Quasi-one-dimensional topological material Bi4X4(X=Br,I)

Topological insulators (TIs) are promising for achieving dissipationless transport devices due to the robust gapless states inside the insulating bulk gap. However, currently realized two-dimensional (2D) TIs, quantum spin Hall (QSH) insulators, suffer from ultrahigh vacuum and extremely low temperature. Thus, seeking for desirable QSH insulators with high feasibility of experimental preparation and large nontrivial gap is of great importance for wide applications in spintronics. On the basis of the first-principles calculations, we predict a novel family of 2D QSH insulators in transition-metal halide MX (M = Zr, Hf; X = Cl, Br, and I) monolayers, especially, which is the first case based on transition-metal halide-based QSH insulators. MX family has the large nontrivial gaps of 0.12-0.4 eV, comparable with bismuth (111) bilayer (0.2 eV), stanene (0.3 eV), and larger than ZrTe5 (0.1 eV) monolayers and graphene-based sandwiched heterstructures (30-70 meV). Their corresponding 3D bulk materials are weak topological insulators from stacking QSH layers, and some of bulk compounds have already been synthesized in experiment. The mechanism for 2D QSH effect in this system originates from a novel d-d band inversion, significantly different from conventional band inversion between s-p, p-p, or d-p orbitals. The realization of pure layered MX monolayers may be prepared by exfoliation from their 3D bulk phases, thus holding great promise for nanoscale device applications and stimulating further efforts on transition metal-based QSH materials.

A quantum spin Hall (QSH) insulator hosts topological states at the one-dimensional (1D) edge, along which backscattering by nonmagnetic impurities is strictly prohibited. Its 3D analogue, a weak topological insulator (WTI), possesses similar quasi-1D topological states confined at side surfaces. The enhanced confinement could provide a route for dissipationless current and better advantages for applications relative to strong topological insulators (STIs). However, the topological side surface is usually not cleavable and is thus hard to observe. Here, we visualize the topological states of the WTI candidate ZrTe5 by spin and angle-resolved photoemission spectroscopy (ARPES): a quasi-1D band with spin-momentum locking was revealed on the side surface. We further demonstrate that the bulk band gap is controlled by external strain, realizing a more stable WTI state or an ideal Dirac semimetal (DS) state. The highly directional spin-current and the tunable band gap in ZrTe5 will provide an excellent platform for applications.

We study the effects of temperature on the topological nature of ZrTe$_5$, which sits near the phase boundary between strong and weak topological insulating orders. Using first-principles calculations, we show that the band gap and the topological indices of ZrTe$_5$ are extremely sensitive to thermal expansion, and that the electron-phonon interaction accounts for more than a third of the total temperature-dependent effects in ZrTe$_5$. We find that the temperature dependence of the band gap has an opposite sign in the strong and weak topological insulator phases. Based on this insight, we propose a robust and unambiguous method to determine the topological nature of any given sample of ZrTe$_5$: if the band gap decreases with temperature it is a strong topological insulator, and if it increases with temperature it is a weak topological insulator. An analogous strategy is expected to be generally applicable to other materials and to become particularly important in the vicinity of topological phase boundaries where other methods provide ambiguous results.

Topologically protected edge states of 2D quantum spin Hall (QSH) insulators have paved the way for dissipationless transport. In this regard, one of the key challenges is to find suitable QSH insulators with large bandgaps. Group IV analogues of graphene such as silicene, germanene, stanene, plumbene etc. are promising materials for QSH insulators. This is because their high spin–orbit coupling (SOC) and large bandgap opening may be possible by chemically decorating these group IV graphene analogues. However, finding suitable chemical groups for such decoration has always been a demanding task. In this work, we investigate the performance of a plumbene monolayer with –CX3 (X = H, F, Cl) chemical decoration and report very large bandgaps in the range of 0.8414 eV to 0.9818 eV with spin–orbit coupling, which is much higher compared to most other topological insulators and realizable at room temperature. The topological invariants of the samples are calculated to confirm their topologically nontrivial properties. The existence of edge states and topological nontrivial property are illustrated by investigating PbCX3 nanoribbons with zigzag edges. Lastly, the structural and electronic stability of the topological materials against strain are demonstrated to extend the scope of using these materials. Our findings suggest that these derivatives are promising materials for spintronic and future high performance nanoelectronic devices.

The measurement of topological insulators is usually achieved by measuring spectra, which is inadequate in higher-order topological insulators because the separation of bandgaps may be blurred, such as body boundaries and corner state hybridization, and the above measurement requires that the energy of the topological mode must be located in the bulk bandgap to be measured. Recent experiments have shown a strong relationship between the fractional mode charge and the location of the Wannier center in the crystal body. The fractional quantization of the corner charge originates from the filling anomaly, and the fractional mode charge can be calculated from experimental measurements of the local density of states to describe the topological state of the system, which has been demonstrated in optical systems. We have achieved a similar effect in acoustics by utilizing a coupled acoustic-cavity system featuring C3, C4, and C6 symmetries for simulation. Furthermore, we extended our study to 3D third-order topological crystalline insulators to validate the fractional-corner charge approach. The simulation and theoretical results demonstrate that the fractional mode charge can clearly identify the high-order topology, even without gap states.

Quantum spin Hall (QSH) insulator with large gap is highly desirable for potential spintronics application. Here we realize electrically tunable QSH insulator with large gap in van der Waals heterobilayer of monolayer transition metal dichalcogenide (TMD) and hexagonal boron arsenide (BAs), in particular the WSe2/BAs heterobilayer. When the type II band alignment gets inverted in an electric field, the hybridization by interlayer hopping between the spin-valley locked valence band edges in TMD and the BAs conduction band edges leads to a stacking-configuration dependent topological band inversion. In the non-interacting limit, the double spin degeneracy of BAs leaves an un-hybridized conduction band inside the gap, so the heterobilayer is a spin-valley locked metal instead of a QSH insulator. With the Coulomb interaction accounted in the double-layer geometry, the interaction with the hybridization induced electric dipole shifts this un-hybridized conduction band upwards in energy, giving rise to a sizable global QSH gap. Consequently, this heterobilayer provides a platform for engineering electrically tunable QSH insulator with sizable band gap. In the long-period moiré pattern with the spatial variation of local stacking-configurations, the competition between Coulomb interaction and interlayer hopping leads to superstructures of QSH insulators and excitonic insulators.

The interplay between non-Hermiticity and topology opens an exciting avenue for engineering novel topological matter with unprecedented properties. While previous studies have mainly focused on one-dimensional systems or Chern insulators, here we investigate topological phase transitions to/from quantum spin Hall (QSH) insulators driven by non-Hermiticity. We show that a trivial to QSH insulator phase transition can be induced by solely varying non-Hermitian terms, and there exists exceptional edge arcs in QSH phases. We establish two topological invariants for characterizing the non-Hermitian phase transitions: i) with time-reversal symmetry, the biorthogonal $\mathbb{Z}_2$ invariant based on non-Hermitian Wilson loops, and ii) without time-reversal symmetry, a biorthogonal spin Chern number through biorthogonal decompositions of the Bloch bundle of the occupied bands. These topological invariants can be applied to a wide class of non-Hermitian topological phases beyond Chern classes, and provides a powerful tool for exploring novel non-Hermitian topological matter and their device applications.

Dual topological materials are unique topological phases that host coexisting surface states of different topological nature on the same or on different material facets. Here, we show that Bi2TeI is a dual topological insulator. It exhibits band inversions at two time reversal symmetry points of the bulk band, which classify it as a weak topological insulator with metallic states on its 'side' surfaces. The mirror symmetry of the crystal structure concurrently classifies it as a topological crystalline insulator. We investigated Bi2TeI spectroscopically to show the existence of both two-dimensional Dirac surface states, which are susceptible to mirror symmetry breaking, and one-dimensional channels that reside along the step edges. Their mutual coexistence on the step edge, where both facets join, is facilitated by momentum and energy segregation. Our observation of a dual topological insulator should stimulate investigations of other dual topology classes with distinct surface manifestations coexisting at their boundaries.

At the heart of the study of topological insulators lies a fundamental\ndichotomy: topological invariants are defined in infinite systems, but surface\nstates as their main footprint only exist in finite systems. In the slab\ngeometry, namely infinite in two planar directions and finite in the\nperpendicular direction, the 2D topological invariant was shown to display\nthree different types of behaviour. The perpendicular Dirac velocity turns out\nto be a critical control parameter discerning between different qualitative\nsituations. When it is zero, the three types of behaviour extrapolate to the\nthree 3D topologically distinct phases: trivial, weak and strong topological\ninsulators. We show analytically that the boundaries between types of behaviour\nare topological phase transitions of particular significance since they allow\nto predict the 3D topological invariants from finite-thickness transitions.\nWhen the perpendicular Dirac velocity is not zero, we identify a new phase with\nsurface states but no band inversion at any finite thickness, disentangling\nthese two concepts which are closely linked in 3D. We also show that at zero\nperpendicular Dirac velocity the system is gapless in the 3D bulk and therefore\nnot a topological insulating state, even though the slab geometry extrapolates\nto the 3D topological phases. Finally, in a parameter regime with strong\ndispersion perpendicular to the surface of the slab, we encounter the unusual\ncase that the slab physics displays non-trivial phases with surface states but\nnevertheless extrapolates to a 3D trivial state.\n

Quantum Spin Hall (QSH) insulators with a large topologically nontrivial bulk gap are crucial for future applications of the QSH effect. Among these, group III-V monolayers and their halides with chair structure (regular hexagonal framework, RHF) were widely studied. Using first-principles calculations, we propose a new structure model for the functionalized group III-V monolayers, which consist of rectangular GaBi-X2 (X=I, Br, Cl) monolayers with a distorted hexagonal framework (DHF). These structures have a much lower energy than the GaBi-X2 monolayers with chair structure. Remarkably, the DHF GaBi-X2 monolayers are all QSH insulators, which exhibit sizeable nontrivial band gaps ranging from 0.17 eV to 0.39 eV. Those band gaps can be widely tuned by applying different spin-orbit coupling (SOC) strengths, resulting in a distorted Dirac cone.

Spectroscopic evidence for the gapless electronic structure in bulk ZrTe5

Subwavelength path-switchable wave routing and topological corner states for a higher-order topological insulator

Bismuth is gaining importance as a key element of functional quantum materials. The effects of spin-orbit coupling (SOC) are at the heart of many exciting proposals for next-generation quantum technologies, including topological materials for efficient information transmission and energy-saving applications. The "heavy" element bismuth and its compounds are predestined for SOC-induced topological properties, but materials design is challenged by a complex link between them and the chemical composition and crystal structure. Nevertheless, a lot can be learned about a certain property by testing its limits with compositional and/or structure modifications. We survey a handful of topological bismuth-based materials that bear structural and chemical semblance to the early topological insulators, antimony-doped elemental bismuth, Bi2Se3 and Bi2Te3. Chemical bonding via p orbitals and modular structure underlie all considered bismuth chalcogenides, subhalides, and chalcogenide halides and allow us to correlate the evolution of chemical bonding and structure with variability of the topological properties, although materials design should not be regarded as a building blocks set. Over the past decade, material discoveries have unearthed a plethora of topological properties, and bismuth is very fertile as a progenitor of a rich palette of exotic quantum materials, ranging from strong and weak 3D and crystalline topological insulators over topological metals and semimetals to magnetic topological insulators, while preserving the general layered structure motif.

Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducting states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on $\mathrm{Hg}\mathrm{Te}∕\mathrm{Cd}\mathrm{Te}$ quantum wells are described that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. Experiments on ${\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x}$, ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, and ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.

Higher-order topological insulators (HOTIs) are unique materials hosting topologically protected states, whose dimensionality is at least by 2 lower than that of the bulk. Topological states in such insulators may be strongly confined in their corners which leads to considerable enhancement of nonlinear processes involving such states. However, all nonlinear HOTIs demonstrated so far were built on periodic bulk lattice materials. Here, we demonstrate the first nonlinear photonic HOTI with the fractal origin. Despite their fractional effective dimensionality, the HOTIs constructed here on two different types of the Sierpiński gasket waveguide arrays, may support topological corner states for unexpectedly wide range of coupling strengths, even in parameter regions where conventional HOTIs become trivial. We demonstrate thresholdless spatial solitons bifurcating from corner states in nonlinear fractal HOTIs and show that their localization can be efficiently controlled by the input beam power. We observe sharp differences in nonlinear light localization on outer and multiple inner corners and edges representative for these fractal materials. Our findings not only represent a new paradigm for nonlinear topological insulators, but also open new avenues for potential applications of fractal materials to control the light flow.