Extended Su Schrieffer Heeger Research Articles

Individually tunable tunnelling coefficients in optical lattices using local periodic driving

Abstract Ultracold atoms in optical lattices have emerged as powerful quantum simulators of translationally invariant systems with many applications in e.g. strongly-correlated and topological systems. However, the ability to locally tune all Hamiltonian parameters remains an outstanding goal that would enable the simulation of a wider range of quantum phenomena. Motivated by recent advances in quantum gas microscopes and optical tweezers, we here show theoretically how local control over individual tunnelling links in an optical lattice can be achieved by incorporating local time-periodic potentials. We propose to periodically modulate the on-site energy of individual lattice sites and employ Floquet theory to demonstrate how this provides full individual control over the tunnelling amplitudes in one dimension. We provide various example configurations realising interesting topological models such as extended Su–Schrieffer–Heeger models that would be challenging to realise by other means. Extending to two dimensions, we demonstrate that local periodic driving in a Lieb lattice engineers a two-dimensional (2D) network with fully controllable tunnelling magnitudes. In a three-site plaquette, we show full simultaneous control over the relative tunnelling amplitudes and the gauge-invariant flux piercing the plaquette, providing a clear stepping stone to building a fully programmable 2D tight-binding model. We also explicitly demonstrate how utilise our technique to generate a magnetic field gradient in 2D. This local modulation scheme is applicable to many different lattice geometries.

Topological Phase and Tunable Quantum State Transfer of Su–Schrieffer–Heeger Chain with an Embedded Quantum Ring

AbstractAn extended Su–Schrieffer–Heeger (SSH) model consisting of an SSH chain with an embedded quantum ring (QR) is investigated. In the case of two SSH chains with a symmetric distribution with respect to QR, by calculating the real‐space winding number and energy spectrum, the hopping amplitude‐induced topological phase is discovered. The probability distributions of gap states and the relationship between energy and magnetic flux prove the existence of the Aharonov–Bohm effect in the QR. Moreover, in the case of asymmetric distribution, the model possesses a zero‐energy mode within the energy gap, and it is found that it can realize quantum state transfer. By adjusting the connection site between the right SSH chain and QR, the direction of the output port can be flexibly engineered. Furthermore, it is shown that high‐fidelity quantum state transfer can still be achieved with the increasing of the system size. The tunable quantum state transfer based on the zero‐energy mode can be equivalent to a topological tunable directional switch with properties of non‐directional transfer. This work provides an approach for studying topological phase transition and tunable topological devices in an SSH chain with an embedded QR.

Effect of lattice distortion on spin admixture and quantum transport in organic devices with spin–orbit coupling

Abstract With an extended Su–Schrieffer–Heeger model and Green’s function method, the spin–orbit coupling (SOC) effects on spin admixture of electronic states and quantum transport in organic devices are investigated. The role of lattice distortion induced by the strong electron–lattice interaction in organics is clarified in contrast with a uniform chain. The results demonstrate an enhanced SOC effect on the spin admixture of frontier eigenstates by the lattice distortion at a larger SOC, which is explained by the perturbation theory. The quantum transport under the SOC is calculated for both nonmagnetic and ferromagnetic electrodes. A more notable SOC effect on total transmission and current is observed for ferromagnetic electrodes, where spin filtering induced by spin-flipped transmission and suppression of magnetoresistance are obtained. Unlike the spin admixture, a stronger SOC effect on transmission exists for the uniform chain rather than the organic lattices with distortion. The reason is attributed to the modified spin-polarized conducting states in the electrodes by lattice configuration, and hence the spin-flip transmission, instead of the spin admixture of eigenstates. This work is helpful to understand the SOC effect in organic spin valves in the presence of lattice distortion.

One-dimensional extended Su–Schrieffer–Heeger models as descendants of a two-dimensional topological model

The topological phase diagrams and finite-size energy spectra of one-dimensional extended Su–Schrieffer–Heeger (SSH) models with long-range hoppings on the trimer lattice are investigated in detail. Due to the long-range hoppings, the band structure of the original SSH model becomes more complicated and new phases with the large Zak phase can emerge. Furthermore, a seeming violation of bulk-edge correspondence occurs in the one-dimensional topological system whose band topology stems from the inversion symmetry. The one-dimensional models are mapped onto a two-dimensional topological model when a parameter of the one-dimensional models is regarded as an additional degree of freedom. As Fourier components of the derived two-dimensional model, phase boudaries and the finite-size spectra of one-dimensional models can be recovered from the model in the higher spatial dimensions. Then the origin of edge modes of one-dimensional models can be understood from two dimensions and we give a reasonable explanation of the violation of bulk-edge correspondence in one spatial dimension. In fact, we may give a general perspective that the topological properties of one-dimensional (lower-dimensional) systems can be found their origin from two-dimensional (higher-dimensional) systems.

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

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

Rigorous analysis of the topologically protected edge states in the quantum spin Hall phase of the armchair ribbon geometry

Studying the edge states of a topological system and extracting their topological properties is of great importance in understanding and characterizing these systems. In this paper, we present a novel analytical approach for obtaining explicit expressions for the edge states in the Kane-Mele model within a ribbon geometry featuring armchair boundaries. Our approach involves a mapping procedure that transforms the system into an extended Su–Schrieffer–Heeger model, specifically a two-leg ladder, in momentum space. Through rigorous derivation, we determine various analytical properties of the edge states, including their wave functions and energy dispersion. Additionally, we investigate the condition for topological transition by solely analyzing the edge states, and we elucidate the underlying reasons for the violation of the bulk-edge correspondence in relatively narrow ribbons. Our findings shed light on the unique characteristics of the edge states in the quantum spin Hall phase of the Kane–Mele model and provide valuable insights into the topological properties of such systems.

Engineering topological phases of any winding and Chern numbers in extended Su–Schrieffer–Heeger models

Simple route of engineering topological phases for any desired value of winding and Chern numbers is found in the Su–Schrieffer–Heeger (SSH) model by adding a further neighbor hopping term of varying distances. It is known that the standard SSH model yields a single topological phase with winding number, ν = 1. In this study it is shown that how one can generate topological phases with any values of winding numbers, for examples, in the presence of a single further neighbor term which preserves inversion, particle-hole and chiral symmetries. Quench dynamics of the topological and trivial phases are studied in the presence of a specific nonlinear term. Another version of SSH model with additional modulating nearest neighbor and next-nearest-neighbor hopping parameters was introduced before which exhibit a single topological phase characterized by Chern number, . Standard form of inversion, particle-hole and chiral symmetries are broken in this model. Here this model has been studied in the presence of several types of parametrization among which, for a special case the system is found to yield a series of phases with Chern numbers, In another parametrization, multiple crossings within the edge states energy lines are found in both trivial and topological phases. Topological phase diagrams are drawn for every case. Emergence of spurious topological phases is also reported.

Spin Hall effect from bipolaron dynamics in organics.

Using an extended Su-Schrieffer-Heeger model and a nonadiabatic dynamics method, we investigate the dynamics of bipolarons in coupled nondegenerate organic chains including the spin-orbit coupling and interchain coupling. By tracing the time-dependent evolution of the charges and spins in each chain, an obvious oscillating spin Hall effect (SHE) from the bipolaron transport is revealed. The results are compared with that from polaron-dominated transport. A reduction of amplitude and an increase of oscillating frequency are observed for the SHE from the bipolaron transport. The mechanism is attributed to the enhanced skew scattering off the larger transient deformations of the chains in the case of the bipolaron. Spectrum analysis by fast Fourier transform of the SHE signal demonstrates a distinct shift of two characteristic peaks to a higher onset frequency compared to the polaron transport. The charge-spin conversion efficiency is also compared, where a larger conversion efficiency is obtained from the bipolaron transport due to the lower saturated velocity. The effects of the strength of the electric field and the interactions are discussed. This work reveals the role of the bipolaron in organic SHE and provides a feasible way to achieve larger conversion efficiency by controlling the species of carriers with the concentration of the dopant.

Engineering topological state transfer in four-period Su–Schrieffer–Heeger chain

An extended Su–Schrieffer–Heeger (SSH) model containing four periods of the hopping coefficients, called SSH4 model, is constructed to explore robust quantum state transfer. The gap state protected by the energy gap plays the role of the topological channel where the particle initially located at the last lattice site has the probability to arise at the first and all even lattice sites equally. Serving those sites as ports, a multi-port router can be realized naturally, and the fidelity reaches unity in a wide range of parameters under the long chain and random disorder. Further, when we reduce the third intracell hopping to a small value, the occupancy probability of the second lattice site in every unit cell will reduce to zero, by which a new topological router can be induced. In addition, our SSH4 model can work as a 1/3 beam splitter. Namely, the particle initially occupies the first lattice site and finally appears with equal probability at three lattice sites. We can also realize a 1/2 beam splitter. Our four-period SSH model provides a novel way for topological quantum information processing and can engineer two kinds of quantum optical devices.

Floquet topological phases with high Chern numbers in a periodically driven extended Su–Schrieffer–Heeger model

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 inter-dimer hopping of the SSH model, an additional next-nearest-neighbor 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 the high Chern number phases. These phase transitions can be controlled by the strength and frequency of the periodic driving. Instead of applying perturbation theory, the whole analysis is done by Floquet replica technique. This gives a freedom to study high as well as low-frequency effects on the system by considering less or more number of photon sectors. This system can be experimentally realized through a pulse sequence scheme in the optical lattice setup.

Rationalizing charge carrier transport in ternary organic solar cells

Ternary bulk heterojunction (BHJ) organic solar cells have energy offsets between multiple donors and acceptors. In such bi-continuous percolating films, electron carriers mainly transport in acceptor materials, and hole carriers typically transport in donor materials. Changing the third component of additional donors or acceptors is a common method to fine-tune the transport properties in ternary BHJs. Experimentally, although there are some empirical guidelines for the mobility evaluation, a clear charge transporting model has still not been fully established in multi-component BHJ films. Herein, we observed a general regularity about charge transport that the charge carriers have higher possibilities to transport in polymeric materials even if only low dosage (<10% in weight) polymeric materials are introduced for both the hole and electron cases. From the extended Su–Schrieffer–Heeger model, a polymer “bridge” assisted charge transport mode in small molecules is proposed after the energy offset exceeds the threshold (ΔEe > 0.15 eV). This work provides a perspective for fine tuning the charge transport property in high-performance ternary organic solar cells.

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.

Extended SSH Model in Non-Hermitian Waveguides with Alternating Real and Imaginary Couplings

We studied the topological properties of an extended Su–Schrieffer–Heeger (SSH) model composed of a binary waveguide array with alternating real and imaginary couplings. The topological invariant of the periodic structures remained quantized with chiral symmetry even though the system was non-Hermitian. The numerical results indicated that phase transition arose when the absolute values of the two couplings were equal. The system supported a topological zero mode at the boundary of nontrivial structures when chiral symmetry was preserved. By adding onsite gain and loss to break chiral symmetry, the topological modes dominated in all supermodes with maximum absolute value of imaginary energy. This study enriches research on the SSH model in non-Hermitian systems and may find applications in optical routers and switches.

Spin-dependent transport and functional design in organic ferromagnetic devices.

Organic ferromagnets are intriguing materials in that they combine ferromagnetic and organic properties. Although challenges in their synthesis still remain, the development of organic spintronics has triggered strong interest in high-performance organic ferromagnetic devices. This review first introduces our theory for spin-dependent electron transport through organic ferromagnetic devices, which combines an extended Su–Schrieffer–Heeger model with the Green’s function method. The effects of the intrinsic interactions in the organic ferromagnets, including strong electron–lattice interaction and spin–spin correlation between π-electrons and radicals, are highlighted. Several interesting functional designs of organic ferromagnetic devices are discussed, specifically the concepts of a spin filter, multi-state magnetoresistance, and spin-current rectification. The mechanism of each phenomenon is explained by transmission and orbital analysis. These works show that organic ferromagnets are promising components for spintronic devices that deserve to be designed and examined in future experiments.

Effect of interchain coupling on the excited polaron in conjugated polymers

Based on the one-dimensional extended Su–Schrieffer–Heeger model, we theoretically investigate the effect of interchain coupling on the formation and polarization of the single-excited state of polaron in conjugated polymers. It is found that there exists a turnover value of the coupling strength, over which the excited polaron could not be formed in either of the two coupled chains. Instead, a polaron-like particle is localized at the center of each chain. In addition, we also find that the reverse polarization of the excited polaron could be enhanced for some cases in polymer when the interchain coupling becomes strong until it exceeds the critical value.

Spin polarization of polaron in quasi-one dimensional organic system

The spin polarization of polarons in quasi-1D organic materials has been investigated by using the extended Su–Schrieffer–Heeger (SSH) model with spin-orbit coupling. Results show that the polaron is partly spin polarized, and that the electron–electron interaction and spin-orbit coupling compete with each other during the formation of spin polarization. The dependence of spin polarization on electron–phonon coupling is also revealed. Our results demonstrate that spin polarization is well correlated with polaron localization, thus providing useful guidance for exploring magnetic effects in organic materials.

A NOT operation on Majorana qubits with mobilizable solitons in an extended Su–Schrieffer–Heeger model

Coupling Majorana qubits with other qubits is absolutely essential for storing, manipulating and transferring information for topological quantum computing. We theoretically propose a manner to couple Majorana qubits with solitons, another kind of topological impurity, which was first studied in the spinless Su–Schrieffer–Heeger model. We present a NOT operation on the Majorana qubit by moving the soliton through a heterostructure adiabatically. Based on these two topological impurities, the operation is robust against local disorder. Furthermore, we find that the soliton may carry nonuniversal fractional electric charge instead of fractional charge , because of the breaking of gauge invariance induced by superconducting proximity.

The effect of interchain coupling on the reverse polarization of a high-energy exciton in conjugated polymers

Abstract By using the one-dimensional extended Su–Schrieffer–Heeger model, we investigate the effect of interchain coupling on the reverse polarization of a high-energy exciton in conjugated polymers. Our results show that, without interchain coupling, the high-energy exciton is completely localized in one chain, which could be reversely polarized by an electric field. However, when taking into account interchain coupling, the high-energy exciton tends to extend between two chains. Its reverse polarization would be reduced accordingly. We also find that the interchain coupling effect is more obvious for polymers with a stronger non-degenerate confinement. Besides this, the effect of electron–electron interaction is also discussed.

Thermoelectric properties of metal/molecule/metal junction for different lengths of polythiophene

Thermoelectric properties of metal/molecule/metal junction is investigated for different lengths of polythiophene connected to three dimensional metallic electrodes, using Green function formalism in linear response regime. An extended Su–Schrieffer–Heeger model is used to describe the Hamiltonian of polythiophene molecule and a tight-binding model is chosen to study the effect of length change on thermoelectric coefficients of the molecular junction. Increasing the molecular length results in decrease of HOMO–LUMO gap, decrease of the electrical conductance and increase of the value and oscillation of the thermopower. In addition, our results show that the temperature dependence of the thermoelectric coefficients is strongly related to the molecule length.

Spontaneous spin polarization in organic thiophene oligomers

We studied the spin polarization phenomenon of injected charges in organic thiophene oligomer by using extended Su–Schrieffer–Heeger (SSH) model including electron–electron interaction, spin–orbit coupling as well as spin-flip effect. Our simulation shows that a charged carrier is spontaneously spin polarized, which has a lower energy than the non-polarized one. This polarization is related with the amount of injected charges and the polymerization of the molecule.