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Owing to the strong spin-orbit coupling and the related fascinating physical properties, heavy 5d transition-metals exhibit desirable application prospects. However, up to now, the 5d magnetic materials are still very limited, especially very rare for tungsten. In this work, we theoretically predict a two-dimensional multiferroic W3Cl8 monolayer. Intrinsic 5d magnetism of tungsten is activated by the W ions' fractional valence in a breathing Kagome lattice of reduced effective dimension. A coplanar Y-type antiferromagnetism composed by ferromagnetic W3 trimers is confirmed as the magnetic ground state. The spontaneous ferroelectric polarization mainly originates from the ion displacement induced by the breathing distortion of Kagome lattice. An intrinsic magneto-optical Kerr effect with sizable Kerr angle can be observed to detect this trimeric Y-type antiferromagnetism, and it depends strongly on the detailed magnetic order. Thereby, we propose a general scheme for realizing more 5d magnetism in two-dimensional multiferroic systems.

The spin-fermion model was previously successful to describe the complex phase diagrams of colossal magnetoresistive manganites and iron-based superconductors. In recent years, two-dimensional magnets have rapidly raised up as a new attractive branch of quantum materials, which are theoretically described based on classical spin models in most studies. Alternatively, here the two-orbital spin-fermion model is established as a uniform scenario to describe the ferromagnetism in a two-dimensional honeycomb lattice. This model connects the magnetic interactions with the electronic structures. Then the continuous tuning of magnetism in these honeycomb lattices can be predicted, based on a general phase diagram. The electron/hole doping, from the empty $e_{g}$ to half-filled $e_{g}$ limit, is studied as a benchmark. Our Monte Carlo result finds that the ferromagnetic $T_{C}$ reaches the maximum at the quarter-filled case. In other regions, the linear relationship between $T_{C}$ and doping concentration provides a theoretical guideline for the experimental modulations of two-dimensional ferromagnetism tuned by ionic liquid or electrical gating.

Low-dimensional ferroelectricity and polar materials have attracted considerable attentions for their fascinating physics and potential applications. Based on first-principles calculations, here we investigate the stacking modes and polar properties of a typical series of quasi-one-dimensional ferroelectrics: double-chain oxyhalides NbO$X_3$ ($X$=Cl, Br, I). The geometry of their double-chains allows both the interchain/intrachain permutation. Thus, different stacking modes of double-chains lead to a variety of ferroelectric and antiferroelectric phases in both the tetragonal and monoclinic crystals. The proximate energies of these phases may lead to multiphase coexistence in real materials, as well as the hydrostatic pressure driving structural phase transition. Their spontaneous polarizations and piezoelectricity of the ferroelectric phases are prominent, comparable to commercially used ferroelectric BaTiO$_3$ and piezoelectric ZnO, respectively. Our work demonstrates that the van der Waals NbO$X_3$ are promising materials for exploring quasi-one-dimensional ferroelectricity and antiferroelectricity., Comment: 7 pages, 5 figures

Considerable research interest has been attracted to noncollinear magnetic structures for their intriguing physics and promising applications. In this work, based on relativistic density functional theory, we reveal the interesting magnetic order and relevant properties in monolayer RuF$_4$, which can be exfoliated from its bulk phase. Although the spins on Ru ions are almost antiferromagnetically aligned between nearest-neighbors, weak ferromagnetism is generated because of the antisymmetric Dzyaloshinskii-Moriya interaction as well as the single-ion anisotropy. A prominent magneto-optical Kerr effect can be observed for this antiferromagnet, similar to those of regular strong ferromagnets. In addition, a uniaxial strain can induce a ferroelastic switching together with the in-plane rotation of spin direction, giving rise to a strong intrinsic magnetoelasticity. Our work not only suggests an alternative direction for two-dimensional magnetic materials, but also provides hints to future devices based on antiferromagnetic magnetoelastic or magneto-optical materials., Comment: 7 pages, 5 figures

Two dimensional multiferroics inherit prominent physical properties from both low dimensional materials and magnetoelectric materials, and can go beyond their three dimensional counterparts for their unique structures. Here, based on density functional theory calculations, a MXene derivative, i.e., i-MXene (Ta$_{2/3}$Fe$_{1/3}$)$_2$CO$_2$, is predicted to be a type-I multiferroic material. Originated from the reliable $5d^0$ rule, its ferroelectricity is robust, with a moderate polarization up to $\sim12.33$ $\mu$C/cm$^2$ along the a-axis, which can be easily switched and may persist above room temperature. Its magnetic ground state is layered antiferromagnetism. Although it is a type-I multiferroic material, its N\'eel temperature can be significantly tuned by the paraelectric-ferroelectric transition, manifesting a kind of intrinsic magnetoelectric coupling. Such magnetoelectric effect is originated from the conventional magnetostriction, but unexpectedly magnified by the exchange frustration. Our work not only reveals a nontrivial magnetoelectric mechanism, but also provides a strategy to search for more multiferroics in the two dimensional limit., Comment: 15 pages, 5 figures

In semi-supervised fuzzy clustering, this paper extends the traditional pairwise constraint (i.e., must-link or cannot-link) to fuzzy pairwise constraint. The fuzzy pairwise constraint allows a supervisor to provide the grade of similarity or dissimilarity between the implicit fuzzy vectors of a pair of samples. This constraint can present more complicated relationship between the pair of samples and avoid eliminating the fuzzy characteristics. We propose a fuzzy discriminant clustering model (FDC) to fuse the fuzzy pairwise constraints. The nonconvex optimization problem in our FDC is solved by a modified expectation-maximization algorithm, involving to solve several indefinite quadratic programming problems (IQPPs). Further, a diagonal block coordinate decent (DBCD) algorithm is proposed for these IQPPs, whose stationary points are guaranteed, and the global solutions can be obtained under certain conditions. To suit for different applications, the FDC is extended into various metric spaces, e.g., the Reproducing Kernel Hilbert Space. Experimental results on several benchmark datasets and facial expression database demonstrate the outperformance of our FDC compared with some state-of-the-art clustering models., Comment: 15 pages,41 figures

In solids, charge polarity can one-to-one correspond to spin polarity phenomenologically, e.g. ferroelectricity/ferromagnetism, antiferroelectricity/antiferromagnetism, and even dipole-vortex/magnetic-vortex, but ferrielectricity/ferrimagnetism kept telling a disparate story in microscopic level. Since the definition of a charge dipole involves more than one ion, there may be multiple choices for a dipole unit, which makes most ferrielectric orders equivalent to ferroelectric ones, i.e. this ferrielectricity is not necessary to be a real independent branch of polarity. In this work, by using the spherical aberration-corrected scanning transmission electron microscope, we visualize a nontrivial ferrielectric structural evolution in BaFe2Se3, in which the development of two polar sub-lattices is out-of-sync, for which we term it as irreducible ferrielectricity. Such irreducible ferrielectricity leads to a non-monotonic behavior for the temperature-dependent polarization, and even a compensation point in the ordered state. Our finding unambiguously distinguishes ferrielectrics from ferroelectrics in solids., Comment: 15 figures

Layered materials with robust magnetic ordering have been attracting significant research interest. In recent experiments, a new layered material 1$T$-CrTe$_{2}$ has been synthesized and exhibits ferromagnetism above the room temperature. Here, based on first-principles calculations, we investigate the electronic, magnetic, and transport properties of 1$T$-CrTe$_{2}$, both in the bulk and in the two-dimensional (2D) limit. We show that 1$T$-CrTe$_{2}$ can be stable in the monolayer form, and has a low exfoliation energy. The monolayer structure is an intrinsic ferromagnetic metal, which maintains a high Curie temperature above the room temperature. Particularly, we reveal interesting features in the anomalous Hall transport. We show that in the ground state, both bulk and monolayer 1$T$-CrTe$_{2}$ possess vanishing anomalous Hall effect, because the magnetization preserves one vertical mirror symmetry. The anomalous Hall conductivity can be made sizable by tuning the magnetization direction or by uniaxial strains that break the mirror symmetry. The room-temperature 2D ferromagnetism and the tunable anomalous Hall effect make the material a promising platform for nanoscale device applications., Comment: 8 pages, 9 figures

Two-dimensional (2D) materials with nodal-loop band crossing have been attracting great research interest. However, it remains a challenge to find 2D nodal loops that are robust against spin-orbit coupling (SOC) and realized in magnetic states. Here, based on first-principles calculations and theoretical analysis, we predict that monolayer MnN is a 2D nodal-loop half metal with fully spin polarized nodal loops. We show that monolayer MnN has a ferromagnetic ground state with out-of-plane magnetization. Its band structure shows half metallicity with three low-energy bands belonging to the same spin channel. The crossing between these bands forms two concentric nodal loops centered around the $\Gamma$ point near the Fermi level. Remarkably, the nodal loops and their spin polarization are robust under SOC, due to the protection of a mirror symmetry. We construct an effective model to characterize the fully polarized emergent nodal-loop fermions. We also find that a uniaxial strain can induce a loop transformation from a localized single loop circling around $\Gamma$ to a pair of extended loops penetrating the Brillouin zone., Comment: 7 pages, 5 fgures