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Topology can manifest topological magnetism (e.g., skyrmion and bimeron) in real space and quantum anomalous Hall (QAH) state in momentum space, which have changed the modern conceptions of matter phase. While the topologies in different spaces are widely studied separately, their coexistence and coupling in single phase is seldomly explored. Here, we report a novel phenomenon that arises from the interaction of topological magnetism and band topology, the multi-space topology, in 2D ferromagnetic lattice. Based on continuum theory and tight-binding model, we reveal that the interconnection between skyrmion/bimeron and QAH state generates distinctive localized chiral bound states (CBSs). With moderating topological magnetism through magnetic field, the multi-space topologies accompanied with different CBSs can be reversed, facilitating the coupling of multi-space topologies. By performing firstprinciples and atomic spin model simulations, we further demonstrate such multi-space topologies and their coupling in monolayer Cr2NSb. These results represent an important step towards the development of multispace topological phenomena in 2D lattice.

Topological magnetism in low-dimensional systems is of fundamental and practical importance in condensed-matter physics and material science. Here, using first-principles and Monte-Carlo simulations, we present that multiple topological magnetism (i.e., skyrmion and bimeron) can survive in van der Waals Heterostructure of MnTe2ZrS2. Arising from interlayer coupling, MnTe2ZrS2 can harbor a large Dzyaloshinskii-Moriya interaction. This, combined with ferromagnetic exchange interaction, yields an intriguing skyrmion phase consisting of sub-10 nm magnetic skyrmions under a tiny magnetic field of 75 mT. Meanwhile, upon harnessing a small electric field, magnetic bimeron can be observed in MnTe2ZrS2 as well, suggesting the existence of multiple topological magnetism. Through interlayer sliding, both topological spin textures can be switched on-off, suggesting their stacking-dependent character. In addition, the impacts of d and Keff on these spin textures are revealed, and a dimensionless parameter is utilized to describe their joint effect. These explored phenomena and insights not only are useful for fundamental research in topological magnetism, but also enable novel applications in nanodevices.

Layer-polarized anomalous Hall effect (LP-AHE) is an attractive phenomenon in condensed-matter physics from the standpoints of both fundamental interest and device applications. The current LP-AHE research is based on the extrinsic paradigm of using external electric fields, in which the generation and control of LP-AHE are not straightforward. Here, we propose a novel mechanism that realizes intrinsic LP-AHE in bilayer lattices, through the mediation of sliding physics and Berry curvature. Moreover, this mechanism could render the LP-AHE in a controllable and reversable fashion. We analyze the symmetry requirements for a system to host such intrinsic LP-AHE. Its validity is further demonstrated in a real material of bilayer MnBi2Te4. By stacking with broken inversion symmetry, the layer-locked Berry curvature enables the intrinsic LP-AHE in bilayer MnBi2Te4, and the switchable control of its LP-AHE is achieved by sliding ferroelectricity. Our work opens a significant new direction for LP-AHE and two-dimensional (2D) materials research.

Antiferromagnetic skyrmion crystal (AF-SkX), a regular array of antiferromagnetic skyrmions, is a fundamental phenomenon in the field of condensed-matter physics. So far, only very few proposals have been made to realize the AF-SkX, and most based on three-dimensional (3D) materials. Herein, using first-principles calculations and Monte-Carlo simulations, we report the identification of AF-SkX in two-dimensional lattice of Janus monolayer CrSi2N2As2. Arising from the broken inversion symmetry and strong spin-orbit coupling, large Dzyaloshinskii-Moriya interaction is obtained in Janus monolayer CrSi2N2As2. This, combined with the geometric frustration of its triangular lattice, gives rise to the skyrmion physics and long-sought AF-SkX in the presence of external magnetic field. More intriguingly, this system presents two different antiferromagnetic skyrmion phases, and such phenomenon is distinct from those reported in 3D systems. Furthermore, by contacting with Sc2CO2, the creation and annihilation of AF-SkX in Janus monolayer CrSi2N2As2 can be achieved through ferroelectricity. These findings greatly enrich the research on antiferromagnetic skyrmions.

The realization of magnetic skyrmions in nanostructures holds great promise for both fundamental research and device applications. Despite recent progress, intrinsic magnetic skyrmions in two-dimensional lattice are still rarely explored. Here, using first-principles calculations and Monte-Carlo simulations, we report the identification of spontaneous magnetic skyrmions in single-layer CrInX3 (X = Te, Se). Due to the joint effect of broken inversion symmetry and strong spin-orbit coupling, inherent large Dzyaloshinskii-Moriya interaction occurs in both systems, endowing the intriguing Neel-type skyrmions in the absence of magnetic field. By further imposing moderate magnetic field, the skyrmion phase can be obtained and is stable within a wide temperature range. Particularly for single-layer CrInTe3, the size of skyrmions is sub-10 nm and the skyrmion phase can be maintained at an elevated temperature of 180 K. In addition, the phase diagrams of their topological spin textures under the variation of magnetic parameters of D, J, and K are mapped out. Our results greatly enrich the research of 2D skyrmionics physics.

Valley-related multiple Hall effect in two-dimensional lattice is of notable interest both for its fundamental physics and for its potential applications. In this work, by means of a low energy kp model analysis, a mechanism of producing valley-related multiple Hall effect in hexagonal lattice via strain engineering is proposed, and a general picture of valley-contrasted band inversion is developed. Through first-principles calculations, this mechanism is further established in a ferromagnetic hexagonal lattice of single-layer ScI2. Single-layer ScI2 prefers in-plane magnetization and exhibits neither anomalous valley Hall effect nor valley-polarized quantum anomalous Hall effect in nature. Remarkably, these two Hall effects emerge simultaneously in this system under 4.705% tensile strain and disappear simultaneously when further increasing strain, suggesting the exotic valley-related multiple Hall effect. The underlying physical mechanism is revealed using a model analysis and is generally applicable. Our work greatly enriches the valley-related physics.

Valley-related multiple Hall effect in 2D lattice is a fundamental transport phenomenon in the fields of condensed-matter physics and material science. So far, most proposals for its realization are limited to toy models or extrinsic effects. Here, based on tight-binding model and first-principles calculations, we report the discovery of intrinsic valley-related multiple Hall effect in 2D organometallic lattice of NbTa-benzene. Protected by the breaking of both time-reversal and inversion symmetry, NbTa-benzene exhibits large valley polarization spontaneously in both the conduction and valence bands, guaranteeing the anomalous valley Hall effect. Simultaneously, because of the exchange interaction and strong spin-orbit coupling, intrinsic band inversion occurs at one valley, which ensures the valley-polarized quantum anomalous Hall effect, thus presenting the extraordinary valley-related multiple Hall effect in nature. In addition, it can be transformed into the phase with ferrovalley or quantum anomalous Hall effect solely through strain engineering. These insights not only are useful for the fundamental research in valley-related physics, but also enable a wide range of novel device applications.

Anomalous valley Hall (AVH) effect is a fundamental transport phenomenon in the field of condensed-matter physics. Usually, the research on AVH effect is mainly focused on 2D lattices with ferromagnetic order. Here, by means of model analysis, we present a general design principle for realizing AVH effect in antiferromagnetic monolayers, which involves the introduction of nonequilibrium potentials to break of PT symmetry. Using first-principles calculations, we further demonstrate this design principle by stacking antiferromagnetic monolayer MnPSe3 on ferroelectric monolayer Sc2CO2 and achieve the AVH effect. The AVH effect can be well controlled by modulating the stacking pattern. In addition, by reversing the ferroelectric polarization of Sc2CO2 via electric field, the AVH effect in monolayer MnPSe3 can be readily switched on or off. The underlying physics are revealed in detail. Our findings open up a new direction of research on exploring AVH effect.

2D materials with valley-related multiple Hall effect are both fundamentally intriguing and practically appealing to explore novel phenomena and applications, but have been largely overlooked up to date. Here, using first-principles calculations, we present that valley related multiple Hall effect can exist in single-layer VSi2P4. We identify single-layer VSi2P4 as a ferromagnetic semiconductor with out-of-plane magnetization and valley physics. Arising from the joint effect of inversion symmetry breaking and time reversal symmetry breaking, the exotic spontaneous valley polarization occurs in single-layer VSi2P4, thus facilitating the observation of anomalous valley Hall effect. Moreover, under external strain, band inversion can occur at only one of the valleys of single-layer VSi2P4, enabling the long-sought valley-polarized quantum anomalous Hall effect, and meanwhile the anomalous valley Hall effect is well preserved.. Our work not only enriches the research on valley-related multiple Hall effect, but also opens a new avenue for exploring valley-polarized quantum anomalous Hall effect.

2D ferrovalley materials that exhibit spontaneous valley polarization are both fundamentally intriguing and practically appealing to be used in valleytronic devices. Usually, the research on 2D ferrovalley materials is mainly focused on inorganic systems, severely suffering from in-plane magnetization. Here, we alternatively show by kp model analysis and high-throughput first-principles calculations that ideal spontaneous valley polarization is present in 2D organometallic lattice. We explore the design principle for organic 2D ferrovalley materials composed of (quasi-)planer molecules and transition-metal atoms in hexagonal lattice, and identify twelve promising candidates. These systems have a ferromagnetic or antiferromagnetic semiconducting state, and importantly they exhibit robust out-of-plane magnetization. The interplay between spin and valley, together with strong spin-orbit coupling of transition-metal atoms, guarantee the spontaneous valley polarization in these systems, facilitating the anomalous valley Hall effect. Our findings significantly broaden the scientific and technological impact of ferrovalley physics.