question:What is the scope of the variables in this function? What do we mean by scope? What are the types of scope and explain how this applies to the above function.

question:What are the reasons for the scoping? One is to prevent naming conflicts. are there any other reasons? What is naming conflict? Give an example of naming conflict?

question:Find the maximum student age and the maximum score in a class given the list of student records. The list contains dictionaries with student names, ages, and scores.

question:Mixed topological semimetals driven by orbital complexity in two-dimensional ferromagnets Chengwang Niu Jan-Philipp Hanke Patrick M. Buhl Hongbin Zhang Lukasz Plucinski Daniel Wortmann Stefan Blügel Gustav Bihlmayer Yuriy Mokrousov 1 2 Two-dimensional (2D) materials are in the focus of intensive research in chemistry, materials science and physics, owing to their wide range of prominent properties that include superconductivity, magnetotransport, magneto- and thermoelectricity. The observations of quantum Hall and quantum spin Hall effects are manifestly associated with 2D materials, and they ignited comprehensive research in the area of topological condensed matter, resulting in the discovery of topological insulators (TIs) and topological crystalline insulators (TCIs) both in 2D and in three spatial dimensions (3D). Recently, research in the area of topological materials has extended to the class of topological semimetals, which notably include Dirac, Weyl, and nodal-line semimetals. These materials have been theoretically proposed and experimentally confirmed in 3D, revealing remarkable properties such as ultrahigh mobility, anomalous magnetoresistance, and nonlinear optical response. However, in 2D films the material realization of topological semimetals has been elusive so far. Although in some situations a gap closing was argued to occur due to symmetries leading to the realization of 2D Dirac and nodal-line states, a gap is usually introduced once the spin-orbit coupling (SOC) comes into play. While magnets have been successfully fabricated in 2D, combining 2D magnetism with non-trivial topological properties holds great opportunities for topological transport phenomena and technological applications in magneto-electric, magneto-optic, and topological spintronics. Thus, studying the unique interplay of topological phases with the dynamic magnetization of solids currently matures into a significant burgeoning research field of condensed-matter physics. In this context, magnetic interfaces with topological insulators and layered van der Waals crystals, which can exhibit ferromagnetism at room temperature, constitute compelling and experimentally feasible classes of 2D quantum materials. Here, we demonstrate the emergence of zero-dimensional and one-dimensional semimetallic topological states which arise at the boundary between distinct topological phases when the direction of the magnetization in a 2D magnet is varied. We show that by including the direction of the magnetization into the topological analysis, one arrives at a natural classification of such mixed Weyl and nodal-line semimetallic phases, which paves the way to scrutinizing their stability with respect to perturbations. We uncover that the appearance of semimetallic phases is typically enforced by the drastic variation of the orbital band character upon changing the magnetization direction, which arises commonly in 2D ferromagnets, and we proclaim that emergent semimetals can be experimentally detected by measuring the current-induced orbital response, e.g., via XMCD. Besides providing realistic material candidates in which the discussed semimetals could be observed, we suggest possible applications of these states in shaping the magnetic properties of the edges and current-induced domain-wall motion. Topological phase transitions constitute a pervasive concept that necessitates the occurrence of metallic points in the electronic structure. Acting as prominent microscopic sources of geometrical curvature of momentum space, such band crossings are currently discussed in three-dimensional Weyl semimetals, where they mediate a plethora of fascinating properties. Analogously, it was suggested that large magneto-electric effects in two-dimensional ferromagnets coined mixed Weyl semimetals (MWSMs) originate from emergent nodal points in the mixed phase space of the crystal momentum bm{mathrm {k}}=(k_x,k_y) and the magnetization direction hat{bm{mathrm {m}}} (see Fig. [fig:overview]b). Discovering material candidates and advancing our understanding of topological states in this novel class of semimetallic systems is invaluable for the interpretation of physical phenomena that root in the global properties of the underlying complex phase space. The emergence of nodal points in MWSMs correlates with drastic changes in the mixed topology and it is accompanied by discrete jumps of the momentum Chern number mathcal C=1/(2pi) int Omega_{xy}^{bm{mathrm {k}} bm{mathrm {k}}} dk_x dk_y with respect to the magnetization direction, as well as of the mixed Chern number mathcal Z=1/(2pi)intOmega_{yx}^{hat{bm{mathrm {m}}}bm{mathrm {k}}} dk_x dtheta with respect to the crystal momentum. Here, the momentum Berry curvature of all occupied states |u_{bm{mathrm {k}}n}^thetarangle is denoted by Omega_{xy}^{bm{mathrm {k}} bm{mathrm {k}}}=2,mathrm{Im}sum_n^mathrm{occ}langle partial_{k_x} u_{bm{mathrm {k}}n}^theta | partial_{k_y} u_{bm{mathrm {k}}n}^theta rangle, the mixed Berry curvature is Omega_{yi}^{hat{bm{mathrm {m}}}bm{mathrm {k}}}=2,mathrm{Im}sum_n^mathrm{occ}langle partial_{theta} u_{bm{mathrm {k}}n}^theta | partial_{k_i} u_{bm{mathrm {k}}n}^thetarangle, and theta is the angle that the magnetization hat{bm{mathrm {m}}}=(sintheta,0,costheta) makes with the z-axis as depicted in Fig. [fig:overview]a. To fully characterize the properties of nodal points in the composite phase space spanned by k_x, k_y, and theta, we introduce the integer topological charge Q = frac{1}{2pi}int_S bm{mathrm {Omega }}cdot dbm{mathrm {S}} ,, label{eq:charge} which describes the non-zero flux of the generalized Berry curvature field bm{mathrm {Omega }}= (-Omega^{hat{bm{mathrm {m}}}bm{mathrm {k}}}_{yy}, Omega^{hat{bm{mathrm {m}}}bm{mathrm {k}}}_{yx}, Omega^{bm{mathrm {k}} bm{mathrm {k}}}_{xy}) through a closed surface S that encompasses the nodal point (see Fig. [fig:overview]b). We classify in the following two different types of such mixed Weyl points in the composite phase space: First, the symmorphic combination of time reversal and mirror symmetries can enforce topological phase transitions accompanied by a closing of the band gap as the magnetization direction is varied. As we discuss below, such type-(i) nodal points are robust against perturbations that preserve the protective symmetry, as long as the magnetization direction is fixed. Second, generic band crossings may arise due to the complex interplay of exchange interaction and SOC in systems of low symmetry. In this case, when the underlying electronic structure is modified, such type-(ii) nodal points disappear as long as the direction of the magnetization is fixed, but they reappear if the magnetization direction is adjusted. In addition, as we demonstrate below, nodal points in mixed topological semimetals can form closed lines in the higher-dimensional phase space of momentum and magnetization direction (see Fig. [fig:overview]c,d). It is tempting to interpret these one-dimensional manifolds of topological states as mixed nodal lines in analogy to their conventional sisters in three-dimensional topological semimetals. While crystalline mirror symmetry underlies the emergence of the nodal line in momentum space shown in Fig. [fig:overview]c, mixed topological semimetals host additionally a distinct type of nodal lines as depicted in Fig. [fig:overview]d. Owing to the subtle balance of spin-orbit and exchange interactions, these topological states can be thought of as series of nodal points that evolve also with the magnetization direction theta. As a direct consequence, this type of mixed nodal line is not protected by crystalline symmetries but stems purely from a non-trivial Berry phase gamma=oint_c bm{mathrm {A}} cdot dbm{mathrm {ell}}, where bm{mathrm {A}}=isum_n^mathrm{occ}langle u_{bm{mathrm {k}}n}^theta | bm{mathrm {nabla }}u_{bm{mathrm {k}}n}^thetarangle is the generalized Berry connection in the complex phase space, bm{mathrm {nabla}} stands for (partial_{k_x},partial_{k_y},partial_theta), and the closed path c encircles the nodal line as shown in Fig. [fig:overview]d. To establish the existence of the predicted mixed topological semimetals, we begin our discussion with a simple insightful model of p-electrons on a 2D honeycomb lattice depicted in Fig. [fig:model]b. The tight-binding Hamiltonian assumes the form H = sum_{ij} t^{vphantom{dagger}}_{ij} c_i^dagger c^{vphantom{dagger}}_j + sum_i (epsilon^{vphantom{dagger}}_i mathbb{1} + bm{mathrm {B}} cdot boldsymbol{sigma} ) c_i^dagger c^{vphantom{dagger}}_i + H_mathrm{soc},, label{eq:model} where the first term is the hopping with t_{ij} between orbitals i,j=p_x,p_y,p_z on different sublattices, the orbital-dependent epsilon_i is an on-site energy, the exchange field is bm{mathrm {B}}=B(sintheta,0,costheta), the SOC reads H_mathrm{soc}=xi bm{mathrm {l}}cdot boldsymbol{sigma}, and boldsymbol{sigma} is the vector of Pauli matrices (see also Methods). While the reflection mathcal M with respect to the film plane is a symmetry of the planar lattice, buckling breaks this mirror symmetry. We consider first the mathcal M-broken model, known to be a quantum anomalous Hall insulator over a wide range of model parameters. As exemplified in Fig. [fig:model]a for strong exchange, valence and conduction bands approach each other as the magnetization direction theta is tuned, which results in an emergent band crossing slightly off the K point for thetaapprox 60^circ. Using our classification scheme, we identify this single mixed Weyl point as type-(ii) since it occurs for a generic magnetization direction, the value of which is controlled by the magnitude of SOC and exchange coupling. The effective Hamiltonian close to the linear crossing is governed by three tunable parameters, entangling momentum, magnetization direction, and the interactions of the model, which facilitates a degenerate point in the spectrum following the von Neumann-Wigner theorem. Figure [fig:model]c suggests that the isolated nodal point manifests in a characteristic distribution of the Berry curvature, whereby it mediates a topological phase transition from the non-trivial (mathcal C=-1) to the trivial regime (mathcal C=0) as shown in Fig. [fig:model]d. The unique orbital signatures of such a mixed Weyl point, summarized in Fig. [fig:model]d–f, will be discussed later. To uncover the non-trivial topology of the metallic point in the mixed space of Bloch vector and magnetization direction, we evaluate the flux of the Berry curvature field bm{mathrm {Omega}} through an enclosing surface in (bm{mathrm {k}},theta)-space, which amounts to the topological charge Q of that point according to Eq. [eq:charge]. The mixed Weyl point emerging at thetaapprox60^circ carries a negative unit charge, which is consistent with the distribution of the Berry curvature field bm{mathrm {Omega}} in Fig. [fig:model]c. Figure [fig:model]d illustrates the presence of another nodal point located near K^prime with the very same topological charge if the magnetization direction is tuned to thetaapprox 120^circ. However, the net topological charge over the full Brillouin zone of the combined phase space is zero since the two mixed Weyl points with negative unit charge are complemented by partners of opposite topological charge for thetaapprox 240^circ and thetaapprox 300^circ, respectively. Owing to their topological protection, these generic mixed Weyl points feature a unique property: if the Hamiltonian is perturbed, they may only move to a different position in (bm{mathrm {k}},theta)-space but cannot gap out easily. In the context of angle-resolved photoemission spectroscopy (ARPES) performed at a fixed magnetization direction, this means that although the generic nodal points might appear non-robust with respect to strain, chemical deposition of adsorbates, alloying etc., it should generally be possible to recover them again upon adjusting the magnetization direction. We anticipate that the topological charge can be measured by experiments that sweep the magnetization close to the nodal point: during the phase transition both the quantized anomalous Hall transport and the currents that are pumped by the magnetization dynamics, relating to the mixed Berry curvature, change uniquely as they are sensitive to the magnetic orientation of the ferromagnet. While tuning the ratio between exchange and SOC expands or shrinks the extent of the topologically non-trivial phases, symmetries can enforce the appearance of the trivial state with mathcal C=0 under certain magnetization directions, resulting in a band crossing. To illustrate this, we turn to the planar honeycomb lattice that respects the mirror symmetry mathcal M with respect to the film plane. If the magnetization points along any in-plane direction, the combined symmetry mathcal T otimes mathcal M of time-reversal and mirror operation requires that the Chern number vanishes. Therefore, starting from a non-trivial phase as induced by the model’s interactions for finite out-of-plane magnetization, the system undergoes a topological phase transition from mathcal C=-1 to mathcal C=1 exactly at theta=90^circ (see Supplementary Figure 1). Contrary to the buckled case, this transition is mediated by two nodal points located at K and K^prime, respectively, each of which carries a negative unit topological charge. Thus, while the minimal number of type-(ii) nodal points is one for a given theta, the symmetry-related type-(i) mixed Weyl points come at least in pairs of the very same charge. This correlates with a distinct nature of the topological phase transition in terms of the minimal change of the Chern number by Deltamathcal C=pm 1 and Delta mathcal C=pm 2, respectively. We anticipate that type-(i) mixed Weyl points may share formal analogies with symmetry-constrained counterparts in 3D solids. Remarkably, whereas adjacent nodal points in momentum space of conventional 3D Weyl semimetals must have opposite charges and thus annihilate under certain conditions if pushed towards each other, this is not necessarily the case in 2D MWSMs. We speculate that it might be more difficult to destroy the nodal points in MWSMs by realistic perturbations of the Hamiltonian as they can have the same topological charge for a given magnetization direction, while their counterparts of opposite charge may be very far from them in theta. Generally this does not imply that the direction under which the mixed Weyl points occur is constant as the Hamiltonian is perturbed, although this is the case in the planar model if the perturbation preserves mathcal T otimes mathcal M symmetry. On the other hand, the two nodal points of negative charge, emerging originally at theta=90^circ, split into two distinct entities that manifest for generic directions of the magnetization if the restrictive symmetry is broken, e.g., due to buckling of the lattice. According to our model analysis, the interplay between magnetism and topology in 2D materials offers the potential to realize mixed Weyl points with non-zero topological charges. We apply electronic-structure methods to uncover these nodal points in first candidates of single-layer ferromagnets with SOC. As prototypical examples that are very susceptible to external magnetic fields, and display a rich topological phase diagram, we choose TlSe, Na_3Bi, and GaBi (see Supplementary Note 1), which are originally TCIs and/or TIs with large energy gaps (for the unit cells see insets in Fig. [fig:phasediagram]). To study systematically the mixed topology, we use an additional exchange field term bm{mathrm {B}}cdotboldsymbol{sigma} on top of the non-magnetic Hamiltonian. We start by considering the case of planar TlSe, which is a TCI if no exchange field is applied. Introducing an exchange field with an in-plane component breaks both time-reversal and mathcal M-mirror symmetry, provides an exchange splitting between spin-up and spin-down states, and brings conduction and valence bands closer together. As follows from our topological analysis, the TCI character is kept even under sufficiently small exchange fields (see Supplementary Note 2). In analogy to the mathcal T-broken quantum spin Hall insulator, we refer to this phase as a 2D mathcal M-broken TCI. Increasing the magnitude B results in a gap closure and gives rise to a non-trivial semimetallic state at the critical value B_mathrm{c}. If the exchange field exceeds this value, the reopening of the energy gap is accompanied by the emergence of the quantum anomalous Hall phase for any magnetization direction with finite out-of-plane component (see Fig. [fig:phasediagram]a and Supplementary Note 3). Now, we turn to the in-plane magnetized system that exhibits mathcal T otimes mathcal M symmetry, and for which the gap closes over a wide range of fields B>B_mathrm{c}, see Fig. [fig:phasediagram]a. As exemplified in Fig. 1a, the electronic structure for B>B_mathrm{c} reveals that the gap closing is mediated by four isolated metallic points around the Fermi energy, where bands of opposite spin cross slightly off the X and Y points. Owing to their characteristic Berry curvature field, Fig. 1c, each of these mixed Weyl points occuring for theta=90^circ carries a positive unit charge, which corresponds to a change of the Chern number mathcal C from +2 to -2. Consequently, the Berry phase gamma evaluated along a closed loop in momentum space around one of the points acquires a value of pi. In total, the topological charge over the full phase space vanishes as the individual charges of the mixed Weyl points at theta=90^circ are compensated by four nodal points that emerge during a second topological phase transition at theta=270^circ, Fig. 1c. Analogously to the planar model, we classify these objects as type-(i) nodal points since their emergence is enforced by the symmorphic symmetry mathcal T otimes mathcal M, contrary to the Dirac nodes in 2D Dirac semimetals that are protected by non-symmorphic symmetries. The non-trivial mixed topology further leads to exotic boundary solutions in finite ribbons of TlSe, Fig. 1b. Electronic properties of mixed topological semimetals. (a) Spin-resolved band structure and (b) energy dispersion of a finite ribbon, where the localization at the edge is indicated by colors ranging from blue (weak) to red (strong), of the TlSe monolayer with an in-plane exchange field of magnitude B=0.5,eV. (c) Around the X point, for example, the emergence of nodal points with opposite topological charge (red and blue balls) for reversed in-plane directions theta of the magnetization imprints characteristic features on the phase-space distribution of the Berry curvature field bm{mathrm {Omega}}=(-Omega_{yy}^{hat{bm{mathrm {m}}}bm{mathrm {k}}},Omega_{yx}^{hat{bm{mathrm {m}}}bm{mathrm {k}}},Omega_{xy}^{bm{mathrm {k}}bm{mathrm {k}}}) as indicated by the arrows. A logarithmic color scale from blue (negative) to red (positive) is used to illustrate the mixed Berry curvature Omega^{hat{bm{mathrm {m}}}bm{mathrm {k}}}_{yx} in the complex phase space of bm{mathrm {k}} and theta. (d) Spin-resolved band structure of one-layer Na_3Bi with an exchange field of magnitude B=0.5,eV applied perpendicular to the film. Owing to the mirror symmetry of the system, the band crossings around Gamma form a mixed nodal line in momentum space, which disperses as illustrated in the inset, where colors indicate energy differences between the crossing points and the Fermi level in meV. Breaking the underlying mathcal T otimes mathcal M symmetry, e.g., by buckling of the lattice (see Supplementary Note 2), splits the four nodal points in TlSe, which originally appeared at theta=90^circ, into two distinct groups that manifest for generic magnetization directions. To elucidate this transition more clearly, we consider the monolayers Na_3Bi and GaBi, where this symmetry is absent. As visible from the phase diagrams in Fig. [fig:phasediagram](b,c), the single Weyl points in these systems emerge at the boundaries between the mathcal{T}-broken quantum spin Hall phase and Chern insulator phases with different Chern numbers. Analogously to TlSe, we identify the mixed topological charge of such points to be Q=pm 1, depending on the position in (bm{mathrm {k}},theta)-space. However, in contrast to TlSe, for which both the number as well as the position of mixed Weyl points is determined by symmetry, the single mixed Weyl point in Na_3Bi and GaBi appears for a given generic direction of the magnetization, and we thus classify it as type-(ii) nodal point. It can occur that the mixed Weyl point is realized accidentally for a range of theta in the 2D ferromagnet, as it is exemplified in TlSe at a fixed value of exchange field of about 0.29 eV, see Fig. [fig:phasediagram]a. In the spirit of Fig. [fig:overview]d, this presents a truly mixed nodal line, a 1D manifold of states which evolves not only in bm{mathrm {k}}-space but also in theta. The topological character of the line is reflected in the Berry phase that is the line integral of the Berry connection along a path in bm{mathrm {k}}-space which encloses the corresponding point that the mixed nodal line pinches in the Brillouin zone at a given theta, see Fig. [fig:overview]d. The occurrence of such mixed nodal lines is purely accidental and does not rely on symmetries, while perturbing the system (i.e. by changing the magnitude of bm{mathrm {B}}) may result in the mixed nodal line’s destruction, as we discuss below. Owing to the subtle interplay of exchange interactions and relativistic effects which underlies their emergence, the realization of such mixed nodal lines in real materials sets an exciting challenge, where 2D magnets are advantageous for semimetallic states that are robust against variations of the magnetization direction. Another distinct type of a mixed nodal line is the 1D nodal line which evolves in bm{mathrm {k}}-space for a fixed direction of the magnetization, see Fig. [fig:overview]c, similarly to the nodal-line semimetals which exhibit nodal lines in high-symmetry planes corresponding to the crystalline mirror symmetry. While the mathcal M symmetry is broken by an in-plane exchange field, it survives when mathbf{hat{m}} is perpendicular to the film. As shown in Fig. [fig:phasediagram]b, the energy gap remains closed in Na_3Bi with theta= 0^{circ} above the critical magnitude B_mathrm{c}. To gain insights into the topological properties in this case, we take B = 0.5 eV and present the spin-resolved band structure of the system in Fig. 1d. In absence of inversion and time-reversal symmetries, all bands are generically non-degenerate. Taking into account the mirror symmetry mathcal M, bands in the xy-plane can be marked by mirror eigenvalues pm i, and those with opposite mirror eigenvalues can cross each other without opening a gap. As the highest occupied and lowest unoccupied bands in Na_3Bi cross each other around the Gamma point, a nodal line is formed as shown in the inset of Fig. 1d. To validate the mixed topological character of this nodal line in Na_3Bi, we compute its non-trivial Berry phase according to Fig. [fig:overview]c. However, the overall flux of the generalized Berry curvature field through any surface in (bm{mathrm {k}},theta)-space surrounding the mixed nodal line vanishes, which confirms the zero topological charge of the mixed nodal line as an object in 3D space of momentum and mathbf{hat{m}}. Accordingly, there is no variation in the Chern number mathcal{C}=+1 as the magnetization crosses theta=0^{circ} (see Supplementary Figure 5b), which signifies the lack of topological protection of the mixed nodal line and its disappearance as the mirror symmetry is broken upon turning mathbf{hat{m}} away from the z-axis. To prove that the nodal line originates from the mirror mathcal M, we perform for theta=0^circ various distortions of the lattice that break the three-fold rotational axis perpendicular to the film but preserve mathcal M, resulting still in the mixed nodal line though for different strengths of the exchange field (see Supplementary Figure 7). Having established the existence of mixed topological semimetals in a simple model and by applying an external exchange field to TIs/TCIs, an important question to ask is whether the proposed mixed topological semimetals can be realized in stable 2D ferromagnets. While the existence of mixed Weyl points in several 2D magnets such as doped graphene or semihydrogenated bismuth has been shown, in this work we demonstrate the possibility of their realization in other realistic systems, aiming especially at van der Waals crystals. Bulk VOI_2 has a layered structure characterized by the orthorhombic space group Immm, and has already been synthesized and investigated. We focus on a VOI_2 monolayer, the unit cell of which contains two I, one O, and one V atom that is coordinated in the center as shown in Fig. [fig:abinitio]a. The electronic structure of the single layer represents the in-plane electronic structure of its bulk parent compound quite well, and one-layer fabrication could be realized experimentally, e.g., by mechanical exfoliation from the layered bulk due to the low cleavage energy of 0.7 meV/Å^2, which is much smaller than for graphite (12 meV/Å^2) or MoS_2 (26 meV/Å^2). As verified by our explicit calculations of the phonon spectrum, the monolayer is dynamically stable and difficult to destroy once formed. The ground state of the system is ferromagnetic with a spin magnetization of about 1 mu_B per unit cell and an easy in-plane anisotropy. Supplementary Note 4 presents further details on the electronic structure. As illustrated in Fig. [fig:abinitio]b, the band structure of one-layer VOI_2 with in-plane magnetization and SOC reveals band crossings along the M-Y and Gamma-X paths near the Fermi level. There are four semimetallic points in the 2D Brillouin zone as can be seen from the bm{mathrm {k}}-resolved energy difference between top valence band and lowest conduction band around Gamma and Y (see Fig. [fig:abinitio]b). To demonstrate the topological nature of these points we analyze the distribution of the Berry curvature in the complex phase space shown in Fig. [fig:abinitio]b, and find that all four crossings are mixed Weyl points with a charge of +1. Similarly to magnetized TlSe discussed before, the predicted mixed Weyl points in VOI_2 monolayer are protected by the mathcal T otimes mathcal M operation as can be confirmed explicitly by breaking this symmetry, which gaps out the nodal points. When constructing a semi-infinite 1D ribbon of the material along the Gamma-Y direction, we observe that the four mixed Weyl points project onto two pairs of distinct points that are connected by emergent edge states close to bar X and bar Gamma as illustrated by the edge dispersion in Fig. [fig:abinitio]c. In addition, to prove the emergence of mixed nodal lines in realistic 2D magnets, we start from Na_3Bi in its hexagonal P6_3/mmc phase, which is one of the first established realizations of Dirac semimetals. This material has been synthesized both in bulk and film form. Replacing one Na atom with Cr, we focus here on a monolayer of Na_2CrBi (see Fig. [fig:abinitio]d for a sketch of the unit cell), which is a strong ferromagnet that is energetically and dynamically stable according to our calculations of cleavage energy (25.5 meV/Å^2) and phonon spectrum (see Supplementary Figure 8). Including SOC, the band structure of perpendicularly magnetized Na_2CrBi in Fig. [fig:abinitio]d exhibits prominent band crossings around the Gamma point. These metallic points form due to the mirror symmetry mathcal M, which allows two bands with opposite mirror eigenvalues to cross close to the Fermi level. Extending the analysis to the full Brillouin zone reveals that these band crossings forge a nodal loop (see Fig. [fig:abinitio]e). In analogy to the previous case of the Na_3Bi monolayer, the nodal line is gapped out as soon as the mirror symmetry is broken, e.g., by tilting the magnetization direction, and we verify its non-trivial mixed topology by evaluating the Berry phase around the nodal line as outlined in Fig. [fig:overview]c. As in the case of VOI_2, a key manifestation of the complex topology of the mixed nodal line is the emergence of characteristic edge states in a semi-infinite ribbon of Na_2CrBi, which can be clearly distinguished from the projected bulk states in Fig. [fig:abinitio]f. Finally, we elucidate one of the universal physical mechanisms that triggers magnetically induced topological phase transitions and gives rise to non-trivial band crossings. We refer again to the above model, which contains elementary ingredients that govern the appearance of mixed nodal points, including exchange and SOC (see Eq. [eq:model]). As illustrated in Fig. [fig:model]d, given an initial spin-orbit driven energy splitting for out-of-plane magnetization, bands with different orbital character are guaranteed to cross as the direction of the magnetization is reversed, owing to the fact that the orbital momentum of the system is dragged by the magnetization via SOC. This observation is well known in molecular physics as well as in the band theory of ferromagnets. In the studied model such crossing points of orbitally polarized states appear either for theta=90^circ enforced by the symmetry, or under generic directions of the magnetization if the symmetry in the orbitally complex system is reduced. Although the non-trivial mixed topology originates here primarily from p-states, we point out that materials with s-electrons on a bipartite lattice can offer similar prospects by exploiting the valley degree of freedom. Since the inversion of the orbital chemistry mediates the level crossing, we argue that this transition imprints general magnetic properties. A representative example of a real material where the crossing emerges at general theta is the semi-hydrogenated bismuth film H-Bi, which has been shown to host single mixed Weyl points at thetaapprox pm 43^{circ} and pm 137^{circ} due to a magnetically induced topological phase transition from a Chern insulator with mathcal C=pm 1 to a trivial phase. In Fig. [fig:orbmag]a we plot the evolution of the orbitally resolved electronic structure of H-Bi around the Fermi energy, where the states originate mainly from p orbitals of bismuth. In accordance with the model scenario, the emergent nodal point correlates with a reordering of the p_xpm ip_y states, which underlines the role of SOC in mediating the inversion of energy bands in terms of their orbital character and stabilizing generic mixed nodal points. The changes in the orbital character of the states across mixed Weyl points manifest in prominent changes in the local orbital magnetization (OM) near the mixed Weyl points as the magnetization is varied. According to its modern theory, the OM as a genuine bulk property of the ground-state wave functions |u_{bm{mathrm {k}}n}^thetarangle is given by bm{mathrm {m}}=int bm{mathrm {m}}(bm{mathrm {k}}) , mathrm d bm{mathrm {k}}, with momentum-resolved contributions from all occupied bands bm{mathrm {m}}(bm{mathrm {k}}) = frac{e}{2hbar}mathrm{Im}sum_n^mathrm{occ}langle partial_{bm{mathrm {k}}}^{vphantom{theta}} u_{bm{mathrm {k}}n}^theta | times [ H_{bm{mathrm {k}}}^{vphantom{theta}} + mathcal E_{bm{mathrm {k}}n}^{vphantom{theta}} - 2mathcal E_mathrm{F}^{vphantom{theta}}] | partial_{bm{mathrm {k}}}^{vphantom{theta}} u_{bm{mathrm {k}}n}^theta rangle, label{eq:orbmag} with H_{bm{mathrm {k}}}=mathrm e^{-ibm{mathrm {k}} cdot bm{mathrm {r}}} H mathrm e^{i bm{mathrm {k}} cdot bm{mathrm {r}}} as the lattice-periodic Hamiltonian, mathcal E_{bm{mathrm {k}}n} as the energy of band n, and mathcal E_mathrm{F} as the Fermi level. Equation [eq:orbmag] underlines the deep relation of the OM to the local geometry in bm{mathrm {k}}-space, and it is thus expected that, in accord with the strongly modified geometry of Bloch states in (bm{mathrm {k}},theta)-space in the vicinity of mixed Weyl points, the OM may also experience a pronounced variation both in mathbf{k} and theta. Therefore, we anticipate that the non-trivial topology of the mixed Weyl points enhance the variation of the orbital character of the states. Indeed, our calculations verify the validity of this line of thought: Fig. [fig:orbmag](d,e) reflects unique local fingerprints and colossal magnitude of the orbital magnetization m_z(bm{mathrm {k}}) in momentum space, which correlate with the emergence of magnetic monopoles in two of the predicted mixed topological semimetals. These features are present for both types of nodal points, i.e., the generic and symmetry-enforced ones. Remarkably, the pronounced but competing local contributions to the OM for fixed theta nearly cancel each other, rendering the net effect of the mixed Weyl points on the total OM rather small. However, the microscopic response of the orbital chemistry to magnetically controlled band crossings opens up bright avenues for generating large orbital magnetization by applying an electric field that repopulates the occupied states (see Fig. [fig:orbmag]b,c). Such a giant current-induced orbital Edelstein effect can have a strong impact on phenomena that rely sensitively on the orbital moment at the Fermi surface. Moreover, the drastic change in the local OM with theta may be used to detect experimentally the presence of mixed Weyl points in the electronic structure by detecting large variations in the current-induced orbital properties. To demonstrate the feasibility of our proposal, we evaluate the orbital Edelstein effect m_i = alpha_{ij} E_j for the buckled p-model with broken inversion symmetry within a Boltzmann theory : alpha_{ij} = etau sum_nint frac{mathrm dbm{mathrm {k}}}{(2pi)^2} frac{mathrm df}{mathrm dmathcal E_{bm{mathrm {k}}n}}, m_{n,i}^mathrm{loc}(bm{mathrm {k}}) ,v_{n,j}(bm{mathrm {k}}) ,, label{eq:edelstein} where tau is the relaxation time, f is the Fermi distribution function, and v_{n,i}(bm{mathrm {k}}) and m^mathrm{loc}_{n,i}(bm{mathrm {k}}) correspond to the ith components of the state’s group velocity and its local orbital moment bm{mathrm {m}}_{n}^mathrm{loc}(bm{mathrm {k}})=(e/2hbar), mathrm{Im}langle partial_{bm{mathrm {k}}}^{vphantom{theta}} u_{bm{mathrm {k}}n}^theta | times[H_{bm{mathrm {k}}}^{vphantom{theta}}-mathcal E_{bm{mathrm {k}}n}^{vphantom{theta}}]|partial_{bm{mathrm {k}}}^{vphantom{theta}} u_{bm{mathrm {k}}n}^thetarangle, respectively. While the equilibrium OM hardly changes as a function of the direction theta (see Fig. [fig:model]e), the sharply peaked current-induced response alpha_{ij} is an immediate orbital signature of the emergent mixed Weyl points with complex topology, Fig. [fig:model]f. Owing to the nature of mixed semimetals incorporating the magnetization direction as an integral variable, we expect pronounced topological magneto-electric effects to which these materials should give rise. Apart from their substantial relevance for technological applications based on magnetic solids, we anticipate that these coupling phenomena can play a key role even in finite systems such as quantum dots. Analogously, we envisage that complementing the topological classification of matter by magnetic information rooting in the electronic degrees of freedom will be valuable for other research fields as well, e.g., for topological magnon semimetals. In addition to the prospects for current-induced orbital magnetism, current-induced spin-orbit torques that can be used to efficiently realize topological phase transitions, and possible giant influences on anisotropic magnetotransport, we would like to emphasize in particular the promises of mixed semimetals for chiral magnetism. While it is known that MWSMs may exhibit a distinct tendency towards chiral magnetism, we speculate that chiral spin textures such as magnetic skyrmions or domain walls can effectively unravel the topological features of mixed semimetals in real space, which can have profound consequences on, e.g., orbital magnetism and transport properties of these textures. To illustrate this point more clearly, we return to the emergent nodal points in H-Bi. In complete analogy to 3D Weyl semimetals in momentum space, the complex mixed topology of MWSMs results generally in the emergence of mixed Fermi arcs at the surfaces of these systems. By following the theta-evolution of the electronic structure of 1D ribbons of H-Bi that are periodic along the x-axis, we effectively construct such a 2D surface for which we present in Fig. [fig:orbmag]f the states at the Fermi energy as a function of k_x and theta. As clearly evident, the emergent surface states connect the projections of the mixed Weyl points with opposite charge, realizing mixed Fermi arcs. Imagining now a long-wavelength chiral domain wall running along the x-axis of a H-Bi ribbon, where theta(x) describes the local variation of the magnetization, we recognize that the mixed Fermi arcs will manifest in topological metallic states in certain regions of the domain wall as a consequence of the non-trivial mixed topology. In chiral spin textures hosting mixed Weyl semimetallic states, we anticipate that such electronic puddles will result in topologically distinct contributions to the current-driven spin torques acting on these spin structures, a prominent variation of the texture-induced Hall signal in real space, chiral and topological orbital magnetism, as well as topological contributions to the longitudinal transport properties of domain walls and chiral magnetic skyrmions made of MWSMs. Interfaces between topological insulators and dynamic magnetization structures present further compelling examples of such complex mixed topologies. We thereby proclaim that exploring the avenues associated with the exotic electronic, transport, and response phenomena in textured mixed semimetals presents one of the most exciting challenges in topological chiral spintronics. In order to arrive at the model [eq:model] of the mixed Weyl semimetal, we extended the tight-binding Hamiltonian of Ref. to describe arbitrary magnetization directions. Throughout this work we incorporated only the nearest-neighbor hopping of sigma-type on the honeycomb lattice with t_sigma=1.854,mathrm{eV} but our general conclusions remain valid even beyond nearest-neighbor hoppings. In addition, we chose B=8.0,mathrm{eV} to fully spin-polarize the bands, xi=1.0,mathrm{eV} for the SOC, and we shifted the p_z states to higher energies. By introducing a relative shift of the on-site energies we further imitated the buckling of the honeycomb lattice. Diagonalizing at every (bm{mathrm {k}},theta)-point the Fourier transform of the 12times12 matrix that results from Eq. [eq:model] grants access to the wave functions and the band energies. Based on density functional theory as implemented in the full-potential linearized augmented-plane-wave code FLEUR (see http://www.flapw.de), we converged the electronic structure of the studied systems including SOC self-consistently. Exchange and correlation effects were treated in the generalized gradient approximation of the PBE functional. To represent the electronic Hamiltonian efficiently, we subsequently constructed so-called maximally localized Wannier functions using the wannier90 program. In this tight-binding basis, the Hamiltonian of the non-magnetic TI/TCI systems was supplemented by an exchange term {bm{mathrm {B}}}cdot bm{mathrm {sigma}}, where bm{mathrm {sigma}} is the vector of Pauli matrices and bm{mathrm {B}}=B(sintheta,0,costheta). In the ferromagnetic candidate materials, we obtained an efficient description of the electronic structure in the complex phase space of bm{mathrm {k}} and theta by constructing a single set of higher-dimensional Wannier functions. The structural relaxations of the ferromagnetic systems were carried out in the Vienna Ab initio Simulation Package. The tight-binding code and the data that support the findings of this study are available from the corresponding authors on reasonable request. This work was supported by the Priority Program 1666 of the Deutsche Forschungsgemeinschaft (DFG), the Virtual Institute for Topological Insulators (VITI), the Natural Science Foundation of Shandong Province under Grant No. ZR2019QA019, and the Qilu Young Scholar Program of Shandong University. This work has been also supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1238. We acknowledge computing time on the supercomputers JUQUEEN and JURECA at Jülich Supercomputing Centre and JARA-HPC of RWTH Aachen University. C.N. and J.-P.H. performed the first-principles calculations and model analysis. Y.M. conceived the concept and designed the research with contributions from C.N., J.-P.H. and G.B.. C.N., J.-P.H., and Y.M. wrote the manuscript with contributions from P.M.B., H.Z., L.P., D.W., G.B., and S.B.. accompanies this paper at XXX. The authors declare no competing interests. 1. These authors contributed equally to this work. Correspondence should be addressed to: [email protected]↩ 2. These authors contributed equally to this work. Correspondence should be addressed to: [email protected]↩ How can the complex mixed topology of the mixed Weyl semimetals lead to distinct electronic, transport and response phenomena in systems with chiral spin textures?