Tunable chirality of noncentrosymmetric magnetic Weyl semimetals

Even if Weyl semimetals are characterized by quasiparticles with well-defined chirality, exploiting this experimentally is severely hampered by Weyl lattice-fermions coming in pairs with opposite chirality, typically causing the net chirality picked up by experimental probes to vanish. Here we show this issue can be circumvented in a controlled manner when both time-reversal- and inversion- symmetry are broken. To this end, we investigate chirality-disbalance in the carbide family RMC$_2$ (R a rare-earth and M a transition metal), showing several members to be Weyl semimetals. Using the noncentrosymmetric ferromagnet NdRhC$_2$ as an illustrating example, we show that an odd number of Weyl nodes can be stabilized at its Fermi surface by properly tilting its magnetization. The tilt direction determines the sign of the resulting net chirality, opening up a simple route to control it.

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 properties [1] 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 lowenergy effective theory around such a point corresponds to the Weyl equation [2], identifying the topological invariant with the corresponding Weyl fermion chirality. In lattice systems, however, Weyl nodes must come in pairs of opposite chirality [3]. 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.
Systems belonging to class III are of particular interest as they allow for a specific demonstration of the unique interplay between topology and magnetism offered by I-broken symmetry. As a proof of principle, we show for NdRhC 2 how in ferromagnetic non-centrosymmetric 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. Further, 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.
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) [51], as implemented in FPLO-18 [52,53] and considering for the treatment of the 4f -shell both the open-core approximation (OC) with Rspecific 4f spin moment but spherical orbital occupation [54] and the GGA+U method with the full-localized limit for the double counting correction [55] and parameters U = 7 eV and J = 1 eV. We study compounds which  [40]; b Ref. [50]; c Ref. [33]; d Ref. [31]; e Ref. [39]; f Ref. [30]; g Ref. [28]; h Ref. [29] have already been synthesized, including R = Y, La, Nd, Pr, Gd or Lu, and M = Co, Rh, Ni or Ru [28,29,31,40], and use the lattice parameters from The Materials Project [56] (see the Supplementary Material (SM) [57] for further details). Table I includes 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 thê in-plane 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 experiment [29,40,50]. Among these, LaRhC 2 is the only system that crystallizes in the noncentrosymmetric tetragonal space group P 4 1 [29] and displays an insulating electronic structure (see SM [57]). Opposed to this, YCoC 2 and LuCoC 2 grow in the noncentrosymmetric orthorhombic space group Amm2 [40,50] and exhibit a semi-metallic band structure [see Fig. 2(a) and SM [57]]. Four bands dominate the energy spectrum of LuCoC 2 near the Fermi energy. These arise from hybridized Co-d, C-p 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. 2 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. RudermanKittelKasuyaYosida 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. 1(a), (c)] belong to Class III, breaking both I and Θ. For these compounds, different magnetic states have been reported. GdCoC 2 was first described as antiferromagnetic (AFM) with in-plane moments canted at 48 • from the a-axis and T N = 15 For (b), the chosen high-symmetry points differ due to the AFM doubling of the unit-cell [49]. (g) Energy of the Weyl nodes vs. unit cell volume for all compounds considered in this work. [31]. However, a more recent study by Meng et al. finds a ferromagnetic state (FM) below ∼ 15 K [33]. GdNiC 2 , on the other hand, as most of the Ni-based members in the family [32], presents an AFM ground-state [31]. For PrRhC 2 and NdRhC 2 , experimental data is limited to high-temperature susceptibility measurements [29]. These indicate small Curie-Weiss temperatures (θ CW ∼ −13 K and ∼ 0 K, respectively) and magnetic moments in good agreement with the values expected for the trivalent R ions. GdRuC 2 exhibits an interesting contrast to the above, since it presents a transition to a FM phase but crystallizes in the I-symmetric Cmcm space group [ Fig.  1(b)], thus belonging to Class II [30].
Our DFT results correctly capture the NM states in YCoC 2 and LuCoC 2 and the FM states in GdCoC 2 and GdRuC 2 . For NdRhC 2 , we find that both GGA+OC and GGA+U predict an FM ground state. Results for PrRhC 2 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 GdNiC 2 , 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 phase [39,43,45,47]. Specifically in GdNiC 2 , different metamagnetic transitions have been observed under moderate external magnetic fields, yielding an interesting and complex phase diagram [39,42,48].
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 [53]. Figure 2(g) 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  I includes the position and energy of the respective lowest energy Weyl nodes (for a complete list, see SM [57]). We also include YCoC 2 as a reference [49]. 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.
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 YCoC 2 , which lie in the k z = 0 plane, are also present in LuCoC 2 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 k z = 0 plane and to a lower energy. To confirm this, we carried out explicit computation of Weyl points in YCoC 2 under external magnetic field acting on the spin degrees of freedom. Indeed, we find that the field endows the nodes a finite k z component.
Regarding GdNiC 2 and GdRuC 2 , the Fermi energy shift associated with their different number of valence electrons with respect to GdCoC 2 is naturally in opposite directions and therefore has different consequences. On GdRuC 2 , 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 [49]. 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 LuCoC 2 features twofold degenerate bands along the line X-T , forming a Dirac cone ∼1 eV above the Fermi level [ Fig. 2(a)]. The band degeneracy is lifted in the isolectronic FM compounds [ Fig. 2(c),(e)] and 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. 2(d)]. While the FM phase in GdNiC 2 has only been stabilized with an external magnetic field [39,42], it could be interesting to study this phenomenology in SmNiC 2 , where the competing CDW is suppressed [37] leading to a FM ground-state [32].
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 NdRhC 2 and PrRhC 2 . Also, low-energy Weyl nodes are present in these compounds [ Fig. 2(g)].
Pumping chirality to and through the Fermi surface -Due to the Nielsen-Ninomiya "no-go" theorem for chiral lattice fermions [3,58], Weyl nodes come in pairs of opposite chirality [59] (which can be broken up by very strong electron-electron interactions [60,61]). 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. Thus, the Weyl node degeneracy need not be even. 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 I-broken material to reduce the symmetry such that the Shubnikov group contains only the identity. As a proof of principle, we consider NdRhC 2 , 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, C 2 (z)}. 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 C 2 (z) but also the symmetries involving mirrors. Thus, starting from m along [001], a perturbation that cants m towards, e.g.,  ergy splittings between Weyl nodes of opposite chirality, of the order of tens of meV. Similar results for GdCoC 2 (see SM [57]) indicate that such splitting need not scale proportionally to the magnetic anisotropy field, expected to be smaller in the Gd compound than in NdRhC 2 due to the S = J = 7/2 state of Gd 3+ .
The tuning of Weyl nodes to the Fermi surface at specific angles resembles the prediction for Co 3 Sn 2 S 2 [17], with the difference that in the latter case the crystal structure is centrosymmetric and each Weyl node is degenerate. This difference is crucial as 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 dynamically broken by external magnetic fields, breaking 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. 1(d)], 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 • , NdRhC 2 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 towards [111] ([−111]). This may generate ideal conditions to experimentally explore the role of Weyl nodes in gyrotropic and chiral magnetic effects [62,63], in the nonlinear electric response of magnetic materials [64,65], and in effects recently associated with the chiral anomaly like the planar Hall effect [66,67].
Conclusions -Combination of broken inversion and broken time-reversal symmetry with a general orientation of the magnetization can lift all degeneracies among Weyl points. We have demonstrated that this allows to tune an odd number of Weyl nodes to the Fermi surface for a specific member of the RMC 2 family, which we have also shown to host various Weyl semi-metals. Our ideas can be naturally tested in all noncentrosymmetric compounds exhibiting magnetic order and provide straightforward experimental access to chiral transport.

SUPPLEMENTAL INFORMATION
Section S1 contains details about the crystal structures used in the calculations. Section S2 explains the methodological aspects of the calculations. Section S3 presents the results of canting the magnetization in GdCoC 2 . Section S4 exhibits, in Figs. S3 − S9, the bandstructures of the considered materials on a larger Brillouin zone path and larger energy window than in the main text. Tables SII − SVII contain further information about the Weyl nodes found in the considered semimetals.

S1. CRYSTAL STRUCTURES
The structural parameters for all the compounds considered in this study were obtained from The Materials Project [56]. Table SI lists the space groups, lattice constants (a, b, c), primitive unit cell volumes (Vol), corresponding formula units (f.u.) per unit cell (Z), and selected bond lengths (d) for these compounds.

S2. METHODS
Density functional theory (DFT) calculations were carried out using the Perdew-Burke-Ernzerhof (PBE) implementation [51] of the generalized gradient approximation (GGA) using the full-potential local-orbital (FPLO) code [52], version 18.00-57 [53]. A k-mesh with 12 × 12 × 12 subdivisions was used for numerical integration in the Brillouin zone (BZ) 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. To study the relative stability of different collinear long-range ordered states (applicable to materials belonging to classes II & III), different lower-symmetry antiferromagnetic (AFM) configurations were generated: -AF1, with AFM interlayer and ferromagnetic (FM) intralayer couplings, -AF2, with FM interlayer and AFM intralayer interaction, -AF3, with AFM interlayer as well as AFM intralayer interaction. Fig. S1 shows these AFM configurations for compounds belonging to classes II and III.
The description of the 4f elements within the framework of DFT is still a subtle problem [54]. In this work, we considered both of the widely-used approaches to treat the 4f -shell: the open-core (OC) approximation, and the GGA+U scheme. In the first approach, we set the spin for R 3+ ions according to the Hund's first rule while maintaining a spherically averaged distribution of electrons in the 4f shell [54]. In the second approach, to circumvent the usual problem of multiple metastable solutions, we used the occupation matrix control [76,77], 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 F 0 = 7.0 eV, F 2 = 11.92 eV, F 4 = 7.96 eV, and F 6 = 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-plane. 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.
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 eV to 10 eV were considered. The Wannier basis set typically included the 5d and 6s states for R (respectively, 3d and 4s 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 bandstructures.  FIG. S1. AFM configurations considered for compounds belonging to the symmetry classes II and III. The R atoms, transition metal atoms and carbon atoms are, respectively, shown with blue, yellow and grey spheres. The relative spin directions on the R atoms are represented by arrows. The choice of quantization axis is only schematic. For each case, the resulting lower symmetry space groups are also indicated.

S3. WEYL NODES IN GdCoC2
Fig . S2 shows the splitting in energy of the Weyl node closest to the Fermi energy in GdCoC 2 as the magnetization is canted from [001] to [111]. Notice that energy splittings of tens of meV are obtained, similar to the results for NdRhC 2 in the main text, regardless of the smaller magnetic anisotropy energy expected in GdCoC 2 due to the half-filled 4f shell.
It is important to realize, as these results indicate, that the energy scale associated with the splitting in energy of the Weyl nodes need not scale with the magnetic anisotropy energy. Thus, sizeable splittings can occur in materials which require significantly different external magnetic field to cant the magnetization.

S4. ELECTRONIC PROPERTIES
For completeness, below we provide the bandstructures and element-resolved density of states (DOS) for the different compounds considered in this work and list all the respective Weyl nodes found in the energy window between ±120 meV.        , as obtained within the OC approach and spin-orbit effects included. The BZ and the high-symmetry points for both the AF1 magnetic configuration is similar to FM, since the AF1 state corresponds to doubling of the unit cell along a (see Fig. S1), shown in (e).