One-dimensional Topological Insulators Research Articles

Longitudinal one-dimensional mechanical topological insulator

Abstract We present a study of a longitudinal one-dimensional mechanical topological insulator based on a slinky spring in the Su–Schrieffer–Heeger (SSH) configuration. The system demonstrates key characteristics of topological insulators, including the existence of edge states in the bandgap, exponential decay of amplitude, and a winding number of 1 for topological phases. By manipulating the stiffness of the spring through the placement of masses, we transition between trivial, metallic, and topological phases. Our findings also show that the edge states are robust against perturbations, and we observe a critical phase transition where the coherence length follows a critical exponent of -1, as predicted by theory. This simple mechanical system provides an accessible platform for studying the special properties of topological insulators and opens up new possibilities for exploring topological phenomena in classical systems.

Long-Range Gating in Single-Molecule One-Dimensional Topological Insulators.

Single-molecule one-dimensional topological insulator (1D TI) is a class of molecular wires that exhibit increasing conductance with wire length. This unique trend is due to the coupling between the two low-lying topological edge states of 1D TIs described by the Su-Schrieffer-Heeger model. In principle, this quantum phenomenon within 1D TIs can be utilized to achieve long-range gating in molecular conductors. Here, we study electron transport through a single-edge state of doubly oxidized oligophenylene bis(triarylamine) to understand the effect of the edge state coupling on conductance. We find that conductance is elevated by approximately 1 order of magnitude compared to a control molecule with the same conductance pathway. Density function theory calculations further support that the increase in conductance is due to the interaction between the edge states of 1D TIs. This work demonstrates a new gating paradigm in molecular electronics, while also providing a deeper understanding of how edge states interact and affect electron transport within 1D TIs.

Weakly interacting one-dimensional topological insulators: A bosonization approach

Weakly interacting one-dimensional topological insulators: A bosonization approach

Entanglement and particle fluctuations of one-dimensional chiral topological insulators

Entanglement and particle fluctuations of one-dimensional chiral topological insulators

Rainbow trapping for sound waves in one-dimensional topological insulator

Over the recent decade, topological insulators, originating from the condensed matter physics, have resided at the frontier in the field of acoustics owing to their novel topological properties for manipulating robust wave propagation, which have also opened an intriguing landscape for potential applications. At the meantime, gradually slowing down acoustic waves with metamaterials allows temporary storage of sound, leading to the exploration of so-called trapped rainbow. However, most of the current studies are reported in a topological trivial context with complex structures, and it is hitherto still a challenge to obtain the high-efficient acoustic rainbow trapping effect in a straightforward setup. Here, we propose an acoustic gradient topological insulator in the one-dimensional system to realize a highly efficient rainbow trapping device. Based on the acoustic analogous Su–Schrieffer–Heeger model, we tune the eigenfrequencies of the topological interface states through modulating the neck widths of Helmholtz resonators. The experimentally measured pressure spectra clearly show that the proposed structure could tightly trap the broad-band sound waves at the target spatial positions. Our proposal may provide versatile possibilities for the design of topological acoustic devices.

Programmable dual-band acoustic topological insulator with dynamically movable interface states

Topological acoustic interface states in one-dimensional (1D) acoustic topological insulators (ATIs) are zero-dimensional (0D) topological states localized at an interface. Unlike topological edge states that can propagate to deliver information in acoustic waveguides, the 0D topological interface states generally cannot serve as information carriers to deliver information from one location to another due to their intrinsic localization. Here, we design and demonstrate a 1D ATI with a movable interface, enabling the 0D topological acoustic interface states to deliver information from one location to another. The ATI design is based on two types of elemental building blocks—denoted as “1” and “0”—which are programmable. These elements of 1 and 0, when periodically arranged, can form topologically distinct crystals, whose interface hosts acoustic topological interface states in two bandgaps simultaneously. Since these two types of elements can switch from each other with external control, a programmable 1D dual-band ATI can be constructed. By programming coding sequences of 1 and 0 elements, we can observe dynamically movable 0D topological interface states riding on a moving interface along the 1D ATI in both bandgaps. Our work opens an avenue to develop topological acoustic devices with programmable and dynamic functions, which may have a variety of potential applications in the fields of energy trapping, topological pumping, information processing, and sound communication.

Designing Long and Highly Conducting Molecular Wires with Multiple Nontrivial Topological States.

Molecular one-dimensional topological insulators (1D TIs), described by the Su-Schrieffer-Heeger (SSH) model, are a new class of molecular electronic wires whose low-energy topological edge states endow them with high electrical conductivity. However, when these 1D TIs become long, the high conductance is not sustained because the coupling between the edge states decreases with increasing length. Here, we present a new design where we connect multiple short 1D SSH TI units linearly or in a cycle to create molecular wires with a continuous topological state density. Using a tight-binding method, we show that the linear system gives a length-independent conductance. The cyclic systems show an interesting odd-even effect, with unit transmission in the topological limit, but zero transmission in the trivial limit. Furthermore, based on our calculations, we predict that these systems can support resonant transmission with a quantum of conductance. We can further expand these results to phenylene-based linear and cyclic 1D TI systems and confirm the length-dependent conductance in such systems.

Recent progress on non-Abelian anyons: from Majorana zero modes to topological Dirac fermionic modes

Majorana zero modes (MZMs) have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation (TQC). In addition to the Majorana scheme, some non-Majorana quasiparticles obeying non-Abelian statistics, including topological Dirac fermionic modes, have also been proposed and therefore become new candidates for TQC. In this review, we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes, emphasizing the recent progress on topological Dirac fermionic modes. A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry, which can be realized in a number of platforms, including the one-dimensional topological insulator, higher-order topological insulator, and spin superconductor. This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin, such as superconductivity-free and larger gaps. Therefore, it provides a new avenue for investigating non-Abelian physics and possible TQC.

Squashed entanglement in one-dimensional quantum matter

Squashed entanglement and its universal upper bound, the quantum conditional mutual information, are faithful measures of bipartite quantum correlations defined in terms of multipartitions. As such, they are sensitive to the fine-grain structure of quantum systems. Building on this observation, we introduce the concept of quantum conditional mutual information between the edges of quantum many-body systems. We show that this quantity characterizes unambiguously one-dimensional topological insulators and superconductors, being equal to Bell-state entanglement in the former and to half Bell-state entanglement in the latter, mirroring the different statistics of the edge modes in the two systems. The edge-to-edge quantum conditional mutual information is robust in the presence of disorder or local perturbations, converges exponentially with the system size to a quantized topological invariant, even in the presence of interactions, and vanishes in the trivial phase. We thus conjecture that it coincides with the edge-to-edge squashed entanglement in the entire ground-state phase diagram of symmetry-protected topological systems, and we provide some analytical evidence supporting the claim. By comparing them with the entanglement negativity, we collect further indications that the quantum conditional mutual information and the squashed entanglement provide a very accurate characterization of nonlocal correlation patterns in one-dimensional quantum matter.

Topological edge states in photonic decorated trimer lattices.

In recent years, topological insulators have been extensively studied in one-dimensional periodic systems, such as Su-Schrieffer-Heeger and trimer lattices. The remarkable feature of these one-dimensional models is that they support topological edge states, which are protected by lattice symmetry. To further study the role of lattice symmetry in one-dimensional topological insulators, here we design a modified version of the conventional trimer lattices, i.e., decorated trimer lattices. Using the femtosecond laser writing technique, we experimentally establish a series of one-dimensional photonic decorated trimer lattices with and without inversion symmetry, thereby directly observing three kinds of topological edge state. Interestingly, we demonstrate that the additional vertical intracell coupling strength in our model can change the energy band spectrum, thereby generating unconventional topological edge states with a longer localization length in another boundary. This work offers novel insight into topological insulators in one-dimensional photonic lattices.

One-dimensional noninteracting topological insulators with chiral symmetry

We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the $\mathbb{Z}$ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. The freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled -- in one case, the total winding number is given by the sum of the winding number of individual chains while in the second case, the difference is taken. We find that the chiral models that belong to $\mathbb{Z}$ classes, AIII, BDI and CII are topologically equivalent, so they can be adiabatically deformed into one another so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localised on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in $\mathbb{Z}_2$ classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We discuss applications of our results to the case of an arbitrary number of coupled chains, construct possible chiral symmetry operators for the multiple chain case, and briefly discuss the generalisation to any odd number of dimensions.

Bismuth antiphase domain wall: A three-dimensional manifestation of the Su-Schrieffer-Heeger model

The Su, Schrieffer and Heeger (SSH) model, describing the soliton excitations in polyacetylene due to the formation of antiphase domain walls (DW) from the alternating bond pattern, has served as a paradigmatic example of one-dimensional (1D) chiral topological insulators. While the SSH model has been realized in photonic and plasmonic systems, there have been limited analogues in three-dimensional (3D) electronic systems, especially regarding the formation of antiphase DWs. Here, we propose that pristine bulk Bi, in which the dimerization of $(111)$ atomic layers renders alternating covalent and van der Waals bonding within and between successive $(111)$ bilayers, respectively, serves as a 3D analogue of the SSH model. First, we confirm that the two dimerized Bi structures belong to different Zak phases of 0 and $\pi$ by considering the parity eigenvalues and Wannier charge centers, while the previously reported bulk topological phases of Bi remain invariant under the dimerization reversal. Next, we demonstrate the existence of topologically non-trivial $(111)$ and trivial $(11\bar{2})$ DWs in which the number of in-gap DW states (ignoring spin) is odd and even respectively, and show how this controls the interlinking of the Zak phases of the two adjacent domains. Finally, we derive general criteria specifying when a DW of arbitrary orientation exhibits a $\pi$ Zak phase based on the flip of parity eigenvalues. An experimental realization of dimerization in Bi and the formation of DWs may be achieved via intense femtosecond laser excitations that can alter the interatomic forces and bond lengths.

Topological Radical Pairs Produce Ultrahigh Conductance in Long Molecular Wires.

Molecular one-dimensional topological insulators (1D TIs), which conduct through energetically low-lying topological edge states, can be extremely highly conducting and exhibit a reversed conductance decay, affording them great potential as building blocks for nanoelectronic devices. However, these properties can only be observed at the short length limit. To extend the length at which these anomalous effects can be observed, we design topological oligo[n]emeraldine wires using short 1D TIs as building blocks. As the wire length increases, the number of topological states increases, enabling an increased electronic transmission along the wire; specifically, we show that we can drive over a microampere current through a single ∼5 nm molecular wire, appreciably more than what has been observed in other long wires reported to date. Calculations and experiments show that the longest oligo[7]emeraldine with doped topological states has over 106 enhancements in the transmission compared to its pristine form. The discovery of these highly conductive, long organic wires helps overcome a fundamental hurdle to implementing molecules in complex, nanoscale circuitry: their structures become too insulating at lengths that are useful in designing nanoscale circuits.

Experimental Realization of Nonreciprocal Adiabatic Transfer of Phonons in a Dynamically Modulated Nanomechanical Topological Insulator.

High quality nanomechanical oscillators are promising platforms for quantum entanglement and quantum technology with phonons. Realizing coherent transfer of phonons between distant oscillators is a key challenge in phononic quantum information processing. Here, we report on the realization of robust unidirectional adiabatic pumping of phonons in a parametrically coupled nanomechanical system engineered as a one-dimensional phononic topological insulator. By exploiting three nearly degenerate local modes-two edge states and an interface state between them-and the dynamic modulation of their mutual couplings, we achieve nonreciprocal adiabatic transfer of phononic excitations from one edge to the other with near unit fidelity. We further demonstrate the robustness of such adiabatic transfer of phonons in the presence of various noises in the control signals. Our experiment paves the way toward nonreciprocal phonon dynamics via adiabatic pumping and is valuable for phononic quantum information processing.

Quasi-one-dimensional characters in topological semimetal TaNiTe5

One-dimensional (1D) topological insulators are superior for low-dissipation applications owing to the 1D character of surface states where scatterings other than prohibited backscattering are further restricted. Among the proposed candidates for 1D topological materials, TaNiTe5 has attracted intensive attention for its quasi-one-dimensional (quasi-1D) crystalline structure. In this study, we identify the chain-like construction and anisotropic electronic states on TaNiTe5 surface with scanning tunneling microscopy. The electron scatterings are largely suppressed even with chromium impurities deposited on the surface and magnetic field applied normal to the surface, which endows TaNiTe5 great potential for low-dissipation spintronic applications.

Reversed Conductance Decay of 1D Topological Insulators by Tight-Binding Analysis.

Reversed conductance decay describes increasing conductance of a molecular chain series with increasing chain length. Realizing reversed conductance decay is an important step toward making long and highly conducting molecular wires. Recent work has shown that one-dimensional topological insulators (1D TIs) can exhibit reversed conductance decay due to their nontrivial edge states. The Su-Schrieffer-Heeger (SSH) model for 1D TIs relates to the electronic structure of these isolated molecules but not their electron transport properties as single-molecule junctions. Herein, we use a tight-binding approach to demonstrate that polyacetylene and other diradicaloid 1D TIs show a reversed conductance decay at the short chain limit. We explain these conductance trends by analyzing the impact of the edge states in these 1D systems on the single-molecule junction transmission. Additionally, we discuss how the self-energy from the electrode-molecule coupling and the on-site energy of the edge sites can be tuned to create longer wires with reversed conductance decays.

Topological edge and interface states in phoxonic crystal cavity chains

Topological states of classic waves are fascinating for both fundamental and practical purposes due to their unprecedented wave characteristics. In this paper, we realize the one-dimensional topological insulators for electromagnetic and mechanical waves simultaneously based on phoxonic crystal (PxC) cavity chains. The proposed PxC cavity chains are the analogs to the Su-Schrieffer-Heeger model in condensed-matter physics. The interaction between the neighboring PxC cavities can be described by the tight-binding model, where the coupling or hopping strength between the adjacent cavities can be tuned by varying their distance. Thus, the PxC cavity chains with topologically nontrivial and trivial phases are achieved by changing the cavity distances directly. The topological edge and interface states for the electromagnetic and elastic (or acoustic) waves are observed simultaneously in the finite-sized PxC cavity chains, which exhibit robustness against geometrical imperfections. The topological states of the proposed PxC cavity chains hold promise for designing robust devices in signal processing, sensing, and optomechanics.

Highly conducting single-molecule topological insulators based on mono- and di-radical cations.

Single-molecule topological insulators are promising candidates as conducting wires over nanometre length scales. A key advantage is their ability to exhibit quasi-metallic transport, in contrast to conjugated molecular wires which typically exhibit a low conductance that decays as the wire length increases. Here, we study a family of oligophenylene-bridged bis(triarylamines) with tunable and stable mono- or di-radicaloid character. These wires can undergo one- and two-electron chemical oxidations to the corresponding mono-cation and di-cation, respectively. We show that the oxidized wires exhibit reversed conductance decay with increasing length, consistent with the expectation for Su-Schrieffer-Heeger-type one-dimensional topological insulators. The 2.6-nm-long di-cation reported here displays a conductance greater than 0.1G0, where G0 is the conductance quantum, a factor of 5,400 greater than the neutral form. The observed conductance-length relationship is similar between the mono-cation and di-cation series. Density functional theory calculations elucidate how the frontier orbitals and delocalization of radicals facilitate the observed non-classical quasi-metallic behaviour.

Hubbard model for spin-1 Haldane chains

The Haldane phase for antiferromagnetic spin-1 chains is a celebrated topological state of matter, featuring gapped excitations and fractional spin-1/2 edge states. Here, we provide numerical evidence that this phase can be realized with a Hubbard model at half filling, where each $s=1$ spin is stored in a four-site fermionic structure. We find that the noninteracting limit of our proposed model describes a one-dimensional topological insulator, and we conjecture it to be adiabatically connected to the Haldane phase. Our work suggests a route to build spin-1 networks using Hubbard model quantum simulators.

Experimental observation of topological phase transitions in a mechanical 1D-SSH model

In this work, we report the experimental observation of the topological phase transition in a mechanical one-dimensional topological insulator using the Su-Schrieffer-Heeger (SSH) model. Our mechanical system was composed of an elastic string with metallic masses emulating atomic sites and the intra- and inter-cell interaction strengths controlled by the distances, d 1 and d 2 respectively, between the masses. We observed a trivial phase with a band gap of ∼9.2 Hz ± 0.5 Hz for d 2/d 1 = 3, while the metallic phase closed the gap, d 1 = d 2, to be reopened in the topological phase as ∼8.7 Hz ± 0.5 Hz, for d 2/d 1 = 1/3, with two edge states located inside the band gap at ∼15.6 Hz ± 0.5 Hz. Our experimental observations were supported with numerical and theoretical models. Our work creates the ability to study topological phase transitions in mechanical systems with materials available in any research lab and allows an understanding of phase transitions in a visual way.