the hidden Chern number C.

Breaking and resurgence of symmetry in the non-Hermitian Su-Schrieffer-Heeger model in photonic waveguides

Symmetry is one of the cornerstones of modern physics and has profound implications in different areas. In symmetry-protected topological systems, symmetries are responsible for protecting surface states, which are at the heart of the fascinating properties exhibited by these materials. When the symmetry protecting the edge mode is broken, the topological phase becomes trivial. By engineering losses that break the symmetry protecting a topological Hermitian phase, we show that a new genuinely non-Hermitian symmetry emerges, which protects and selects one of the boundary modes: the topological monomode. Moreover, the topology of the non-Hermitian system can be characterized by an effective Hermitian Hamiltonian in a higher dimension. To corroborate the theory, we experimentally investigated the non-Hermitian one- and two-dimensional SSH models using photonic lattices and observed dynamically generated monomodes in both cases. We classify the systems in terms of the (non-Hermitian) symmetries that are present and calculate the corresponding topological invariants. Published by the American Physical Society 2024

Implementation of the SSH model in an optical ring resonator

The implementation of physical models with topological features in optical systems has garnered much attention in recent times. In particular, on-chip integrated photonics platforms are promising platforms enabling us to take advantage of the promise of topologically robust modes against inevitable fabrication defects. Here, we propose to study the SSH model superimposed in an optical ring resonator in a quantitative way using electromagnetic simulations. We are interested in the localized states that appear when a topological phase transition is introduced into the ring. In particular, we examine the extent to which topologically protected modes maintain their properties in the presence of random deformations in the surrounding lattice. We find that the modes maintain their properties when small amounts of disorder are introduced into the system. We also study loss mechanisms in the localized states, distinguishing between losses to the adjacent waveguide and to radiation, finding that the topological protection only applies to the former.

Entanglement of edge modes in (very) strongly correlated topological insulators

Identifying topological phases for a strongly correlated theory remains a non-trivial task, as defining order parameters, such as Berry phases, is not straightforward. Quantum information theory is capable of identifying topological phases for a theory that exhibits quantum phase transition with a suitable definition of order parameters that are related to different entanglement measures for the system. In this work, we study entanglement entropy for a coupled SSH model, both in the presence and absence of Hubbard interaction and at varying interaction strengths. For the free theory, edge entanglement acts as an order parameter, which is supported by analytic calculations and numerical (DMRG) studies. We calculate the symmetry-resolved entanglement and demonstrate the equipartition of entanglement for this model which itself acts as an order parameter when calculated for the edge modes. As the DMRG calculation allows one to go beyond the free theory, we study the entanglement structure of the edge modes in the presence of on-site Hubbard interaction for the same model. A sudden reduction of edge entanglement is obtained as interaction is switched on. The explanation for this lies in the change in the size of the degenerate subspaces in the presence and absence of interaction. We also study the signature of entanglement when the interaction strength becomes extremely strong and demonstrate that the edge entanglement remains protected. In this limit, the energy eigenstates essentially become a tensor product state, implying zero entanglement. However, a remnant entropy survives in the non-trivial topological phase, which is exactly due to the entanglement of the edge modes.

Quantum quenches in driven-dissipative quadratic fermionic systems with parity-time symmetry

We study the quench dynamics of noninteracting fermionic quantum many-body systems that are subjected to Markovian drive and dissipation and are described by a quadratic Liouvillian which has parity-time (PT) symmetry. In recent work, we have shown that such systems relax locally to a maximum entropy ensemble that we have dubbed the PT-symmetric generalized Gibbs ensemble (PTGGE), in analogy to the generalized Gibbs ensemble that describes the steady state of isolated integrable quantum many-body systems after a quench. Here, using driven-dissipative versions of the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain as paradigmatic model systems, we corroborate and substantially expand upon our previous results. In particular, we confirm the validity of a dissipative quasiparticle picture at finite dissipation by demonstrating light cone spreading of correlations and the linear growth and saturation to the PTGGE prediction of the quasiparticle-pair contribution to the subsystem entropy in the PT-symmetric phase. Further, we introduce the concept of directional pumping phases, which is related to the non-Hermitian topology of the Liouvillian and based upon qualitatively different dynamics of the dual string order parameter and the subsystem fermion parity in the SSH model and the Kitaev chain, respectively: depending on the postquench parameters, there can be pumping of string order and fermion parity through both ends of a subsystem corresponding to a finite segment of the one-dimensional lattice, through only one end, or there can be no pumping at all. We show that transitions between dynamical pumping phases give rise to a new and independent type of dynamical critical behavior of the rates of directional pumping, which are determined by the soft modes of the PTGGE. Published by the American Physical Society 2024

Higher-Order Topological Peierls Insulator in a Two-Dimensional Atom-Cavity System.

In this work, we investigate a two-dimensional system of ultracold bosonic atoms inside an optical cavity, and show how photon-mediated interactions give rise to a plaquette-ordered bond pattern in the atomic ground state. The latter corresponds to a 2D Peierls transition, generalizing the spontaneous bond dimerization driven by phonon-electron interactions in the 1D Su-Schrieffer-Heeger (SSH) model. Here the bosonic nature of the atoms plays a crucial role to generate the phase, as similar generalizations with fermionic matter do not lead to a plaquette structure. Similar to the SSH model, we show how this pattern opens a nontrivial topological gap in 2D, resulting in a higher-order topological phase hosting corner states, that we characterize by means of a many-body topological invariant and through its entanglement structure. Finally, we demonstrate how this higher-order topological Peierls insulator can be readily prepared in atomic experiments through adiabatic protocols. Our work thus shows how atomic quantum simulators can be harnessed to investigate novel strongly correlated topological phenomena beyond those observed in natural materials.

Multi-topological state via the Brillouin zone overlap for nonlinear frequency conversion.

Multiband topological edge states (TESs) or topological corner states (TCSs) in photonic crystals provide effective ways to manipulate the nonlinear frequency conversions. However, the deliberate design and the limited number of multibands lead to the difficulty of experimental realization of the topological nonlinear frequency conversion or higher harmonic generation. Here, we propose an effective method to achieve multiple TESs and TCSs by combining the Brillouin zones of multiple different systems. It is shown that the spectra of the subsystems disperse into different energy levels due to the inter-system hopping. Based on this approach, we construct a topological photonic crystal based on the Brillouin zone overlapped SSH model, which enables the overlapped TCSs to participate in nonlinear frequency conversion. Our scheme can provide a significant way to realize the topological nonlinear frequency conversion with double resonances or multiple resonances.

Nonlinear control of topological corner and edge states in a loop resonator

The modulation and transformation of topological states play a crucial role in controlling light transport in optical communication system. Herein, we construct a Su-Schrieffer-Heeger (SSH) lattice beyond the linear optical regime. The SSH model has resulted in the topological edge states (TESs) and the topological corner states (TCSs). It is demonstrated that the second-harmonic generation (SHG) and third-harmonic generation (THG) can occur in a topological edge loop based on the frequency matching in all-dielectric photonic crystals. Different from previous studies in which the excitation of TESs and TCSs is achieved through the selection of frequencies, in this paper the excitation of TESs and TCSs has been achieved through the harmonic generations with different orders. More importantly, there is a transformation from the trapping of optical signal on the TCS to the releasing of optical signal on the TES. The mode transformation between TESs and TCSs has important applications in optical flow control, optical signal storing and releasing.

Localization and topological transitions in generalized non-Hermitian SSH models

We study the localization and topological transitions of the generalized non-Hermitian SSH models. Under the open boundary condition, two delocalization transitions are found due to the competition between the Anderson localization and the boundary localization from the nontrivial edge states and the non-Hermitian skin effect. Under the periodic boundary condition, only one delocalization transition is found due to the disappearance of the non-Hermitian skin effect. The winding numbers of energy and the Lyapunov exponents in analytical form are obtained to exactly characterize the two delocalization transitions. It finds that the delocalization transitions don't accompany the topological transition. Furthermore, the large on-site non-Hermiticity is detrimental to the topological transitions, delocalization and localization transitions. However, the large nonreciprocal hopping enhances the Anderson localizations but is detrimental to the delocalization transition. The above analyses are verified by calculating the energy gap and the inverse of the participation ratio numerically.

Zoology of non-Hermitian spectra and their graph topology

We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configuration of complex spectra are characterized by the algebraic structures of their corresponding energy dispersions, drawing new intimate links between combinatorial graph theory, algebraic geometry, and non-Hermitian band topology. Spectral graphs that are conformally related belong to the same equivalence class, and are characterized by emergent symmetries not necessarily present in the physical Hamiltonian. The simplest class encompasses well-known examples such as the Hatano-Nelson and non-Hermitian SSH models, while more sophisticated classes represent novel multicomponent models with interesting spectral graphs resembling stars, flowers, and insects. With recent rapid advancements in metamaterials, ultracold atomic lattices, and quantum circuits, it is now feasible to not only experimentally realize such esoteric spectra, but also investigate the non-Hermitian flat bands and anomalous responses straddling transitions between different spectral graph topologies.

Phonon-induced instabilities in correlated electron Hamiltonians

Studies of Hamiltonians modeling the coupling between electrons as well as to local phonon excitations have been fundamental in capturing the novel ordering seen in many quasi-one dimensional condensed matter systems. Extending studies of such Hamiltonians to quasi-two dimensional systems is of great current interest, as electron-phonon couplings are predicted to play a major role in the stabilization or enhancement of novel phases in 2D material systems. In this work, we study model systems that describe the interplay between the Hubbard coupling and the phonon modes in the Holstein (H) and Su-Schrieffer-Heeger (SSH) Hamiltonians using the functional renormalization group (fRG). For both types of electron phonon couplings, we find the predicted charge density wave phases in competition with anti-ferromagnetic ($AF$) ordering. As the system is doped, the transition shifts, with both orders showing incommensurate peaks. We compare the evolution of the quasiparticle weight for the Holstein model with that of the SSH model as the systems transition from antiferromagnetic to charge-ordered ground states. Finally, we calculate the self-energy of the phonon and capture the impact of charge ordering on the phonon modes.