


\section*{General considerations}
Nearly a century ago Wigner, von Neumann, and Herring \cite{neumann_wigner_z_physik, herring_PR_37} were first to address the conditions under which Bloch states form degenerate band crossings, but their topological character and technological relevance was only recognized recently\cite{chiu_RMP_16, armitage_mele_vishwanath_review, burkov_review_Weyl}.  To be useful\cite{armitage_mele_vishwanath_review, burkov_review_Weyl, wang_hechang_nat_comm_intrinsic_AHE, huang_chiral_anomaly_PRX_15, ong_chiral_anomaly_PRX_18, huang_hasan_TaAs_Fermi_Arc_nat_comm_15} 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 to date. This raises the question, if topological band crossings exist that are, (i) generically at the Fermi level, (ii) separated sufficiently in the BZ, and, (iii) easy to control. 

Natural candidates are systems with non-symmorphic symmetries, e.g., screw rotations, generating positions in reciprocal space at which band-crossings are symmetry-enforced. The associated key characteristics include: \cite{MICHEL2001377, young_kane_rappe_PRL_12, FURUSAKI2017788, zhao_schnyder_PRB_16,zhangPRMat2018, 2019_PRB_Yuxin, PRB_Yuxin, turker_moroz_PRB_18} 
(i) the crossings are due to symmetry alone, i.e., they occur on all bands independent of details such as chemical composition,
(ii) pairs of band-crossings with opposite chirality are separated in $k$-space by about half a reciprocal lattice vector, 
(iii)~the band crossings may be enforced on entire planes~\cite{young_kane_rappe_PRL_12, FURUSAKI2017788},  forming so-called nodal planes (NPs) with non-zero topological charge, and 
(iv) 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 material-specific details and, moreover, may be controlled by symmetry breaking.  
The putative existence of topological NPs was studied in phononic metamaterials \cite{2017_Xiao_arXiv,2019_Yang_NatComm, 2020_Xiao_SciAdv}, and mentioned in a study of non-magnetic chiral systems focussing on Kramers-Weyl fermions \cite{2018_Chang_NatMat}. 

To demonstrate the formation of symmetry-enforced TPs at the intersection of NPs with the FS, we decided to study the ferromagnetic state of MnSi, which attracts great interest for its itinerant-electron magnetism \cite{1988_Lonzarich_JMMM}, helimagnetism, skyrmion lattice\cite{2009_Muehlbauer_Science}, and quantum phase transition\cite{1997_Pfleiderer_PRB}. Crystallizing in space group (SG) 198, MnSi is a magnetic sibling of nonmagnetic RhSi\cite{RhSi_Nature_hasan_19}, CoSi \cite{CoSi_Nature_hong_19}, and PdGa \cite{2020_Schroeter_Science}, in which sizeable Fermi arcs and multi-fold fermions were recently inferred from ARPES. 
MnSi is ideally suited for our study as magnetic fields exceeding $\sim {0.7}$~T stabilize ferromagnetism with magnetic screw rotation symmetries enforcing NPs. 

\section*{Initial Assessment}
A first theoretical assessment establishes that a ferromagnetic spin-polarization along a high-symmetry direction, e.g., $[010]$, reduces the symmetries from space group (SG) 198 (P2$_1$3) of paramagnetic MnSi to the magnetic SG 19.27 (P$2_1 2'_1 2'_1$) (Supplementary Note~S1, Extended Data Fig.~\ref{EDI:Theory:Fig1}). This SG contains two magnetic screw rotations $\theta  \tilde{C}^x_2$ and $\theta  \tilde{C}^z_2$ (Fig.~\ref{fig:1}a), i.e., 180$^\circ$ screw rotations around the $x$ and $z$ axes combined with time-reversal symmetry $\theta$. These rotations act like mirror symmetries, since 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 $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$ invariant. 
Squaring $\theta  \tilde{C}^x_2$ and $\theta  \tilde{C}^z_2$ and letting them operate on the Bloch state $| \psi ( {\bf k}) \rangle$ one finds that $(\theta  \tilde{C}^x_2 )^2  | \psi ( {\bf k}) \rangle  =   e^{i k_x} | \psi ( {\bf k}) \rangle$  and $(\theta  \tilde{C}^z_2 )^2  | \psi ( {\bf k}) \rangle  =   e^{i k_z} | \psi ( {\bf k}) \rangle$. Hence, by Kramers theorem~\cite{kramers_theorem_paper}, all Bloch states on planes with $k_x=\pm$\pi$$ or $k_z=\pm$\pi$$ are two-fold degenerate. Moving away from these BZ boundaries, the symmetries are lowered such that the Bloch states become non-degenerate.  Therefore, all bands in ferromagnetic MnSi are forced to cross at $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$, representing a duo of NPs. 
	
The topological charge $\nu$ of this duo of NPs (Fig.~\ref{fig:1}b) may be determined with the fermion doubling theorem (FDT) \cite{nielsen_no_go}, which states that $\nu$ summed over all band crossings must be zero. We note that besides the NPs there is an odd number of symmetry-enforced band crossings on the Y$_1-\Gamma-$Y and R$_1-$U$-$R lines forming Weyl points (WPs, $\nu = \pm 1$) and fourfold points (FPs, $\nu = \pm 2$), respectively (Figs.~\ref{fig:1}c and \ref{fig:1}d, Extended Data Fig.~\ref{EDI:Theory:Fig2}, Supplementary Note~S1). Moreover, due to the effective mirror symmetries accidental Weyl points away from these high-symmetry lines must form pairs or quadruplets with the same $\nu$. Because the sum over $\nu$ of all of these Weyl and fourfold points is odd, the duo of NPs must carry a nonzero topological charge to satisfy the FDT. Hence, the duo of NPs at the BZ boundary is the topological partner of a single Weyl point on the Y$_1-\Gamma-$Y line (Fig.~\ref{fig:1}b). This is a counter-example to Weyl semimetals, in which Weyl points occur always in pairs. 
	
Shown in Fig.~\ref{fig:1}d is the band structure of a generic tight-binding model satisfying SG 19.27 (Supplementary Note~S2), where pairs of bands form NPs on the BZ boundaries $k_x =\pm $\pi$$ and $k_z = \pm $\pi$$, while on the Y$_1-\Gamma-$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 nodal planes, 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.~\ref{EDI:Theory:Fig3}). By the bulk boundary correspondence~\cite{chiu_RMP_16,armitage_mele_vishwanath_review}, 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.~\ref{EDI:Theory:Fig4}).
These arguments may be extended to 254 of the 1651 magnetic SGs of which 33 have nodal planes whose topological charges are enforced to be non-zero by symmetry alone (Supplementary Note~S3).  

\section*{Calculated Electronic Structure}
Shown in Fig.~\ref{fig:1}e is the Density Functional Theory (DFT) band structure of MnSi taking into account spin-orbit coupling (SOC), for the experimental moment of $0.41\mu_B/$Mn along $[010]$ (Methods, Extended Data Fig.~\ref{EDI:Exp:Fig6}). Ten bands are found to cross the Fermi level (Fig.~1e). In agreement with our symmetry analysis and the tight-binding model (Fig.\,\ref{fig:1}d), we find the same generic band crossings, namely: (i) NPs on the BZ boundaries $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$, (ii) an odd number of Weyl points along Y$_1-\Gamma-$Y, and, (iii) an odd number of fourfold points along R$_1-$U$ -$R. 

The calculated FSs as matched to experiment are shown in Fig.~\ref{fig:1}f highlighting the NPs at the BZ boundaries at $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$ (see Extended Data Table~\ref{EDI:Exp:tab1} for key parameters and Extended Data Fig.~\ref{EDI:Exp:Fig6}). Eight FS sheets centred at $\Gamma$ comprise two small isolated hole pockets (sheets $1$ and $2$), two intersecting hole pockets with avoided crossings and magnetic breakdown due to SOC (sheets $3$ and $4$), and two pairs of jungle-gym type sheets, ($5$,$6$) and ($7$,$8$). Sheets $9$ and $10$ are centered at $R$, comprising eight three-fingered 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 $\theta  \tilde{C}^x_2$ and $\theta  \tilde{C}^z_2$. In contrast, sheets $5$ through $10$ do not form TPs at the BZ boundary $k_y=\pm$\pi$$, because the moment pointing along $[010]$ breaks $\theta  \tilde{C}^y_2$.

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 (cf. Supplementary Video). For instance, rotating the moments within the $x-y$ plane away from $[010]$ breaks the magnetic screw rotation $\theta  \tilde{C}^x_2$, but keeps $\theta  \tilde{C}^z_2$ intact. In turn the TPs gap out on the  $k_y=\pm$\pi$$ and $k_x=\pm$\pi$$ planes, while they remain degenerate at $k_z=\pm$\pi$$ planes (cf. Extended Data Fig.~\ref{EDI:Theory:Fig1}, Suppl. Note~S1).  

\section*{Experimental Results}
To experimentally prove the mechanism causing generic TPs at the intersection of the FS with symmetry-enforced NPs and their dependence on the direction of the magnetization, we mapped out the FS by means of the dHvA effect using capacitive cantilever magnetometry (Methods, Extended Data Fig.~\ref{EDI:Exp:Fig6} and Suppl. Note~S4). 
In the following we focus on magnetic field rotations in the $(001)$ plane, where $\varphi$ 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 $(\bar 1 \bar 1 0)$ planes are presented in the Extended Data Fig.\,\ref{EDI:Exp:Fig6}. 
Typical torque data at different temperatures for $\varphi=82.5^{\circ}$ (Fig.~\ref{fig:2}a,b) display pronounced dHvA oscillations for fields exceeding $B \sim 0.7$\,T. The hysteretic behavior below $ \sim 0.7$\,T (inset of Fig.~\ref{fig:2}a) originates from the well-understood helimagnetic and conical phases \cite{2017_Bauer_PRB}.  Figure~\ref{fig:2}b shows the oscillatory high-field part of the torque $\tau (1/B)$ at $T=35$\,mK with the low-frequency components removed for clarity. To extract the dHvA frequencies an FFT analysis of $\tau (1/B)$ 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 cross-sections in low effective fields of $\sim 0.7-1.9\,{\rm T}$ (Methods).

Typical dHvA frequencies and FFT amplitudes, shown for $\varphi=82.5^{\circ}$ in Fig.~\ref{fig:2}c, display 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.~\ref{fig:2}c and Extended Data Table~\ref{EDI:Exp:tab2}). In our data analysis we delineated artefacts due to the finite FFT window, such as the side lobes between $\kappa_2$ and $2\kappa_1$, or $3\kappa_2$ and $\xi_1$ (cf. Methods).
Fitting the temperature dependence of the FFT amplitudes within Lifshitz-Kosevich theory \cite{Shoenberg1984}, the effective masses for each of the orbits were deduced ranging from $m^*=0.4\,m_e$ to $17\,m_e$, where $m_e$ is the bare electron mass (Fig.~\ref{fig:2}d). 

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 (cf. Methods and Extended Data Table~\ref{EDI:Exp:tab1}). 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~\ref{EDI:Exp:tab2}, Extended Data Fig.\,\ref{EDI:Exp:Fig9} and \ref{EDI:Exp:Fig10}, and Supplementary Note~S5. 

Shown in Fig.~\ref{fig:3}a1 and \ref{fig:3}a2 is an intensity maps of the experimental data of the $(001)$ plane as a function of $\varphi$, where the theoretical dHvA branches are depicted by colored lines (colors correspond to the FS sheets in Fig.~\ref{fig:1}). For comparison,  Fig.~\ref{fig:3}b1 and \ref{fig:3}b2 displays an intensity map of the calculated dHvA spectra, where the experimental frequencies are marked by grey crosses. 

For regimes I through IV, featuring contributions of the large FS sheets (5,6) and (7,8), all frequencies may be assigned unambiguously (Extended Data Fig.\,\ref{EDI:Exp:Fig9} and \ref{EDI:Exp:Fig10}, Suppl. Note~S5). 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  $\Gamma$-$Y$-$\Gamma$ 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 $[010]$ high symmetry 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~S5). Regime III arises from both, pairs (5,6) and (7,8), i.e., neck orbits around $\Gamma$-$Y$-$\Gamma$ 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 $1^{\rm{st}}$ harmonic of sheet 2.

As the magnetic torque generically vanishes at high-symmetry directions, which corresponds to the $\langle100\rangle$ axes in regimes I through IV, the associated FS sheets are centered at the $\Gamma$-point. Likewise, the lowest frequency in regime V corresponds to a $\Gamma$-centered FS sheet, which can be assigned to the small hole pocket of sheet 1. In stark contrast, for regime V above $\sim 0.05\,{\rm kT}$ the high-symmetry directions correspond to the $\langle111\rangle$ axes, while the torque for the $\langle100\rangle$ axes is finite (see also Fig.~\ref{fig:3}a2 and \ref{fig:3}b2 and Extended Data Fig.\,\ref{EDI:Exp:Fig6}g).  Hence, regime V is related to FS pockets in the vicinity of the R-point that may be assigned to FS sheets (9,10) without the need for a detailed account of their shape, completing the assignment. The calculations demonstrate the presence of symmetry-enforced crossings of sheets (9,10) if they intersect the NPs (cf. Fig.~\ref{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~\ref{EDI:Exp:tab1}). Excellent agreement is observed within a few percent of experiment\cite{Bauer2010}, $\gamma\approx28$~mJmol$^{-1}$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. 

\section*{Topological Nodal Planes}
Spectroscopic evidence of the symmetry-enforced topological band degeneracies at the BZ boundaries may be inferred from FS sheets (5,6). Identical characteristics are observed for FS sheets (7,8) (cf Extended Data Fig.\,\ref{EDI:Exp:Fig10} and Supplementary Note~S5). 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.~\ref{fig:4}a, a magnetic field parallel to $[010]$ leads to extremal cross-sections for FS sheets (5,6) supporting cyclotron orbits in the vicinity of the U and the Y$_1$ point 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 $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$. In the absence of the non-symmorphic symmetries, these intersections would exhibit anticrossing and magnetic breakdown leading to several orbits with different cross-sections and hence several dHvA frequencies. Instead the behaviour is distinctly different to magnetic breakdown or Klein tunneling \cite{2017_Alexandradinata_PRL,2018_vanDelft_PRL}.

As the BZ boundaries at $k_x=\pm$\pi$$ and $k_z=\pm$\pi$$ represent symmetry-enforced NPs, the crossing-points of sheets 5 and 6 at TP1 to TP4 are, hence, protected band-degeneracies at which the wave-functions are orthogonal, i.e., TP1 to TP4 are part of the topological protectorates 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 (Figs.~\ref{fig:4}b1 and \ref{fig:4}b2). This is in excellent agreement with experiment, which shows a single dHvA frequency for field parallel $[010]$ ($\varphi = 90^{\circ}$ in Fig.~4c).
Rotating the direction of the magnetic field within the $xy$ plane away from $[010]$, the NP at $k_x = \pm $\pi$$ gaps out, while the NP at $k_z = \pm $\pi$$ remains protected. Thus, the associated loop orbits around U (Figs.\,\ref{fig:4}b3 and \ref{fig:4}b4) continue to include two points on the FS at $k_z = \pm $\pi$$ (TP3 and TP4), leading to two additional topological orbits (topological orbits 3 and 4) of identical cross-section with the same dHvA frequency, in perfect agreement with the observed spectra (Fig.\,\ref{fig:4}c). 

Comparing the extremal cross sections of the neck orbits around $\Gamma$-Y$_1$-$\Gamma$ with those around $\Gamma$-{X}-$\Gamma$, the latter crosses a NP while the former does not. With respect to $\Gamma$-{X}-$\Gamma$  there would be two extremal cross sections with identical areas, positioned symmetrically with respect to X (Fig.~\ref{fig:4}d1), whereas for the cross sections with respect to $\Gamma$-Y$_1$-$\Gamma$ there are two extremal orbits with different areas positioned asymmetrically with respect to Y$_1$ (Fig.~\ref{fig:4}d2). Thus within our symmetry analysis and our DFT calculations we expect a single dHvA branch for neck orbits parallel to a NP as compared to two dHvA branches for neck orbits that are not parallel to a NP (Fig.~\ref{fig:4}d3). 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 = \pm $\pi$$ BZ boundary (Fig.~\ref{fig:4}e). 

\section*{Concluding Remarks}
The symmetry-enforced 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 ~\cite{niu_review_RMP_10} or the nonlinear optical responses~\cite{morimoto_nagaosa_science_advances_16}. Indeed, large anomalous contributions to the Hall response are in excellent \textit{quantitative} agreement with ab initio calculations, where the calculated FS and Berry curvatures were essentially identical to the FS we report here \cite{2014_Franz_PRL}.
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 NP directly with a Weyl point (cf. Extended Data Fig.\,\ref{EDI:Theory:Fig4}). These Fermi arcs reflect the presence of duos of NPs. Analogous Fermi arcs will not exist in non-magnetic materials with SG 198\cite{RhSi_Nature_hasan_19, CoSi_Nature_hong_19, 2020_Schroeter_Science} which support trios of NPs (Supplementary Note\,S1). 

In systems with symmetry-enforced 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 \cite{2006_Smith_PRB} and enable exotic states of matter \cite{2014_Grover_JSM}. Extending the analysis presented here to all 1651 magnetic SGs, we find that there is a large number of candidate materials, such as CoNb$_3$S$_6$ \cite{2020_Tenasini_PRM} or Nd$_5$Si$_3$ \cite{2001_Boulet_JAC} with similar topological protectorates (Extended Data Table~\ref{EDI:Theory:tab1}, Supplementary Note~S3), which await to be explored from a fundamental point of view and harnessed for future technologies. 



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\section*{References}

