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We show that H-phase transition metal dichalcogenides (TMDs) monolayers such as MoS2 and WSe2, are orbital Hall insulators. They present very large orbital Hall conductivity plateaus in their semiconducting gap, where the spin Hall conductivity vanishes. This novel effect does not rely on conducting edge channels as in the case of quantum spin Hall insulators. Our results open the possibility of using TMDs for orbital current injection and orbital torque transfers that surpass their spin-counterparts in spin-orbitronics devices. The orbital Hall effect (OHE) in TMD monolayers occurs even in the absence of spin-orbit coupling. It can be linked to exotic momentum-space orbital textures, analogous to the spin-momentum locking in 2D Dirac fermions that arise from a combination of orbital attributes and lattice symmetry.

The orbital-Hall effect (OHE), similarly to the spin-Hall effect (SHE), refers to the creation of a transverse flow of orbital angular momentum that is induced by a longitudinally applied electric field [1]. For systems in which the spin-orbit coupling (SOC) is sizeable, the orbital and spin angular momentum degrees of freedom are coupled, and an interrelationship between charge, spin and orbital angular momentum excitations is naturally established. The OHE has been explored mostly in metallic systems, where it can be quite strong [2][3][4][5]. However, several of its features remain unexplored in two-dimensional (2D) materials. Here, we investigate the role of orbital textures for the OHE displayed by multiorbital 2D materials. We predict the appearance of rather large orbital Hall effect in these systems both in their metallic and insulating phases. In some cases the orbital Hall currents are larger than the spin Hall ones, and their use as information carriers widens the development possibilities of novel spin-orbitronic devices.In our analyses, we consider a minimal tight-binding model Hamiltonian, which involves only two atomic orbitals (p x and p y ) per atom in a honeycomb lattice [6,7]:(1) where i and j denote the honeycomb lattice sites positioned at R i and R j , respectively. The symbol ij indicates that the sum is restricted to the nearest neighbour (n.n) sites only. The operator p † iµs creates an electron of spin s in the atomic orbitals p µ = p ± = 1 √ 2 (p x ± ip y ) centred at R i . Here, s = ↑, ↓ labels the two electronic spin states, and i is the atomic energy at site i, which may symbolise a staggered on-site potential that takes values i = ±V AB , when site i belongs to the A and B sub-lattices of the honeycomb arrangement, respectively. The transfer integrals t µν ij between the p µ orbitals centred on n.n atoms are parametrised according to the standard Slater-Koster tight-binding formalism [8]. They depend on the direction cosines of the n.n. interatomic directions, and may be approximately expressed as linear combinations of two other integrals (V ppσ and V ppπ ) involving the p σ and p π orbitals, where σ and π refer to the usual components of the angular momentum around these axes. Since our model does not include the atomic orbital p z , it is restricted to a sector of the = 1 angular momentum vector space spanned only by the eigenstates of z p ± associated with m = ±1, respectively. Within this sector it is useful to introduce a pseudo angular momentum SU (2)-algebra where the Pauli matrices act on p ± . In this case, there is a one-to-one correspondence between the representations of the Cartesian components of the orbital angular momentum operators in this basis and the usual Pauli matrices, and z is not conserved. Using this approach, the third term may describe either an intrinsic atomic SOC given by h z µs = λ I z µµ σ z ss , or an exchange coupling in a spinless system, where h z µs = λ ex z µµ σ 0 ss . Figure 1: (a) Schematic representation of the OHE in our 2Dmodel material. ...

The fabrication of bismuthene on top of SiC paved the way for substrate engineering of room temperature quantum spin Hall insulators made of group V atoms. We perform large-scale quantum transport calculations in these 2d materials to analyse the rich phenomenology that arises from the interplay between topology, disorder, valley and spin degrees of freedom. For this purpose, we consider a minimal multi-orbital real-space tight-binding Hamiltonian and use a Chebyshev polynomial expansion technique. We discuss how the quantum spin Hall states are affected by disorder, sublattice resolved potential and Rashba spin-orbit coupling.

The outline presented by Du in the Comment [1] on our Letter [2] partially agrees with the pictorial view we have used to explain the orbital-Hall effect (OHE) in transition metal dichalcogenide (TMD) bilayers. In fact, most of the phenomenological description of the OHE effect in the TMD bilayer that he brought up in the Comment [1] is discussed in the third section-Low energy calculations-of our Letter [2]. Notwithstanding, according to Du [1], the OHE that we have shown to occur in 2H-TMD bilayers should be termed the "hidden valley-Hall effect," and the Berry curvatures of the two valence bands as "hidden Berry curvatures." We do not believe that such terminology brings new insights to our Letter.

Profiles of the spin and orbital angular momentum accumulations induced by a longitudinally applied electric field are explored in nanoribbons of p-band systems with a honeycomb lattice. We show that nanoribbons with zigzag borders can exhibit orbital magnetoelectric effects. More specifically, we have found that purely orbital magnetisation may be induced in these systems by means of this external electric field. The effect is rather general and may occur in other twodimensional multi-orbital materials. Here, it requires the presence of both spin-orbit interaction and sub-lattice symmetry breaking to manifest.

The Haldane model on a honeycomb lattice is a paradigmatic example of a system featuring quantized Hall conductivity in the absence of an external magnetic field, that is, a quantum anomalous Hall effect. Recent theoretical work predicted that the anomalous Hall conductivity of massive Dirac fermions can display Shubnikov-de Haas (SdH) oscillations, which could be observed in topological insulators and honeycomb layers with strong spin-orbit coupling. Here, we investigate the electronic transport properties of Chern insulators subject to high magnetic fields by means of accurate spectral expansions of lattice Green's functions. We find that the anomalous component of the Hall conductivity displays visible SdH oscillations at low temperature. The effect is shown to result from the modulation of the next-nearest neighbour flux accumulation due to the Haldane term, which removes the electron-hole symmetry from the Landau spectrum. To support our numerical findings, we derive a long-wavelength description beyond the linear ('Dirac cone') approximation. Finally, we discuss the dependence of the energy spectra shift for reversed magnetic fields with the topological gap and the lattice bandwidth.

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