Tunable ferromagnetic weyl fermions from a hybrid nodal ring

ABSTRACT Realization of nontrivial band topology in condensed matter systems is of great interest in recent years. Using first-principles calculations and symmetry analysis, we propose an

exotic topological phase with tunable ferromagnetic Weyl fermions in a half-metallic oxide CrP2O7. In the absence of spin–orbit coupling (SOC), we reveal that CrP2O7 possesses a hybrid nodal

ring. When SOC is present, the spin-rotation symmetry is broken. As a result, the hybrid nodal ring shrinks to discrete nodal points and forms different types of Weyl points. The Fermi arcs

projected on the (100) surface are clearly visible, which can contribute to the experimental study for the topological properties of CrP2O7. In addition, the calculated quasiparticle

interference patterns are also highly desirable for the experimental study of CrP2O7. Our findings provide a good candidate of ferromagnetic Weyl semimetals, and are expected to realize

related topological applications with their attracted features. SIMILAR CONTENT BEING VIEWED BY OTHERS ISING FERROMAGNETISM AND ROBUST HALF-METALLICITY IN TWO-DIMENSIONAL HONEYCOMB-KAGOME

CR2O3 LAYER Article Open access 09 November 2020 SYNTHESIS OF A SEMIMETALLIC WEYL FERROMAGNET WITH POINT FERMI SURFACE Article 22 January 2025 GIANT ROOM TEMPERATURE ANOMALOUS HALL EFFECT

AND TUNABLE TOPOLOGY IN A FERROMAGNETIC TOPOLOGICAL SEMIMETAL CO2MNAL Article Open access 10 July 2020 INTRODUCTION Following the great advancements of topological insulators,1,2 many

efforts have been devoted to the studies of topological semimetals (TSMs) that also present topologically nontrivial band structures. TSMs mainly include Dirac,3,4 Weyl,5,6,7,8,9,10 and

nodal line semimetals,11,12,13,14,15,16,17 and so on, in which the valence and the conduction bands cross discretely or continuously in momentum space. The band crossings of TSMs can be

topologically protected and thus are robust against the perturbations. Physically, the low-energy excitations near the crossing points can be regarded as the corresponding quasiparticles,

namely Dirac, Weyl, and nodal line fermions, and so on. Contrary to fourfold-degenerate Dirac fermions, Weyl fermions, a concept derived from the high energy particle physics and finally

realized in Weyl semimetal (WSM) systems,18 are twofold degenerate at the crossing points (i.e., Weyl points) with breaking either time-reversal or inversion symmetries. Owing to the

Nielsen–Ninomiya “no-go theorem”,19,20 Weyl points severely appear in pairs with opposite chirality, and are merely annihilated with another Weyl point that has opposite chirality or by

violating charge conservation.21 There are two types of Weyl fermions in general. A type-I Weyl cone is only Lorentz invariant in the case of no tilting of the Weyl cone whatsoever. However,

there is, in general, some tilting in condensed matter systems, just not enough to correspond to electron and hole pockets. In the case of a type-II Weyl cone, in contrast, tilting is

severe enough that electron and hole pockets form.10 WSMs that host unique Weyl fermions present nontrivial electronic structures, which leads to their topologically protected Fermi arc

surface states, the negative magnetoresistance associated with the chiral anomaly, and anomalous Hall effect. These properties hold some potential applications for future electronic devices.

Therefore, searching for WSMs with exotic topological properties is of great interest. The research of WSMs started from the magnetic pyrochlore iridates Y2Ir2O75 and spinel HgCr2Se4,6 in

which the time-reversal symmetry \({\cal{T}}\) is broken. Subsequently, most of the reported WSMs are proposed in non-magnetic materials by breaking inversion symmetry

\({\cal{P}}\).7,10,22,23,24 Nevertheless, it is well known that the magnetism has the influence on the band topology and the topologically protected surface states. First, ferromagnetic WSMs

can provide an intrinsic source of magnetic field to realize the enhanced anomalous Hall effect associated with the Weyl point near the Fermi level.25 Second, the magnetic space group of

crystal is sensitive to the magnetization directions, which leads to tunable Weyl points. Therefore, focusing on ferromagnetic WSMs is essential to study their exotic quantum phenomena

related to magnetism. However, there are few candidates of magnetic WSMs,26,27,28 even though they exerted an important influence on early work of WSMs. In particular, the candidates of

ferromagnetic WSMs that coexist in type-I and type-II Weyl points are highly desirable, because they exhibit rich topological physics. Recently, the concept of hybrid nodal ring that

includes both type-I and type-II band-crossing points was proposed.29,30 Zhang et al.31 showed that the nonmagnetic material Ca2As hosts the hybrid nodal ring, and revealed that the hybrid

nodal ring is characterized with unique signatures in magnetic response. In general, the nodal ring can be topologically protected by a combined \({\cal{P}}{\cal{T}}\) symmetry or a mirror

reflection symmetry. If these symmetries are broken, the hybrid nodal ring could shrink to several discrete crossing points, leading to the emergence of type-I and type-II Weyl points.32

Therefore, one of the possible routes to realizing ferromagnetic WSMs coexisting in type-I and type-II Weyl points is to introduce spin–orbit coupling (SOC) for the hybrid nodal ring in

ferromagnetic materials. In this work, we propose that the ferromagnetic CrP2O7 possesses a hybrid nodal ring when SOC is absent. After introducing SOC, we found that CrP2O7 is ferromagnetic

WSM coexisting type-I and type-II Weyl points. Moreover, the band topology of CrP2O7 can be manipulated with magnetization directions. To facilitate the experimental investigation, we also

calculated the surface states and the joint density of states (JDOSs) associated with quasiparticle interference (QPI) pattern of CrP2O7. RESULTS AND DISCUSSION The crystal structure of

monoclinic CrP2O7 is illustrated in Fig. 1a, which belongs to the non-symmorphic space group of _P_21/_c_ (_C_2_h_) with four formula units in the primitive cell. Its optimized lattice

parameters and atomic Wyckoff positions are summarized in Supplementary Table 1. Clearly, the structure consists of the distorted PO4 tetrahedrons, with P being their center and CrO6

octahedrons with Cr occupying their center. The phonon spectrum (see Supplementary Fig. 1) indicates that the monoclinic CrP2O7 is dynamically stable. In addition, to assess its

thermodynamic stability, we also calculated the formation enthalpy of CrP2O7. The chemical composition of CrP2O7 can be decomposed to chromium dioxide (CrO2) and diphosphorus pentaoxide

(P2O5), and therefore we chose rutile CrO233 and orthorhombic P2O534 as the reference phases. As a result, the formation energy of CrP2O7 can be defined by Δ_E_f = _E_(CrP2O7) − _E_(CrO2) − 

_E_(P2O5), in which _E_(CrP2O7), _E_(CrO2), and _E_(P2O5) are the total energies of CrP2O7, rutile CrO2, and orthorhombic P2O5, respectively. The negative formation energy of −1.396 eV per

unit cell indicates that the CrP2O7 is thermodynamically stable and is feasible to be experimentally synthesized. To determine the magnetic configuration of CrP2O7, we performed total energy

calculations on various magnetic phases for the monoclinic CrP2O7. Our calculations show that the ferromagnetic phases are the most energetically favorable (at least 45.2 meV per Cr atom

lower in energy than the collinear and noncollinear antiferromagnetic phases, see details in Supplementary Table 2). The calculated magnetic moment per Cr atom is 2_μ_B. Moreover, the

ferromagnetic phases with different magnetization directions are almost degenerate, suggesting that the magnetization directions of soft magnetic CrP2O7 can be easily manipulated by external

magnetic field. To study the band topology, we first calculated the band structure of CrP2O7 without SOC, as shown in Fig. 2a. Clearly, CrP2O7 presents the half-metallic behavior. The

majority-spin channel is metallic, while there is a gap of 4.03 eV for the minority-spin channel. Along the _Y_–Γ direction in momentum space, there exists one type-II band crossing with the

electron- and hole-like states coexisting in energy. Along the other high symmetry lines, we do not find any band-crossing points associated with the valence and conduction bands. To

further find all band crossings in the whole Brillouin zone (BZ), we constructed the tight-binding Hamiltonian by fitting the band structures. Then, all the nodes are obtained based on the

tight-binding Hamiltonian. As a result, we found that the nodes in the BZ form a closed nodal ring, which is centered around the Γ point, as shown in Fig. 2b, c. Owing to the existence of

glide mirror for CrP2O7, the nodal ring satisfies the mirror reflection with respect to the (010) plane. From the projection of nodal ring onto the (010) plane, we can clearly see that the

nodal ring goes across the first BZ and partly locates in the second BZ (see Fig. 2c). The calculated Berry phases _γ_ along a closed loop \({\cal{L}}\) that encircles the nodal ring,

defined by \(\gamma = {\oint}_{\cal{L}} {\cal{A}} ({\mathbf{k}})\cdot{d}{\mathbf{k}}\) [\({\cal{A}}({\mathbf{k}})\) is Berry connection], are plotted in Fig. 2d along the high-symmetry lines

_Y_–Γ–_Y_. The obtained results confirm the nontrivial topology of the nodal ring in ferromagnetic CrP2O7. In addition to the type-II band crossings, we also found that the nodal ring

contains type-I band crossings. For example, the band structure around the crossing point (0.350, 0.0, −0.518) shows that the two bands cross linearly with electron- and hole-like states

occupying different energy ranges (see Fig. 2e), characteristic of the type-I band crossings. The coexistence of type-I and type-II band crossings suggests that the hybrid nodal ring exists

in the ferromagnetic CrP2O7. Next, we show the nodal ring solution based on the two-band _k_ · _p_ effective Hamiltonian, which can be described as $$H({\mathbf{k}}) = \mathop

{\sum}\limits_{i = 0}^3 {d_i} ({\mathbf{k}})\sigma _i,$$ (1) where _σ_0 is 2 × 2 identity matrix and _σ_1, _σ_2, _σ_3 represent the Pauli matrices, respectively, _d__i_(K) are real

functions, in which K = (_k__x_, _k__y_, _k__z_) are three components of the momentum K with respect to the Γ point. Because the kinetic term _d_0(K)_σ_0 is independent with the band

crossings, we ignore it in the following discussions. Consider the inversion symmetry of the crystal structure, we can choose it as \({\cal{P}} = \sigma _z\). Under this symmetry, we can

obtain $$H({\cal{P}}{\mathbf{k}}) = {\cal{P}}H({\mathbf{k}}){\cal{P}}^{ - 1},$$ (2) which translates into $$d_{1,2}({\mathbf{k}}) = - d_{1,2}( - {\mathbf{k}}),\;\;\;\;d_3({\mathbf{k}}) =

d_3( - {\mathbf{k}}).$$ (3) When SOC is absent, the majority spin and the minority spin can be clearly distinguished because they cannot hybridize with each other. Because this nodal ring

locates near the Fermi level, there exists only the states of one spin. The spin flip can only occur by acting on the time-reversal operation \({\cal{T}}\), which assures that the two-band

_k_ · _p_ effective Hamiltonian is invariant under \({\cal{T}}\) symmetry, namely, $$H({\cal{T}}{\mathbf{k}}) = {\cal{T}}H({\mathbf{k}}){\cal{T}}^{ - 1}.$$ (4) In the absence of SOC, the

time-reversal operation can be chosen as \({\cal{T}} = K\) in terms of \({\cal{T}}^2 = + 1\), where _K_ is the complex conjugation. Thus, Eq. (4) can be simplified into

$$d_{1,3}({\mathbf{k}}) = d_{1,3}( - {\mathbf{k}}),\;\;\;\;d_2({\mathbf{k}}) = - d_2( - {\mathbf{k}}).$$ (5) From Eqs. (3) and (5), we can obtain the following expressions constrained by the

inversion and time-reversal symmetries: $$\begin{array}{l}d_1({\mathbf{k}}) = 0,\\ d_2({\mathbf{k}}) = \mathop {\sum}\limits_{i = x,y,z} {A_i} k_i + \mathop {\sum}\limits_{i,j,l = x,y,z}

{A_{ijl}} k_ik_jk_l,\\ d_3({\mathbf{k}}) = B + \mathop {\sum}\limits_{i,j = x,y,z} {B_{ij}} k_ik_j + \mathop {\sum}\limits_{i,j,l,m = x,y,z} {B_{ijlm}} k_ik_jk_lk_m,\end{array}$$ (6) where

the parameters _A__i_, _A__ijl_, _B_, _B__ij_, and _B__ijlm_ derived from the _k_ · _p_ model can be determined by the first-principles calculations. For simplicity, we can ignore the

high-order terms of _d_2(K) and _d_3(K) in Eq. (6). Meanwhile, considering that the little group is _C_s in the _k__x_–_k__z_ plane, we can simplify _d_2(K) and _d_3(K) of Eq. (6) into

$$\begin{array}{l}d_2({\mathbf{k}}) = A_xk_x + A_yk_y + A_zk_z,\\ d_3({\mathbf{k}}) = B + B_{xx}k_x^2 + B_{yy}k_y^2 + B_{zz}k_z^2 + B_{xz}k_xk_z.\end{array}$$ (7) Based on above discussions,

we can determine the energy dispersion as $$E({\mathbf{k}}) = \pm \sqrt {d_2({\mathbf{k}})^2 + d_3({\mathbf{k}})^2} .$$ (8) Thus, the valence and conduction bands can touch only when the

conditions of _d_2(K) = 0 and _d_3(K) = 0 are satisfied simultaneously, which allow the codimension of the band crossing that is one, resulting in the formation of a nodal ring in momentum

space. From Eq. (7), we can see that _d_2(K) = 0 is the equation of a plane across the Γ point. In addition, if the conditions of _B__xx_ > 0, _B__yy_ > 0, _B__zz_ > 0, and _B_ <

 0 are satisfied, _d_3(K) = 0 can represent the equation of a closed ellipsoidal surface centered at the Γ point. Certainly, _B__xx_ < 0, _B__yy_ < 0, _B__zz_ < 0, and _B_ > 0

can also result in the solution of a closed ellipsoidal surface centered at the Γ point for _d_3(K) = 0. As a consequence, the intersection of the plane _d_2(K) = 0 and the closed

ellipsoidal surface _d_3(K) = 0 forms the closed nodal ring. Due to existence of only _k__x__k__z_ cross-term, the nodal ring must cross the _k__y_ axis and is located in the quadrants 1 and

3 of _k__x_–_k__z_ plane, as shown in Fig. 2b, c. Note that the detailed shape of the closed nodal ring is determined by the high-order terms in Eq. (6). When SOC is included, the two spin

channels couple together and the spin-rotation symmetry that protects the nodal ring is broken. Thus, the hybrid nodal ring shall shrink to discrete points and form different types of Weyl

points. We firstly provide a parity analysis for the band topology because of the presence of \({\cal{P}}\) symmetry for CrP2O7. The calculated parity products of occupied Bloch states at

the time-reversal invariant momenta (TRIM) points are +1 except the Γ points, which leads to the fact that the product of parity eigenvalues at eight TRIM points is −1, suggesting that there

exists odd number of pairs of Weyl points in the bulk.35 Consider that the magnetic symmetry is closely related to the magnetization direction and hence has a significant impact on the

topological properties of ferromagnetic materials. Here, we mainly focus on two typical magnetic directions, that is, [0 1 0] and [1 0 0] magnetization directions. As listed in Table 1,

there exist one pair of Weyl fermions along the [0 1 0] magnetization direction and three pairs of Weyl fermions along the [1 0 0] magnetization direction, respectively, which is consistent

with our parity analysis. With the magnetization along the [0 1 0] direction, the magnetic double point group _C_2_h_ is preserved. Along the _Y_–Γ direction in momentum space, the little

group is _C_2, which can allow for the band crossing along this direction, thus resulting in type-II Weyl point W1 (0.0, 0.1126, 0.0) below the Fermi level, as listed in Table 1. The two

crossing bands belong to different irreducible representations of _C_2 operation, namely Γ3 and Γ4, as shown in Fig. 3a, b. Certainly, another type-II Weyl point must appear at W1′ (0.0,

−0.1126, 0.0) with opposite Chern number in view of the \({\cal{P}}\) symmetry. It is well known that the type-II Weyl point appears as the contact point between electron and hole pockets,

characteristic of the open constant energy surfaces. The 3D representation of band structure is shown in Fig. 3c, implying that the Weyl fermion with breaking Lorentz invariance (i.e.,

type-II Weyl fermion) appears. In addition, we can observe the electron and hole pockets from the Fermi surface at the (001) plane (see Supplementary Fig. 2), which further confirm that

these points are type-II. Along the [1 0 0] magnetization direction, the magnetic double point group is reduced to _C__i_. Consequently, the screw-rotation symmetry is broken, which leads to

the fact that the Weyl points along _Y_–Γ without SOC open a tiny gap when the magnetization is along the [1 0 0] direction, as shown in Fig. 3d. However, we find other three pairs of Weyl

points that do not locate along the high symmetry lines or planes (see Table 1), implying that these band crossings are accidental. The three pairs of Weyl points contain one pair of type-I

(W2) and two pairs of type-II (W3 and W4) Weyl points. The band structures and their 3D representations around the type-I W2 and type-II W3 Weyl points are shown in Supplementary Fig. 3. The

corresponding Chern numbers are calculated by using Wilson-loop method as implemented in Z2Pack package,36 which are also summarized in Table 1. To check the topological stability of the

coexistence of the type-I and type-II Weyl points, we tuned the the strength of SOC with 0.8_λ_0 and 1.2_λ_0 (here _λ_0 denotes the actual SOC strength). The calculated results suggest that

the SOC strength can only slightly change the positions of Weyl nodes. Along the [0 1 0] and [1 0 0] magnetization directions, the type-I and type-II Weyl points remain present, dictating

that the topological properties of CrP2O7 are robust against perturbation. The evident manifestation of WSMs is the existence of the topologically protected surface states and the visible

Fermi arcs. As shown in Fig. 4, the obtained surface states of CrP2O7 within the [0 1 0] magnetization direction show clear connection between the valence and conduction bands on the (100)

surface. The type-II Weyl point W1 located along the _Y_–Γ direction is projected onto the \(\tilde X\)–\({\tilde{\mathrm \Gamma }}\) direction of the (100) surface, and the tilted cone is

therefore observed (see Fig. 4c). Along \(\tilde Y\)–\({\tilde{\mathrm \Gamma }}\) and \(\tilde S\)–\({\tilde{\mathrm \Gamma }}\) directions, the small topologically nontrivial gaps are

observed. When magnetization is along the [1 0 0] direction, the _C_2 symmetry is broken. Thus, the surface states on the (100) plane show a nontrivial gap along the projected \(\tilde

X\)–\({\tilde{\mathrm \Gamma }}\) direction. However, we can clearly see a type-I cone along \(\tilde Y\)–\({\tilde{\mathrm \Gamma }}\) direction. This cone originates from the type-I Weyl

point W2, which can be projected on \(\tilde Y\)–\({\tilde{\mathrm \Gamma }}\) direction of the (100) plane. The accidentally degenerate W3 and W4, forming tilted cons, can also be found in

the surface states on the (100) plane if the suitable _k_ paths are selected in the projected BZ. In addition, the nontrivial Fermi arc states are visible in the projected (100) surface (see

Fig. 4b), which contributes to be experimentally observed in angle-resolved photoemission spectroscopy (ARPES). However, we cannot find the Fermi arc on the projected (010) surface, which

is ascribed to the fact that the two Weyl points along the _Y_–Γ direction are projected to one point on the (010) surface. Along the [1 0 0] magnetization direction, the visible Fermi arc

states projected on the (010) plane are also observed (see Supplementary Fig. 4). QPI pattern can directly signify the momentum transformation across the Fermi arc surface state, which

provides deep insight into the surface states of WSMs in momentum space based on spectroscopic-imaging scanning tunneling microscopy, and is considered to be good complementarity to ARPES.37

The QPI pattern corresponds to the Fourier transform of a real-space local density of states map and can be approximately interpreted in terms of JDOSs:38,39 $$J({\mathbf{q}},\omega ) =

{\int} {d^2} {\mathbf{k}}\rho ^0({\mathbf{k}},\omega )\rho ^0({\mathbf{k}} - {\mathbf{q}},\omega ),$$ (9) and \(\rho ^0({\mathbf{k}},\omega ) = - \frac{1}{\pi

}{\mathbf{Im}}G^0({\mathbf{k}},\omega )\) is K-resolved surface density of states, where _G_0(_K_,_ω_) denotes surface Green function at surface momentum K and a given energy _ω_, and Q is

the scattering vector. As plotted in Fig. 5, the calculated weighted Fermi arcs show an approximately constant spectral density. From the Fermi arcs, we can identify three independent

scattering vectors (labeled as _q_1, _q_2, and _q_3) arising from arc–arc scattering. Figure 5b shows visible QPI pattern of CrP2O7. The highest intensity patterns appear at the Γ point,

which can be ascribed to the fact that _q_1, _q_2, and _q_3 scatterings are all contributed to the QPI pattern of the Γ point. In summary, multiple nontrivial fermions are explored in the

ferromagnetic CrP2O7 employing the first-principles calculations and symmetry analysis. This ferromagnetic material possesses the hybrid nodal ring with coexistence of type-I and type-II

band crossings. The nodal ring is demonstrated by employing a two-band _k_ · _p_ effective Hamiltonian. Moreover, the hybrid nodal ring becomes the discrete type-I and type-II Weyl fermions

when SOC is considered. The interplay between the magnetization directions and crystal symmetries in CrP2O7 is discussed in detail using symmetry analysis. The calculated Fermi arc surface

states and QPI patterns are highly desirable for the experimental study. We expect that our results can be helpful in understanding the ferromagnetic WSMs and realizing the topological

properties of CrP2O7 in experiments. METHODS We performed first-principles calculations within the framework of density functional theory40,41 as implemented in the Vienna Ab initio

Simulation Package.42 The generalized gradient approximation (GGA) combined with the Perdew–Burke–Ernzerhof functional43 was chosen to describe the exchange correlation interactions. The

electron and core interactions were described by the projector augmented-wave method,44 in which the 3_p_63_d_54_s_1, 3_s_23_p_3 and 2_s_22_p_4 were treated as the valence electrons for Cr,

P, and O, respectively. The kinetic-energy cutoff for plane-wave basis set used was 600 eV. Considering the strong correlation effect of 3_d_ electron of Cr, we employed the GGA + _U_

scheme45 with an effective on-site Coulomb energy _U_eff = _U_ − _J_ = 3.6 eV. It is worth noting that the topological properties of CrP2O7 can be obtained in a _U_eff range from 2.6 to 4 

eV, which works well on other compounds including Cr atoms.46,47,48 The influence of the exchange coupling _J_ on the band structure of CrP2O7 is also discussed in Supplementary Fig. 5. The

tight-binding Hamiltonian was constructed by the maximally localized Wannier functions basis as implemented in Wannier90 code.49 Based on the tight-binding Hamiltonian, the iterative Green

function method as implemented in WannierTools package50 was employed to calculate the surface states, Fermi arcs, and the joint density of states. DATA AVAILABILITY The data that support

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Natural Science Foundation of China (NSFC, Grant Nos. 11674148, 11334003, and 11847301), the Guangdong Natural Science Funds for Distinguished Young Scholars (No. 2017B030306008), the

Fundamental Research Funds for the Central Universities (Nos. cqu2018CDHB1B01, 2019CDXYWL0029, and 2019CDJDWL0005), the China Postdoctoral Science Foundation (No. 2019M652686), the Natural

Science Basic Research plan in Shaanxi Province of China (Grant No. 2018JQ1083), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Grant No. 17JK0041), the

Baoji University of Arts and Sciences Key Research (Grant No.: ZK2017009), and the Science, Technology, and Innovation Commission of Shenzhen Municipality (No. ZDSYS20170303165926217).

AUTHOR INFORMATION Author notes * These authors contributed equally: Baobing Zheng, Bowen Xia, Rui Wang AUTHORS AND AFFILIATIONS * Department of Physics and Shenzhen Key Laboratory of

Quantum Science and Engineering, Southern University of Science and Technology, 518055, Shenzhen, P. R. China Baobing Zheng, Bowen Xia, Rui Wang, Jinzhu Zhao, Zhongjia Chen & Hu Xu *

School of Physics and Technology, Wuhan University, 430072, Wuhan, P. R. China Baobing Zheng * College of Physics and Optoelectronic Technology and Advanced Titanium Alloys and Functional

Coatings Cooperative Innovation Center, Baoji University of Arts and Sciences, 721016, Baoji, P. R. China Baobing Zheng * Institute for Structure and Function and Department of physics,

Chongqing University, 400044, Chongqing, P. R. China Rui Wang * Department of Physics, South China University of Technology, 510640, Guangzhou, P. R. China Zhongjia Chen & Yujun Zhao *

Center for Quantum Computing, Peng Cheng Laboratory, 518055, Shenzhen, P. R. China Hu Xu Authors * Baobing Zheng View author publications You can also search for this author inPubMed Google

Scholar * Bowen Xia View author publications You can also search for this author inPubMed Google Scholar * Rui Wang View author publications You can also search for this author inPubMed 

Google Scholar * Jinzhu Zhao View author publications You can also search for this author inPubMed Google Scholar * Zhongjia Chen View author publications You can also search for this author

inPubMed Google Scholar * Yujun Zhao View author publications You can also search for this author inPubMed Google Scholar * Hu Xu View author publications You can also search for this

author inPubMed Google Scholar CONTRIBUTIONS H.X. directed and designed the whole research. B.B.Z. did all calculations and wrote the paper. B.W.X. initiated the research and did early

calculations. R.W. provided scientific discussions. All authors discussed the results and provided inputs to the manuscript. CORRESPONDING AUTHOR Correspondence to Hu Xu. ETHICS DECLARATIONS

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http://creativecommons.org/licenses/by/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Zheng, B., Xia, B., Wang, R. _et al._ Tunable ferromagnetic Weyl fermions from a

hybrid nodal ring. _npj Comput Mater_ 5, 74 (2019). https://doi.org/10.1038/s41524-019-0214-z Download citation * Received: 20 March 2019 * Accepted: 26 June 2019 * Published: 15 July 2019 *

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