Protected Edge Modes Research Articles

Peierls-Nabarro barrier effect in nonlinear Floquet topological insulators

The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically driven waveguides, which serves as a paradigmatic Floquet topological insulator. Here it is shown that discrete topologically protected edge modes do not suffer from the typical slowdown associated with the Peierls-Nabarro barrier. Instead, as a result of their topological nature, the modes always move forward and redistribute their energy: a narrow (discrete) mode transforms into a wide effectively continuous mode. On the other hand, a discrete edge mode that is not topologically protected does eventually slow down and stop propagating. Topological modes that are initially narrow naturally tend to wide envelope states that are described by a generalized nonlinear Schrödinger equation. These results provide insight into the nature of nonlinear topological insulators and their application.

Fractional edge reconstruction in integer quantum Hall phases

Protected edge modes are the cornerstone of topological states of matter. The simplest example is provided by the integer quantum Hall state at Landau level filling unity, which should feature a single chiral mode carrying electronic excitations. In the presence of a smooth confining potential it was hitherto believed that this picture may only be partially modified by the appearance of additional counterpropagating integer-charge modes. Here, we demonstrate the breakdown of this paradigm: The system favors the formation of edge modes supporting fractional excitations. This accounts for previous observations, and leads to additional predictions amenable to experimental tests.

Conductance in a Nanoribbon of Topologically Insulating MoS2 in the 1T’ Phase

The use of new materials with advanced properties has become mandatory to meet the needs for higher electronics performance at reduced power. Topological insulators (TIs) possess highly conductive topologically protected edge states which are insensitive to scattering and thus suitable for energy-efficient high-speed devices. Here, we evaluate the subband structure in a narrow nanoribbon of 1 $\text {T}^\prime$ molybdenum disulphide using an effective k.p Hamiltonian. Highly conductive topologically protected edge modes whose energies lie within the bulk band gap are investigated on equal footing with traditional electron and hole subbands. Due to the interaction between the edge modes at opposite sides, a small gap in their linear spectrum opens up in a narrow nanoribbon. This gap is shown to sharply increase with the perpendicular out-of-plane electric field, in contrast to the behavior in a wide nanoribbon with negligible edge modes’ interaction. This increase leads to a rapid decrease in the ballistic nanoribbon conductance and current with the electric field, which can be used for designing molybdenum disulphide nanoribbon-based current switches.

Topologically protected edge modes in one-dimensional chains of subwavelength resonators

The goal of this paper is to advance the development of wave-guiding subwavelength crystals by developing designs whose properties are stable with respect to imperfections in their construction. In particular, we make use of a locally resonant subwavelength structure, composed of a chain of high-contrast resonators, to trap waves at deep subwavelength scales. We first study an infinite chain of subwavelength resonator dimers and define topological quantities that capture the structure's wave transmission properties. Using this for guidance, we design a finite crystal that is shown to have wave localization properties, at subwavelength scales, that are robust with respect to random imperfections.

Inaccessible entanglement in symmetry protected topological phases

We study the entanglement structure of symmetry-protected topological (SPT) phases from an operational point of view by considering entanglement distillation in the presence of symmetries. We demonstrate that non-trivial SPT phases in one-dimension necessarily contain some entanglement which is inaccessible if the symmetry is enforced. More precisely, we consider the setting of local operations and classical communication (LOCC) where the local operations commute with a global onsite symmetry group G, which we call G-LOCC, and we define the inaccessible entanglement Einacc as the entanglement that cannot be used for distillation under G-LOCC. We derive a tight bound on Einacc which demonstrates a direct relation between inaccessible entanglement and the SPT phase, namely , where Dω is the topologically protected edge mode degeneracy of the SPT phase ω with symmetry G. For particular phases such as the Haldane phase, so the bound becomes an equality. We numerically investigate the distribution of states throughout the bound, and show that typically the region near the upper bound is highly populated, and also determine the nature of those states lying on the upper and lower bounds. We then discuss the relation of Einacc to string order parameters, and also the extent to which it can be used to distinguish different SPT phases of matter.

Tailoring and probing the quantum states of matter of 2D Dirac materials with a buckled honeycomb structure

The quantum state of matter of a two-dimensional Dirac material with a buckled honeycomb structure can be tuned by an electric field. In the absence of an external electric field the material is a topological insulator owing to the spin-orbit coupling, which opens a band gap at the K and K ’ points of the Brillouin zone. The size of this band gap decreases with increasing electric field until eventually the band gap completely closes at a critical electric field E c and the material becomes a semi-metal. For electric fields exceeding E c the band gaps reopens again and the material undergoes a topological phase transition from a semi-metal to a normal band insulator. The electric field in a tunnel junction depends on the applied voltage bias across the junction as well as the difference in work function of the two electrodes. Here we show how scanning tunneling microscopy can be employed to simultaneously apply an electric field and study the electronic structure of a two-dimensional Dirac material with a buckled honeycomb structure. The electric field applied by the scanning tunneling microscope offers the possibility to locally alter the quantum state of matter of two-dimensional topological insulator to a semi-metal or normal band insulator. This results in the development of topologically protected spin polarized edge states within the material. We present a spectroscopic method to probe these topologically protected edge states. • Electric field induces topological phase transitions in 2D materials. •A method to probe topologically protected edge modes in a 2D Dirac material is proposed. • The band gap in a 2D Dirac material with a buckled honeycomb lattice can be tuned by an electric field.

Discrete approximation of topologically protected modes in magneto-optical media

Topologically protected waves in the linearly polarized Maxwell's equations with gyrotropic, magneto-optic media were studied a decade ago both computationally and experimentally. This paper develops a robust tight-binding model for this system that makes careful use of Wannier function representations. The model provides very good approximations to the underlying band structure. When solved on a semi-infinite strip, it produces exponentially localized edge modes whose corresponding eigenvalues span the frequency band gaps. A set of coupled differential equations are derived which allows one to find how the electromagnetic field propagates unidirectionally, without backscatter from defects. Furthermore, the discrete model predicts topologically protected edge modes with nontrivial Chern number which are consistent with direct simulation.

Interacting symmetry-protected topological phases out of equilibrium

The topological features of quantum many-body wave functions are known to have profound consequences for the physics of ground-states and their low-energy excitations. We describe how topology influences the dynamics of many-body systems when driven far from equilibrium. Our results are succinctly captured by a nonequilibrium topological classification that can be used to predict universal aspects of generic isolated quantum systems as they evolve unitarily in time. By analogy to the classifications used to describe systems in equilibrium, we consider two short-ranged entangled wave functions to be topologically equivalent if they can be interconverted via finite-time unitary evolution governed by a symmetry-respecting Hamiltonian. We demonstrate that this definition captures the salient features of these systems in a broad range of nonequilibrium scenarios. As well as providing conceptual insights into the constraints imposed by topology on many-body dynamics, we discuss the practical implications of our findings. In particular, we show that the characteristic zero-frequency spectroscopic peaks associated with topologically protected edge modes will be broadened by external noise only when the system is trivial in the nonequilibrium classification.

Actively controllable topological phase transition in homogeneous piezoelectric rod system

By employing periodic electrical boundary conditions, an innovative method to generate actively tunable topologically protected interface mode in a “homogeneous” piezoelectric rod system is proposed. Made of homogeneous material and with uniform structure, each unit cell of the piezoelectric rod system consists of three sub-rods forming an A-B-A structure, where the two A sub-rods have the same geometry and electric boundary conditions. It is discovered that the switch of electrical boundary conditions from A-closed and B-open to A-open and B-closed will yield topological phase inversion, based on which topologically protected interface mode is realized. When capacitors CA and CB are connected to the electrodes of sub-rods A and B, respectively, a variety of physical phenomena is observed. On one hand, varying the capacitance in a certain path leads to topological phase transition. On the other hand, different variation paths of the capacitors give rise to different locations of topological phase transition points. This discovery allows the eigenfrequency of the topologically protected edge mode thus formed be actively controlled by appropriately varying the capacitance. The active topological protected interface mode may find wide engineering applications that require high sensitivity sensing, nondestructive testing, reinforcing energy harvesting, information processing, and others.

Many-body localization induced protection of topological order in a XXZ spin model

There is a counter-intuitive expectation proposed by Huse {\it et al} [Phys. Rev. B {\bf 88}, 014206 (2013)], and Chandran {\it et al} [Phys. Rev. B 89,144201 (2014)]: Localization protects quantum order of groundstate even in high excited eigenstates. In this work, we numerically investigate the localization protection for symmetry-protected-topological (SPT) order by considering a modified XXZ spin model, related to an interacting Su-Schrieffer-Heeger (SSH) model. We systematically study how Many-body-localization (MBL) protects SPT for different disorder types and different forms of interactions. A certain disorder leads the clear degeneracy of the low-lying entanglement spectrum of each excited many-body eigenstates. This fact indicates that existence of edge modes, which is a hallmark of the SPT order, is protected by the MBL in excited many-body eigenstates. In addition, we also report how the MBL protected edge modes in excited many-body eigenstates fade away for a diagonal type disorder.

Topological kink plasmons on magnetic-domain boundaries

Two-dimensional topological materials bearing time reversal-breaking magnetic fields support protected one-way edge modes. Normally, these edge modes adhere to physical edges where material properties change abruptly. However, even in homogeneous materials, topology still permits a unique form of edge modes – kink modes – residing at the domain boundaries of magnetic fields within the materials. This scenario, despite being predicted in theory, has rarely been demonstrated experimentally. Here, we report our observation of topologically-protected high-frequency kink modes – kink magnetoplasmons (KMPs) – in a GaAs/AlGaAs two-dimensional electron gas (2DEG) system. These KMPs arise at a domain boundary projected from an externally-patterned magnetic field onto a uniform 2DEG. They propagate unidirectionally along the boundary, protected by a difference of gap Chern numbers (pm1) in the two domains. They exhibit large tunability under an applied magnetic field or gate voltage, and clear signatures of nonreciprocity even under weak-coupling to evanescent photons.

Inverse design of discrete mechanical metamaterials

Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete phononic metamaterial structures. The underlying algorithm optimizes the linear response of elastic networks directly, is applicable to ordered and disordered structures, scales efficiently in 2D and 3D, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical bandgap switches that open and close pre-programmed spectral gaps in response to an externally applied stimulus such as shear or compression. We further show that the designed structures can host topologically protected edge modes, and validate the numerical predictions through explicit 3D finite element simulations of continuum elastica with experimentally relevant material parameters. Generally, this network-based inverse design paradigm offers a direct pathway towards manufacturing phononic metamaterials, DNA origami structures and topolectric circuits that can realize a wide range of static and dynamic target functionalities.

Topological proximity effects in a Haldane graphene bilayer system

We reveal a proximity effect between a topological band (Chern) insulator described by a Haldane model and spin-polarized Dirac particles of a graphene layer. Coupling weakly the two systems through a tunneling term in the bulk, the topological Chern insulator induces a gap and an opposite Chern number on the Dirac particles at half-filling resulting in a sign flip of the Berry curvature at one Dirac point. We study different aspects of the bulk-edge correspondence and present protocols to observe the evolution of the Berry curvature as well as two counter-propagating (protected) edge modes with different velocities. In the strong-coupling limit, the energy spectrum shows flat bands. Therefore we build a perturbation theory and address further the bulk-edge correspondence. We also show the occurrence of a topological insulating phase with Chern number one when only the lowest band is filled. We generalize the effect to Haldane bilayer systems with asymmetric Semenoff masses. We propose an alternative definition of the topological invariant on the Bloch sphere.

Bulk-edge correspondence and long-range hopping in the topological plasmonic chain

Abstract The existence of topologically protected edge modes is often cited as a highly desirable trait of topological insulators. However, these edge states are not always present. A realistic physical treatment of long-range hopping in a one-dimensional dipolar system can break the symmetry that protects the edge modes without affecting the bulk topological number, leading to a breakdown in bulk-edge correspondence (BEC). Hence, it is important to gain a better understanding of where and how this occurs, as well as how to measure it. Here we examine the behaviour of the bulk and edge modes in a dimerised chain of metallic nanoparticles and in a simpler non-Hermitian next-nearest-neighbour model to provide some insights into the phenomena of bulk-edge breakdown. We construct BEC phase diagrams for the simpler case and use these ideas to devise a measure of symmetry-breaking for the plasmonic system based on its bulk properties. This provides a parameter regime in which BEC is preserved in the topological plasmonic chain, as well as a framework for assessing this phenomenon in other systems.

Photonic topological phase transition on demand

Abstract On-demand, switchable phase transitions between topologically nontrivial and trivial photonic states are demonstrated. Specifically, it is shown that the integration of a two-dimensional array of coupled ring resonators within a thermal heater array enables unparalleled control over topological protection of photonic modes. Importantly, auxiliary control over a spatial phase modulation opens up a way to guide topologically protected edge modes along generated virtual boundaries. The proposed approach can lead to practical realizations of topological phase transitions in many photonic applications, including topologically protected photonic memory/logic devices, robust optical modulators, and switches.

Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study

Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the Fractional Quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by Exact Diagonalisation of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral Projected Entangled Pair States (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with Density Matrix Renormalisation Group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range "gossamer tails". We argue these tails are finite-$D$ artifacts of the chiral PEPS, which become irrelevant when the PEPS bond dimension $D$ is increased. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)$_2$ Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge $c=3/2$, as expected for a Moore-Read chiral spin liquid. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.

Topological phonon polaritons in one-dimensional non-Hermitian silicon carbide nanoparticle chains

Topological phonon polaritons (TPhPs) are highly protected and localized edge modes that are capable of achieving a strong confinement of electromagnetic waves and immune to impurities and disorder. Here we realize TPhPs by constructing one-dimensional dimerized silicon carbide nanoparticle chains, which mimic the topological property of the well-known Su-Schrieffer-Heeger (SSH) model. We analytically calculate the complex band structure of such chains by taking all near-field and far-field dipole-dipole interactions into account. For longitudinal modes, we demonstrate that, despite the non-Hermiticity and breaking of the chiral symmetry, the band topology can be still characterized by the complex Zak phase, which is quantized and indicates a topological phase transition when the dimerization parameter $\beta$ changes from less than 0.5 to larger than 0.5, like the conventional Hermitian SSH model. By calculating the eigenmodes of a finite chain, we find such a dimerized chain with $\beta>0.5$ supports nontrivial topological eigenmodes localized over both of its edges, indicating the validity of the bulk-boundary correspondence. On the other hand, for transverse modes, we discover a topological phase transition by increasing the lattice constant, which is due to the presence of strong long-range far-field transverse dipole-dipole interactions decaying with the distance $r$ as $1/r$ for an infinitely long chain. However, we surprisingly find the emergence of non-Hermitian skin effect in a finite chain, which leads to the breakdown of the bulk-boundary correspondence. Furthermore, by incorporating the effect of localized bulk eigenmodes and proposing a modified complex Zak phase for a finite lattice, we still recover the topological behavior of the conventional SSH model. We also demonstrate the excitation of topological phonon polaritons and show their enhancement to the photonic LDOS.

Quantum computation via Floquet topological edge modes

Floquet topological matter has emerged as one exciting platform to explore rich physics and game-changing applications of topological phases. As one remarkable and recently discovered feature of Floquet symmetry protected topological (SPT) phases, in principle a simple periodically driven system can host an arbitrary number of topological protected zero edge modes and pi edge modes, with Majorana zero modes and Majorana pi modes as examples protected by the particle-hole symmetry. This work advocates a new route to holonomic quantum computation by exploiting the co-existence of many Floquet SPT edge modes, all of which have trivial dynamical phases during a computation protocol. As compelling evidence supporting this ambitious goal, three pairs of Majorana edge modes, hosted by a periodically driven one-dimensional (1D) superconducting superlattice, are shown to suffice to encode two logical qubits, realize quantum gate operations, and execute two simple quantum algorithms through adiabatic lattice deformation. When compared with early studies on quantum computation based on Majorana zero modes of topological quantum wires, significant resource saving is now made possible by use of Floquet SPT phases. This paper is thus hoped to motivate a series of future studies on the potential of Floquet topological matter in quantum computation.

Chiral flow in one-dimensional Floquet topological insulators

We propose a bulk topological invariant for one-dimensional Floquet systems with chiral symmetry which quantifies the particle transport on each sublattice during the evolution. This chiral flow is physically motivated, locally computable, and improves on existing topological invariants by being applicable to systems with disorder. We derive a bulk-edge correspondence which relates the chiral flow to the number of protected dynamical edge modes present on a boundary at the end of the evolution. In the process, we introduce two real-space edge invariants which classify the dynamical topological boundary behavior at various points during the evolution. Our results provide an explicit bulk-boundary correspondence for Floquet systems in this symmetry class.

Criteria for protected edge modes with Z2 symmetry

We derive a necessary and sufficient criterion for when a two dimensional gapped many-body system with Abelian anyons and a unitary $\mathbb{Z}_2$ symmetry has a protected gapless edge mode. Our criterion is phrased in terms of edge theories --- or more specifically, chiral boson edge theories with $\mathbb{Z}_2$ symmetry --- and it applies to any bosonic or fermionic system whose boundary can be described by such an edge theory. At an operational level, our criterion takes as input a chiral boson edge theory with $\mathbb{Z}_2$ symmetry, and then produces as output a prediction as to whether this edge theory can be gapped without breaking the symmetry. Like previous work, much of our derivation involves constructing explicit perturbations that gap chiral boson edge theories. Interestingly, however, we find that the standard class of gapping perturbations --- namely cosine terms constructed from null-vectors --- is not sufficient to gap some edge theories with $\mathbb{Z}_2$ symmetry, and thus we are forced to go beyond the usual null-vector analysis to establish our results.