## Introduction

Since their experimental discovery in the TaAs family, Weyl semimetals continue to gain interest. The non-trivial topology of this electronic phase follows from the geometrical properties1 associated with electronic bands. Specifically, for band crossing points, a topological invariant can be defined as the flux of Berry curvature through a surface enclosing the point. The low-energy effective theory around such a point corresponds to the Weyl equation2, identifying the topological invariant with the corresponding Weyl fermion chirality. In lattice systems, however, Weyl nodes must come in pairs of opposite chirality3. Each pair having zero net chirality severely hampers experimental probes sensitive to Brillouin zone integrated quantities from picking up the Weyl node chirality: for this to work, one should create an overall chirality imbalance.

The essential condition to enable the existence of Weyl nodes in the first place is broken spin degeneracy of Bloch states at a generic crystal momentum. This requires time-reversal symmetry (Θ) or inversion symmetry ($${{{\mathcal{I}}}}$$) to be broken. While the first experimental confirmations of topological semimetals were achieved in Θ-symmetric compounds4,5,6,7,8,9,10,11, magnetic compounds are naturally appealing due to the broad prospects that the interplay between the electronic structure and external magnetic fields can offer12,13,14,15,16,17,18,19,20,21,22. Still, experimentally confirmed magnetic topological semimetals are rather scarce. Two recently investigated cases are the nodal-line semimetal Co2MnGa23,24 and the Weyl semimetal Co3Sn2S225,26,27,28, both centrosymmetric ferromagnets with non-trivial topology. In EuCd2As229 and EuCd2Sb230, on the other hand, a Weyl semi-metallic phase is realized for the fully spin-polarized state induced by an external magnetic field.

Here we focus on the rare-earth carbides RMC2, with R a rare-earth metal and M a transition metal. This broad family of compounds exhibits a diversity of Θ-breaking or $${{{\mathcal{I}}}}$$-breaking phenomena31,32,33,34,35,36,37,38,39,40,41,42,43,44,45 both in long-range ordered magnetic compounds31,32,33,34,35,36,37 as well as in superconducting phases38,39,40,41 and a complex interplay between these phases and charge density waves (CDW)42,43,44,45,46,47,48,49,50,51. We show by consideration of available experimental information and own density-functional calculations (DFT) that RMC2 compounds can be categorized in four classes: (I) Θ-symmetric and $${{{\mathcal{I}}}}$$-broken semimetals (YCoC2 and LuCoC2); (II) Θ-broken and $${{{\mathcal{I}}}}$$-symmetric metals (GdRuC2); (III) both Θ- and $${{{\mathcal{I}}}}$$-broken semimetals (PrRhC2, NdRhC2, GdCoC2 and GdNiC2); and (IV) insulators (LaRhC2), the latter being of secondary interest for this work. We find that all the mentioned semimetals possess Weyl nodes close to the Fermi energy, as summarized in Table 1, where they also tend to have relatively simple and uncluttered band structures. Figure 1 illustrates the two observed structure types of the (semi-)metallic compounds.

Systems belonging to class III are of particular interest as they allow for specific demonstrations of the unique interplay between topology and magnetism offered by $${{{\mathcal{I}}}}$$-broken symmetry. As a proof of principle, we show for NdRhC2 and GdCoC2 how in ferromagnetic (FM) noncentrosymmetric phases tilting of the magnetization (m) along a low-symmetry direction produces a disbalance in the number of opposite chirality Weyl fermions near the Fermi surface: of all Weyl nodes the degeneracy is lifted. Noteworthy, the direction in which the magnetization is canted controls the sign of the chirality disbalance, allowing therefore to switch the dominant low-energy chirality of the electrons. Such chirality imbalanced configuration naturally leads to a variety of physical consequences, some simply related with the chiral symmetry induced on the electronic structure, others with the possibility—unique of chiral systems—of topological phase transitions in which a single Weyl node crosses the Fermi level and moves between regions enclosed by different Fermi surface sheets. We discuss the ramifications of chirality imbalance and the related topological transitions for existence and observation of chiral and gyrotropic magnetic effects (GME) in these compounds.

## Results and discussions

### From Y to Lu

For compounds based on different R elements, we first present the main structural, electronic and magnetic properties of interest for this work. Our DFT results were obtained using the generalized gradient approximation (GGA)52, as implemented in FPLO-1853 (https://www.fplo.de/) and considering for the treatment of the occupied 4f-states both the open-core approximation (OC) with R-specific 4f spin moment but spherical orbital occupation54 and the GGA + U method with the full-localized limit for the double counting correction55 and parameters U = 7 eV and J = 1 eV. We study compounds which have already been synthesized, including R = Y, La, Nd, Pr, Gd or Lu, and M = Co, Rh, Ni or Ru31,32,34,43, and use the lattice parameters from The Materials Project56 (see the Supplementary Note I for further details). Table 1 includes the available experimental data. For most of the compounds, we find that the DFT calculations agree with the reported magnetic properties. In order to analyze the electronic properties of the whole family on the same footing, we will fix in the following the quantization axis along the direction [001].

We find that the compounds involving rare earths with empty or completely filled f-shells, R = Y, La or Lu, result in Θ-symmetric non-magnetic (NM) states, in agreement with experiment32,43,57. Among these, LaRhC2 is the only system that crystallizes in the noncentrosymmetric tetragonal space group P4132 and displays an insulating electronic structure (see Supplementary Note III). Opposed to this, YCoC2 and LuCoC2 grow in the noncentrosymmetric orthorhombic space group Amm243,57 and exhibit a semi-metallic band structure (see Fig. 2a and Supplementary Note III). Four bands dominate the energy spectrum of LuCoC2 near the Fermi energy. These arise from hybridized Co-3d, C-2p and R-5d states (referred to as pd states from now on). The same characteristic four bands are present at low energy for other compounds with M = Co or Rh (Fig. 2b, c, e). On the other hand, compounds based on transition metals not isoelectronic to Co, namely, Ru or Ni, exhibit a shift of the Fermi energy of roughly ±1 eV (Fig. 2d). This shift naturally changes the nature of the bands around the Fermi energy and, as we will see, their electronic properties.

Members of the family based on magnetic R ions are characterized as a lattice of localized moments on the R-4f states coupled with the pd states via an onsite (Kondo) exchange. Ruderman–Kittel–Kasuya–Yosida exchange interactions between the 4f moments can lead to long-range magnetic order including the induced moments on the pd states. Consequently, these compounds are an interesting platform to explore how external magnetic fields, which couple primarily with the R-4f states, can tune Weyl node properties of the low-energy electronic structure associated with the pd states.

Among the magnetic compounds, those obtained by replacing Lu by magnetic rare earths while retaining the space group Amm2 (Fig. 1a, c) belong to Class III. The crystal structure in this case can be regarded as a stack of rare-earth planes, responsible for the breaking of Θ at low temperatures, and planes of MC2, responsible for the breaking of $${{{\mathcal{I}}}}$$.

For these compounds, different magnetic states have been reported. GdCoC2 was first described as antiferromagnetic (AFM) with in-plane moments canted at 48° from the a-axis and TN = 15.6 K34. However, a more recent study by Meng et al. finds a FM state below ~15 K36. GdNiC2, on the other hand, as most of the Ni-based members in the family35, presents an AFM ground state34. For PrRhC2 and NdRhC2, experimental data is limited to high-temperature susceptibility measurements32. These indicate small Curie–Weiss temperatures (θCW ~ −13 K and ~0 K, respectively) and magnetic moments in good agreement with the values expected for the trivalent R ions. GdRuC2 exhibits an interesting contrast to the above, since it presents a transition to a FM phase but the $${{{\mathcal{I}}}}$$-breaking distortion in the MC2 layers is absent and the space group is Cmcm (Fig. 1b), thus belonging to Class II33.

Our DFT results correctly capture the NM states in YCoC2 and LuCoC2 and the FM states in GdCoC2 and GdRuC2. For NdRhC2, we find that both GGA+OC and GGA + U predict an FM ground state. Results for PrRhC2 are somewhat more complex, as GGA+OC and GGA + U predict as ground state AFM and FM order, respectively. The only clear deviation from experiment is found in GdNiC2, where our calculations predict the ground state to be FM. One should, however, keep in mind that in the Ni-based carbides, a strong interplay between the magnetic state and a CDW (not explored in our calculations) has been established. In fact, there are indications that the CDW tends to compete against the FM phase42,46,48,50. Specifically in GdNiC2, different metamagnetic transitions have been observed under moderate external magnetic fields, yielding an interesting and complex phase diagram42,45,51.

### Weyl nodes

We now turn our attention to the topological properties of the electronic structure. For this, we will focus on the GGA + OC calculations. As a common reference for the following, we define for each compound the number of valence electrons as N and search for Weyl points in a relevant low-energy window [−120, 120] meV, using the Pyfplo module of the FPLO package ().

Figure 2g shows the energy of the identified Weyl nodes for all the compounds considered in this work, ordered as a function of their unit cell volume (V). Table 1 includes the position and energy of the respective lowest energy Weyl nodes (for a complete list, see Supplementary Note III). Weyl nodes at low energy are found both for Θ-symmetric and Θ-breaking cases and, among the latter, both for compounds with FM or AFM orderings. YCoC2 has been previously studied in ref. 57 and our results agree with the addition that the search for crossings between bands N − 1 and N discloses Weyl nodes at even lower energy, indicated in Fig. 2g.

Relations between the Weyl node structure of different compounds can also be established. For instance, an inspection of the node coordinates reveals that the fourfold degenerate Weyl nodes of lowest energy in YCoC2, which lie in the kz = 0 plane, are also present in LuCoC2 but are higher in energy. It further suggests that on replacing Y by Gd, the large exchange field induced by the Gd-4f spins moves these Weyl nodes away from the kz = 0 plane and to a lower energy. To confirm this, we carried out explicit computation of Weyl points in YCoC2 under external magnetic field acting on the spin degrees of freedom. Indeed, we find that the field endows the nodes a finite kz component.

Regarding GdNiC2 and GdRuC2, the Fermi energy shift associated with their different number of valence electrons with respect to GdCoC2 is naturally in opposite directions and therefore has different consequences. On GdRuC2, it increases the density of states and the complexity of the low-energy band structure. While it is interesting that it presents Weyl nodes close to the Fermi surface, it should not be considered as a semimetal. On the other hand, the upward shift of Fermi energy makes the Ni-based compound of strong interest. Indeed, upon this shift, a single pair of Weyl nodes—the minimum possible in a periodic system—lies ~100 meV above the Fermi energy. Our calculations performed on different compounds neatly explain the origin of these Weyl nodes: the isostructural but NM LuCoC2 features twofold degenerate bands along the line X-T forming a massive Dirac cone ~1 eV above the Fermi level (Fig. 2a). The band degeneracy is protected by a reflection symmetry and is lifted in the isolectronic FM compounds (Fig. 2c, e), where the Dirac cone is split into Weyl cones. Substituting Co by Ni shifts these Weyl nodes closer to the Fermi level, while keeping them along X-T (Fig. 2d). While the FM phase in GdNiC2 has only been stabilized with an external magnetic field42,45, it could be interesting to study this phenomenology in SmNiC2, where the competing CDW is suppressed40 leading to a FM ground state35.

The rather large volume change caused by the substitutions Gd → Nd or Pr and Co → Rh naturally induces sizable changes in the electronic structure which remains, however, semi-metallic in NdRhC2 and PrRhC2. Also, low-energy Weyl nodes are present in these compounds (Fig. 2g). While there are already examples of noncollinear AFM Weyl semimetals58, should the predicted magnetic structure of PrRhC2 be confirmed, this compound would be (to the best of our knowledge) the first realization of a collinear AFM Weyl semimetal. In addition, it would allow for the presence of the recently uncovered crystal Hall effect59. This effect is characterized by a finite anomalous Hall conductivity (AHC) in a collinear antiferromagnet due to the asymmetric local environment of the magnetic sublattices. Our AHC calculations (see Table 1) show that this effect is indeed present in PrRhC2.

### Pumping chirality to and through the Fermi surface

Due to the Nielsen-Ninomiya “no-go” theorem for chiral lattice fermions3,60, Weyl nodes come in pairs of opposite chirality61 (which can be broken up by very strong electron–electron interactions62,63). In a crystal, Weyl nodes occur in multiplets, their degeneracy being dictated by the Shubnikov group of the material which relates Weyl nodes of same or different chirality. Depending on the specific Shubnikov group, the Weyl node degeneracy need not be even. If the degeneracy is odd, pairs of opposite chirality are necessarily energy-split. The energy splitting between nodes of opposite chirality, a key quantity for the magnitude of different electromagnetic responses sensitive to the Weyl node chirality, actually depends on the material and external conditions. Low symmetry, in particular the absence of inversion and mirror symmetries is of the essence. Here we build on the idea of using the magnetic degrees of freedom in a $${{{\mathcal{I}}}}$$-broken material to reduce the symmetry such that the Shubnikov group contains only the identity.

As a proof of principle, we consider NdRhC2, although the physics we discuss can be readily extended to other noncentrosymmetric magnetic compounds. In the FM ground state, m points along the [001] direction and the Shubnikov group contains {E, m(x)Θ, m(y)Θ, C2(z)}, where m and C2 represent the mirror symmetries and the twofold rotation, respectively. Therefore, for each Weyl node away from high-symmetry lines, there are three degenerate symmetry-related partners. Any component of m along a low-symmetry direction does not only break the rotation C2(z) but also the symmetries involving mirrors. Thus, starting from m along [001], a perturbation that cants m toward, e.g., [111] leaves {E} as the only symmetry element, removing all degeneracies among the Weyl nodes. The natural question is how large this effect is.

Figure 3a shows the energy of Weyl nodes in NdRhC2 as m is canted at an angle θ toward [−111] (on the left) or [111] (right). At θ = 0, there are two sets of fourfold degenerate Weyl nodes in the energy range [−120, 80] meV. Remarkably, even a moderate canting of m is enough to produce experimentally meaningful energy splittings between Weyl nodes of opposite chirality, of the order of tens of meV. We have verified that similar results are obtained using either the experimentally reported crystal structures32 or crystal structures relaxed keeping the ratio of the lattice constants as in the experimental structure (see Supplementary Note II).

The tuning of Weyl nodes to the Fermi surface at specific angles resembles the prediction for Co3Sn2S217, with the difference that in the latter case the crystal structure is centrosymmetric. This difference is crucial as $${{{\mathcal{I}}}}$$ always connects Weyl nodes of opposite chirality and, therefore, enforces a vanishing total chirality of the Weyl nodes at the Fermi surface. While mirror symmetries can be reversibly broken by external magnetic fields, breaking $${{{\mathcal{I}}}}$$ is the structural prerequisite for tuning an odd number of Weyl nodes to the Fermi surface.

The two chosen canting directions are related by a crystal mirror symmetry (Fig. 1d), and therefore, the chirality of the Weyl nodes reaching the Fermi surface at a certain θ is opposite for [−111] and [111]. Thus, the canting direction provides an experimentally viable way of discarding effects not originated in the chirality disbalance. It is worth noting that this chirality valve is effective not only at the specific angle at which a Weyl node crosses the Fermi surface. Indeed, due to the particularly large energy trajectory of one of the nodes below the Fermi energy, in a wide range of angles for θ 20°, NdRhC2 exhibits two Weyl fermions of negative (positive) and only one of positive (negative) chirality in a transport-relevant energy window of ±40 meV around the Fermi energy for θ toward [111] ([−111]).

### Magnetism-induced bulk topological phase transitions

The dynamics of a single Weyl node going from above to below the Fermi level is a topological phase transition which can be described by the flux of the Berry curvature through the different Fermi surface sheets:

$${C}_{n\alpha }=\frac{1}{2\pi }{\int}_{\alpha }dS\cdot {{{\Omega }}}_{nk},$$
(1)

where α denotes different sheets of the Fermi surface associated with the band n. Cnα can be computed based on the momentum position of the Weyl nodes connecting the bands n and n + 1 and the bands n and n − 164. The latter are found to play no role in the energy and angle range analyzed here. In the following we will focus on only one band so we drop the n index. In NdRhC2, two electron-like and two hole-like bands conform the Fermi surface, shown in Fig. 3b. For θ < θc1 ~ 20°, the four Weyl nodes above the Fermi energy at θ = 0 lie inside the outer hole pocket which has, therefore, Chole = 0 (Fig. 3c). At θ = θc1, one Weyl node sits exactly on the Fermi surface. This is a type-II node connecting the outer hole pocket with an electron Fermi surface (Fig. 3d). Further increasing θ, this Weyl node goes below the Fermi level and leaves the hole pocket. In this situation, Chole = 1. This holds for an angle range between θc1 and θc2 ~ 27°, since at θc2 another Weyl node of opposite chirality to the previous one transitions out of the hole pocket. As a result, in the parameter space of canting angle θ and chemical potential, the system presents a series of topological phase transitions in which the topological invariant Chole changes (Fig. 3e).

### Magnetic anisotropy and Weyl node dynamics

A relevant question is how large is the magnetic field required to cant the magnetization in order to enable the Weyl node dynamics described above. Within the OC approximation employed here, the contribution of the 4f states to the magnetic anisotropy energy (MAE) is not considered. Our estimates for the MAE arising from the remaining valence electrons suggest that the first topological phase transition is accessible at ~5 T. An order of magnitude estimate for the 4f contributions can be obtained from the experimental anisotropy field of the well-known permanent magnet compound Nd2Fe14B in comparison with the anisotropy field of Y2Fe14B. The former has an anisotropy field of about ~7 T at room temperature65, the latter somewhat <3 T66, implying similar magnitude of the Nd-4f and the non-4f contributions.

Figure 4 shows the energy of the Weyl nodes closest to the Fermi energy in GdCoC2 as the magnetization is canted. As in the Nd-based compound, the induced energy splitting between Weyl nodes can reach tens of meV. This is interesting since in this compound the 4f-contribution to the MAE vanishes to first-order in the spin-orbit coupling. These results are in line with previous work where it has been shown that the energy scale associated with the change in energy of the Weyl nodes due to canting of the magnetization in general does not scale with the MAE17. Last, notice that the larger magnetic moment of Gd3+ contributes to lowering the required magnetic field to cant the magnetization.

### Discussion

The predicted Weyl nodes dynamics has consequences on a number of physical observables. Some of them are related to the effects that canting of the magnetization has on the symmetry of the electronic structure, others follow from the topological phase transitions associated with having a single Weyl node going through the Fermi surface. We now briefly outline these open possibilities.

The bulk Fermi surface topological phase transitions lead to visible consequences on the surface electronic structure. To illustrate this, we consider the (100)-surface of NdRhC2 terminated on a RhC2 plane. In addition to the corresponding projection of the bulk Fermi surface, the surface Fermi surface exhibits clear Fermi arcs connecting the projection of opposite chirality Weyl nodes, as shown in Fig. 5. For θ = 0, Fig. 5b shows the Fermi arcs fixing the Fermi energy at the Weyl node energy (εF = 0.043 meV), while Fig. 5c corresponds to the charge-neutrality chemical potential (εF = 0). As one of the Weyl nodes moves out from the hole pocket, the upper part of the Fermi arc, which at θ = 0 starts in the hole pocket (Fig. 5d), at θ = 25°, starts in the electron Fermi surface (Fig. 5d). We notice that similar results are obtained for an Nd-terminated surface.

The GME accounts for the generation of an electrical current by a time-dependent magnetic field and can be regarded as the low-frequency limit of the natural optical activity in inversion-broken materials67,68:

$${j}_{i}={\alpha }_{ij}^{\,{{\mathrm{GME}}}\,}{B}_{j}.$$
(2)

The tensor $${\alpha }_{ij}^{\,{{\mathrm{GME}}}\,}$$ is determined by the angular momentum of the Bloch states at the Fermi surface, and the chiral symmetry is a sufficient condition for it to be nonzero67. Thus, as explained earlier, canting of the magnetization along a low-symmetry direction, which breaks all the mirror symmetries, together with the inversion-broken lattice structure enable the GME in the carbides family. When evaluated for simplified models for Weyl semimetals, the GME is found to scale linearly with the energy splitting between Weyl nodes67. While the question of how such results are modified when having more realistic band structures deserves further investigation, the sizable and controllable energy splitting here predicted both for NdRhC2 and GdCoC2 makes them interesting platforms to explore the GME.

The chiral magnetic effect (CME) also accounts for the generation of an electrical current under applied magnetic field but is fundamentally different to the GME in that it is driven by topological properties of the Fermi surface69. Specifically, the relevant topological invariants are the Chern numbers associated with the Fermi surfaces (Eq. (1)). In the presence of electric and magnetic fields having a common projection E B, the Cα invariants break the conservation of the electron density ni of individual Fermi surface sheets. Specifically, the continuity equation describing the conservation of charge reads:

$$\frac{\partial {n}_{\alpha }}{\partial t}+\nabla \cdot {{{{\bf{j}}}}}_{\alpha }=\frac{{C}_{\alpha }}{4{\pi }^{2}}{{{\bf{E}}}}\cdot {{{\bf{B}}}}$$
(3)

As shown in ref. 69, a natural consequence of nonzero Cα invariants is the CME. Specifically, turning on a E B for a finite time causing a difference Δμ between the chemical potential of sheets having opposite Chern number C leads to a finite current j = CΔμB/4π2. Remarkably, as indicated by Fig. 3e, canting of the magnetization in noncentrosymmetric magnets can provide a way to turn on/off the CME. In particular, when a type-II Weyl node crosses the Fermi energy, as predicted here for NdRhC2 and GdCoC2, the switching on/off of the CME directly exposes the underlying Weyl node dynamics.

Last, we emphasize that although both the GME and the CME rely on the system having a chiral Shubnikov group, the requirements for the CME are stronger as also a finite Berry curvature flux trough the Fermi surfaces sheets is necessary. This distinction suggests that the dependence of the GME and of the CME on the canting angle may be qualitatively different, the latter including rather abrupt changes according to the phase diagram in Fig. 3e.

To summarize, realization of a chirality imbalance configuration in real materials is of paramount importance for exploiting the most essential property of Weyl fermions, their chirality. Our findings not only show that the family of carbides with broken $${{{\mathcal{I}}}}$$ and broken Θ are natural candidates to that end, but also that canting of the magnetization toward a low-symmetry direction in inversion-broken magnets can be generally considered as a knob to measure and control various chiral electromagnetic responses, particularly of gyrotropic and chiral magnetic type.

These effects were recently shown to harbor particular application potential: as our asymmetric Weyl semimetals from the RMC2 carbide family naturally have a chirality dependent Fermi velocity of Weyl cones, time-dependent pumping of electrons from a non-chiral external source can be used to generate a non-vanishing chiral chemical potential. This again generates, via the chiral anomaly, a current along the direction of an applied magnetic field even in the absence of an external electric field so that the material acts as a rectifying element70. Moreover, chirality imbalanced has been predicted to trigger an interesting photoresponse. The simultaneously broken particle-hole and spatial inversion symmetry can generate giant photocurrents, suggesting a potential application of these asymmetric Weyl semimetals for creating tuneable THz photosensors71.

## Methods

### Band structure calculations

DFT calculations were carried out using the Perdew–Burke–Ernzerhof implementation52 of the GGA using the full-potential local-orbital (FPLO) code53, version 18.00-57 (https://www.fplo.de). A k-mesh with 12 × 12 × 12 subdivisions was used for numerical integration in the Brillouin zone along with a linear tetrahedron method. Spin-orbit effects were included in the self-consistent calculations via the 4-spinor formalism implemented in the FPLO code.

The description of the 4f elements within the framework of DFT is still a subtle problem54. In this work, we considered both of the widely-used approaches to treat the 4f-shell: the OC approximation, and the GGA + U scheme. In the first approach, we set the spin for R3+ ions according to the Hund’s first rule while maintaining a spherically averaged distribution of electrons in the 4f shell54. The trivalent configuration of the rare-earth atoms is found in the related Ni-based carbides RNiC2 (R = Pr, Nd, Gd)34,72 and GdCoC236. In the second approach, to circumvent the usual problem of multiple metastable solutions, we used the occupation matrix control73,74, whereby several different initial 4f density matrices, corresponding to +3 valence of R ions, were considered to explore the energy landscape. We used the full-localized limit of the double counting scheme with F0 = 7.0 eV, F2 = 11.92 eV, F4 = 7.96 eV, and F6 = 5.89 eV, leading to U = 7 eV and J = 1 eV. For most of the compounds, both GGA + OC and GGA + U favors the magnetic moments to be in the rare-earth planes. Therefore, we consider m[001] for all the compounds for brevity and comparison. While we can anticipate that the existence of Weyl nodes at low energies is a finding robust to these explored choices, we will discuss to what extent details such as position in energy and momentum can be affected in a forthcoming publication.

### Topological electronic properties

To study the topological properties, we constructed a tight-binding model based on maximally projected Wannier functions. In the basis set, states lying in the energy window −9.5–10 eV were considered. The Wannier basis set typically included the 5d and 6s states for R (respectively, 4d and 5s for Y), valence d and s states for M, and 2s and 2p states for C. The accuracy of the resulting tight-binding models was typically 15 meV compared to the self-consistent band structures.

### Magnetic configurations

To study the relative stability of different collinear long-range ordered states (applicable to materials belonging to classes II and III), different lower-symmetry AFM configurations were generated:

• AF1, with AFM interlayer and ferromagnetic (FM) intralayer couplings.

• AF2, with FM interlayer and AFM intralayer interactions.

• AF3, with AFM interlayer as well as AFM intralayer interactions.

Figure 6 shows these AFM configurations for compounds belonging to classes II and III.