The quantum anomalous Hall (QAH) effect is a quintessential consequence of non-zero Berry curvature in momentum-space. The QAH insulator harbors dissipation-free chiral edge states in the absence of an external magnetic field. On the other hand, the topological Hall (TH) effect, a transport hallmark of the chiral spin textures, is a consequence of realspace Berry curvature. While both the QAH and TH effects have been reported separately, their coexistence, a manifestation of entangled chiral edge states and chiral spin textures, has not been reported. Here, by inserting a TI layer between two magnetic TI layers to form a sandwich heterostructure, we realized a concurrence of the TH effect and the QAH effect through electric field gating. The TH effect is probed by bulk carriers, while the QAH effect is characterized by chiral edge states. The appearance of TH effect in the QAH insulating regime is the consequence of chiral magnetic domain walls that result from the gate-induced Dzyaloshinskii-Moriya interaction and occur during the magnetization reversal process in the magnetic TI sandwich samples. The coexistence of chiral edge states and chiral spin textures potentially provides a unique platform for proof-of-concept dissipationless spintextured spintronic applications. Electronic band structures of nontrivial topology in momentum-space and magnetic chiral spin textures in real-space have attracted enormous attention in the past decade since they harbor elegant Berry curvature physics 1, 2, 3 . The intrinsic anomalous Hall (AH) effect is such an example: it is induced by the Berry curvature in momentum-space in ferromagnetic (FM) materials 4 and can even be quantized under certain circumstances, leading to the quantum anomalous Hall (QAH)effect. The QAH effect has been theoretically proposed 5, 6, 7, 8 and experimentally realized 9,10,11,12,13 in magnetically doped topological insulator (TI) films. On the other hand, chiral spin textures (e.g. skyrmions) provide another example of nontrivial topology, but in real-space. It has been shown that chiral spin textures can also induce a Hall current: this is known as the topological Hall (TH) effect and is generally regarded as the transport signature of non-zero spin chirality 3 . The TH effect has been experimentally observed in many metallic systems, such as MnSi 14, 15 , MnGe 16 , FeGe 17 , and SrIrO3/SrRuO3 interface 18,19 as well as magnetically doped TI films and heterostructures 20, 21 . The TH effect in these systems is usually observed accompanied by the AH effect. However, there is no conclusive evidence to date that the AH effect found in these metallic systems to be intrinsic, i.e., exclusively induced by the momentum-space Berry curvature 4 .The QAH and TH effects have been separately observed in magnetically doped TI 9, 10, 20, 21 , with distinctly different sample geometries. The QAH effect can be realized only in the insulating regime of a magnetic TI 9,10,11,12,13 while the TH effect is usually seen in metallic systems 20, 21 .In this Article, we re...

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At the interface between two-dimensional materials with different topologies, topologically protected one-dimensional states (also named as zero-line modes) arise. Here, we focus on the quantum anomalous Hall effect based zero-line modes formed at the interface between regimes with different Chern numbers. We find that, these zero-line modes are chiral and unilaterally conductive due to the breaking of time-reversal invariance. For a beam splitter consisting of two intersecting zero lines, the chirality ensures that current can only be injected from two of the four terminals. Our numerical results further show that, in the absence of contact resistance, the (anti-)clockwise partitions of currents from these two terminals are the same owing to the current conservation, which effectively simplifies the partition laws. We find that the partition is robust against relative shift of Fermi energy, but can be effectively adjusted by tuning the relative magnetization strengths at different regimes or relative angles between zero lines. PACS numbers:Introduction-. The presence of edge states that are topologically protected from backscattering is one of the striking hall-marks of topologically nontrivial insulators [1][2][3][4][5]. According to the rigorous bulk-edge correspondence rule [6,7], edge states appear at the boundary of two-dimensional topological systems, like quantum Hall effect [8], quantum anomalous Hall effect [9,10], and quantum spin-Hall effect (or two-dimensional topological insulators) [11,12]. These edge states are localized at the boundaries that are interfaces between topological materials and topologically trivial vacuum. Thus, these boundaries can be generalized to interfaces between two topologically distinct materials, like the interface between quantum anomalous Hall effect and quantum valley Hall effect [13], quantum spin-Hall effect and quantum valley Hall effect [14], or the graphene nanoroad between two structurally different boron-nitride sheets [15]. One widely explored system is the zero-line modes (ZLMs) occurred at the interface, across which the valley Chern numbers varies [1,16]. These ZLMs are protected from long-range scattering potential by large momentum separation and exhibit zero bend resistance in the absence of atomic defects [15,17]. Such modes are experimentally feasible [19,20] in Bernal stacked multilayer graphenes with out-of-plane electric field [21][22][23] and have attracted much attention from both theoreticians [1,17,18,[24][25][26][27][28][29][30][31][32][33][34] and experimentalists [19,20,35].

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We demonstrate new mechanisms for gate tunable current partition at topological zero-line intersections in a graphene-based current splitter. Based on numerical calculations of the non- a perpendicular magnetic field on the order of a few Teslas allows for carrier sign dependent current routing. In the zero-field limit the control on current routing and partition can be achieved within a range of 10%-90% of the total incoming current by tuning the carrier density at tilted intersections, or by modifying the relative magnitude of the bulk band gaps via gate voltage. We discuss the implications of our findings in the design of topological zero-line networks where finite orbital magnetic moments are expected when the current partition is asymmetric.

Vertex-and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0-resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

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The Economic and Technological Development Zone (ETDZ) is a critical urban economic and functional area. Inefficient land exploitation and insufficient supervision have led to a great waste of land resources. The timely and precise extraction of residential and industrial building type, area and density information is urgently needed and essential for sustainable land use development. This study attempted to discriminate residential and industrial buildings by integrating LiDAR data, a lacunarity algorithm, object-based segmentation and a decision tree classifier. All buildings were identified using a normalized digital surface model (nDSM) and object-based segmentation. Lacunarity features and a decision tree classifier were then adopted to discriminate building types. An accuracy assessment indicated that the proposed method can effectively identify residential and industrial buildings. The overall classification accuracy was greater than 85.67% for both of the two test blocks. A comparison with results derived from the nDSM alone showed a significantly improved classification accuracy using the KHAT statistics from a Kappa analysis. The results not only confirmed the applicability and effectiveness of the combined method but also provided fundamental information for evaluating land use in the ETDZ.