Topological phases with insulating bulk and gapless surface or edge modes have attracted much attention because of their fundamental physics implications and potential applications in dissipationless electronics and spintronics. In this review, we mainly focus on the recent progress in the engineering of topologically nontrivial phases (such as Z2 topological insulators, quantum anomalous Hall effects, quantum valley Hall effects etc.) in two-dimensional material systems, including quantum wells, atomic crystal layers of elements from group III to group VII, and the transition metal compounds.

Symmetry, dimensionality, and interaction are crucial ingredients for phase transitions and quantum states of matter. As a prominent example, the integer quantum Hall effect (QHE) represents a topological phase generally regarded as characteristic for two-dimensional (2D) electronic systems, and its many aspects can be understood without invoking electron-electron interaction. The intriguing possibility of generalizing QHE to three-dimensional (3D) systems was proposed decades ago, yet it remains elusive experimentally. Here, we report for the first time clear experimental evidence for the 3D QHE, observed in bulk ZrTe5 crystals. Owing to the extremely high sample quality, the extreme quantum limit with only the lowest Landau level occupied can be achieved by an applied magnetic field as low as 1.5 T. Remarkably, in this regime, we observe a dissipationless longitudinal resistivity ≅ accompanied with a well-developed Hall resistivity plateau = ( ± . ) ( , ) , where , is the Fermi wavelength along the field direction ( axis). This striking result strongly suggests a Fermi surface instability driven by the enhanced interaction effects in the extreme quantum limit. In addition, with further increasing magnetic field, both and increase dramatically and display an interesting metal-insulator transition, representing another magnetic field driven quantum phase transition. Our findings not only unambiguously reveal a novel quantum state of matter resulting from an intricate interplay among dimensionality, interaction, and symmetry breaking, but also provide a promising platform for further exploration of more exotic quantum phases and transitions in 3D systems.Since its discovery in 1980, the QHE has been established and well understood in a variety of 2D electron systems, including the traditional 2D electron gas 1,2 , and 2D materials like graphene 3,4 , etc. The hallmark of QHE is that the Hall conductivity takes precisely quantized values as 2 /ℎ while the longitudinal conductivity vanishes 1,2 . Here, the prefactor is the filling factor which counts the number of filled Landau levels, is the elementary charge, and ℎ is Plank's constant. Soon after its

The existence of inequivalent valleys K and K′ in the momentum space of two-dimensional hexagonal lattices provides a new electronic degree of freedom, the manipulation of which can potentially lead to new types of electronics, in analogy to the role played by electron spin 1-3 . In materials with broken inversion symmetry, such as an electrically gated bilayer graphene 4,5 , the momentum-space Berry curvature  carries opposite sign in the K and K′ valleys. A sign reversal of  along an internal boundary of the sheet gives rise to counterpropagating one-dimensional conducting modes encoded with opposite valley indices. These metallic states are topologically protected against backscattering in the absence of valley-mixing scattering, and thus can carry current ballistically 1,6-11 . In bilayer graphene, the reversal of  can occur at the domain wall of AB and BA stacked domains 12-14 , or at the line junction of two oppositely gated regions 6 . The latter approach can provide a scalable platform to implement valleytronic operations such as valves and waveguides 9,15 , but is technically challenging to realize. Here we fabricate a dual-split-gate structure in bilayer graphene and demonstrate transport evidence of the predicted metallic states. They possess a mean free path of up to a few hundred nanometers in the absence of a magnet field. The application of perpendicular magnetic field suppresses backscattering significantly and enables a 400-nanometer-long junction to exhibit conductance close to the ballistic limit of 4 e 2 /h at 8 Tesla. Our experiment paves the path to the realization of gate-controlled ballistic valley transport and the development of valleytronic applications in atomically thin materials.Exploiting the valley degree of freedom in hexagonal lattices may offer an alternative pathway to achieving low-power-consumption electronics. Experiments have shown that a net valley polarization in the material can be induced by the use of circularly polarized light 2,16,17 or a net bulk current [18][19][20] . However the use of light is not always desirable in electronics and device proposals using bulk valley polarization often put stringent requirements on the size and edge orientation of the active area 3 . Alternatively, electrically created, valley-polarized topological conducting channels in high-mobility bilayer graphene may offer a robust, scalable platform to realize valleytronic operations 1,6,[8][9][10][11][12][13][14][15] . Figure 1a illustrates the dual-split-gating scheme proposed by Martin et al 6 , where an AB-stacked bilayer graphene (BLG) sheet is controlled by two pairs of top and bottom gates separated by a line junction. The device operates in the regime where both the left and the right regions of the BLG sheet are insulating due to a bulk band gap induced by the independently applied displacement fields DL and DR. In the "odd" field configuration, where DLDR < 0, theory predicts the existence of eight conducting modes (referred to as the "kink" states) propagating along the line ...

An intersection between one-dimensional chiral acts as a topological current splitter. We find that the splitting of a chiral zero-line mode obeys very simple, yet highly counterintuitive, partition laws which relate current paths to the geometry of the intersection. Our results have far reaching implications for device proposals based on chiral zero-line transport in the design of electron beam splitters and interferometers, and for understanding transport properties in systems where multiple topological domains lead to a statistical network of chiral channels.A massive chiral two-dimensional electron gas (C2DEG) has a valley Hall conductivity that has the same sign as its mass. The valley Hall effect leads to conducting edge states and also, when the mass parameter varies spatially, to conducting states localized along mass zero-lines.1-3 Provided that inter-valley scattering is weak, zero-line state properties are closely analogous [1][2][3] to edge state properties of quantum spin-Hall insulators and include both chiral propagation and suppressed backscattering.1 Metallic zero-line modes (ZLMs), or topological 1D kink states, provide a two dimensional realization of Dirac zero energy modes, 4,5 and their existence has been proposed in a wide variety of systems including graphene mono and bilayers, 1-3,6-8 topological insulators with lattice dislocations, 9 boron nitride crystals with grain boundaries, 10 superfluid 3 He, 11 and photonic crystals.12,13 In the present Letter we examine current partition properties at zero-lines intersections, 1,3 which are expected to be ubiquitous in systems in which the mass term results from a disorder potential or from spontaneous symmetry breaking.ZLMs in C2DEGs are centered on zero-lines of the mass 1-3,6,7 , i.e. on lines along which the mass changes sign as illustrated in Figure 1a. A mass term leading to a valley Hall effect 14,15 can be produced by a sublattice staggered external potential in single layer graphene, 6,7 and more practically by a gate controlled interlayer potential difference in Bernal bilayer and ABC stacked multilayer graphene. [1][2][3]8 Mass terms can also be generated by spin-orbit coupling [16][17][18] and by electron-electron interactions. [19][20][21] In this last case ZLMs 22 appear naturally at domain walls separating regions with different local anomalous, spin, or valley Hall conductivities.Chiral propagation implies that ZLMs can travel only in the direction which places negative masses either on their left, or depending on valley, on their right. It follows, as illustrated in Figure 1c, that there is no forward propagation at a zero-lines intersection; a propagating mode is split between a portion that turns clockwise and a portion that turns counterclockwise. These unusual transport properties are potentially valuable for new types of electronic devices. We have therefore carried out quantum transport calculations for an explicit model of intersecting ZLMs in order to discover rules for current partitioning at such a ZLM splitter. The s...

The two inequivalent valleys in graphene are protected against long range scattering potentials due to their large separation in momentum space. In tailored √ 3N × √ 3N or 3N × 3N graphene superlattices, these two valleys are folded into Γ and coupled by Bragg scattering from periodic adsorption. We find that, for top-site adsorption, strong inter-valley coupling closes the bulk gap from inversion symmetry breaking and leads to a single-valley metallic phase with quadratic band crossover. The degeneracy at the crossing point is protected by C3v symmetry. In addition, the emergence of pseudo-Zeeman field and valley-orbit coupling are also proposed, which provide the possibility of tuning valley-polarization coherently in analogy to real spin for spintronics. Such valley manipulation mechanisms can also find applications in honeycomb photonic crystals. We also study the strong geometry-dependent influence of hollow-and bridge-site adatoms in the inter-valley coupling.

We theoretically demonstrate that the second-order topological insulator with robust corner states can be realized in two-dimensional Z2 topological insulators by applying an in-plane Zeeman field. Zeeman field breaks the time-reversal symmetry and thus destroys the Z2 topological phase. Nevertheless, it respects some crystalline symmetries and thus can protect the higher-order topological phase. By taking the Kane-Mele model as a concrete example, we find that spin-helical edge states along zigzag boundaries are gapped out by Zeeman field whereas in-gap corner state at the intersection between two zigzag edges arises, which is independent on the field orientation. We further show that the corner states are robust against the out-of-plane Zeeman field, staggered sublattice potentials, Rashba spin-orbit coupling, and the buckling of honeycomb lattices, making them experimentally feasible. Similar behaviors can also be found in the well-known Bernevig-Hughes-Zhang model.

As the thinnest conductive and elastic material, graphene is expected to play a crucial role in post-Moore era. Besides applications on electronic devices, graphene has shown great potential for nano-electromechanical systems. While interlayer interactions play a key role in modifying the electronic structures of layered materials, no attention has been given to their impact on electromechanical properties. Here we report the positive piezoconductive effect observed in suspended bi- and multi-layer graphene. The effect is highly layer number dependent and shows the most pronounced response for tri-layer graphene. The effect, and its dependence on the layer number, can be understood as resulting from the strain-induced competition between interlayer coupling and intralayer transport, as confirmed by the numerical calculations based on the non-equilibrium Green's function method. Our results enrich the understanding of graphene and point to layer number as a powerful tool for tuning the electromechanical properties of graphene for future applications.

We theoretically report that, with in-plane magnetization, the quantum anomalous Hall effect (QAHE) can be realized in two-dimensional atomic crystal layers with preserved inversion symmetry but broken out-of-plane mirror reflection symmetry. We take the honeycomb lattice as an example, where we find that the low-buckled structure, which makes the system satisfy the symmetric criteria, is crucial to induce QAHE. The topologically nontrivial bulk gap carrying a Chern number of C = ±1 opens in the vicinity of the saddle points M , where the band dispersion exhibits strong anisotropy. We further show that the QAHE with electrically tunable Chern number can be achieved in Bernalstacked multilayer systems, and the applied interlayer potential differences can dramatically decrease the critical magnetization to make the QAHE experimentally feasible.