Class Of Topological Insulators Research Articles

Observation of Boundary‐Obstructed Topological Insulators in 2D Asymmetric Su–Schrieffer–Heeger Topological Electric Circuits

Topological insulators that support unique symmetry‐protected boundary states gain popularity in the field of electric circuit systems. Boundary‐obstructed topological (BOT) insulators, as a new class of topological insulators, do not utilize the closure of the bulk band during the topological phase transition, which displays as an obstruction of boundary states. In this work, the experimental realization of BOT insulators based on electric circuit systems are reported and it is further demonstrated experimentally that the corresponding hollow structure consisting of only edges and corners can offer corner states in the edge gap. In this research, a highly controllable platform is offered for investigating unique topological phenomena and the way is paved for observing boundary‐obstructed topology by electric circuit systems.

Electronic Band Structures of the Possible Topological Insulator Pb2BiBrO6 and Pb2SeTeO6 Double Perovskite: An Ab Initio Study

Using the frameworks of density functional theory, we found a new class of three-dimensional (3D) topological insulators (TIs) in Pb2BiBrO6 and Pb2SeTeO6 double perovskites. Our ab initio theoretical calculations show that Pb2BiBrO6 and Pb2SeTeO6 are Z2 nontrivial, and their bandgaps are 0.390 eV and 0.181 eV, respectively. The topology comes from two mechanisms. Firstly, the band inversion occurs at Γ point in the absence of spin-orbit interactions and secondly, the bandgap is induced by the SOC. This results in a larger bandgap for this new class of topological insulators than conventional TI. In Pb2BiBrO6 double perovskites, our slab calculations confirm that the topology-protected surface metallic bands come from the BiBrO4 surface which means that one can build a transport device using Pb2BiBrO6 double perovskites with a PbO layer as an outmost protection layer. The mechanical stabilities such as bulk, shear, Young’s moduli, Poisson’s and Pugh’s ratio, longitudinal, transverse, and average sound velocity, together with Debye temperature are also studied. Our results show that these Pb2AA’O6 (A = Sb and Bi; A’ = Br and I) and Pb2SeTeO6 are mechanically stable.

Symmetric Fermi projections and Kitaev’s table: Topological phases of matter in low dimensions

We review Kitaev’s celebrated “periodic table” for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections that depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for zero- and one-dimensional systems and recover the (weak and strong) topological invariants proposed by Kitaev and others.

Discrete-Time Quantum-Walk & Floquet Topological Insulators via Distance-Selective Rydberg-Interaction

This article proposes the first discrete-time implementation of Rydberg quantum walk in multi-dimensional spatial space that could ideally simulate different classes of topological insulators. Using distance-selective exchange-interaction between Rydberg excited atoms in an atomic-array with dual lattice-constants, the new setup operates both coined and coin-less models of discrete-time quantum walk (DTQW). Here, complicated coupling tessellations are performed by global laser that exclusively excite the site at the anti-blockade region. The long-range interaction provides a new feature of designing different topologically ordered periodic boundary conditions. Limiting the Rydberg population to two excitations, coherent QW over hundreds of lattice sites and steps are achievable with the current technology. These features would improve the performance of this quantum machine in running the quantum search algorithm over topologically ordered databases as well as diversifying the range of topological insulators that could be simulated.

Takagi topological insulator with odd PT pairs of corner states

We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for the TTIs can be realized on any bipartite lattice where the inversion exchanges sublattices. The protecting symmetries lead to the classifying space of Hamiltonians being unitary symmetric matrices, and therefore Takagi's factorization can be performed. Particularly, the global Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D, there is a $\mathbb{Z}_2$ topological invariant corresponding to the parity of the winding number of Takagi's unitary-matrix factor over the entire Brillouin zone, where the $\mathbb Z_2$ nature comes from the $O(N)$ gauge degrees of freedom in Takagi's factorization. In 2D, the obstruction for a global Takagi's factorization is characterized by another $\mathbb{Z}_2$ topological invariant, equivalent to the second Stiefel-Whitney number. For the third-order topological phases, the $3$D TTIs feature a parity condition for corner zero-modes, i.e., there always exist odd $\mathcal P\mathcal T$ pairs of corners with zero-modes. Moreover, for any $\mathcal P\mathcal T$ invariant sample geometry, all configurations of corner zero-modes satisfying the parity condition can exist with the same nontrivial bulk topological invariant. Actually, without closing the bulk gap, the boundary phase diagram have a cellular structure, where each topological boundary phase associated with a particular (cross-order) boundary-mode pattern corresponds to a contractible cell with certain dimension in the parameter space.

Realization of quasicrystalline quadrupole topological insulators in electrical circuits

Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical circuits, and observe the spectrally and spatially localized corner modes. In measurement, the modes appear as topological boundary resonances in the corner impedance spectra. Additionally, we demonstrate the robustness of corner modes on the circuit. Our circuit design may be extended to study topological phases in higher-dimensional aperiodic structures.

Three-dimensional higher-order topological acoustic system with multidimensional topological states

Topologically protected gapless edge/surface states are phases of quantum matter which behave as massless Dirac fermions, immunizing against disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with gapped edge states and in-gap corner states have been theoretically predicted in electric systems 1,2, and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems 3,4, electrical circuits 5, optical and sonic crystals 6-11, and elastic phononic plates 12. Here, we elaborately design a strong three-dimensional (3D) topological acoustic system, by arranging acoustic meta-atoms in a simple cubic lattice. Under the direct field measurements, besides of the 2D surface propagations on all of the six surfaces, the 1D hinge propagations behaving as robust acoustic fibers along the twelve hinges and the 0D corner modes working as robust localized resonances at the eight corners are experimentally confirmed. As these multidimensional topological states are activated in different frequencies and independent spaces, our works pave feasible ways for applications in the topological acoustic cavities, communications and signal-processing.

Elastic phononic topological plate with edge and corner sates based on pseudospin-valley-coupling

Topologically protected gapless edge states are phases of quantum matter which behave as massless Dirac fermions, immunizing against disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with topological corner states have been theoretically predicted in electric systems, and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems, electrical circuits, optical and sonic crystals, and elastic phononic plates. Here, we demonstrate a pseudospin-valley-coupled phononic TI, which simultaneously exhibits gapped edge states and topological corner states. Pseudospin-orbit coupling edge states and valley-polarized edge state are respectively induced by the lattice deformation and the symmetry breaking. When both of them coexist, these topological edge states will be greatly gapped and the topological corner state emerges. Under direct field measurements, the robust edge propagation behaving as an elastic waveguide and the topological corner mode working as a robust localized resonance are experimentally confirmed. The pseudospin-valley coupling in our phononic TIs can be well-controlled which provides a reconfigurable platform for the multiple edge and corner states, and exhibits well applications in the topological elastic energy recovery and the highly sensitive sensing.

Appearance of hinge states in second-order topological insulators via the cutting procedure

In recent years, second-order topological insulators have been proposed as a new class of topological insulators. Second-order topological insulators are materials with gapped bulk and surfaces, but with topologically protected gapless states at the intersection of two surfaces. These gapless states are called hinge states. In this paper, we give a general proof that any insulators with inversion symmetry and gapped surface in class A always have hinge states when the $\mathbb{Z}_{4}$ topological index $\mu_{1}$ is $\mu_{1}=2$. We consider a three-dimensional insulator whose boundary conditions along two directions change by changing the hopping amplitudes across the boundaries. We study behaviors of gapless states through continuously changing boundary conditions along the two directions, and reveal that the behaviors of gapless states result from the $\mathbb{Z}_{4}$ strong topological index. From this discussion, we show that gapless states inevitably appear at the hinge of a three-dimensional insulator with gapped surfaces when the strong topological index is $\mathbb{Z}_{4}=2$ and the weak topological indices are $\nu_{1}=\nu_{2}=\nu_{3}=0$.

Classification of topological insulators and superconductors out of equilibrium

We establish the existence of a topological classification of many-particle quantum systems undergoing unitary time evolution. The classification naturally inherits phenomenology familiar from equilibrium -- it is robust against disorder and interactions, and exhibits a non-equilibrium bulk-boundary correspondence, which connects bulk topological properties to the entanglement spectrum. We explicitly construct a non-equilibrium classification of non-interacting fermionic systems with non-spatial symmetries in all spatial dimensions (the `ten-fold way'), which differs from its equilibrium counterpart. Direct physical consequences of our classification are discussed, including important ramifications for the use of topological zero-energy bound states in quantum information technologies.

Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology

We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic nonmagnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead of the $K$-theory in the momentum space, both of which give the same topological classification. We apply the Atiyah-Hirzebruch spectral sequence (AHSS) for computation of the $K$-homology, which is a mathematical tool for generalized (co)homology. In the real-space picture, the AHSS naturally gives the classification of higher-order topological insulators at the same time. By solving the group extension problem in the AHSS on the basis of physical arguments, we completely determine possible topological phases including higher-order ones for each point group. Relationships among different higher-order topological phases are argued in terms of the AHSS in the $K$-homology. We find that in some nonmagnetic and magnetic point groups, a stack of two ${\mathbb{Z}}_{2}$ second-order topological insulators can be smoothly deformed into nontrivial fourth-order topological insulators, which implies nontrivial group extensions in the AHSS.

Measurement of the atomic orbital composition of the near-fermi-level electronic states in the lanthanum monopnictides LaBi, LaSb, and LaAs

Recent debates in the literature over the relationship between topology and Extreme Magnetoresistance (XMR) have drawn attention to the Lanthanum Monopnictide family of binary compounds. Angle resolved photoemission spectroscopy (ARPES) is used to measure the electronic structure of the XMR topological semimetal candidates LaBi, LaSb, and LaAs. The orbital content of the near-EF states in LaBi and LaSb are extracted using varying photon polarizations and both dominant d and p bands are observed near X. The measured bulk bands are shifted in energy when compared to the results of Density Functional Calculations. This disagreement is minor in LaBi, but large in LaSb and LaAs. The measured bulk band structure of LaBi shows a clear band inversion and puts LaBi in the υ = 1 class of Topological Insulators (or semimetals), as predicted by calculations and consistent with the measured Dirac-like surface states. LaSb is on the verge of a band inversion with a less-clear case for any distinctly topological surface states and in disagreement with calculations. Lastly, these same bands in LaAs are clearly non-inverted implying its topological triviality and demonstrating a topological phase transition in the Lanthanum monopnictides. Using a wide range of photon energies the true bulk states are cleanly disentangled from the various types of surface states which are present. These surface states exist due to surface projections of bulk states in LaSb and for topological reasons in LaBi.

Index of Dirac operators and classification of topological insulators

Real and complex Clifford bundles and Dirac operators defined on them are considered. By using the index theorems of Dirac operators, table of topological invariants is constructed from the Clifford chessboard. Through the relations between K-theory groups, Grothendieck groups and symmetric spaces, the periodic table of topological insulators and superconductors is obtained. This gives the result that the periodic table of real and complex topological phases is originated from the Clifford chessboard and index theorems.

Theoretical construction of weak topological crystalline insulators

Inspired by the discovery of topological crystalline insulators (TCIs) in three-dimensional materials such as Pb[Formula: see text]Sn[Formula: see text]Se(Te), the classification of topological insulators has been extended to other discrete symmetry classes such as crystal point group symmetries. In this paper, we construct and study a simple model of weak TCIs, which will serve as a more viable project in the experimental probe for such new type of topological phases.

Plasmonics of topological insulators at optical frequencies

The development of nanoplasmonic devices, such as plasmonic circuits and metamaterial superlenses in the visible to ultraviolet frequency range, is hampered by the lack of low-loss plasmonic media. Recently, strong plasmonic response was reported in a certain class of topological insulators. Here, we present a first-principles density functional theory analysis of the dielectric functions of topologically insulating quaternary (Bi,Sb)2(Te,Se)3 trichalcogenide compounds. Bulk plasmonic properties, dominated by interband transitions, are observed from 2 to 3 eV and extend to higher frequencies. Moreover, trichalcogenide compounds are better plasmonic media than gold and silver at blue and UV wavelengths. By analyzing thin slabs, we also show that these materials exhibit topologically protected surface states, which are capable of supporting propagating plasmon polariton modes over an extremely broad spectral range, from the visible to the mid-infrared and beyond, owing to a combination of inter- and intra-surface band transitions.

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator**This work was supported by the grants from the Ministry of Science and Technology of China and the Ministry of Education. L.M.D. and S.T.W. acknowledge in addition support from the ARL and the AFOSR MURI programs. D.L.D. is supported by JQI-NSF-PFC and LPS-MPO-CMTC.

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including nontrivial topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.

Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

We consider the effects of impurities on topological insulators and superconductors. We start by identifying the general conditions under which the eigenenergies of an arbitrary Hamiltonian H belonging to one of the Altland-Zirnbauer symmetry classes undergo a robust zero energy crossing as a function of an external parameter which can be, for example, the impurity strength. We define a generalized root of \det H, and use it to predict or rule out robust zero-energy crossings in all symmetry classes. We complement this result with an analysis based on almost degenerate perturbation theory, which allows a derivation of the asymptotic low-energy behavior of the ensemble averaged density of states $\rho \sim E^\alpha$ for all symmetry classes, and makes it transparent that the exponent \alpha\ does not depend on the choice of the random matrix ensemble. Finally, we show that a lattice of impurities can drive a topologically trivial system into a nontrivial phase, and in particular we demonstrate that impurity bands carrying extremely large Chern numbers can appear in different symmetry classes of two-dimensional topological insulators and superconductors. We use the generalized root of \det H(k) to reveal a spiderweblike momentum space structure of the energy gap closings that separate the topologically distinct phases in p_x + i p_y superconductors in the presence of an impurity lattice.

Thermoelectric properties of half-Heusler topological insulators MPtBi (M = Sc, Y, La) induced by strain

Thermoelectric (TE) materials and topological insulators (TIs) were recently known to exhibit close connection, which offers new prospects in improving the TE performance. However, currently known TE materials from TIs mostly belong to the early Bi2Te3 family. In order to extend TE materials to other classes of TIs, we use the first-principles combined with Boltzmann transport theory to study the electronic and TE properties of experimental half-Heusler compounds MPtBi (M = Sc, Y, La). We find that all MPtBi are topological semimetals at equilibrium lattices while TIs under a stretched uniaxial strain, which is in agreement with previous works. We further predict that comparable TE performance with Bi2Te3 can be realized in half-Heusler TI LaPtBi by an 8% stretched uniaxial strain. We also reveal that the lattice thermal conductivity of LaPtBi is unprecedented low compared with those of traditional half-Heusler compounds (not TIs). These findings indicate the potential of half-Heusler TIs as TE materials.

Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond

After briefly recalling the quantum entanglement-based view of topological phases of matter in order to outline the general context, we give an overview of different approaches to the classification problem of topological insulators and superconductors of non-interacting fermions. In particular, we review in some detail general symmetry aspects of the ‘ten-fold way’ which forms the foundation of the classification, and put different approaches to the classification in relationship with each other. We end by briefly mentioning some of the results obtained on the effect of interactions, mainly in three spatial dimensions.

Monolayer Topological Insulators: Silicene, Germanene, and Stanene

We report the recent progress on the theoretical aspects of monolayer topological insulators including silicene, germanene and stanene, which are monolayer honeycomb structures of silicon, germanium and tin, respectively. They show quantum spin Hall effects in nature due to the spin orbit interaction. The band gap can be tuned by applying perpendicular electric field, which induces a topological phase transition. We also analyze the topological properties of generic honeycomb systems together with the classification of topological insulators. Phase diagram of topological insulators and superconductors in honeycomb systems are explicitly determined. We also investigate topological electronics including a topological field-effect transistor, the topological Kirchhoff's law and the topological spin-valleytronics.