The layered transition metal dichalcogenides (TMDs) MX 2 (M = Mo, W; X = S, Se, Te), a class of graphene-like two-dimensional materials, have attracted significant interest because they demonstrate quantum confinement at the single-layer limit 13 . As with graphene, these layered materials can be easily exfoliated mechanically to provide monolayers 3-7,14-16 and assume a hexagonal honeycomb structure in which the M and X atoms are located at alternating corners of the hexagons. However, unlike graphene, which has a gapless Dirac cone band structure, MX 2 has a rather large bandgap, making these materials more versatile as candidates for thin, flexible device applications and useful for a variety of other applications including lubrication 16 , catalysis 17 , transistors 18 and lithium-ion batteries 19 . Most interestingly, an indirect to direct bandgap transition in the monolayer limit has been predicted theoretically and supported experimentally by optical measurements [3][4][5]9,12 . Because of the direct bandgap, monolayer MX 2 is favourable for optoelectronic applications5 and field-effect transistors 15,16,18 . Furthermore, both the conduction and valence bands have two energy degenerate valleys at corners of the first Brillouin zone, making it viable to optically control the charge carriers in these valleys and suggesting the possibility of valley-based electronic and optoelectronic applications 3,6-8 .Despite these exciting developments, direct experimental verification of the novel band structure at the monolayer limit remains lacking. Furthermore, for many applications, it is vital to manufacture high-quality epitaxial films with controllable methods such as chemical vapour deposition (CVD) or molecular beam epitaxy (MBE) 20,21 .

2The electrical Hall effect is the production of a transverse voltage under an out-of-plane magnetic field [1]. Historically, studies of the Hall effect have led to major breakthroughs including the discoveries of Berry curvature and the topological Chern invariants [2, 3]. In magnets, the internal magnetization allows Hall conductivity in the absence of external magnetic field [3]. This anomalous Hall effect (AHE) has become an important tool to study quantum magnets [3][4][5][6][7][8]. In nonmagnetic materials without external magnetic fields, the electrical Hall effect is rarely explored because of the constraint by time-reversal symmetry.However, strictly speaking, only the Hall effect in the linear response regime, i.e., the Hall voltage linearly proportional to the external electric field, identically vanishes due to time-reversal symmetry [9]. The Hall effect in the nonlinear response regime, on the other hand, may not be subject to such symmetry constraints [10][11][12]. Here, we report the observation of the nonlinear Hall effect (NLHE) [12] in the electrical transport of the nonmagnetic 2D quantum material, bilayer WTe 2 . Specifically, flowing an electrical current in bilayer WTe 2 leads to a nonlinear Hall voltage in the absence of magnetic field. The NLHE exhibits unusual properties sharply distinct from the AHE in metals: The NLHE shows a quadratic I -V characteristic; It strongly dominates the nonlinear longitudinal response, leading to a Hall angle of ∼ 90 • . We further show that the NLHE directly measures the "dipole moment" [12] of the Berry curvature, which arises from layer-polarized Dirac fermions in bilayer WTe 2 . Our results demonstrate a new Hall effect and provide a powerful methodology to detect Berry curvature in a wide range of nonmagnetic quantum materials in an energy-resolved way.In 1879 Edwin H. Hall observed that, when an electrical current passes through a gold film under a magnetic field, a transverse voltage develops [1]. This effect, known as the Hall effect, forms the basis of both fundamental research and practical applications such as magnetic field measurements and motion detectors. In contrast to the classical Hall effect where the Lorentz force bends the trajectory of the charge carriers, quantum mechanics describes the "bending" by the intrinsic geometry of the quantum electron wavefunctions under time-reversal symmetry breaking. This crucial theoretical understanding eventually led to the seminal discoveries of the Berry curvature and the topological Chern number, which have become pillars of modern condensed matter physics [2, 3]. One important cur-3 rent frontier is to identify AHE with quantized or topological character in unconventional magnetic quantum materials, where spin-orbit coupling (SOC), geometrical frustration and electronic correlations coexist [3][4][5][6][7][8]. These extensive studies [1,[3][4][5][6][7][8] have established a paradigm for the electrical Hall effect: (1) A non-vanishing Hall conductivity arises from the momentum-integrated Berry curva...

The Kondo insulator SmB 6 has long been known to exhibit low temperature transport anomalies whose origin is of great interest. Here we uniquely access the surface electronic structure of the anomalous transport regime by combining state-of-the-art laser-and synchrotron-based angle-resolved photoemission techniques. We observe clear in-gap states (up to ∼ 4 meV), whose temperature dependence is contingent upon the Kondo gap formation. In addition, our observed in-gap Fermi surface oddness tied with the Kramers' points topology, their coexistence with the two-dimensional transport anomaly in the Kondo hybridization regime, as well as their robustness against thermal recycling, taken together, collectively provide by-far the strongest evidence for protected surface metallicity with a Fermi surface whose topology is consistent with the theoretically predicted topological surface Fermi surface. Our observations of systematic surface electronic structure provide the fundamental electronic parameters for the anomalous Kondo ground state of the correlated electron material SmB 6 .

Silicene is a one-atom-thick two-dimensional crystal of silicon with a hexagonal lattice structure that is related to that of graphene but with atomic bonds that are buckled rather than flat. This buckling confers advantages on silicene over graphene, because it should, in principle, generate both a band gap and polarized spin-states that can be controlled with a perpendicular electric field. Here we use first-principles calculations to show that field-gated silicene possesses two gapped Dirac cones exhibiting nearly 100% spin-polarization, situated at the corners of the Brillouin zone. Using this fact, we propose a design for a silicene-based spin-filter that should enable the spin-polarization of an output current to be switched electrically, without switching external magnetic fields. Our quantum transport calculations indicate that the proposed designs will be highly efficient (nearly 100% spin-polarization) and robust against weak disorder and edge imperfections. We also propose a Y-shaped spin/valley separator that produces spin-polarized current at two output terminals with opposite spins.

Owing to the unusual geometry of kagome lattices-lattices made of corner-sharing triangles-their electrons are useful for studying the physics of frustrated, correlated and topological quantum electronic states. In the presence of strong spin-orbit coupling, the magnetic and electronic structures of kagome lattices are further entangled, which can lead to hitherto unknown spin-orbit phenomena. Here we use a combination of vector-magnetic-field capability and scanning tunnelling microscopy to elucidate the spin-orbit nature of the kagome ferromagnet FeSn and explore the associated exotic correlated phenomena. We discover that a many-body electronic state from the kagome lattice couples strongly to the vector field with three-dimensional anisotropy, exhibiting a magnetization-driven giant nematic (two-fold-symmetric) energy shift. Probing the fermionic quasi-particle interference reveals consistent spontaneous nematicity-a clear indication of electron correlation-and vector magnetization is capable of altering this state, thus controlling the many-body electronic symmetry. These spin-driven giant electronic responses go well beyond Zeeman physics and point to the realization of an underlying correlated magnetic topological phase. The tunability of this kagome magnet reveals a strong interplay between an externally applied field, electronic excitations and nematicity, providing new ways of controlling spin-orbit properties and exploring emergent phenomena in topological or quantum materials.