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Quantum anomalous Hall effect (QAHE),

The axion insulator which may exhibit an exotic quantized magnetoelectric effect is one of the most interesting quantum phases predicted for the three-dimensional topological insulator (TI). The axion insulator state is expected to show up in magnetically doped TIs with magnetizations pointing inwards and outwards from the respective surfaces. Towards the realization of the axion insulator, we here engineered a TI heterostructure in which magnetic ions (Cr) are modulation-doped only in the vicinity of the top and bottom surfaces of the TI ((Bi,Sb)Te) film. A separation layer between the two magnetic layers weakens interlayer coupling between them, enabling the magnetization reversal of individual layers. We demonstrate the realization of the axion insulator by observing a zero Hall plateau (ZHP) (where both the Hall and longitudinal conductivity become zero) in the electric transport properties, excluding the other possible origins for the ZHP. The manifestation of the axion insulator can lead to a new stage of research on novel magnetoelectric responses in topological matter.

Geometry, both in momentum and in real space, plays an important role in the electronic dynamics of condensed matter systems. Among them, the Berry phase associated with nontrivial geometry can be an origin of the transverse motion of electrons, giving rise to various geometric e ects such as the anomalous 1 , spin 2 and topological Hall e ects 3-6 . Here, we report two unconventional manifestations of Hall physics: a sign-reversal of the anomalous Hall e ect, and the emergence of a topological Hall e ect in magnetic/non-magnetic topological insulator heterostructures, Cr x (Bi 1−y Sb y ) 2−x Te 3 /(Bi 1−y Sb y ) 2 Te 3 . The sign-reversal in the anomalous Hall e ect is driven by a Rashba splitting at the bulk bands, which is caused by the broken spatial inversion symmetry. Instead, the topological Hall e ect arises in a wide temperature range below the Curie temperature, in a region where the magnetic-field dependence of the Hall resistance largely deviates from the magnetization. Its origin is assigned to the formation of a Néel-type skyrmion induced by the Dzyaloshinskii-Moriya interaction.The geometry and topology in Hilbert space constitute a central issue in quantum physics, which has recently also shed a new light on the electronic states in solids. The wavefunctions are characterized by the Berry connection between two neighbouring points, in both momentum space and real space, which plays the role of the vector potential leading to the concept of emergent electromagnetic field (EEMF). Global topology of the manifold in Hilbert space is represented by topological integers. As integers cannot change continuously, it gives a certain stability to the system. For example, the Chern number, which is given by the integral of the emergent magnetic field over the first Brillouin zone, protects the surface or edge states supporting the dissipationless current flows (that is, bulk-edge correspondence). The one-dimensional chiral edge mode in the quantum Hall effect and the Dirac surface state in three-dimensional topological insulators (TI; refs 7,8) are the representative examples of this phenomenon. In addition, it has recently been proposed 9,10 and observed 11-14 that TI with doped magnetic ions-namely, Cr or V-produces the quantized version of the anomalous Hall effect (AHE) in the absence of an external magnetic field.Topology in real space, on the other hand, is exemplified by skyrmion spin texture 15-17 found in chiral-lattice magnets such as MnSi (ref. 15) or Fe 1−x Co x Si (ref. 16). Here, the solid angle subtended by the spins forms an emergent magnetic field in real space, and its integral over the two-dimensional space defines a topological integer called the skyrmion number. Namely, the skyrmion number counts the number of times the spin direction wraps around a unit sphere. This integer protects the skyrmion from annihilation and allows it to behave as a single particle.Thus far, the EEMF in momentum and real spaces have mostly been studied separately. Recent advances in fabricating artificial stru...