Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet

Sample preparation

For our study, two MnSi samples were prepared from a high-quality single crystalline ingot obtained by optical float-zoning⁴². The samples were oriented by X-ray Laue diffraction and cut into 1 × 1 × 1 mm³ 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 double-beam type cantilevers sketched in Extended Data Fig. 5e were obtained from CuBe foil by standard optical lithography and wet-chemical etching. The cantilever position was read out in terms of the capacitance between the cantilever and a fixed counter electrode using an Andeen-Hagerling AH2700A capacitance bridge, similar to the design described in refs. ^43,44.

Angular rotation studies were performed in a ³He 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 approach⁴⁵. Applying a d.c. voltage, U, to the capacitance C₀ = ε₀A/d₀, defined by the area A, the plate distance d₀ and the vacuum permittivity ε₀, leads to an electrostatic force F = C₀U²/2d₀. This force is equivalent to a torque τ = βFL, where L is the effective beam length and β = 0.78 is a geometry-dependent 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 field-polarized state generated a strongly hysteretic behaviour. At higher fields, magnetic quantum oscillations on different amplitude and frequency scales could be readily resolved. The first low-frequency components appeared at magnetic fields as low as B ≈ 4 T, whereas several high-frequency components, corresponding to larger extremal cross-sections, could only be resolved in high fields (Fig. 2b). Consequently, the data acquisition and evaluation was optimized by treating low- and high-frequency components separately.

To eliminate the non-oscillatory component of the signal, low-order 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 ³He 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 field³¹B_int = μ₀H_ext + μ₀(1 − N_d)M, where μ₀ is the vacuum permeability, H_ext is the applied magnetic field and M is the magnetization. Taking into account the demagnetization factor⁴⁶N_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 low-field value of the magnetization in the field-polarized 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 cross-sectional areas are enlarged with increasing B for the majority electron orbits and minority hole orbits. Cross-sectional areas shift downwards for majority hole and minority electron orbits.

This change in cross-sectional area is not directly proportional to the change in the observed dHvA frequencies f, that is, the dHvA frequencies deviate from the field-dependent frequency ({f}_{{rm{B}}}(B)=frac{hbar }{2{rm{pi }}e}{A}_{k}(B)) obeying the Onsager relation (here A_k is the extremal cross-sectional area in k-space, ħ is the reduced Planck constant and e is the electron charge). The frequency f observed may be inferred⁴⁷ 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 zero-field 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 f₀ is the hypothetical frequency without exchange splitting. Note, that this model is only meaningful in the field-polarized regime B ≳ 0.7 T. Using the experimental M(B) curve of MnSi³², we estimate that the frequencies f(B) in the windows used for f > 350 T defined above with centre fields B_average = 2B_highB_low/(B_low + B_high) ranging from 11.8 T to 13.2 T correspond to the extremal cross-sections 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 cross-sections at B ≈ 0.7 T. Thus, even under large magnetic fields, the experimental frequency values correspond to a field-polarized 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 formalism³¹.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 cross-sectional 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 non-trivial structures in k-space. The ± holds for maximal and minimal cross-sections, respectively. The torque amplitude is given by τ_osc = M_osc,⊥B. The torque thus vanishes in high-symmetry 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 field-polarized phase were calculated using DFT. The calculations included the effect of spin–orbit coupling. In all calculations, the magnetic part of the exchange-correlation terms was scaled⁴⁹ 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 P2₁3 (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 WIEN2k⁵⁰, ELK⁵¹ and VASP^52,53 using different versions of the local spin density approximation. The results are consistent within the expected reproducibility of current DFT codes⁵⁴. 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 Wang⁵⁵ and a sampling of the full BZ with a 23 × 23 × 23 Γ-centred grid. The results of Extended Data Figs. 1, 2, 4 were obtained using VASP with the PBE functional⁵⁶ and a BZ sampling with a 15 × 15 × 15 k-mesh centred around Γ.

Bands used for the determination of the Fermi surface were calculated with WIEN2k on a 50 × 50 × 50 k-mesh. Owing to the presence of spin–orbit coupling, but the absence of both inversion and time-reversal 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 k-point-wise using third-order splines to obtain band structure information in 1° steps.

For the prediction of the dHvA branches from the DFT results, the Supercell k-space Extremal Area Finder (SKEAF)⁵⁷ was used on interpolated data corresponding to 150 × 150 × 150 k-points 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 field-polarized phase (Extended Data Fig. 4), we first constructed a DFT-derived tight-binding model using the maximally localized Wannier function method as implemented in Wannier90⁵⁸. Using this tight-binding model, we computed the momentum-resolved surface density of states by means of an iterative Green’s function method, using WannierTools⁵⁹. 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 B₀ were calculated from Chamber’s formula

where k_g is the gap in k-space and a and b are the curvatures of the trajectories at the breakdown junction³¹. 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 cross-sections was based on the following criteria: (1) dHvA frequency—determining sheet size in terms of the cross-sectional 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 well-established 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 cross-section, 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 symmetry-enforced band crossings and the band topology follow from the non-trivial 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 four-fold points, which are obtained from generalizations of the Nielsen–Ninomiya theorem²⁹. To illustrate the band topology for ferromagnets in SG 19.27 and SG 4.9, two tight-binding 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 symmetry-enforced NPs. We find that (at least) 33 of these have NPs whose topological charge is guaranteed to be non-zero due to symmetry alone.