Abstract
Despite recent efforts to advance spintronics devices and quantum information technology using materials with nontrivial topological properties, three key challenges are still unresolved^{1,2,3,4,5,6,7,8,9}. First, the identification of topological band degeneracies that are generically rather than accidentally located at the Fermi level. Second, the ability to easily control such topological degeneracies. And third, the identification of generic topological degeneracies in large, multisheeted Fermi surfaces. By combining de Haas–van Alphen spectroscopy with density functional theory and bandtopology calculations, here we show that the nonsymmorphic symmetries^{10,11,12,13,14,15,16,17} in chiral, ferromagnetic manganese silicide (MnSi) generate nodal planes (NPs)^{11,12}, which enforce topological protectorates (TPs) with substantial Berry curvatures at the intersection of the NPs with the Fermi surface (FS) regardless of the complexity of the FS. We predict that these TPs will be accompanied by sizeable Fermi arcs subject to the direction of the magnetization. Deriving the symmetry conditions underlying topological NPs, we show that the 1,651 magnetic space groups comprise 7 grey groups and 26 blackandwhite groups with topological NPs, including the space group of ferromagnetic MnSi. Thus, the identification of symmetryenforced TPs, which can be controlled with a magnetic field, on the FS of MnSi suggests the existence of similar properties—amenable for technological exploitation—in a large number of materials.
Similar content being viewed by others
Main
Nearly a century ago Wigner, von Neumann and Herring^{1,2} addressed the conditions under which Bloch states form degenerate band crossings, but their topological character and technological relevance has been recognized only recently^{3,4,5}. To be useful^{4,5,6,7,8,9}, tiny changes of a control parameter must generate a large response, underscoring the lack of control over the band filling as the unresolved key challenge in materials with band crossings known so far. This raises the question whether topological band crossings exist that are (1) generically located at the Fermi level, (2) separated sufficiently in the Brillouin zone (BZ) and (3) easy to control.
Natural candidates are systems with nonsymmorphic symmetries—for example, screw rotations—that generate positions in reciprocal space at which bandcrossings are symmetryenforced. The associated key characteristics include^{10,11,12,13,14,15,16,17}: (1) the crossings are due to symmetry alone, that is, they occur on all bands independent of details such as chemical composition; (2) pairs of band crossings with opposite chirality are separated in kspace by about half a reciprocal lattice vector; (3) the band crossings may be enforced on entire planes^{11,12}, forming socalled nodal planes (NPs) with nonzero topological charge; and (4) their existence may be controlled by means of symmetry breaking. Thus, if in a material the Fermi surfaces (FSs) cross such topological NPs, they enforce pairwise FS degeneracies with large Berry curvatures. The topology of these FS degeneracies, which we refer to as topological protectorates (TPs), will be independent of materialspecific details and, moreover, may be controlled by symmetry breaking. The putative existence of topological NPs has been studied in phononic metamaterials^{18,19,20}, and mentioned in a study of nonmagnetic chiral systems focusing on Kramers–Weyl fermions^{21}.
To demonstrate the formation of symmetryenforced TPs at the intersection of NPs with the FS, we decided to study the ferromagnetic state of manganese silicide (MnSi), which has attracted great interest for its itinerantelectron magnetism^{22}, helimagnetism, skyrmion lattice^{23} and quantum phase transition^{24}. Crystallizing in space group (SG) 198, MnSi is a magnetic sibling of nonmagnetic RhSi (ref. ^{25}), CoSi (ref. ^{26}) and PdGa (ref. ^{27}), in which sizeable Fermi arcs and multifold fermions were recently inferred from angleresolved photoemission spectroscopy. MnSi is ideally suited for our study, as magnetic fields exceeding around 0.7 T stabilize ferromagnetism with magnetic screwrotation symmetries enforcing NPs.
Initial assessment
A first theoretical assessment establishes that a ferromagnetic spin polarization along a highsymmetry direction, for example, [010], reduces the symmetries from SG 198 (P2_{1}3) of paramagnetic MnSi to the magnetic SG 19.27 (P2_{1}2′_{1}2′_{1}) (Supplementary Note 1, Extended Data Fig. 1). This SG contains two magnetic screw rotations \(\theta {\tilde{C}}_{2}^{x}\) and \(\theta {\tilde{C}}_{2}^{z}\) (Fig. 1a), that is, 180° screw rotations around the x and z axes combined with timereversal symmetry θ. These rotations act like mirror symmetries, as they relate Bloch wave functions at (k_{x}, k_{y}, k_{z}) to those at (−k_{x}, k_{y}, k_{z}) and (k_{x}, k_{y}, −k_{z}), respectively, leaving the planes k_{x} = 0 and k_{z} = 0 and the BZ boundaries k_{x} = ±π and k_{z} = ±π invariant. Squaring \(\theta {\tilde{C}}_{2}^{x}\) and \(\theta {\tilde{C}}_{2}^{z}\) and letting them operate on the Bloch state ψ(k)⟩, one finds that \({(\theta {\tilde{C}}_{2}^{x})}^{2}\psi ({\bf{k}})\rangle ={{\rm{e}}}^{{\rm{i}}{k}_{x}}\psi ({\bf{k}})\rangle \) and \({(\theta {\tilde{C}}_{2}^{z})}^{2}\psi ({\bf{k}})\rangle ={{\rm{e}}}^{{\rm{i}}{k}_{z}}\psi ({\bf{k}})\rangle \). Hence, by Kramers theorem^{28}, all Bloch states on planes with k_{x} = ±π or k_{z} = ±π are twofold degenerate. Moving away from these BZ boundaries, the symmetries are lowered such that the Bloch states become nondegenerate. Therefore, all bands in ferromagnetic MnSi are forced to cross at k_{x} = ±π and k_{z} = ±π, representing a duo of NPs.
The topological charge ν of this duo of NPs (Fig. 1b) may be determined with the fermion doubling theorem^{29}, which states that ν summed over all band crossings must be zero. We note that besides the NPs, there is an odd number of symmetryenforced band crossings on the Y_{1}–Γ–Y and R_{1}–U–R lines forming Weyl points (ν = ±1) and fourfold points (ν = ±2), respectively (Fig. 1c, d, Extended Data Fig. 2, Supplementary Note 1). Moreover, due to the effective mirror symmetries, accidental Weyl points away from these highsymmetry lines must form pairs or quadruplets with the same ν. As the sum over ν of all of these Weyl and fourfold points is odd, the duo of NPs must carry a nonzero topological charge to satisfy the fermion doubling theorem. Hence, the duo of NPs at the BZ boundary is the topological partner of a single Weyl point on the Y_{1}–Γ–Y line (Fig. 1b). This is a counterexample to Weyl semimetals, in which Weyl points occur always in pairs.
Shown in Fig. 1d is the band structure of a generic tightbinding model satisfying SG 19.27 (Supplementary Note 2), where pairs of bands form NPs on the BZ boundaries k_{x} = ±π and k_{z} = ±π, whereas on the Y_{1}–Γ–Y and R_{1}–U–R lines there are Weyl and fourfold points, respectively. Explicit calculation of the Chern numbers shows that all of these band crossings, including those at the NPs, exhibit nonzero topological charges as predicted above. In turn, all of the FSs carry substantial Berry curvatures. The numerical analysis shows that these Berry curvatures become extremal at the NPs and close to the fourfold and Weyl points (Extended Data Fig. 3). By the bulk–boundary correspondence^{3,4}, the nontrivial topology of these band crossings generates large Fermi arcs on the surface, which extend over half of the BZ of the surface (Extended Data Fig. 4). These arguments may be extended to 254 of the 1,651 magnetic SGs, of which 33 have NPs whose topological charges are enforced to be nonzero by symmetry alone (Supplementary Note 3).
Calculated electronic structure
Figure 1e shows the density functional theory (DFT) band structure of MnSi, taking into account spin–orbit coupling, for the experimental moment of 0.41 Bohr magnetons (μ_{B}) per Mn atom along the [010] direction (Methods, Extended Data Fig. 5). Ten bands are found to cross the Fermi level (Fig. 1e). In agreement with our symmetry analysis and the tightbinding model (Fig. 1d), we find the same generic band crossings, namely: (1) NPs on the BZ boundaries k_{x} = ±π and k_{z} = ±π; (2) an odd number of Weyl points along Y_{1}–Γ–Y; and (3) an odd number of fourfold points along R_{1}–U–R.
The calculated FSs as matched to experiment are shown in Fig. 1f, highlighting the NPs at the BZ boundaries at k_{x} = ±π and k_{z} = ±π (see Extended Data Table 1 for key parameters and Extended Data Fig. 5). Eight FS sheets centred at Γ comprise two small isolated hole pockets (sheets 1 and 2), two intersecting hole pockets with avoided crossings and magnetic breakdown due to spin–orbit coupling (sheets 3 and 4) and two pairs of junglegymtype sheets (sheets 5 and 6, and sheets 7 and 8). Sheets 9 and 10 are centred at R, comprising eight threefingered electron pockets around the [111] axes and a tiny electron pocket, respectively. The sheet pairs (5, 6), (7, 8) and (9, 10) extend beyond the BZ boundaries with pairwise sticking at the NPs. They represent TPs (marked in red) with extremal Berry curvatures protected by the magnetic screw rotations \(\theta {\tilde{C}}_{2}^{x}\) and \(\theta {\tilde{C}}_{2}^{z}\). In contrast, sheets 5 to 10 do not form TPs at the BZ boundary k_{y} = ±π, because the moment pointing along [010] breaks \(\theta {\tilde{C}}_{2}^{y}\).
Rotating the direction of the magnetization away from [010] distorts the FS sheets, where TPs exist only on those BZ boundaries parallel to the magnetization (Supplementary Videos 1 and 2). For instance, rotating the moments within the x–y plane away from [010] breaks the magnetic screw rotation \(\theta {\tilde{C}}_{2}^{x}\), but keeps \(\theta {\tilde{C}}_{2}^{z}\) intact. In turn, the TPs gap out on the k_{y} = ±π and k_{x} = ±π planes, whereas they remain degenerate at the k_{z} = ±π planes (Extended Data Fig. 1, Supplementary Note 1).
Experimental results
To experimentally prove the mechanism causing generic TPs at the intersection of the FS with symmetryenforced NPs and their dependence on the direction of the magnetization, we mapped out the FS by means of the de Haas–van Alphen (dHvA) effect using capacitive cantilever magnetometry (Methods, Extended Data Fig. 5, Supplementary Note 4). In the following, we focus on magnetic field rotations in the (001) plane, where φ denotes the angle of the field with respect to [100]. This plane proves to be sufficient to infer the main FS features. Complementary data for the (001) and (\(\overline{1}\overline{1}0\)) planes are presented in Extended Data Fig. 5. Typical torque data at different temperatures for φ = 82.5° (Fig. 2a, b) show pronounced dHvA oscillations for magnetic fields exceeding B ≈ 0.7 T. The hysteretic behaviour below about 0.7 T (Fig. 2a, inset) originates from the well understood helimagnetic and conical phases^{30}. Figure 2b shows the oscillatory highfield part of the torque τ(1/B) at temperature T = 35 mK with the lowfrequency components removed for clarity. To extract the dHvA frequencies, a fast Fourier transform (FFT) analysis of τ(1/B) was carried out, where the effects of demagnetizing fields and the unsaturated magnetization were taken into account (Methods). The FFT frequencies correspond to extremal FS crosssections in low effective fields of about 0.7−1.9 T (Methods).
Typical dHvA frequencies and FFT amplitudes, shown for φ = 82.5° in Fig. 2c, show five different regimes of dHvA frequencies labelled I to V. They comprise over 40 dHvA frequencies corresponding to different extremal FS orbits, as denoted by Greek letters (Fig. 2c, Extended Data Table 2). In our data analysis, we delineated artefacts due to the finite FFT window, such as the side lobes between κ_{2} and 2κ_{1}, or 3κ_{2} and ξ_{1} (Methods). Fitting the temperature dependence of the FFT amplitudes within Lifshitz–Kosevich theory^{31}, the effective masses for each of the orbits were deduced ranging from m* = 0.4m_{e} to m* = 17m_{e}, where m_{e} is the bare electron mass (Fig. 2d).
To relate the dHvA frequencies to the calculated FS orbits, the torque amplitude was inferred from the DFT band structure by means of the Lifshitz–Kosevich formalism, using small rigid band shifts of the order of 10 meV to improve the matching following convention (Methods, Extended Data Table 1). The assignment to experiment was based on the consistency between dHvA frequency, angular dispersion, strength of torque signal, field dependence of the dHvA frequencies, effective masses and presence of magnetic breakdown, as explained in Methods, Extended Data Table 2, Extended Data Figs. 6, 7, Supplementary Note 5.
Figure 3a shows an intensity map of the experimental data of the (001) plane as a function of φ, where the theoretical dHvA branches are depicted by coloured lines (colours correspond to the FS sheets in Fig. 1). For comparison, Figure 3b shows an intensity map of the calculated dHvA spectra, where the experimental frequencies are marked by grey crosses.
For regimes I to IV, featuring contributions of the large FS sheets (5, 6) and (7, 8), all frequencies may be assigned unambiguously (Extended Data Figs. 6, 7, Supplementary Note 5). Namely, regime I contains the loop orbits around U associated with pair (5, 6) (blue and orange) and the neck orbit of sheet 8 (yellow). Regime II exhibits the dHvA branches originating from neck orbits around Γ–Y–Γ on sheet 7 (purple). The neck orbits of sheet 8 (yellow), which evades detection because of the large slope of the dispersion, its high mass and the suppression of the magnetic torque near the [010] highsymmetry direction, is consistent with an anomalous frequency splitting at the expected crossing with the loop orbits of pair (5, 6) (blue and orange) around 6.5 kT (Supplementary Note 5). Regime III arises from both pairs (5, 6) and (7, 8), that is, neck orbits around Γ–Y–Γ of (5, 6) and loop orbits around U of (7, 8). The remaining cascade of frequencies in regime III reflects breakdown orbits (translucent yellow) arising from avoided crossings between sheets 3 and 4 (red and green). Regime IV is, finally, dominated by sheet 2 of the isolated hole pocket and the first harmonic of sheet 2.
As the magnetic torque generically vanishes at highsymmetry directions, which corresponds to the ⟨100⟩ axes in regimes I to IV, the associated FS sheets are centred at the Γ point. Likewise, the lowest frequency in regime V corresponds to a Γcentred FS sheet, which can be assigned to the small hole pocket of sheet 1. In stark contrast, for regime V above about 0.05 kT, the highsymmetry directions correspond to the ⟨111⟩ axes, whereas the torque for the ⟨100⟩ axes is finite (see also Fig. 3b, Extended Data Fig. 5g). Hence, regime V is related to FS pockets in the vicinity of the R point that may be assigned to FS sheets (9, 10). This allows for a basic estimate of the size and the effective mass of FS sheets (9, 10) without the need for a detailed account of their shape, completing the assignment. The calculations demonstrate the presence of symmetryenforced crossings of sheets (9, 10) if they intersect the NPs (Fig. 1).
To confirm that we observed the entire FS, we calculated the Sommerfeld coefficient of the specific heat from the density of states at the Fermi level as rescaled by the measured mass enhancements (Extended Data Table 1). Excellent agreement is observed within a few percent of experiment^{32}, γ ≈ 28 mJ mol^{−1} K^{−2} at B = 12 T. This analysis reveals, that sheets (5, 6), (7, 8) and (9, 10), which form TPs, contribute 86% to the total density of states at the Fermi level.
Topological NPs
Spectroscopic evidence of the symmetryenforced topological band degeneracies at the BZ boundaries may be inferred from FS sheets (5, 6). Identical characteristics are observed for FS sheets (7, 8) (Extended Data Fig. 7, Supplementary Note 5). We note that the dHvA cyclotron orbits are perpendicular to the NPs for fundamental reasons, piercing through them at specific points of the TPs. As shown in Fig. 4a, a magnetic field parallel to [010] leads to extremal crosssections for FS sheets (5, 6), supporting cyclotron orbits in the vicinity of the U and the Y_{1} points on planes depicted by blue and green shading, respectively. Centred with respect to the U point are possible cyclotron orbits comprising different segments of FS sheets 5 and 6, which interact at TP1 to TP4 with the BZ boundaries at k_{x} = ±π and k_{z} = ±π. In the absence of the nonsymmorphic symmetries, these intersections would exhibit anticrossing and magnetic breakdown, leading to several orbits with different crosssections and hence several dHvA frequencies. Instead, the behaviour is distinctly different to magnetic breakdown or Klein tunnelling^{33,34}.
As the BZ boundaries at k_{x} = ±π and k_{z} = ±π represent symmetryenforced NPs, the crossing points of sheets 5 and 6 at TP1 to TP4 are, hence, protected band degeneracies at which the wavefunctions are orthogonal, that is, TP1 to TP4 are part of the TPs that suppress transitions between orbits (we call orbits containing at least one TP ‘topological orbits’). In turn, two independent topological orbits (topological orbits 1 and 2) with identical areas and hence the same dHvA frequencies are expected (Fig. 4b, top). This is in excellent agreement with experiment, which shows a single dHvA frequency for field parallel [010] (φ = 90° in Fig. 4c). Rotating the direction of the magnetic field within the x–y plane away from [010], the NP at k_{x} = ±π gaps out, whereas the NP at k_{z} = ±π remains protected. Thus, the associated loop orbits around U (Fig. 4b, bottom) continue to include two points on the FS at k_{z} = ±π (TP3 and TP4), leading to two additional topological orbits (topological orbits 3 and 4) of identical crosssection with the same dHvA frequency, in perfect agreement with the observed spectra (Fig. 4c).
Comparing the extremal crosssections of the neck orbits around Γ–Y_{1}–Γ with those around Γ–X–Γ, the latter crosses an NP whereas the former does not. With respect to Γ–X–Γ, there would be two extremal crosssections with identical areas, positioned symmetrically with respect to X (Fig. 4d, top left), whereas for the crosssections with respect to Γ–Y_{1}–Γ there are two extremal orbits with different areas positioned asymmetrically with respect to Y_{1} (Fig. 4d, bottom left). Thus, within our symmetry analysis and our DFT calculations, we expect a single dHvA branch for neck orbits parallel to a NP compared with two dHvA branches for neck orbits that are not parallel to a NP (Fig. 4d, top right). Keeping in mind that only neck orbits around Y_{1} are accessible experimentally, we clearly observe two branches, giving strong evidence that there are no NPs on the k_{y} = ±π BZ boundary (Fig. 4e).
Concluding remarks
The symmetryenforced NPs and TPs that are generically located at the Fermi level, which support large Berry curvatures, may account for various properties, such as anomalous Hall currents^{35} or the nonlinear optical responses^{36}. Indeed, large anomalous contributions to the Hall response are in excellent quantitative agreement with ab initio calculations, where the calculated FS and Berry curvatures were essentially identical to the FS we report here^{37}. Our calculations imply also sizeable Fermi arcs at the surface of MnSi and related magnetic compounds such as FeGe and Fe_{1−x}Co_{x}Si, connecting the topological charge of the NPs directly with a Weyl point (Extended Data Fig. 4). These Fermi arcs reflect the presence of duos of NPs. Analogous Fermi arcs will not exist in nonmagnetic materials with SG 198^{25,26,27}, which support trios of NPs (Supplementary Note 1).
In systems with symmetryenforced NPs and TPs, tiny changes of the direction of the magnetization will control the topological band crossing in the bulk and the Fermi arcs, causing massive changes of Berry curvature that may be exploited technologically. The formation of TPs irrespective of the complexity of the FS raises the question of whether they affect the transport properties^{38} and enable exotic states of matter^{39}. Extending the analysis presented here to all 1,651 magnetic SGs, we find that there is a large number of candidate materials, such as CoNb_{3}S_{6} (ref. ^{40}) or Nd_{5}Si_{3} (ref. ^{41}) with similar TPs (Extended Data Table 3, Supplementary Note 3), which await to be explored from a fundamental point of view and harnessed for future technologies.
Methods
Sample preparation
For our study, two MnSi samples were prepared from a highquality single crystalline ingot obtained by optical floatzoning^{42}. The samples were oriented by Xray Laue diffraction and cut into 1 × 1 × 1 mm^{3} cubes with faces perpendicular to [100], [110], and [110] and [110], and [111] and [112] cubic equivalent directions, respectively. Both samples exhibited a residual resistivity ratio close to 300.
Experimental methods
Quantum oscillations of the magnetization, that is, the dHvA effect, was measured by means of cantilever magnetometry measuring the magnetic torque τ = m × B. The doublebeam type cantilevers sketched in Extended Data Fig. 5e were obtained from CuBe foil by standard optical lithography and wetchemical etching. The cantilever position was read out in terms of the capacitance between the cantilever and a fixed counter electrode using an AndeenHagerling AH2700A capacitance bridge, similar to the design described in refs. ^{43,44}.
Angular rotation studies were performed in a ^{3}He insert with a manual rotation stage at a base temperature T = 280 mK under magnetic fields up to 15 T. In addition, the effective charge carrier mass was determined using a dilution refrigerator insert with fixed sample stage under magnetic fields up to 14 T (16 T using a Lambda stage) at temperatures down to 35 mK.
We discuss partial rotations in the (001) and (\(\overline{1}\overline{1}0\)) crystallographic planes. The angle φ is measured from [100] in the (001) plane and the angle θ is measured from [001] in the (\(\overline{1}\overline{1}0\)) plane. Corresponding data are shown in Fig. 3a and Extended Data Fig. 5g. Owing to the topology of the FS and the simple cubic BZ, the (001) plane rotation shows most of the extremal orbits and is already sufficient for an assignment to the FS sheets. For this reason, the discussion of the dHvA data in the main text focuses on the rotation in the (001) plane.
The response of the cantilever was calibrated by means of the electrostatic displacement, taking into account the cantilever bending line obtained from an Euler–Bernoulli approach^{45}. Applying a d.c. voltage, U, to the capacitance C_{0} = ε_{0}A/d_{0}, defined by the area A, the plate distance d_{0} and the vacuum permittivity ε_{0}, leads to an electrostatic force F = C_{0}U^{2}/2d_{0}. This force is equivalent to a torque τ = βFL, where L is the effective beam length and β = 0.78 is a geometrydependent prefactor accounting for the different mechanical response of a bending beam to a torque and force, respectively. From this, the calibration constant K(C) = τ/ΔC quantifying the capacitance change ΔC in response to the torque was obtained for different values of C. Changes in K(C) up to 10% were recorded during magnetic field sweeps. The torque was calculated using
Evaluation of the dHvA signal
The dependence of the capacitance, C(B_{ext}), was converted into torque and corrected as described below, where B_{ext} is the applied magnetic field. An exemplary torque curve obtained at T = 280 mK and φ = 82.5° is shown in Fig. 2a. In the regime below B ≈ 0.7 T the transitions from helical to conical and fieldpolarized state generated a strongly hysteretic behaviour. At higher fields, magnetic quantum oscillations on different amplitude and frequency scales could be readily resolved. The first lowfrequency components appeared at magnetic fields as low as B ≈ 4 T, whereas several highfrequency components, corresponding to larger extremal crosssections, could only be resolved in high fields (Fig. 2b). Consequently, the data acquisition and evaluation was optimized by treating low and highfrequency components separately.
To eliminate the nonoscillatory component of the signal, loworder polynomial fits or curves obtained by adjacent averaging over suitable field intervals were subtracted from the data, producing consistent results. FFTs of τ(1/B) were used to determine the frequency components contained in the signal. Field sweeps were performed from 0 T to 15 T at 0.03–0.04 T min^{−1} and from 15 T to 10 T at 0.008 T min^{−1}. FFTs over the range 4 T to 15 T (10 T to 15 T) were performed to evaluate frequency components below (above) f = 350 T for measurements in the ^{3}He insert and from 10 T to 14 T (11 T to 16 T with Lambda stage) in the dilution refrigerator. The values correspond to the applied field before taking into account demagnetization. Rectangular FFT windows were chosen to maximize the ability to resolve closely spaced frequency peaks. See Supplementary Note 4 for details.
Internal magnetic field and dHvA frequency f(B) in a weak itinerant magnet
MnSi is a weak ferromagnet with an unsaturated magnetization up to the largest magnetic fields studied. This results in two different peculiarities concerning the observed dHvA frequencies. (1) The field governing the quantum oscillations is the internal field^{31} B_{int} = μ_{0}H_{ext} + μ_{0}(1 − N_{d})M, where μ_{0} is the vacuum permeability, H_{ext} is the applied magnetic field and M is the magnetization. Taking into account the demagnetization factor^{46} N_{d} = 1/3 for a cubic sample to first order yields a field correction \(\Delta B={B}_{{\rm{i}}{\rm{n}}{\rm{t}}}{B}_{{\rm{e}}{\rm{x}}{\rm{t}}}=\frac{2}{3}{\mu }_{0}{M}_{\exp }\approx 0.131\,{\rm{T}}\), where M_{exp} is the lowfield value of the magnetization in the fieldpolarized phase determined experimentally. The applied field was corrected by this value. The field dependence of the magnetic moment yields only a minor correction of the internal field that may be neglected. (2) The effect of the unsaturated magnetization on the Fermi surface is more prominent and may be described in a good approximation as a rigid Stoner exchange splitting that scales with the magnitude of the magnetization. Consequently, FS crosssectional areas are enlarged with increasing B for the majority electron orbits and minority hole orbits. Crosssectional areas shift downwards for majority hole and minority electron orbits.
This change in crosssectional area is not directly proportional to the change in the observed dHvA frequencies f, that is, the dHvA frequencies deviate from the fielddependent frequency \({f}_{{\rm{B}}}(B)=\frac{\hbar }{2{\rm{\pi }}e}{A}_{k}(B)\) obeying the Onsager relation (here A_{k} is the extremal crosssectional area in kspace, ħ is the reduced Planck constant and e is the electron charge). The frequency f observed may be inferred^{47} from the derivative of the dHvA phase factor \(2{\rm{\pi }}\left(\frac{{f}_{{\rm{B}}}(B)}{B}\gamma \right)\pm \frac{1}{4}\) with respect to 1/B:
Thus, a linear relation f_{B}(B) results in a constant f(B). This may be understood intuitively, because a linear term in f_{B}(B) leads only to a phase shift since the oscillations are periodic in 1/B. Equation (2) shows that f(B) is the zerofield intercept of the tangent to f_{B}(B).
In the Stoner picture of rigidly split bands, f_{B}(B) may be related to the magnetization^{47,48} using
where I is the Stoner exchange parameter, m_{b} is the band mass, the ± is for electron and hole orbits, respectively, s = ±1 is the spin index and f_{0} is the hypothetical frequency without exchange splitting. Note, that this model is only meaningful in the fieldpolarized regime B ≳ 0.7 T. Using the experimental M(B) curve of MnSi^{32}, we estimate that the frequencies f(B) in the windows used for f > 350 T defined above with centre fields B_{average} = 2B_{high}B_{low}/(B_{low} + B_{high}) ranging from 11.8 T to 13.2 T correspond to the extremal crosssections at B ≈ 1.7−1.9 T (Extended Data Fig. 5f). For the window used for frequencies f < 350 T, it is B_{average} = 6.5 T and f(B) corresponds to the extremal crosssections at B ≈ 0.7 T. Thus, even under large magnetic fields, the experimental frequency values correspond to a fieldpolarized state in a low field.
Quantum oscillatory torque and Lifshitz–Kosevich equation
Evaluation and interpretation of the quantum oscillatory torque magnetization was performed using the Lifshitz–Kosevich formalism^{31}.The components of M parallel (∥) and perpendicular (⊥) to the field are given by:
and
where V is the sample volume, p is the harmonic index, A″ is the curvature of the crosssectional area parallel to B, and f is the dHvA frequency observed (see comments above). The phase γ = 1/2 corresponds to a parabolic band. In general, the phase includes also contributions due to Berry phases when the orbit encloses topologically nontrivial structures in kspace. The ± holds for maximal and minimal crosssections, respectively. The torque amplitude is given by τ_{osc} = M_{osc,⊥}B. The torque thus vanishes in highsymmetry directions where f(φ) is stationary. This feature of τ may be used to infer additional information about the symmetry properties of a dHvA branch. R_{T} describes the temperature dependence of the oscillations
from which the effective mass m* including renormalization effects can be extracted, where, k_{B} is the Boltzmann constant. Equation (6) was fitted to the temperature dependence of the FFT peaks using the average fields B_{average} defined above. No systematic changes in the mass values were observed within the standard deviation of the fits when different window sizes were chosen. See Supplementary Note 4 for details. The Dingle factor
describes the influence of a finite scattering time τ. Here, ω_{c} =eB/m* is the cyclotron frequency.
DFT calculations
The band structure and FS sheets of MnSi in the fieldpolarized phase were calculated using DFT. The calculations included the effect of spin–orbit coupling. In all calculations, the magnetic part of the exchangecorrelation terms was scaled^{49} to match the experimental magnetic moment of 0.41μ_{B} per Mn atom at low fields. As input for the DFT calculations, the experimental crystal structure of MnSi was used, that is, space group P2_{1}3 (198) with an experimental lattice constant a = 4.558 Å. Both Mn and Si occupy Wyckoff positions 4a with coordinates (u, u, u), (−u + 1/2, −u, u + 1/2), (−u, u + 1/2, −u + 1/2), (u + 1/2, −u + 1/2, −u) where u_{Mn} = 0.137 and u_{Si} = 0.845 (Extended Data Fig. 5a).
Calculations were carried out using WIEN2k^{50}, ELK^{51} and VASP^{52,53} using different versions of the local spin density approximation. The results are consistent within the expected reproducibility of current DFT codes^{54}. The remaining uncertainties motivate a comprehensive experimental FS determination as reported in this study. In the main text, we focus on the results obtained with WIEN2k, using the local spin density approximation parametrization of Perdew and Wang^{55} and a sampling of the full BZ with a 23 × 23 × 23 Γcentred grid. The results of Extended Data Figs. 1, 2, 4 were obtained using VASP with the PBE functional^{56} and a BZ sampling with a 15 × 15 × 15 kmesh centred around Γ.
Bands used for the determination of the Fermi surface were calculated with WIEN2k on a 50 × 50 × 50 kmesh. Owing to the presence of spin–orbit coupling, but the absence of both inversion and timereversal symmetry, band structure data had to be calculated for different directions of the spin quantization axis. For a given experimental plane of rotation, calculations were performed in angular steps of 10°. The bands were then interpolated kpointwise using thirdorder splines to obtain band structure information in 1° steps.
For the prediction of the dHvA branches from the DFT results, the Supercell kspace Extremal Area Finder (SKEAF)^{57} was used on interpolated data corresponding to 150 × 150 × 150 kpoints in the full BZ. The theoretical torque amplitudes shown in Fig. 3b were calculated directly from the prefactors in equations (4) and (5) convoluted with a suitable distribution function.
To compute the surface states of MnSi in the fieldpolarized phase (Extended Data Fig. 4), we first constructed a DFTderived tightbinding model using the maximally localized Wannier function method as implemented in Wannier90^{58}. Using this tightbinding model, we computed the momentumresolved surface density of states by means of an iterative Green’s function method, using WannierTools^{59}. The symmetry eigenvalues of the DFT bands were computed from expectation values using VASP pseudo wavefunctions, as described in ref. ^{60}.
Magnetic breakdown
The probabilities for magnetic breakdown at a junction i is given by \({p}_{i}={{\rm{e}}}^{\frac{{B}_{0}}{B}}\). The probability for no breakdown to occur is thus q_{i} = 1 − p_{i}. The breakdown fields B_{0} were calculated from Chamber’s formula
where k_{g} is the gap in kspace and a and b are the curvatures of the trajectories at the breakdown junction^{31}. In our study of MnSi, we observed magnetic breakdown in particular between sheets 3 and 4, which exhibit up to eight junctions depending on the magnetic field direction and between FS sheet pairs touching the BZ surfaces on which the NP degeneracy is lifted. Only breakdown orbits that are closed after one cycle are considered in the analysis. Further details can be found in the Supplementary Note 5.
Assignment of dHvA orbits and rigid band shifts
The assignment of the experimental dHvA branches to the corresponding extremal FS crosssections was based on the following criteria: (1) dHvA frequency—determining sheet size in terms of the crosssectional area; (2) angular dispersion—relating to sheet shape, topology and symmetry; (3) torque signal strength—relating to sheet shape and symmetry; (4) direction of f(B) shift—relating to spin orientation and charge carrier type; (5) effective mass—relating to the temperature dependence; (6) magnetic breakdown behaviour—relating to proximity of neighbouring sheets.
The majority of the observed dHvA branches could be related directly to the FS as calculated. In addition, we used the wellestablished procedure of small rigid band shifts to optimize the matching. While this procedure is, in general, neither charge nor spin conserving, it results in a very clear picture of the experimental FS. One has to bear in mind, however, that the deviations between the true FS and the calculated FS are not due to a rigid band shift (this might be justified, for example, in case of unintentional doping, which we rule out here). Rather, it may be attributed to differences in the band dispersions that originate in limitations of our DFT calculations (for example, neglecting electronic correlations and the coupling to the spin fluctuation spectrum).
The dHvA orbits, the assignments to a specific extremal crosssection, the observed and predicted frequencies, the observed and predicted masses and mass enhancements are listed in Extended Data Table 2. Extended Data Table 1 summarizes the resulting characteristic properties of the FS sheets including their contribution to the density of states at the Fermi level.
Symmetry analysis
The symmetryenforced band crossings and the band topology follow from the nontrivial winding of the symmetry eigenvalues through the BZ. This winding of the eigenvalues is derived in Supplementary Note 1, both for the paramagnetic and ferromagnetic phases of MnSi. Supplementary Note 1 also contains the derivation of the topological charges of the NPs, Weyl points and fourfold points, which are obtained from generalizations of the Nielsen–Ninomiya theorem^{29}. To illustrate the band topology for ferromagnets in SG 19.27 and SG 4.9, two tightbinding models are derived in Supplementary Note 2, which includes also a discussion of the Berry curvature and the surface states. The classification of NPs in magnetic materials is given in Supplementary Note 3. It is found that among the 1,651 magnetic SGs, 254 exhibit symmetryenforced NPs. We find that (at least) 33 of these have NPs whose topological charge is guaranteed to be nonzero due to symmetry alone.
Data availability
Materials and additional data related to this paper are available from the corresponding authors upon reasonable request.
References
von Neumann, J. & Wigner, E. Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Z. Phys. 30, 467–470 (1929).
Herring, C. Accidental degeneracy in the energy bands of crystals. Phys. Rev. 52, 365–373 (1937).
Chiu, C.K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in threedimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Burkov, A. Weyl metals. Annu. Rev. Condens. Matter Phys. 9, 359–378 (2018).
Wang, Q. et al. Large intrinsic anomalous Hall effect in halfmetallic ferromagnet Co_{3}Sn_{2}S_{2} with magnetic Weyl fermions. Nat. Commun. 9, 3681 (2018); correction 9, 4212 (2018).
Huang, X. et al. Observation of the chiralanomalyinduced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).
Liang, S. et al. Experimental tests of the chiral anomaly magnetoresistance in the Dirac–Weyl semimetals Na_{3}Bi and GdPtBi. Phys. Rev. X 8, 031002 (2018).
Huang, S.M. et al. A Weyl fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015).
Michel, L. & Zak, J. Elementary energy bands in crystals are connected. Phys. Rep. 341, 377–395 (2001).
Young, S. M. et al. Dirac semimetal in three dimensions. Phys. Rev. Lett. 108, 140405 (2012).
Furusaki, A. Weyl points and Dirac lines protected by multiple screw rotations. Sci. Bull. 62, 788–794 (2017).
Zhao, Y. X. & Schnyder, A. P. Nonsymmorphic symmetryrequired band crossings in topological semimetals. Phys. Rev. B 94, 195109 (2016).
Zhang, J. et al. Topological band crossings in hexagonal materials. Phys. Rev. Mater. 2, 074201 (2018).
Yu, Z.M., Wu, W., Zhao, Y. X. & Yang, S. A. Circumventing the nogo theorem: a single Weyl point without surface Fermi arcs. Phys. Rev. B 100, 041118 (2019).
Wu, W. et al. Nodal surface semimetals: theory and material realization. Phys. Rev. B 97, 115125 (2018).
Türker, O. & Moroz, S. Weyl nodal surfaces. Phys. Rev. B 97, 075120 (2018).
Xiao, M. & Fan, S. Topologically charged nodal surface. Preprint at https://arxiv.org/abs/1709.02363 (2017).
Yang, Y. et al. Observation of a topological nodal surface and its surfacestate arcs in an artificial acoustic crystal. Nat. Commun. 10, 5185 (2019).
Xiao, M. et al. Experimental demonstration of acoustic semimetal with topologically charged nodal surface. Sci. Adv. 6, eaav2360 (2020).
Chang, G. et al. Topological quantum properties of chiral crystals. Nat. Mater. 17, 978–985 (2018).
Lonzarich, G. G. Magnetic oscillations and the quasiparticle bands of heavy electron systems. J. Magn. Magn. Mater. 76–77, 1–10 (1988).
Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).
Pfleiderer, C., McMullan, G. J., Julian, S. R. & Lonzarich, G. G. Magnetic quantum phase transition in MnSi under hydrostatic pressure. Phys. Rev. B 55, 8330–8338 (1997).
Sanchez, D. S. et al. Topological chiral crystals with helicoidarc quantum states. Nature 567, 500–505 (2019).
Rao, Z. et al. Observation of unconventional chiral fermions with long Fermi arcs in CoSi. Nature 567, 496–499 (2019).
Schröter, N. B. M. et al. Observation and control of maximal Chern numbers in a chiral topological semimetal. Science 369, 179–183 (2020).
Kramers, H. A. Théorie générale de la rotation paramagnétique dans les cristaux. Proc. Amsterdam Acad. 33, 959–972 (1930).
Nielsen, H. & Ninomiya, M. A nogo theorem for regularizing chiral fermions. Phys. Lett. B 105, 219–223 (1981).
Bauer, A. et al. Symmetry breaking, slow relaxation dynamics, and topological defects at the fieldinduced helix reorientation in MnSi. Phys. Rev. B 95, 024429 (2017).
Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 1984).
Bauer, A. et al. Quantum phase transitions in singlecrystal Mn_{1−x}Fe_{x}Si and Mn_{1−x}Co_{x}Si: crystal growth, magnetization, ac susceptibility, and specific heat. Phys. Rev. B 82, 064404 (2010).
Alexandradinata, A. & Glazman, L. Geometric phase and orbital moment in quantization rules for magnetic breakdown. Phys. Rev. Lett. 119, 256601 (2017).
van Delft, M. R. et al. Electron–hole tunneling revealed by quantum oscillations in the nodalline semimetal HfSiS. Phys. Rev. Lett. 121, 256602 (2018).
Xiao, D., Chang, M.C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).
Morimoto, T. & Nagaosa, N. Topological nature of nonlinear optical effects in solids. Sci. Adv. 2, e1501524 (2016).
Franz, C. et al. Realspace and reciprocalspace Berry phases in the Hall effect of Mn_{1−x}Fe_{x}Si. Phys. Rev. Lett. 112, 186601 (2014).
Smith, M. F. Smallangle interband scattering as the origin of the T^{3/2} resistivity in MnSi. Phys. Rev. B 74, 172403 (2006).
Grover, T. & Fisher, M. P. A. Quantum disentangled liquids. J. Stat. Mech. 1014, P10010 (2014).
Tenasini, G. et al. Giant anomalous Hall effect in quasitwodimensional layered antiferromagnet Co_{1/3}NbS_{2}. Phys. Rev. Res. 2, 023051 (2020).
Boulet, P., Weizer, F., Hiebl, K. & Noël, H. Structural chemistry, magnetism and electrical properties of binary Nd silicides. J. Alloys Compd. 315, 75–81 (2001).
Neubauer, A. et al. Ultrahigh vacuum compatible image furnace. Rev. Sci. Instrum. 82, 013902 (2011).
Wilde, M. A. et al. Magnetometry on quantum Hall systems: thermodynamic energy gaps and the density of states distribution. Phys. Status Solidi B 245, 344–355 (2008).
Wilde, M., Heitmann, D. & Grundler, D. Magnetization of Interacting Electrons in LowDimensional Systems Ch. 10, 245 (Springer Nanoscience and Technology, 2010).
Wilde, M. Magnetization Measurements on LowDimensional Electron Systems in HighMobility GaAs and SiGe Heterostructures. PhD thesis, Universität Hamburg (2004).
Aharoni, A. Demagnetizing factors for rectangular ferromagnetic prisms. J. Appl. Phys. 83, 3432–3434 (1998).
van Ruitenbeek, J. M. et al. A de Haas–van Alphen study of the field dependence of the Fermi surface in ZrZn_{2}. J. Phys. F 12, 2919–2928 (1982).
Kimura, N. et al. de Haas–van Alphen effect in ZrZn_{2} under pressure: crossover between two magnetic states. Phys. Rev. Lett. 92, 197002 (2004).
Hoshino, T., Zeller, R., Dederichs, P. H. & Weinert, M. Magnetic energy anomalies of 3d systems. Europhys. Lett. 24, 495–500 (1993).
Blaha, P. et al. Wien2k: an apw+lo program for calculating the properties of solids. J. Chem. Phys. 152, 074101 (2020).
The Elk Code (GNU General Public License, 2021); https://elk.sourceforge.io/
Kresse, G. & Furthmüller, J. Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Lejaeghere, K. A. Reproducibility in density functional theory calculations of solids. Science 351, aad3000 (2016).
Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electrongas correlation energy. Phys. Rev. B 45, 13244–13249 (1992).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Rourke, P. & Julian, S. Numerical extraction of de Haas–van Alphen frequencies from calculated band energies. Comput. Phys. Commun. 183, 324–332 (2012).
Mostofi, A. A. et al. Wannier90: a tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).
Wu, Q., Zhang, S., Song, H.F., Troyer, M. & Soluyanov, A. A. Wanniertools: an opensource software package for novel topological materials. Comput. Phys. Commun. 224, 405–416 (2018).
Gao, J., Wu, Q., Persson, C. & Wang, Z. Irvsp: to obtain irreducible representations of electronic states in the VASP. Comput. Phys. Commun. 261, 107760 (2021).
Acknowledgements
We thank D. Grundler, F. Rucker, A. Leonhardt, A. Rosch, T. Rapp, S. G. Albert and S. M. Sauther for support and discussions. Preliminary band structure calculations for a limited number of field orientations using FLEUR and JuDFT KKRGGA were carried out in collaboration with F. Freimuth, B. Zimmermann and Y. Mokrousov in the very early stages of this study. M.A.W., A.B. and C.P. were supported through DFG TRR80 (projectid 107745057, project E1 and E3), DFG SPP 2137 (Skyrmionics) under grant number PF393/19 (projectid 403191981), DFG GACR Projekt WI 3320/31, ERC Advanced Grants number 291079 (TOPFIT) and number 788031 (ExQuiSid), and Germany’s excellence strategy EXC2111 390814868.
Funding
Open access funding provided by Max Planck Society.
Author information
Authors and Affiliations
Contributions
M.A.W. and C.P. conceived the experiment and devised its interpretation together with A.P.S. A.B. prepared and characterized the samples. M.D. and M.A.W. conducted the measurements and analysed the data. M.A.W., A.N. and K.A. performed comprehensive band structure calculations. M.A.W. connected the experimental data with the calculated band structure. M.M.H., K.A. and A.P.S. performed the symmetry analysis and identified the topological properties of the band structure. M.M.H. and K.A. calculated the surface states and the Berry curvatures. M.A.W., A.P.S. and C.P. wrote the manuscript with contributions from M.M.H. and K.A. All authors discussed the data and commented on the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature thanks Zhiqiang Mao, Gang Xu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Magnetic space groups and electronic band structures for different directions of the magnetization.
a, Magnetic subgroups of space group 198 (P2_{1}3) and their group–subgroup relations. The magnetic space groups describing the symmetries for magnetizations along [010] and within the x–y plane are highlighted in green and blue, respectively. b, Orthorhombic BZ for ferromagnetic MnSi (left) and magnetization directions used for the ab initio calculations in c–f (right). c–f, Ab initio electronic band structure of ferromagnetic MnSi along the four highsymmetry paths indicated in b: Y_{1}–Γ–Y (c), X_{1}–Γ–X (d), R_{1}–U–R (e) and R_{2}–S–R (f). In the first, second and third rows, the magnetization is oriented along [010], 10° rotated into the x–y plane and along [110], respectively. Some of the Weyl points and fourfold degenerate points at (or near) the highsymmetry lines are highlighted by violet and brown circles.
Extended Data Fig. 2 Momentum dependence of the screwrotation eigenvalues.
a, Schematic band connectivity diagrams for a minimal set of bands along the Y_{1}–Γ–Y line (a1), the R_{1}–U–R line (a2) and the S_{1}–X–S (a3) line, respectively. The eigenvalues of the screwrotation symmetry \({\tilde{C}}_{2}^{y}\) are indicated by colour. b, Ab initio electronic band structure of MnSi in the [010] FM phase with the \({\tilde{C}}_{2}^{y}\) eigenvalues indicated by colour. The crossings of bands with different colour on the Y_{1}–Γ–Y line (b1) and on the R_{1}–U–R line (b2) are Weyl points and fourfold degenerate points, respectively.
Extended Data Fig. 3 Berry curvature on the Fermi surface.
a, Berry curvature Ω_{μ}(k) on one of the Fermi surfaces of a tightbinding model in SG 19.27, corresponding to ferromagnetic MnSi with the magnetization pointing along [010]. a1 shows the three components of Ω_{μ} as a function of k_{x}, along the direction indicated by the green arrow in a2. The absolute value of the Berry curvature Ω(k) is indicated in a2 by a logarithmic colour code. b, Same as a but for a tightbinding model in SG 4.9, corresponding to ferromagnetic MnSi with the magnetization rotated into the x–y plane.
Extended Data Fig. 4 Topological surface states.
a–c, Density of states (DOS) at the (010) surface of the tightbinding model with SG 19.27 (a, b) and ferromagnetic MnSi with the magnetization aligned along [010] (c). The first and second rows display the DOS at the top and bottom surfaces, respectively. In a, the surface DOS is shown at an energy E = −1.2 of the single Weyl point (WP) on the Y_{1}–Γ–Y line. A single Fermi arc emanates from the projected Weyl point and connects to the k_{z} = π NP. In b, the surface DOS is shown at the energy E = +1.4 of the fourfold point (FP) on the R_{1}–U–R line, whose chirality ν = −2 is compensated by two accidental Weyl points in the bulk. Two Fermi arcs emanate from the projected fourfold point and connect to the accidental Weyl points in the bulk. In c, the DFTderived surface DOS of ferromagnetic MnSi is shown at the Fermi level E = E_{F}. Fermi arcs emanate from the projected Weyl points on the Y_{1}–Γ–Y line and connect with the bulk bands forming NPs on the BZ boundaries.
Extended Data Fig. 5 Crystal structure, calculated Fermi surfaces, experimental methods, and dHvA spectra in the (001) and (\(\overline{{\bf{1}}}\overline{{\bf{1}}}{\bf{0}}\)) planes.
a, Crystal structure of MnSi. b, Fermi surface as calculated within local spin density approximation without rigid band shifts. c, Calculated Fermi surface neglecting spin–orbit coupling. d, Calculated Fermi surface neglecting spin–orbit coupling and highlighting majority and minority spin. e, Sketch of the cantilever magnetometer chip with capacitive readout. f, Magnetic field dependence of the frequency f_{B}(B) tracking the magneticfield dependence of the unsaturated magnetization in the fieldpolarized phase. The frequency f(B) observed corresponds to the zerofield intercept of the tangent to f_{B}(B). g, Experimental dHvA frequency branches (crosses) for rotation in the (001) and (\(\overline{1}\overline{1}0\)) planes together with the theory (lines) matched to the experiment. See Supplementary Note 5 for details.
Extended Data Fig. 6 Details of the assignment of experimental dHvA orbits to FS sheets 1 to 6.
a1, Experimental signature of sheet 2. Colour scale corresponds to experimental FFT amplitude and crosses show positions of maxima. a2, Torque signal predicted from DFT as calculated. Lines show theoretical branches, crosses show experimental positions. a3, FS sheet 2 as calculated. Three extremal orbits 2Γ and 2ΓY(1,2) are present for B close to [010], which are assigned to κ_{1,2,3}. a4, Calculated dHvA branch including a small upward band shift of 20 meV, yielding a good match with experiment (crosses). a5, Comparison of ascalculated (inner) and matched (outer) FS sheet 2. b1, Dispersions of bands 1, 2 and 3 in the k_{x}–k_{z} plane without (transparent) and with spin–orbit coupling (solid), showing a spin1 excitationlike threefold degeneracy that is lifted by spin–orbit coupling. Since band 2 (cyan) crosses the Fermi level, the α branch must originate from band 1 and not band 3. The Fermi level matching the experimental frequency is shown in black. b2, FS sheet 1 as calculated (outer) and matched to experiment (inner). The α branch is assigned to orbit 1Γ. c1, FS sheets 3 and 4 exhibit extremal orbits with 8 breakdown junctions j_{1} to j_{8} for B close to [010] as shown in c2. The inset shows the two extremal orbits that arise when spin–orbit coupling is neglected. c3, 256 breakdown orbit branches originating from sheets 3 and 4. Symbol size reflects orbit probability. The torque amplitude is not considered in this graph. The breakdown orbits group into five sets labelled in red. The branches ρ and H are assigned to the inner and outer orbits 3Γ and 4Γ, respectively. d1, FS sheets 5 and 6 as in Fig. 4a with two neck orbits 5ΓY and 6ΓY and the loop orbits 5U6U assigned to (ξ_{1}, ξ_{2}), π and M_{1}, respectively. d2, Top: neck crosssectional areas of sheet 5 (blue) and 6 (orange) versus k_{∥} neglecting (dashed) and including (solid) spin–orbit coupling for φ = 90° and φ = 180°. Middle: crosssectional area a of sheet 5 versus k_{∥} for field directions 70°−90° and 160°−180°. Dashed line: position of single extremal area around φ = 90°. Shaded grey area: neck being on the verge of developing a second minimum close to 180° but not around 90° that could give rise to ξ_{2}. Bottom: derivative da/dk_{∥}, where zerocrossings correspond to extremal orbits. d3, FFT amplitude of the loop orbits around φ = 90° and φ = 180°. Left: a distinct splitting of the M_{1} branch into M_{1} and M_{2} is observed close to φ = 180° but not around φ = 90°. Right: the FFT amplitude of the M_{1} branch shows unexpected secondary minima (shaded areas) close to φ = 90° on both sides. Both effects may be connected to either the quasidegeneracy of the U5U6 orbits shown in Fig. 4b3, b4 or to a crossing with the 8ΓY branch.
Extended Data Fig. 7 Assignment of experimental dHvA orbits to FS sheets 7 to 10.
a1, Along Γ–Y_{1}–Γ three neck orbits on sheet 7 (purple), one neck orbit on sheet 8 (yellow), and two loop orbits around U are predicted for B∥[010]. a2, Loop orbits are shared between the sheets at TP1 to TP4 in analogy to the sheet pair (5, 6). a3, For B∥[010], the upper two lensshaped orbits exist, while for B in the (001) plane away from [010] the lower two heartshaped orbits are allowed in addition. b, Sheet 9 neglecting spin–orbit coupling in perspective (b1), top (b2) and back (b3) views for φ = 83°, that is, B slightly off the [010] direction. Bands 9 and 10 are shifted upward by 10 meV for an optimal match to the low frequencies as shown in b4, where grey lines correspond to the calculations. In total, 15 orbits are predicted for this specific field direction alone. Without spin–orbit coupling, band 10 does not cross the Fermi level for this shift. c1–c3, Sheets 9 and 10 and predicted dHvA orbits including spin–orbit coupling for φ = 83° and a shift of 11 meV yielding a good match as shown in c4. Sheet 10 resides inside sheet 9 as highlighted by black arrows. It occurs only for field directions where two or more ‘banana bunches’ cross the BZ surface and connect. In the situation depicted here, (001) is an NP, thereby connecting parts of sheets 9 and 10 in such a way that extremal orbits cross from one sheet to the other.
Supplementary information
Supplementary Information
This Supplementary Information file provides comprehensive information on the theoretical framework, theoretical analysis, and magnetic space groups featuring topological nodal planes, as well as the analysis of the experimental data. It is organized in terms of the following five sections: (S1) Band Topology of MnSi; (S2) Tightbinding models, Berry curvature, and surface states; (S3) Catalogue of space groups with symmetryenforced nodal planes; (S4) Technical aspects of the analysis; and (S5) Comprehensive Fermi surface determination.
Supplementary Video 1
This video highlights the evolution of a cutaway view of the Fermi surface akin Fig.1(f) as a function of the direction of the magnetization (blue arrow) following an applied magnetic field. Note the emergence of the topological degeneracies of Fermi surface pairs (5,6), (7,8) and (9,10) perpendicular to the direction of the magnetization.
Supplementary Video 2
This video highlights the evolution of a cutaway view of the Fermi surface akin Fig.1(f) as a function of the direction of the magnetization (blue arrow) following an applied magnetic field. Note the emergence of the topological degeneracies of Fermi surface sheet pairs (5,6), (7,8) and (9,10) perpendicular to the direction of the magnetization.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wilde, M.A., Dodenhöft, M., Niedermayr, A. et al. Symmetryenforced topological nodal planes at the Fermi surface of a chiral magnet. Nature 594, 374–379 (2021). https://doi.org/10.1038/s4158602103543x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4158602103543x
This article is cited by

Kramers nodal lines and Weyl fermions in SmAlSi
Communications Physics (2023)

Quasisymmetryprotected topology in a semimetal
Nature Physics (2022)

Large curvature near a small gap
Nature Physics (2022)

Chirality locking charge density waves in a chiral crystal
Nature Communications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.