Model Of Topological Insulator Research Articles

Non-Abelian Braiding of Topological Edge Bands.

Braiding is a geometric concept that manifests itself in a variety of scientific contexts from biology to physics, and has been employed to classify bulk band topology in topological materials. Topological edge states can also form braiding structures, as demonstrated recently in a type of topological insulators known as Möbius insulators, whose topological edge states form two braided bands exhibiting a Möbius twist. While the formation of Möbius twist is inspiring, it belongs to the simple Abelian braid group B_{2}. The most fascinating features about topological braids rely on the non-Abelianness in the higher-order braid group B_{N} (N≥3), which necessitates multiple edge bands, but so far it has not been discussed. Here, based on the gauge enriched symmetry, we develop a scheme to realize non-Abelian braiding of multiple topological edge bands. We propose tight-binding models of topological insulators that are able to generate topological edge states forming non-Abelian braiding structures. Experimental demonstrations are conducted in two acoustic crystals, which carry three and four braided acoustic edge bands, respectively. The observed braiding structure can correspond to the topological winding in the complex eigenvalue space of projective translation operator, akin to the previously established point-gap winding topology in the bulk of the Hatano-Nelson model. Our Letter also constitutes the realization of non-Abelian braiding topology on an actual crystal platform, but not based on the "virtual" synthetic dimensions.

Evolution of surface conductivity in SmB6 under nonmagnetic (Yb2+) and magnetic (Eu2+) doping

Among of rare-earth (RE) hexaborides only two compounds SmB6 and YbB6 are discussed in literature to be members of a new class of 3D topological insulators. However, their ground states originate due to different physical mechanisms, including Kondo 4f-5d hybridization and 5d-2p band inversion, respectively. Here we report a comparative study of magnetotransport (resistivity and transverse magnetoresistance) measured on high quality single crystals of YbxSm1-xB6 and EuxSm1-xB6 solid solutions (x ≤ 0.05) at temperatures 1.7 − 300 K in magnetic fields up to 82 kOe. The choice of dopant was determined by the fact that the presence of magnetic/nonmagnetic (Eu2+/Yb2+) impurity in parent SmB6 matrix should lift/not lift the topological protection of surface states. Based on the two-gap paradigm the x-evolution of electron spectra in YbxSm1-xB6 and EuxSm1-xB6 was studied. Our data show that both the 4f lattice coherence (Eg) and the intrinsic gap (Ea) related to many-body states survive under RE doping at least for x ≈ 0.02 − 0.024. We also suggest that the point x(Eu) = 0.05 can be treated as an upper limit of the small gap closing in EuxSm1-xB6 materials. In YbxSm1-xB6 family a negative linear transverse magnetoresistance (TMR) was detected for the first time in the regime of surface conductivity (T < T∗ ≈ 5 K). The TMR anomaly at T∗ caused possibly by the topological protection of surface states in SmB6 is found to survive in Eu-doped compounds but disappears almost completely for Yb-doped compositions in the same fixed magnetic fields. This paradoxical observation is not consistent with general predictions of the topological Kondo insulator (TKI) model.

High-order harmonics from elliptically polarized laser for classification of topological states in 2D Chern insulators and 2D topological insulators

High-order harmonics (HH) have drawn attention in the field of condensed matter physics mainly because of the capability of light to encode structural, dynamical, and topological information. In this paper, we address the fundamental question whether HH can map topological information in two-dimensional (2D) quantum materials by studying the interaction between topological materials and an elliptically polarized laser. We use the Haldane model for topological Chern insulators (CIs) and the Kane–Mele model for topological insulators (TIs). In the case of a circularly polarized or nearly circularly polarized driving field in CIs and TIs, the harmonic intensity of the co-rotating orders is increased. This increase in topologically non-trivial materials implies that HH can be used to detect topological transitions in 2D CIs and TIs. Moreover, interference between two spin bands in TIs does not affect the elliptical dependence of co-rotating harmonic orders in the plateau region.

Elementary models of three-dimensional topological insulators with chiral symmetry

Elementary models of three-dimensional topological insulators with chiral symmetry

Asymmetric transport computations in Dirac models of topological insulators

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme.We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover, confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.

Topological synchronization of quantum van der Pol oscillators

To observe synchronization in a large network of classical or quantum systems demands both excellent control of the interactions between the nodes and very accurate preparation of the initial conditions due to the involved nonlinearities and dissipation. This limits the applicability of this phenomenon for future devices. Here, we demonstrate a route towards significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topology. In a lattice of quantum van der Pol oscillators with topologically motivated couplings, boundary synchronization emerges in the classical mean field as well as the quantum model. In addition to its robustness against disorder and initial state perturbations, the observed dynamics is independent of the underlying topological insulator model provided the existence of zero-energy modes. Our work extends the notion of topology to the general nonlinear dynamics and open quantum system realm with applications to networks where specific nodes need special protection like power grids or quantum networks.

Quantum oscillations and three-dimensional quantum Hall effect in ZrTe5

Recent experiments have reported a lot of spectacular transport properties in topological materials, such as quantum oscillations and three-dimensional (3D) quantum Hall effect (QHE) in ${\mathrm{ZrTe}}_{5}$. In this paper, by using a strong topological insulator model to describe ${\mathrm{ZrTe}}_{5}$, we study the magnetotransport property of the 3D system. With fixed carrier density, we find that there exists a deferring effect in the chemical potential, which favors distinguishing the saddle points of the inverted LLs. On the other hand, with fixed chemical potential, the features of 3D QHE are demonstrated and we attribute the underlying mechanisms to the interplay between Dirac fermions, magnetic field, and impurity scatterings.

Higher-order topological insulators in hyperbolic lattices

To explore the non-Euclidean generalization of higher-order topological phenomena, we construct a higher-order topological insulator model in hyperbolic lattices by breaking the time-reversal symmetry (TRS) of quantum spin Hall insulators. We investigate three kinds of hyperbolic lattices, i.e., hyperbolic $\{4,5\}$, $\{8,3\}$ and $\{12,3\}$ lattices, respectively. The non-Euclidean higher-order topological behavior is characterized by zero-energy effective corner states appearing in hyperbolic lattices. By adjusting the variation period of the TRS breaking term, we obtain 4, 8 and 12 zero-energy effective corner states in these three different hyperbolic lattices, respectively. It is found that the number of zero-energy effective corner states of a hyperbolic lattice depends on the variation period of the TRS breaking term. The real-space quadrupole moment is employed to characterize the higher-order topology of the hyperbolic lattice with four zero-energy effective corner states. Via symmetry analysis, it is confirmed that the hyperbolic zero-energy effective corner states are protected by the particle-hole symmetry $P$, the effective chiral symmetry $Sm_{z}$, and combined symmetries $C_{p}T$ and $C_{p}m_{z}$. The hyperbolic zero-energy effective corner states remain stable unless these four symmetries are broken simultaneously. The topological nature of hyperbolic zero-energy effective corner states is further confirmed by checking the robustness of the zero-energy modes in the hyperbolic lattices in the presence of disorder. Our paper provides a route for research on hyperbolic higher-order topological insulators in non-Euclidean geometric systems.

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.

Kollektivnye i kvazilokal'nye mody v opticheskikh spektrakh geksaboridov YB6 i YbB6 s yan-tellerovskoy strukturnoy neustoychivost'yu

The broad-band reflection spectra of YB6 and YbB6 hexaborides with Jahn–Teller instability of the boron cage have been measured at room temperature. An optical conductivity analysis has revealed, along with the Drude electronic components, heavily overdamped collective modes, which are notable in YB6 for high dielectric contributions, Δε = 2000–5700. The fraction of nonequilibrium charge carriers in YB6, which is at the boundary of structural instability in the hexaboride family, reaches 85–90%, whereas this fraction in doped YbB6 semiconductor is not higher than 25%. It has been shown that unlike the predictions of the topological Kondo insulator model, the surface “metallization” in Yb2+B6 crystals can be explained by additional doping of a surface layer with Yb3+ ions.

Non-Hermitian kagome photonic crystal with a totally topological spatial mode selection.

Recently, the study of non-Hermitian topological edge and corner states in sonic crystals (SCs) and photonic crystals (PCs) has drawn much attention. In this paper, we propose a Wannier-type higher-order topological insulator (HOTI) model based on the kagome PC containing dimer units and study its non-Hermitian topological corner states. When balanced gain and loss are introduced into the dimer units with a proper parity-time symmetric setting, the system will show asymmetric Wannier bands and can support two Hermitian corner states and two pairs of complex-conjugate or pseudo complex-conjugate non-Hermitian corner states. These topological corner states are solely confined at three corners of the triangular supercell constructed by the trivial and non-trivial kagome PCs, corresponding to a topological spatial mode selection effect. As compared to the non-Hermitian quadrupole-type HOTIs, the non-Hermitian Wannier-type HOTIs can realize totally topological spatial mode selection by using much lower coefficients of gain and loss. Our results pave the way for the development of novel non-Hermitian photonic topological devices based on Wannier-type HOTIs.

Analytical solutions of topological surface states in a series of lattice models

We derive the analytical solutions of surface states in a series of lattice models for three-dimensional topological insulators and their nontopological counterparts based on an ansatz. A restriction on the spin-flip matrices in nearest-neighbor hopping characterizes the series. This restriction affords the ansatz and favors analytical solvability of surface-state eigenvectors. Despite the restriction, the series retains sufficient designability to describe various types of surface states. We also describe how it can serve as a tractable tool for elucidating unique phenomena on topological surfaces.

Hyperbolic Topological Band Insulators.

Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their nontrivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modeling propagation of edge excitations, and by their robustness against disorder.

Unconventional optical selection rules in ZrTe5 under an in-plane magnetic field

The optical selection rules of an electron system under a magnetic field play key roles in determining its optical properties, from which the band structures and underlying symmetries can be derived. In this Letter, based on a three-dimensional strong topological insulator model describing ZrTe5,we study the Landau levels (LLs) and magneto-optical conductivity under an in-plane magnetic field. We reveal that in the transverse conductivity Re(\sigma_{zz}), the unconventional optical selection rules n\righatarrow n\pm 2 dominate, with n being the LL index. We attribute the unconventional selection rules to the peculiar distribution of parity carried by the LLs, resulting from the chiral symmetry of the sub-Hamiltonians. Moreover, we predict that, if the strong anisotropic system is tuned to be nearly isotropic, the LLs would redistribute and the conventional selection rules n\rightarrow n\pm 1 can be recovered.

Error-correction properties of an interacting topological insulator

We analyze the phase diagram of a topological insulator model including antiferromagnetic interactions in the form of an extended Su-Schrieffer Heeger model. To this end, we employ a recently introduced operational definition of topological order based on the ability of a system to perform topological error correction. We show that the necessary error correction statistics can be obtained efficiently using a Monte-Carlo sampling of a matrix product state representation of the ground state wave function. Specifically, we identify two distinct symmetry-protected topological phases corresponding to two different fully dimerized reference states. Finally, we extend the notion of error correction to classify thermodynamic phases to those exhibiting local order parameters, finding a topologically trivial antiferromagnetic phase for sufficiently strong interactions.

Spin-orbit coupling in transition metal dichalcogenide heterobilayer flat bands

The valence flat bands in transition metal dichalcogenide (TMD) heterobilayers are shown to exhibit strong intralayer spin-orbit coupling. I show that symmetry constrains the spin-dependent complex phase of hopping terms in an effective tight-binding model of the valence flat bands. A perpendicular electric field causes interlayer hybridization, such that the effective model becomes equivalent to the Kane-Mele model of topological insulators. The proposed model can be used as a starting point to understand interactions and the experimentally observed topological transitions.

Large-S and Tensor-Network Methods for Strongly-Interacting Topological Insulators

The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.

Symmetry-allowed nonlinear orbital response across the topological phase transition in centrosymmetric materials

Nonlinear optical responses are commonly used as a probe for studying the electronic properties of materials. For topological materials, studies thus far focused on photogalvanic electric currents, which are forbidden in centrosymmetric materials because they require broken inversion symmetry. In this Letter, we propose a class of symmetry-allowed responses for inversion-symmetric topological insulators with two doubly degenerate bands. We consider a specific example of such a response, the orbital current, and show that the sign of the response reflects the ${\mathbb{Z}}_{2}$ topological index, i.e., the orbital current changes sign at the transition between trivial and topological insulator phases. This is illustrated in two models of topological insulators: the Bernevig-Hughes-Zhang model and the $1{T}^{\ensuremath{'}}$ phase of transition metal dichalcogenides.

Long-range hopping and indexing assumption in one-dimensional topological insulators

In this paper, we show that the introduction of long-range hoppings in 1D topological insulator models implies that different possibilities of site indexing must be considered when determining the bulk topological invariants in order to avoid the existence of hidden symmetries. The particular case of the extended SSH chain is addressed as an example where such behavior occurs. In this model, the introduction of long-range hopping terms breaks the bipartite property and a band inversion occurs in the band structure as the relative values of the hopping terms change, signaling a crossover between hopping parameter regions of "influence" of different chiral symmetries. Furthermore, edge states become a linear combination of edge-like states with different localization lengths and reflect the gradual transition between these different chiral symmetries.

Asymmetric Transport Computations in Dirac Models of Topological Insulators

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme. We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.