Topological quantum computation based on chiral Majorana fermions.

The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.

Learning atoms for materials discovery.

Exciting advances have been made in artificial intelligence (AI) during recent decades. Among them, applications of machine learning (ML) and deep learning techniques brought human-competitive performances in various tasks of fields, including image recognition, speech recognition, and natural language understanding. Even in Go, the ancient game of profound complexity, the AI player has already beat human world champions convincingly with and without learning from the human. In this work, we show that our unsupervised machines (Atom2Vec) can learn the basic properties of atoms by themselves from the extensive database of known compounds and materials. These learned properties are represented in terms of high-dimensional vectors, and clustering of atoms in vector space classifies them into meaningful groups consistent with human knowledge. We use the atom vectors as basic input units for neural networks and other ML models designed and trained to predict materials properties, which demonstrate significant accuracy.

Topological semimetal in honeycomb lattice LnSI.

Recognized as elementary particles in the standard model, Weyl fermions in condensed matter have received growing attention. However, most of the previously reported Weyl semimetals exhibit rather complicated electronic structures that, in turn, may have raised questions regarding the underlying physics. Here, we report promising topological phases that can be realized in specific honeycomb lattices, including ideal Weyl semimetal structures, 3D strong topological insulators, and nodal-line semimetal configurations. In particular, we highlight a semimetal featuring both Weyl nodes and nodal lines. Guided by this model, we showed that GdSI, the long-perceived ideal Weyl semimetal, has two pairs of Weyl nodes residing at the Fermi level and that LuSI (YSI) is a 3D strong topological insulator with the right-handed helical surface states. Our work provides a mechanism to study topological semimetals and proposes a platform for exploring the physics of Weyl semimetals as well as related device designs.

Epitaxial growth of ultraflat stanene with topological band inversion.

Two-dimensional (2D) topological materials, including quantum spin/anomalous Hall insulators, have attracted intense research efforts owing to their promise for applications ranging from low-power electronics and high-performance thermoelectrics to fault-tolerant quantum computation. One key challenge is to fabricate topological materials with a large energy gap for room-temperature use. Stanene-the tin counterpart of graphene-is a promising material candidate distinguished by its tunable topological states and sizeable bandgap. Recent experiments have successfully fabricated stanene, but none of them have yet observed topological states. Here we demonstrate the growth of high-quality stanene on Cu(111) by low-temperature molecular beam epitaxy. Importantly, we discovered an unusually ultraflat stanene showing an in-plane s-p band inversion together with a spin-orbit-coupling-induced topological gap (~0.3 eV) at the Γ point, which represents a foremost group-IV ultraflat graphene-like material displaying topological features in experiment. The finding of ultraflat stanene opens opportunities for exploring two-dimensional topological physics and device applications.

Topological states of condensed matter.

Topological states of quantum matter have been investigated intensively in recent years in materials science and condensed matter physics. The field developed explosively largely because of the precise theoretical predictions, well-controlled materials processing, and novel characterization techniques. In this Perspective, we review recent progress in topological insulators, the quantum anomalous Hall effect, chiral topological superconductors, helical topological superconductors and Weyl semimetals.

Buckled two-dimensional Xene sheets.

Silicene, germanene and stanene are part of a monoelemental class of two-dimensional (2D) crystals termed 2D-Xenes (X = Si, Ge, Sn and so on) which, together with their ligand-functionalized derivatives referred to as Xanes, are comprised of group IVA atoms arranged in a honeycomb lattice - similar to graphene but with varying degrees of buckling. Their electronic structure ranges from trivial insulators, to semiconductors with tunable gaps, to semi-metallic, depending on the substrate, chemical functionalization and strain. More than a dozen different topological insulator states are predicted to emerge, including the quantum spin Hall state at room temperature, which, if realized, would enable new classes of nanoelectronic and spintronic devices, such as the topological field-effect transistor. The electronic structure can be tuned, for example, by changing the group IVA element, the degree of spin-orbit coupling, the functionalization chemistry or the substrate, making the 2D-Xene systems promising multifunctional 2D materials for nanotechnology. This Perspective highlights the current state of the art and future opportunities in the manipulation and stability of these materials, their functions and applications, and novel device concepts.

Wave Function and Emergent SU(2) Symmetry in the ν_{T}=1 Quantum Hall Bilayer.

We propose a trial wave function for the quantum Hall bilayer system of total filling factor ν_{T}=1 at a layer distance d to magnetic length ℓ ratio d/ℓ=κ_{c1}≈1.1, where the lowest charged excitation is known to have a level crossing. The wave function has two-particle correlations, which fit well with those in previous numerical studies, and can be viewed as a Bose-Einstein condensate of free excitons formed by composite bosons and anticomposite bosons in different layers. We show the free nature of these excitons indicating an emergent SU(2) symmetry for the composite bosons at d/ℓ=κ_{c1}, which leads to the level crossing in low-lying charged excitations. We further show the overlap between the trial wave function, and the ground state of a small size exact diagonalization is peaked near d/ℓ=κ_{c1}, which supports our theory.

Conversion Rules for Weyl Points and Nodal Lines in Topological Media.

According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a π rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

Three-Dimensional Chiral Lattice Fermion in Floquet Systems.

We show that the Nielsen-Ninomiya no-go theorem still holds on a Floquet lattice: there is an equal number of right-handed and left-handed Weyl points in a three-dimensional Floquet lattice. However, in the adiabatic limit, where the time evolution of the low-energy subspace is decoupled from the high-energy subspace, we show that the bulk dynamics in the low-energy subspace can be described by Floquet bands with extra left- or right-handed Weyl points, despite the no-go theorem. Assuming adiabatic evolution of two bands, we show that the difference of the number of right-handed and left-handed Weyl points equals twice the winding number of the adiabatic Floquet operator over the Brillouin zone. Based on these findings, we propose a realization of purely left- or right-handed Weyl particles on a 3D lattice using a Hamiltonian obtained through dimensional reduction of a four-dimensional quantum Hall system. We argue that the breakdown of the adiabatic approximation on the surface facilitates unusual closed orbits of wave packets in an applied magnetic field, which traverse alternatively through the low-energy and high-energy sector of the spectrum.

Multiple Types of Topological Fermions in Transition Metal Silicides.

Exotic massless fermionic excitations with nonzero Berry flux, other than the Dirac and Weyl fermions, could exist in condensed matter systems under the protection of crystalline symmetries, such as spin-1 excitations with threefold degeneracy and spin-3/2 Rarita-Schwinger-Weyl fermions. Herein, by using the ab initio density functional theory, we show that these unconventional quasiparticles coexist with type-I and type-II Weyl fermions in a family of transition metal silicides, including CoSi, RhSi, RhGe, and CoGe, when spin-orbit coupling is considered. Their nontrivial topology results in a series of extensive Fermi arcs connecting projections of these bulk excitations on the side surface, which is confirmed by (001) surface electronic spectra of CoSi. In addition, these stable arc states exist within a wide energy window around the Fermi level, which makes them readily accessible in angle-resolved photoemission spectroscopy measurements.

Double Helix Nodal Line Superconductor.

Time-reversal invariant superconductors in three dimensions may contain nodal lines in the Brillouin zone, which behave as Wilson loops of 3D momentum-space Chern-Simons theory of the Berry connection. Here we study the conditions of realizing linked nodal lines (Wilson loops), which yield a topological contribution to the thermal magnetoelectric coefficient that is given by the Chern-Simons action. We find the essential conditions are the existence of torus or higher genus Fermi surfaces and spiral spin textures. We construct such a model with two torus Fermi surfaces, where a generic spin-dependent interaction leads to double-helix-like linked nodal lines as the superconductivity is developed.

Epitaxial growth of two-dimensional stanene.

Following the first experimental realization of graphene, other ultrathin materials with unprecedented electronic properties have been explored, with particular attention given to the heavy group-IV elements Si, Ge and Sn. Two-dimensional buckled Si-based silicene has been recently realized by molecular beam epitaxy growth, whereas Ge-based germanene was obtained by molecular beam epitaxy and mechanical exfoliation. However, the synthesis of Sn-based stanene has proved challenging so far. Here, we report the successful fabrication of 2D stanene by molecular beam epitaxy, confirmed by atomic and electronic characterization using scanning tunnelling microscopy and angle-resolved photoemission spectroscopy, in combination with first-principles calculations. The synthesis of stanene and its derivatives will stimulate further experimental investigation of their theoretically predicted properties, such as a 2D topological insulating behaviour with a very large bandgap, and the capability to support enhanced thermoelectric performance, topological superconductivity and the near-room-temperature quantum anomalous Hall effect.

High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator.

The discovery of the quantum Hall (QH) effect led to the realization of a topological electronic state with dissipationless currents circulating in one direction along the edge of a two-dimensional electron layer under a strong magnetic field. The quantum anomalous Hall (QAH) effect shares a similar physical phenomenon to that of the QH effect, whereas its physical origin relies on the intrinsic spin-orbit coupling and ferromagnetism. Here, we report the experimental observation of the QAH state in V-doped (Bi,Sb)2Te3 films with the zero-field longitudinal resistance down to 0.00013 ± 0.00007h/e(2) (~3.35 ± 1.76 Ω), Hall conductance reaching 0.9998 ± 0.0006e(2)/h and the Hall angle becoming as high as 89.993° ± 0.004° at T = 25 mK. A further advantage of this system comes from the fact that it is a hard ferromagnet with a large coercive field (Hc > 1.0 T) and a relative high Curie temperature. This realization of a robust QAH state in hard ferromagnetic topological insulators (FMTIs) is a major step towards dissipationless electronic applications in the absence of external fields.

Topological Superconductivity on the Surface of Fe-Based Superconductors.

As one of the simplest systems for realizing Majorana fermions, the topological superconductor plays an important role in both condensed matter physics and quantum computations. Based on ab initio calculations and the analysis of an effective 8-band model with superconducting pairing, we demonstrate that the three-dimensional extended s-wave Fe-based superconductors such as Fe_{1+y}Se_{0.5}Te_{0.5} have a metallic topologically nontrivial band structure, and exhibit a normal-topological-normal superconductivity phase transition on the (001) surface by tuning the bulk carrier doping level. In the topological superconductivity (TSC) phase, a Majorana zero mode is trapped at the end of a magnetic vortex line. We further show that the surface TSC phase only exists up to a certain bulk pairing gap, and there is a normal-topological phase transition driven by the temperature, which has not been discussed before. These results pave an effective way to realize the TSC and Majorana fermions in a large class of superconductors.

Ideal Weyl Semimetals in the Chalcopyrites CuTlSe_{2}, AgTlTe_{2}, AuTlTe_{2}, and ZnPbAs_{2}.

Weyl semimetals are new states of matter which feature novel Fermi arcs and exotic transport phenomena. Based on first-principles calculations, we report that the chalcopyrites CuTlSe_{2}, AgTlTe_{2}, AuTlTe_{2}, and ZnPbAs_{2} are ideal Weyl semimetals, having largely separated Weyl points (∼0.05 Å^{-1}) and uncovered Fermi arcs that are amenable to experimental detections. We also construct a minimal effective model to capture the low-energy physics of this class of Weyl semimetals. Our discovery is a major step toward a perfect playground of intriguing Weyl semimetals and potential applications for low-power and high-speed electronics.

Quantum anomalous Hall effect in magnetic insulator heterostructure.

On the basis of ab initio calculations, we predict that a monolayer of Cr-doped (Bi,Sb)2Te3 and GdI2 heterostructure is a quantum anomalous Hall insulator with a nontrivial band gap up to 38 meV. The principle behind our prediction is that the band inversion between two topologically trivial ferromagnetic insulators can result in a nonzero Chern number, which offers a better way to realize the quantum anomalous Hall state without random magnetic doping. In addition, a simple effective model is presented to describe the basic mechanism of spin polarized band inversion in this system. Moreover, we predict that 3D quantum anomalous Hall insulator could be realized in (Bi2/3Cr1/3)2Te3 /GdI2 superlattice.

Intrinsic Quantum Anomalous Hall Effect in the Kagome Lattice Cs_{2}LiMn_{3}F_{12}.

In a kagome lattice, the time reversal symmetry can be broken by a staggered magnetic flux emerging from ferromagnetic ordering and intrinsic spin-orbit coupling, leading to several well-separated nontrivial Chern bands and intrinsic quantum anomalous Hall effect. Based on this idea and ab initio calculations, we propose the realization of the intrinsic quantum anomalous Hall effect in the single layer Cs_{2}Mn_{3}F_{12} kagome lattice and on the (001) surface of a Cs_{2}LiMn_{3}F_{12} single crystal by modifying the carrier coverage on it, where the band gap is around 20 meV. Moreover, a simplified tight binding model based on the in-plane ddσ antibonding states is constructed to understand the topological band structures of the system.

Helical Spin Order from Topological Dirac and Weyl Semimetals.

We study dynamical mass generation and the resultant helical spin orders in topological Dirac and Weyl semimetals, including the edge states of quantum spin Hall insulators, the surface states of weak topological insulators, and the bulk materials of Weyl semimetals. In particular, the helical spin textures of Weyl semimetals manifest the spin-momentum locking of Weyl fermions in a visible manner. The spin-wave fluctuations of the helical order carry electric charge density; therefore, the spin textures can be electrically controlled in a simple and predictable manner.

Electrically Tunable Magnetism in Magnetic Topological Insulators.

The external controllability of the magnetic properties in topological insulators would be important both for fundamental and practical interests. Here we predict the electric-field control of ferromagnetism in a thin film of insulating magnetic topological insulators. The decrease of band inversion by the application of electric fields results in a reduction of magnetic susceptibility, and hence in the modification of magnetism. Remarkably, the electric field could even induce the magnetic quantum phase transition from ferromagnetism to paramagnetism. We further propose a transistor device in which the dissipationless charge transport of chiral edge states is controlled by an electric field. In particular, the field-controlled ferromagnetism in a magnetic topological insulator can be used for voltage based writing of magnetic random access memories in magnetic tunnel junctions. The simultaneous electrical control of magnetic order and chiral edge transport in such devices may lead to electronic and spintronic applications for topological insulators.

Observation of the Zero Hall Plateau in a Quantum Anomalous Hall Insulator.

We report experimental investigations on the quantum phase transition between the two opposite Hall plateaus of a quantum anomalous Hall insulator. We observe a well-defined plateau with zero Hall conductivity over a range of magnetic field around coercivity when the magnetization reverses. The features of the zero Hall plateau are shown to be closely related to that of the quantum anomalous Hall effect, but its temperature evolution exhibits a significant difference from the network model for a conventional quantum Hall plateau transition. We propose that the chiral edge states residing at the magnetic domain boundaries, which are unique to a quantum anomalous Hall insulator, are responsible for the novel features of the zero Hall plateau.