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Floquet engineering with periodic driving as a powerful tool for designing desirable topological states has been the subject of intense recent studies. Here, we present the application of Floquet engineering to investigate evolution of topological triple fermions under irradiation of circularly polarized light (CPL), a phenomenon that currently remains a mystery. By using first-principles calculations and Floquet theorem, we demonstrate that WC-type TiO and its analogues are promising candidates for Floquet engineering of triple fermions. The symmetry analysis reveals that the electric field of CPL can break the specific symmetries, such as the time-reversal symmetry and its combination of spatial symmetries, inducing a transition to a flexibly controllable Weyl semimetallic phase. The survived spatial symmetry, controlled by light, guarantees that the Weyl nodes are located along the high-symmetry line or in high-symmetry planes in momentum space. Our findings focusing on Floquet engineering in realistic materials featured by triple fermions would facilitate both theoretical and experimental interest., Comment: 6 pages, 4 figures

The interaction between light and non-trivial energy band topology allows for the precise manipulation of topological quantum states, which has attracted intensive interest in condensed matter physics. In this work, using first-principles calculations, we studied the topological transition of ferromagnetic (FM) MnBi$_2$Te$_4$ upon irradiation with circularly polarized light (CPL). We revealed that the MnBi$_2$Te$_4$ can be driven from an FM insulator to a Weyl semimetal with a minimum number of Weyl points, i.e., two Weyl points in systems without time-reversal symmetry. More importantly, in FM MnBi$_2$Te$_4$ with out-of-plane easy magnetization axis, we found that the band dispersion of the WP evolves from Type-II to Type-III and finally to Type-I when the light intensity increases. Moreover, we show that the profile of the characteristic Fermi arc of Weyl semimetal phase is sensitive to changes in light intensity, which enables efficient manipulation of the Fermi arc length of FM MnBi$_2$Te$_4$ in experiments. In addition, for FM MnBi$_2$Te$_4$ with in-plane easy magnetization axis, the system becomes a type I Weyl semimetal under CPL irradiation. With controllable band dispersion, length of Fermi arc, and minimum number of WPs, our results indicate that CPL-irradiated FM MnBi$_2$Te$_4$ is an ideal platform to study novel transport phenomena in Weyl semimetals with distinct band dispersion.

Recent studies have attracted widespread attention on magnet-superconductor hybrid systems with emergent topological superconductivity. Here, we present the Floquet engineering of realistic two-dimensional topological nodal-point superconductors that are composed of antiferromagnetic monolayers in proximity to an s-wave superconductor. We show that Floquet chiral topological superconductivity arises naturally due to light-induced breaking of the effective time-reversal symmetry. More strikingly, we find that the Floquet chiral topological superconducting phases can be flexibly controlled by irradiating elliptically polarized light, with the photon-dressed quasi-energy spectrum carrying different Chern numbers. Such optically switchable topological transition is attributed to the simultaneous creations (or annihilations) of valley pairs. Our findings provide a feasible approach for achieving the Floquet chiral topological superconductivity with flexible tunability, which would draw extensive attention in experiments.

The realization of single-pair chiral fermions in Weyl systems remains challenging in topology physics, especially for the systems with higher chiral charges $C$. In this work, based on the symmetry analysis, low-energy effective model, and first-principles calculations, we identify the single-pair high-fold fermions in chiral cubic lattices. We first derive the minimal lattice model that exhibits a single pair of Weyl points with the opposite chiral charges of $C$ = $\pm{2}$. Furthermore, we show the ultralight chiral crystal P4$_3$32-type LiC$_2$ and its mirror enantiomer as high-quality candidate materials, which exhibit large energy windows to surmount the interruption of irrelevant bands. Since two enantiomers are connected by the mirror symmetry, we observe the opposite chiral charges $C$ and the reversal of the Fermi arc velocities, showing the correspondence of chirality in the momentum space and the real space. In addition, we also reveal type-II van Hove singularities on the helicoid surfaces, which may induce chirality-locked charge density waves on the crystal surface. Our work not only provides a promising platform for controlling the sign of topological charge through the structural chirality but also facilitates the exploration of electronic correlations on the surface of ultralight chiral crystals., Comment: 8 pages, 5 figures

Based on first-principles calculations, we predict that nitrogen atoms can assemble into a single-layer double kagome lattice (DKL), which possesses the characteristics of an intrinsic direct band gap semiconductor, boasting a substantial band gap of 3.460 eV. The DKL structure results in a flat valence band with high effective mass and a conduction band with small effective mass comes from Dirac electrons. These distinctive band edges lead to a significant disparity in carrier mobilities, with electron mobility being four orders of magnitude higher than that of holes. The presence of flat band in DKL-nitrogene can be further discerned through the enhanced optical absorption and correlated effects as exemplified by hole-induced ferromagnetism. Interestingly, DKL-nitrogene exhibits inherent second-order topological states, confirmed by a non-trivial second Stiefel-Whitney number and the presence of 1D floating edge states and 0D corner states within the bulk band gap. Additionally, the robust N-N bonds and the lattice's bending structure ensure thermodynamic stability and mechanical stiffness. These attributes make it exceptionally stable for potential applications in nano-devices., Comment: 9 pages, 4 figures

Topological superconductors are a class of unconventional superconducting materials featuring sub-gap zero-energy Majorana bound modes that hold promise as a building block for topological quantum computing. In this work, we study the realization of second-order topology that defines anomalous gapless boundary modes in a two-orbital superconductor with spin-orbital couplings. We reveal a time-reversal symmetry-breaking second-order topological superconducting phase with $d+id$-wave orbital-dependent paring without the need for the external magnetic field. Remarkably, this orbital-active $d$-wave paring gives rise to anomalous zero-energy Majorana corner modes, which is in contrast to conventional chiral $d$-wave pairing, accommodating one-dimensional Majorana edge modes. Our work not only reveals a unique mechanism of time-reversal symmetry breaking second-order topological superconductors but also bridges the gap between second-order topology and orbital-dependent pairings., Comment: 5+ pages, 5 figures

Xenes, two-dimensional (2D) monolayers composed of a single element, with graphene as a typical representative, have attracted widespread attention. Most of the previous Xenes, X from group-IIIA to group-VIA elements have bonding characteristics of covalent bonds. In this work, we for the first time unveil the pivotal role of a halogen bond, which is a distinctive type of bonding with interaction strength between that of a covalent bond and a van der Waals interaction, in 2D group-VIIA monolayers. Combing the ingenious non-edge-to-edge tiling theory and state-of-art ab initio method with refined local density functional M06-L, we provide a precise and effective bottom-up construction of 2D iodine monolayer sheets, iodinenes, primarily governed by halogen bonds, and successfully design a category of stable iodinenes, encompassing herringbone, Pythagorean, gyrated truncated hexagonal, i.e. diatomic-kagome, and gyrated hexagonal tiling pattern. These iodinene structures exhibit a wealth of properties, such as flat bands, nontrivial topology, and fascinating optical characteristics, offering valuable insights and guidance for future experimental investigations. Our work not only unveils the unexplored halogen bonding mechanism in 2D materials but also opens a new avenue for designing other non-covalent bonding 2D materials., Comment: 6 pages, 4 figures

The existence of fractionally quantized topological corner states serves as a key indicator for two-dimensional second-order topological insulators (SOTIs), yet has not been experimentally observed in realistic materials. Here, based on effective model analysis and symmetry arguments, we propose a strategy for achieving SOTI phases with in-gap corner states in two dimensional systems with antiferromagnetic (AFM) order. We uncover by a minimum lattice model that the band topology originates from the interplay between intrinsic spin-orbital coupling and interlayer AFM exchange interactions. Using first principles calculations, we show that the 2D AFM SOTI phases can be realized in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_{m}$ films. Moreover, we demonstrate that the nontrivial corner states are linked to rotation topological invariants under three-fold rotation symmetry $C_3$, resulting in $C_3$-symmetric SOTIs with corner charges fractionally quantized to $\frac{n}{3} \lvert e \rvert $ (mod $e$). Due to the great recent achievements in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_{m}$ systems, our results providing reliable material candidates for experimentally accessible AFM higher-order band topology would draw intense attentions., Comment: 7 pages, 4 figures

Two-dimensional checkerboard lattice, the simplest line-graph lattice, has been intensively studied as a toy model, while material design and synthesis remain elusive. Here, we report theoretical prediction and experimental realization of the checkerboard lattice in monolayer Cu$_2$N. Experimentally, monolayer Cu$_2$N can be realized in the well-known N/Cu(100) and N/Cu(111) systems that were previously mistakenly believed to be insulators. Combined angle-resolved photoemission spectroscopy measurements, first-principles calculations, and tight-binding analysis show that both systems host checkerboard-derived hole pockets near the Fermi level. In addition, monolayer Cu$_2$N has outstanding stability in air and organic solvents, which is crucial for further device applications., Comment: Nano Letters, in press

Three-dimensional obstructed atomic insulators (OAIs) are characterized by the appearance of floating surface states (FSSs) at specific surfaces. Benefiting from this feature, our study here shows the presence of abundant surface-semimetal phases in 3D OAIs. The symmetries of obstructed Wannier charge centers ensure the degeneracy of such FSSs at high-symmetry points or invariant lines in the surface Brillouin zone. Utilizing topological quantum chemistry theory, we identify a carbon allotrope with a body-centered tetragonal structure, named bct-C20, as an ideal candidate for realizing different kinds of surface-semimetal phases. For the (001)surface of bct-C20, there are four in-gap FSSs, and these four FSSs form two kinds of surface Dirac cones, i.e., topological Dirac cones with linear dispersion and symmetry-enforced quadratic Dirac cones. The band topology of a surface Dirac cone is captured by the effective surface Hamiltonian and the emergence of hinge states. Moreover, the existence of the surface-nodal-line state is also discussed. This work reports an approach to obtain d-dimensional semimetal phases from the surface states of (d + 1)-dimensional systems, which is of great significance for the studies in revealing topological states and their practical applications in high-dimensional crystals., Comment: 6 pages, 4 figures

Topological phonon states in crystalline materials have attracted significant research interests due to their importance for fundamental physical phenomena, yet their implication on phonon thermal transport remains largely unexplored. Here, we use density functional theory calculations and symmetry analyses to explore topological phonon phase transitions under uniaxial strains and their tuning effects on thermal transport in titanium monoxide (TiO). Our calculation shows that application of 10% tension significantly diminishes lattice thermal conductivity of TiO by 77% and 66% along the a and c axes, respectively, at room temperature. This suppression is found to result largely from the breaking of symmetry protected degeneracy of acoustic branches, which induces a substantial enhancement of phonon scattering phase space due to the easier fulfillment of scattering selection rules. Our study provides evidence for the importance of phononic band topology in modulating thermal conductivity and offers a promising route towards controlling solid-state heat transport.

Hydrogen, a simple and magic element, has attracted increasing attention for its effective incorporation within solids and powerful manipulation of electronic states. Here, we show that hydrogenation tackles common problems in two-dimensional borophene, e.g., stability and applicability. As a prominent example, a ladder-like boron hydride sheet, named as 2D ladder polyborane, achieves the desired outcome, enjoying the cleanest scenario with an anisotropic and tilted Dirac cone, that can be fully depicted by a minimal two-band tight-binding model. Introducing external fields, such as an electric field or a circularly-polarized light field can effectively induce distinctive massive Dirac fermions, whereupon four types of multi-field-driven topological domain walls hosting tunable chirality and valley indexes are further established. Moreover, the 2D ladder polyborane is thermodynamically stable at room temperature and supports highly switchable Dirac fermions, providing an ideal platform for realizing and exploring the various multi-field-tunable electronic states., Comment: 25 pages, 5 figures

The distinct over-tilting of band crossings in topological semimetal generates the type-I and typeII classification of Dirac/Weyl and nodal-line fermions, accompanied by the exotic electronic and magnetic transport properties. In this work, we propose a concept of hybrid nodal-chain semimetal, which is identified by the linked type-I and type-II nodal rings in the Brillouin zone. Based on first-principles calculations and swarm-intelligence structure search technique, a new ternary nitride MgCaN2 crystal is proposed as the first candidate to realize a novel 3D hybrid nodal-chain state. Remarkably, a flat band is emergent as a characteristic signature of such a hybrid nodal-chain along certain direction in the momentum space, thereby serving a platform to explore the interplay between topological semimetal state and flat band. Moreover, the underlying protection mechanism of the hybrid nodal-chain is revealed by calculating the mirror Z2 invariant and developing a k.p effective Hamiltonian. Additionally, considerable drumhead-like surface states with unique connection patterns are illustrated to identify the non-trivial band topology, which may be measured by future experiments., Comment: 9 pages, 6 figures

Phonons play a crucial role in many properties of solid state systems, such as thermal and electrical conductivity, neutron scattering and associated effects or superconductivity. Hence, it is expected that topological phonons will also lead to rich and unconventional physics and the search of materials hosting topological phonons becomes a priority in the field. In electronic crystalline materials, a large part of the topological properties of Bloch states can be indicated by their symmetry eigenvalues in reciprocal space. This has been adapted to the high-throughput calculations of topological materials, and more than half of the stoichiometric materials on the databases are found to be topological insulators or semi-metals. Based on the existing phonon materials databases, here we have performed the first catalogue of topological phonon bands for more than ten thousand three-dimensional crystalline materials. Using topological quantum chemistry, we calculate the band representations, compatibility relations, and band topologies of each isolated set of phonon bands for the materials in the phonon databases. We have also calculated the real space invariants for all the topologically trivial bands and classified them as atomic and obstructed atomic bands. In particular, surface phonon modes (dispersion) are calculated on different cleavage planes for all the materials. Remarkably, we select more than one thousand "ideal" non-trivial phonon materials to fascinate the future experimental studies. All the data-sets obtained in the the high-throughput calculations are used to build a Topological Phonon Database., Comment: 8+535 pages, 187 figures, 21 tables. The Topological Phonon Database is available at https://www.topologicalquantumchemistry.com/topophonons or https://www.topologicalquantumchemistry.fr/topophonons

Higher-order topological insulators (HOTIs) are described by symmetric exponentially decayed Wannier functions at some $necessary$ unoccupied Wyckoff positions and classified as obstructed atomic insulators (OAIs) in the topological quantum chemistry (TQC) theory. The boundary states in HOTIs reported so far are often fragile, manifested as strongly depending on crystalline symmetries and cleavage terminations in the disk or cylinder geometry. Here, using the TQC theory, we present an intuitive argument about the connection between the obstructed Wannier charge centers of OAIs and the emergence of robust corner states in two-dimensional systems. Based on first-principles calculations and Real Space Invariant theory, we extend the concept of OAIs to phonon systems and thereby predict that the robust corner states can be realized in the phonon spectra of $MX_3$ ($M$=Bi, Sb, As, Sc, Y; $X$=I, Br, Cl) monolayers. The phonon corner modes in different shapes of nano-disks are investigated, and their robustness facilitates the detection in experiments and further applications. This work suggests a promising avenue to explore more attractive features of higher-order band topology.

More is left to do in the field of flat bands besides proposing theoretical models. One unexplored area is the flat bands featured in the van der Waals (vdW) materials. Exploring more flat-band material candidates and moving the promising materials toward applications have been well recognized as the cornerstones for the next-generation high-efficiency devices. Here, we utilize a powerful high-throughput tool to screen desired vdW materials based on the Inorganic Crystal Structure Database. Through layers of filtration, we obtained 861 potential monolayers from 4997 vdW materials. Significantly, it is the first example to introduce flat-band electronic properties in the vdW materials and propose three families of representative flat-band materials by mapping two-dimensional (2D) flat-band lattice models. Unlike existing screening schemes, a simple, universal rule, i.e., 2D flat-band score criterion, is first proposed to efficiently identify 229 high-quality flat-band candidates, and guidance is provided to diagnose the quality of 2D flat bands. All these efforts to screen experimental available flat-band candidates will certainly motivate continuing exploration towards the realization of this class of special materials and their applications in material science.

Topological band insulators and (semi-) metals can arise out of atomic insulators when the hopping strength between electrons increases. Such topological phases are separated from the atomic insulator by a bulk gap closing. In this work, we show that in many (magnetic) space groups, the crystals with certain Wyckoff positions and orbitals being occupied must be semimetal or metals in the atomic limit, e.g. the hopping strength between electrons is infinite weak but not vanishing, which then are termed atomic (semi-)metals (ASMs). We derive a sufficient condition for realizing ASMs in spinless and spinful systems. Remarkably, we find that increasing the hopping strength between electrons may transform an ASM into an insulator with both symmetries and electron fillings of crystal are preserved. The induced insulators inevitably are topologically non-trivial and at least are obstructed atomic insulators (OAIs) that are labeled as trivial insulator in topological quantum chemistry website. Particularly, using silicon as an example, we show ASM criterion can discover the OAIs missed by the recently proposed criterion of filling enforced OAI. Our work not only establishes an efficient way to identify and design non-trivial insulators but also predicts that the group-IV elemental semiconductors are ideal candidate materials for OAI., Comment: 4 pages, 4 figures

Topological electronic flatten bands near or at the Fermi level are a promising avenue towards unconventional superconductivity and correlated insulating states. However, the related experiments are mostly limited to the engineered materials, such as moire systems. Here we present a catalogue of all the three-dimensional stoichiometric materials with flat bands around the Fermi level that exist in nature. We consider 55,206 materials from the Inorganic Crystal Structure Database catalogued using the Topological Quantum Chemistry website which provides their structural parameters, space group (SG), band structure, density of states and topological characterization. We combine several direct signatures and properties of band flatness to a high-throughput analysis of all crystal structures. In particular, we identify materials hosting line-graph or bipartite sublattices - either in two or three dimensions - likely leading to flat bands. From this trove of information, we create the Materials Flatband Database website, a powerful search engine for future theoretical and experimental studies. We use it to extract a curated list of 2,379 materials, with among them 345 promising candidates, potentially hosting flat bands whose charge centers are not strongly localized on the atomic sites. We showcase five representative materials: KAg[CN]2 in SG 163 $(P\bar{3}1c)$, Pb2Sb2O7 in SG 227 $(Fd\bar{3}m)$, Rb2CaH4 in SG 139 $(I4/mmm)$, Ca2NCl in SG 166 $(R\bar{3}m)$ and WO3 in SG 221 $(Pm\bar{3}m)$. We provide a theoretical explanation for the origin of their flat bands close to the Fermi energy using the $S$-matrix method introduced in a parallel work [Calugaru et al., Nature Physics 18, 185 (2022)]., Comment: 16+162 pages, 21 tables, 71 figures, published version. This updated version contains an additional appendix providing ferromagnetic calculations for four flat-band materials. Acknowledgements have been fixed

Topological Weyl semimetals have been attracting broad interest. Recently, a new type of Weyl point with topological charge of $4$, termed as charge-4 Weyl point (C-4 WP), was proposed in spinless systems. Here, we show the minimum symmetry requirement for C-4 WP is point group $T$ together with ${\cal T}$ symmetry or point group $O$. We establish a minimum tight-binding model for C-4 WP on a cubic lattice with time-reversal symmetry and without spin-orbit coupling effect. This lattice model is a two-band one, ontaining only one pair of C-4 WPs with opposite chirality around Fermi level. Based on both the low-energy effective Hamiltonian and the minimum lattice model, we investigate the electronic, optical and magnetic properties of C-4 WP. Several chirality-dependent properties are revealed, such as chiral Landau bands, quantized circular photogalvanic effect and quadruple-helicoid surface arc states. Furthermore, we predict that under symmetry breaking, various exotic topological phases can evolve out of C-4 WPs. Our work not only reveals several interesting phenomena associate to C-4 WPs, but also provides a simple and ideal lattice model of C-4 WP, which will be helpful for the subsequent study on C-4 WPs.

We propose a new concept of two-dimensional (2D) Dirac semiconductor which is characterized by the emergence of fourfold degenerate band crossings near the band edge and provide a generic approach to realize this novel semiconductor in the community of material science. Based on the first-principle calculations and symmetry analysis, we discover recently synthesised triple-layer (TL)-BiOS2 is such Dirac semiconductor that features Dirac cone at X/Y point, protected by nonsymmorphic symmetry. Due to sandwich-like structure, each Dirac fermion in TL-BiOS2 can be regarded as a combination of two Weyl fermions with opposite chiralities, degenerate in momentum-energy space but separated in real space. Such Dirac semiconductor carries layer-dependent helical spin textures that never been reported before. Moreover, novel topological phase transitions are flexibly achieved in TL-BiOS2: (i) an vertical electric field can drive it into Weyl semiconductor with switchable spin polarization direction, (ii) an extensive strain is able to generate ferroelectric polarization and actuate it into Weyl nodal ring around X point and into another type of four-fold degenerate point at Y point. Our work extends the Dirac fermion into semiconductor systems and provides a promising avenue to integrate spintronics and optoelectronics in topological materials., Comment: 5 pages, 3 figures

Manipulating the spin degrees of freedom of electrons affords an excellent platform for exploring novel quantum states in condensed-matter physics and material science. Based on first-principles calculations and analysis of crystal symmetries, we propose a fully spin-polarized composite semimetal state, which is combined with the one-dimensional nodal lines and two-dimensional nodal surfaces, in the half-metal material CaFeO$_3$. In the nodal line-surface states, the Baguenaudier-like nodal lines feature six rings linked together, which are protected by the three independent symmetry operations:$\mathcal{PT}$, $\mathcal{M}_{y}$, and $\mathcal{\widetilde{M}}_{z}$. Near the Fermi level, the spin-polarized nodal surface states are guaranteed by the joint operation $\mathcal{T}\mathcal{S}_{2i}$ in the $k_{i(i=x,y,z)}=\pi$ plane. Furthermore, high-quality CaFeO$_3$ harbors ultra-clean energy dispersion, which is rather robust against strong triaxial compressional strain and correlation effect. The realization of the Weyl nodal line-surface half-metal presents great potential for spintronics applications with high speed and low power consumption., Comment: 5 figures + 1 table

We propose a class of nodal line semimetals that host an eight-fold degenerate double Dirac nodal line (DDNL) with negligible spin-orbit coupling. We find only 5 of the 230 space groups host the DDNL. The DDNL can be considered as a combination of two Dirac nodal lines, and has a trivial Berry phase. This leads to two possible but completely different surface states, namely, a torus surface state covering the whole surface Brillouin zone and no surface state at all. Based on first-principles calculations, we predict that the hydrogen storage material LiBH is an ideal DDNL semimetal, where the line resides at Fermi level, is relatively flat in energy, and exhibits a large linear energy range. Interestingly, both the two novel surface states of DDNL can be realized in LiBH. Further, we predict that with a magnetic field parallel to DDNL, the Landau levels of DDNL are doubly degenerate due to Kramers-like degeneracy and have a doubly degenerate zero-mode.

The geometric properties of a lattice can have profound consequences on its band spectrum. For example, symmetry constraints and geometric frustration can give rise to topologicially nontrivial and dispersionless bands, respectively. Line-graph lattices are a perfect example of both of these features: their lowest energy bands are perfectly flat, and here we develop a formalism to connect some of their geometric properties with the presence or absence of fragile topology in their flat bands. This theoretical work will enable experimental studies of fragile topology in several types of line-graph lattices, most naturally suited to superconducting circuits., Comment: 8+25 pages, 3+19 figures, 2+3 tables

Topological flat bands, such as the band in twisted bilayer graphene, are becoming a promising platform to study topics such as correlation physics, superconductivity, and transport. In this work, we introduce a generic approach to construct two-dimensional (2D) topological quasi-flat bands from line graphs and split graphs of bipartite lattices. A line graph or split graph of a bipartite lattice exhibits a set of flat bands and a set of dispersive bands. The flat band connects to the dispersive bands through a degenerate state at some momentum. We find that, with spin-orbit coupling (SOC), the flat band becomes quasi-flat and gapped from the dispersive bands. By studying a series of specific line graphs and split graphs of bipartite lattices, we find that (i) if the flat band (without SOC) has inversion or $C_2$ symmetry and is non-degenerate, then the resulting quasi-flat band must be topologically nontrivial, and (ii) if the flat band (without SOC) is degenerate, then there exists an SOC potential such that the resulting quasi-flat band is topologically nontrivial. This generic mechanism serves as a paradigm for finding topological quasi-flat bands in 2D crystalline materials and meta-materials.

Topological semimetals in ferromagnetic materials have attracted enormous attention due to the potential applications in spintronics. Using the first-principles density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline-metal monochalcogenide $MX$ ($M$ = Li, Na, K, Rb, Cs; $X$ = S, Se, Te) monolayers. The half-metallic ferromagnetism can be established in $MX$ monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin-orbit coupling owing to the symmetry protection provided by the $\mathcal{M}_{z}$ mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The $MX$ monolayers hosting rich topological phases thus offer an excellent materials platform for realizing the advanced spintronics concepts.

Based on $ab$ $initio$ calculations and low-energy effective $k{\cdot}p$ model, we propose a type of Weyl nodal point-line fermion, composed of 0D Weyl points and 1D Weyl nodal line, in ferromagnetic material Eu$_5$Bi$_3$. In the absence of spin-orbital coupling (SOC), the spin-up bands host a pair of triply degenerate points together with a unique bird-cage like node structure. In the presence of SOC with (001) magnetization, each triplet point splits into a double Weyl point and a single Weyl point accompanied by two nodal rings, forming two sets of Weyl nodal point-line fermions near the Fermi level. The novel properties of Weyl nodal point-line fermion are explored by revealing the unusual Berry curvature field and demonstrating the pinned chiral surface states with exotic Fermi arcs at different planes. Moreover, a large anomalous Hall conductivity of -260 ($\hbar$/$e$)($\Omega$cm)$^{-1}$ parallel to [001] direction is predicted. Our work offers a new perspective for exploring novel topological semimetal states with diverse band-crossing dimensions, and provides an ideal material candidate for future experimental realiztion., Comment: 19 pages, 4 figures

Based on first-principles calculations and symmetry analysis, we report that the three-dimensional (3D) nodal line (NL) semimetal phases can be realized in the lead dioxide family (\emph{$\alpha$}-PbO$_2$ and \emph{$\beta$}-PbO$_2$) and its derivatives. The \emph{$\beta$}-PbO$_2$ features two orthogonal nodal rings around the Fermi level, protected by the mirror reflection symmetry. The effective model is developed and the related parameters are given by fitting with the HSE06 band structures. The NLs mainly come from the $p$ orbitals of the light element O and are rather robust against such tiny spin-orbit coupling. The NL phase of the \emph{$\alpha$}-PbO$_2$ can be effectively tailored by strain, making a topological phase transition between a semiconductor phase and a NL phase. In addition, the exploration of \emph{$\beta$}-PbO$_2$ derivatives (i.e. \emph{$\beta$}-PbS$_2$ and \emph{$\beta$}-PbSe$_2$) and confirmation of their topological semimetallicity greatly enrich the NL semimetal family. These findings pave a route for designing topological NL semimetals and spintronic devices based on realistic PbO$_2$ family., Comment: 7 pages, 7 figures

The conventional k.p method fails to capture the full and essential physics of many symmetry enriched multiple nodal line structures in the three dimensional Brillouin zone. Here we present a new and systematical method to construct the effective lattice model of mirror symmetry protected three-dimensional multiple nodal line semimetals, when the spin-orbit interaction is ignored. For systems with a given pair of perpendicular nodal rings, we obtain all the effective lattice models and eleven inequivalent nodal line Fermi surfaces together with their related constraints. By means of first-principles calculations, we first propose a family of real materials, beta phase of ternary nitrides X2GeN2 (X = Ca; Sr; Ba), that support one kind of these novel Fermi surfaces. Therefore, our work deepens the understanding of the nodal line structures and promotes the experimental progress of topological nodal line semimetals.

Electrical tunability of spin polarization has been a focus in spintronics. Here, we report that the trigonal warping (TW) effect, together with spin-orbit coupling (SOC), can lead to two distinct magnetoelectric effects in low-dimensional systems. Taking graphene with Rashba SOC as example, we study the electronic properties and spin-resolved scattering of system. It is found that the TW effect gives rise to a terraced spin texture in low-energy bands and can render significant spin polarization in the scattering, both resulting in an efficient electric control of spin polarization. Our work unveils not only SOC but also the TW effect is important for low-dimensional spintronics.

Two-dimensional (2D) topological insulators (TIs) have attracted tremendous research interest from both theoretical and experimental fields in recent years. However, it is much less investigated in realizing node line (NL) semimetals in 2D materials.Combining first-principles calculations and $k \cdot p$ model, we find that NL phases emerge in p-CS$_2$ and p-SiS$_2$, as well as other pentagonal IVX$_2$ films, i.e. p-IVX$_2$ (IV= C, Si, Ge, Sn, Pb; X=S, Se, Te) in the absence of spin-orbital coupling (SOC). The NLs in p-IVX$_2$ form symbolic Fermi loops centered around the $\Gamma$ point and are protected by mirror reflection symmetry. As the atomic number is downward shifted, the NL semimetals are driven into 2D TIs with the large bulk gap up to 0.715 eV induced by the remarkable SOC effect.The nontrivial bulk gap can be tunable under external biaxial and uniaxial strain. Moreover, we also propose a quantum well by sandwiching p-PbTe$_2$ crystal between two NaI sheets, in which p-PbTe$_2$ still keeps its nontrivial topology with a sizable band gap ($\sim$ 0.5 eV). These findings provide a new 2D materials family for future design and fabrication of NL semimetals and TIs., Comment: 6 pages, 5 figures,2 tables