# Generation of a nodal line and Weyl points by magnetization reorientation in Co3Sn2S2

F. Schilberth, M.-C. Jiang, F. Le Mardelé, L. Papp, I. Mohelsky, M. Kassem, Y. Tabata, T. Waki, H. Nakamura, G.-Y. Guo, et al.

► To cite this version:

F. Schilberth, M.-C. Jiang, F. Le Mardelé, L. Papp, I. Mohelsky, et al. Generation of a nodal line and Weyl points by magnetization reorientation in Co3Sn2S2. Npj Quantum Materials, 2025, 10 (1), pp.67. 10.1038/s41535-025-00785-0. hal-05345326

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# Generation of a nodal line and Weyl points by magnetization reorientation in Co₃Sn₂S₂

F. Schilberth,¹,²,* M.-C. Jiang,³,⁴,* F. Le Mardelé,⁵ L. B. Papp,² I. Mohelsky,⁵ M. A. Kassem,⁶,⁷ Y. Tabata,⁶
T. Waki,⁶ H. Nakamura,⁶ G.-Y. Guo,³,⁸ M. Orlita,⁵,⁹ R. Arita,⁴,¹⁰ I. Kézsmárki,¹ and S. Bordács¹,²,¹¹

¹Experimentalphysik V. Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany

²Department of Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary†

³Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan

⁴RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako 351-0198, Japan‡

⁵Laboratoire National des Champs Magnétiques Intenses, LNCMI-EMFL,
CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse,
Univ. Toulouse 3, INSA-T, Grenoble and Toulouse, France

⁶Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan

⁷Department of Physics, Faculty of Science, Assiut University, 71516 Assiut, Egypt

⁸Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan

⁹Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic

¹⁰Department of Physics, University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-0033, Japan

¹¹HUN-REN-BME Condensed Matter Physics Research Group,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary§

(Dated: May 16, 2025)

Topological magnets exhibit fascinating physics like topologically protected surface states and anomalous transport. Although these states and phenomena are expected to strongly depend on the magnetic order, experimental evidence for their magnetic manipulation remains challenging. Here, we demonstrate the magnetic field control of the topological band structure in Co₃Sn₂S₂ by magneto-optical spectroscopy. We resolve a magnetic field-induced redshift of the nodal loop resonance as the magnetization is rotated into the kagome plane. Our material-specific theory, capturing the observed field-induced spectral reconstruction, reveals the emergence of a gapless nodal loop for one of the in-plane magnetization directions. The calculations show that the additionally created Weyl points for in-plane fields marginally contribute to the optical response. These findings demonstrate that breaking underlying crystal symmetries with external fields provides an efficient way to manipulate topological band features. Moreover, our results highlight the potential of low-energy magneto-optical spectroscopy in probing variations of quantum geometry.

## I. INTRODUCTION

In recent years, intriguing topological states of matter were predicted and experimentally confirmed, among them topological insulators, Dirac and Weyl semimetals [1–3]. As the interplay between magnetism and topology may fundamentally shape the band structure through symmetries dictated by the magnetic order, magnetic materials potentially allowing field control of the topological states are particularly appealing for applications in spintronics and quantum information technology [4, 5]. While several theoretical studies indeed predict that magnetic fields can create or annihilate Weyl nodes and move their position in the band structure [6–9], experimental demonstrations of these effects are scarce.

Magnetotransport experiments suggest the transformation of a nodal ring to Weyl points in EuP₃ [10] and in Co₂MnAl [11], whereas Shubnikov-de Haas oscillations propose a field-induced reconstruction of the Fermi surface in Co₃Sn₂S₂ [12]. In addition, it was shown in MnSi that nodal planes can emerge only on *k* planes parallel to an applied magnetic field [13]. While these examples prove that indeed the band structure in

topological magnets can be highly sensitive to the magnetic state, they also highlight the difficulties of magnetotransport and magnetometry experiments to provide direct band-specific information. Low-energy spectroscopy may help to overcome these challenges, and may additionally provide direct information on the quantum geometry [14–17]. However, perhaps the only spectroscopic study on the magnetic control of band topology so far is the scanning tunneling spectroscopy (STS) work of Yin *et al.*, implying a reconstruction of massive Dirac bands upon the rotation of a magnetic field in the kagome metal Fe₃Sn₂ [18]. Due to the shortcomings of other spectroscopic methods in magnetic fields, here we use magneto-optical spectroscopy to reveal the magnetic field control of the band topology in Co₃Sn₂S₂ up to 34 T.

As a proof of this concept, here we investigate the evolution of the topological features of the band structure in Co₃Sn₂S₂ by measuring magneto-reflectance upon the gradual rotation of the magnetization from the *c* axis to two perpendicular directions in the kagome plane. We observe a strong redshift of the previously reported giant magneto-optical resonance [19] upon this magnetization reorientation. As only few bands lie around the Fermi energy, our material-specific theory can unambiguously reveal the origin of this shift: Due to the lower symmetry of the in-plane magnetized case, various Weyl nodes emerge by closing the spin-orbit coupling (SOC) induced gap at specific *k* points or even recover a continuous gapless nodal line, depending on the orientation of the magnetization within

<footnote>* These two authors contributed equally</footnote>

<footnote>† felix.schilberth@physik.uni-augsburg.de</footnote>

<footnote>‡ ming-chun.jiang@riken.jp</footnote>

<footnote>§ bordacs.sandor@ttk.bme.hu</footnote>
the kagome plane. Our showcase study provides a deep insight
into the mechanism of magnetically driven reconstructions of
electronic band topology.

We choose the archetypical magnetic Weyl semimetal
Co₃Sn₂S₂ due to its simple band structure close to the Fermi
level. The crystal structure is built up by kagome layers of Co
stacked along the *c* axis in an ABC-fashion. Below *T*_c ≈ 177
K, it becomes a collinear ferromagnet with magnetic moments
pointing along the stacking direction [20] with extraordinarily
large uniaxial anisotropy, where fields as high as 23 T are re-
quired to fully turn the magnetization into the kagome plane
[21]. Nonrelativistic density functional theory (DFT calcula-
tions predicted nodal loops, located around the Fermi energy
on high symmetry planes of the Brillouin zone (BZ), which
are gapped by SOC except for six Weyl points [22]. Angu-
lar resolved photoemission studies confirmed the existence of
the nodal lines and traced their evolution across the magnetic
transition [23]. In the paramagnetic state, these results suggest
the existence of a 4-fold degenerate Dirac loop, which upon
breaking time-reversal by the magnetic order is split into two
Weyl loops of opposite spin polarisation. One of them moves
to higher energies upon increasing magnetic moment, while
its partner stays close to the Fermi energy and was shown to
be responsible for large anomalous Hall effect, Nernst effect
and magneto-optical activity [9, 19, 22–25]. These results in-
dicate that the topological band structure features are strongly
coupled to the magnetic order in this kagome metal.

## II. RESULTS

### A. Symmetry analysis

In order to motivate our detailed experimental and *ab initio*
studies, we consider the modification of symmetries upon the
field-induced magnetization reorientation. For the ground state
magnetization **M** || **c**, all nodal loops are equivalent due to the
C₃ rotational symmetry, as shown in Fig. 1(d-f). In contrast,
for magnetization in the kagome plane, the 3-fold rotational
symmetry is broken, leaving mirror symmetries when the axial
vector of the magnetic moment is perpendicular to the mirror
plane. Therefore, for **M** || **a**, this results in the magnetic point
group *C*2/*m*, which hosts a mirror plane σ<sub>*a*</sub>. As shown in
Fig. 1(g-i), this plane contains a nodal loop NL1, which
will therefore be protected from being gapped by SOC (thick
yellow). The remaining loops, NL2 (light blue) are related
by the mirror operation, but are not protected, hence they are
gapped except for 6 Weyl points on each loop. Additionally, we
find 6 Weyl points away from the nodal lines (dark blue). So
in total we obtain 18 Weyl points and a protected nodal loop in
the BZ. For the perpendicular in-plane direction **M** ⊥ **a**, shown
in panels (a-c), we obtain a similar situation. However, in this
case, the magnetic point group is *C*2′/*m*′, e.g. the mirror plane
is a symmetry only in combination with time-reversal. As this
does not protect a pair of loops (NL1), these are gapped to 6
Weyl points by SOC. The remaining loops are again related
by this operation and gapped to 6 Weyl points. Here we find
8 additional Weyl points away from the loops, so we obtain a

total of 26 Weyl points distributed in the BZ.

Some of these features were predicted earlier, e.g. Ghimire
et al. [9] theoretically investigated the detailed evolution of the
26 Weyl points for **M** || **a** upon canting of the magnetization,
but did not comment on the gap closing of the fully degenerate
loop. Similarly, Ozawa et al. initially investigated the topolog-
ical properties of a minimal tight-binding model for Co₃Sn₂S₂
[26], which was later extended to a more complete analysis of
the magnetization direction dependent band structure includ-
ing the fully degenerate nodal loop [27]. While both groups
comment on implications for the Hall conductivity, a direct
experimental investigation of field-induced reconstructions of
band topology has been lacking.

Along this line, the central question of this work is whether
changes in the nodal lines and the emergence of additional
Weyl nodes can be resolved by optical spectroscopy. Addi-
tionally, a very recent theoretical study by Nakazawa et al.
[28] developed for kagome monolayers, predicts an anisotropy
of the band structure for different in-plane directions of the
magnetization, which manifests in magnetotransport and, as
we show, also in optical properties of bulk crystals.

### B. Magneto-optical spectroscopy

Fig. 2(a) and (d) show magneto-reflectance spectra over
the energy range of 12-200 meV in different magnetic fields
applied in the kagome plane along and perpendicular to the **a**
axis, respectively. Due to the limited measurement time at the
high-field (HF) facility, the spectra for the latter configuration
could only be collected in a superconducting (SC) setup up
to 16 T. For both configurations, we obtain similar magneto-
reflectance spectra, $[(R(B) - R(0))]/R(0)$, although the detailed
lineshapes are slightly different. With increasing magnetic
field, a peak develops at low energies, followed by a minimum
at the high-energy side. For $\mathbf{H} \perp \mathbf{a}$, these two features are
centered around 29 meV and 85 meV, while for $\mathbf{H} \parallel \mathbf{a}$ the
peak is shifted to lower energies and the minimum broadens
considerably. For both configurations, the magneto-reflectance
vanishes towards higher energies, indicating that field-induced
band reconstructions are limited to the $\lesssim 200$ meV vicinity of
the Fermi energy.

The detailed field dependence of the low-energy peak for the
two directions is shown in Fig. S1 for the SC and HF datasets,
where error bars are determined from the noise level of the
spectra. For $\mathbf{H} \perp \mathbf{a}$ in panel (a), the magnitude increases up
to 22 T and then saturates, following the same behavior as the
magnetization for in-plane fields [21]. The field dependence
of the peak for $\mathbf{H} \parallel \mathbf{a}$ is shown in panel (b), resulting in the
same trend and similar magnitude up to 16 T. For $\mathbf{H} \perp \mathbf{a}$, the
additional spectral weight due to this peak pushes the reflec-
tivity very close to 1 for high fields, as shown in Fig. S2. The
observed changes with a peak magnitude of 3.7% at 34 T can
thus be considered remarkably large for an itinerant magnet.

In order to quantify and interpret the magnetic field-induced
changes of the band structure, we calculate the field depen-
dence of the optical conductivity. The corresponding spectra
are shown in Fig. 2(b) and (e) for the two field directions, ex-
hibiting a very similar behavior in magnetic field. The most
prominent features are the tail of the Drude response domi-
nating the spectra below 15 meV and a peak around 29 meV,
which shifts to smaller energies at higher fields. The position
of the latter as a function of the field is shown in the respective
inset. It again follows the same trend as the magnetization for
both in-plane field directions, although the magnitude of the
shift is larger for **H** || **a**.

As demonstrated in Ref. 19, this peak is caused by transi-
tions between bands associated with the partially gapped nodal
lines, which can be excited almost exclusively by one of the
two circular polarization states of light. In panels (c) and (f),
the spectra measured at 0 T (magnetization along *c*) and 16 T
(magnetization almost in plane), after subtraction of the Drude
contribution, are compared to the spectra obtained from *ab initio*
calculations for the two magnetic configurations. The
theory spectra reproduce the experimental features on a broad
energy scale (compare Ref. 19). The shift of the nodal line
resonance is captured well as the onset of optical weight ap-
pears at lower energies for both directions of the in-plane fields,
though the peak in the theory spectra is not as pronounced as
in the experiment. The difference may be caused by electronic
correlations, or the magnitude of the matrix elements for these
transitions may be underestimated by the theory. Especially
the former is a likely scenario, as correlations would compress
the band structure around the Fermi level. Hence, the energy
difference in the denominator of the Kubo-Greenwood for-
mula would strongly enhance the optical conductivity around
resonances, specifically concentrating optical weight at low-
energy interband excitations. Beyond this simple picture, Xu
et al. showed in their DFT+DMFT calculation, that indeed
the optical conductivity at the 30 meV peak can be strongly
enhanced by many-body effects [29]. Due to the quantitative
agreement of the experimentally observed and theoretically
calculated redshifts, we nevertheless believe that our current
DFT treatment is sufficient to explain the mechanism of the
gap shrink.

We directly compare the calculated optical conductivity for different in-plane orientations of the magnetic field in Fig. S3. The in-plane spectra are very similar at high energies but show small differences around the resonance at 30 meV. While for **M** || **a** the slope of the peak is shifted parallel compared to the **M** || **c** spectrum, the peak for **M** ⊥ **a** is broadened and the slope reduced. In addition, the redshift is larger for **M** || **a**. All of these observations are consistent with the experimental data measured for the two directions.

C. Ab initio analysis

In order to evaluate the origin of the peak shift, we directly
compare the band structures for the in- and out-of-plane config-
urations. Here, we only show the **M** || **a** case with the protected
nodal line, while the analysis for **M** ⊥ **a** can be found in Fig.
S4, which yields very similar conclusions. Firstly, we analyze
the differences between the nodal lines. Fig. 3(a-c) plot the
gap size of the loops for **M** || **c**, NL1 and NL2 for **M** || **a** re-
spectively. The position of Weyl points is shown in red and the

Fermi surface with dark green lines. For both in-plane loops
in panels (b) and (c), the average gap size is smaller than for
out-of-plane magnetization in panel (a), not only for the pro-
tected loop. This is clearly reflected in the band structure plot
in panel (g), where we show the bands for **M** || **c** and **M** || **a** in
green and red, respectively. Along A–B and C–D, the gapless
nature of NL1 is highlighted, while the gap for NL2 is also
smaller than for the bands with **M** || **c**.

Due to the smaller gap, these nodal lines contribute in a
different energy range to the optical conductivity. Therefore, in
panel (d), we plot the spectral weight distribution *H*<sub>xx</sub>(*ω*,*k*) (for
a definition see Eq. 2 in the Methods section) for out-of-plane
magnetization at a photon energy of 42 - 45.6 meV, whereas
for in-plane field in (e) and (f) we show *H*<sub>xx</sub> at 24.4 - 30.8
meV. Qualitatively, all plots show similar features with a small
hotspot along Γ–L and several hotspots where the nodal lines
lie around the Fermi energy, e.g. on the A–B and C–D lines.
This comparison shows that the onset of optical weight of the
optical conductivity for in- and out-of-plane magnetization is
generated by the same band structure regions, but at different
photon energies, explaining the redshift of the peak observed
as the moments cant into the kagome plane. For the **M** ⊥ **a** case
in Fig. S4, the spectral weight is plotted at energies between
28.8 - 33.6 meV, also explaining the redshift found for this
configuration, although it is smaller than for **M** || **a** which is
also observed by the experiment.

Interestingly, if we infer the position of the Weyl nodes from
the left column with the gap energy plots in Figs. 3 and S4,
we do not find any optical weight associated with transitions
around these points for either magnetization direction. As
was shown previously for **M** || **c**, the Weyl nodes are located
around 60 meV above the Fermi level and therefore cannot
contribute to the low-energy optical response [19]. For in-plane
fields, while the gap changes significantly, the energy of the
crossing point with respect to the Fermi level does not change
upon reorientation of the magnetization. Therefore, the same
argument also accounts for the absence of optical weight from
the Weyl points for in-plane field.

III. DISCUSSION

First, let us discuss the possible origin of the nodal line recon-
struction. Since the gap is initially produced by SOC, it has the
same nature as the orbital magnetic moment [30] or the magne-
tocrystalline anisotropy [31, 32]. The origin of such anisotropy
can be explained through the relative SOC strength by com-
paring the SOC matrix elements in the *p*- [32] or the *d*-orbital
basis [33]. For the same spin channel, the SOC matrix elements
⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>xy,x2−y2</sub>⟩ and ⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>z2</sub>⟩ prefer the in-plane
anisotropy and ⟨d<sub>xy</sub>|H<sub>SOC</sub>|d<sub>x2−y2</sub>⟩ and ⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>xz</sub>⟩ prefer
the out-of-plane anisotropy with the magnitude ratio being
⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>xy,x2−y2</sub>⟩² : ⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>z2</sub>⟩² : ⟨d<sub>xy</sub>|H<sub>SOC</sub>|d<sub>x2−y2</sub>⟩² :
⟨d<sub>yz</sub>|H<sub>SOC</sub>|d<sub>xz</sub>⟩² = 1 : 3 : 4 : 1 [33]. Fig. S5 shows the pro-
jected band structures of Co<sub>3</sub>Sn<sub>2</sub>S<sub>2</sub> without the SOC. We no-
tice that the nodal lines consist mainly of Co *d*<sub>xy,x2−y2</sub> orbitals
with less contributions from the Co *d*<sub>xz,yz</sub> orbitals. Thus, from
the above relation, we note that the SOC matrix element of
$\langle d_{xy} | H_{SOC} | d_{x^2-y^2} \rangle$ dominates, which vanishes for in-plane spin orientation. This not only implies that out-of-plane magnetism is energetically preferred, but also that the SOC strength is weaker when the in-plane field is applied. Hence, we observe a shrinking of the gap for in-plane magnetization in $Co_3Sn_2S_2$.

Next, we discuss the relation of our observation to the quantum geometry. The first moment of the absorptive optical conductivity $W^1 = \text{Re}(\sigma_{xx})/\omega$ is proportional to the quantum metric of the electronic states. The quantum geometry theory for the interband optical conductivity is developed considering a multiband manifold in the reciprocal space, leading to the definition of the multiband quantum metric, also known as the quantum geometric tensor, given as the product of the Berry connections [15, 16, 34]. Note that from Eq. 2, the evaluation of the optical conductivity sums over all bands, which results in the quantum metric $g_{\alpha\beta}$ (known as the Fubini-Study metric [15]). Importantly, through the delta function, we can probe the quantum metric for a certain transition energy. Accordingly, the quantum weight $K_{xx}$ has been introduced as an integrated measure of the quantum metric up to a specific cutoff frequency $\omega_c$. This relation has been formalized in recent works of Onishi et al. [16].

$$ \text{Re} W_{xx}^1 (\omega_c \to \infty) = \int_0^{\omega_c \to \infty} d\omega \frac{\text{Re} \sigma_{xx}}{\omega} = \frac{e^2}{2\hbar} K_{xx}. \quad (1) $$

In the following, we analyze the optical conductivity and quantum weights for in-plane, $M \parallel a$ and out-of-plane, $M \parallel c$ magnetization as displayed in Fig. 4(a). We find that the redshift of the dominant optical transitions enhances the energy-resolved quantum metric $\text{Re}(\sigma_{xx})/\omega$ through the denominator, subsequently enhancing the quantum metric, as evidenced by the larger quantum weight $K_{xx}$.

Quantum weight is always positive and would eventually saturate at a large enough cutoff frequency $\omega_c$. In Fig. 4(b), we show the difference of quantum weight between the two magnetic configurations by defining $\Delta K_{xx} = (K_{xx}^{M\parallel a} - K_{xx}^{M\parallel c})$. On a larger energy scale shown in the inset, we can see that the quantum weight saturates at a cutoff frequency around 1 eV, with a difference in quantum weight being $(e^2/2\hbar)\Delta K_{xx} \sim 109$ S/cm. Interestingly, we find that the saturated value of $\Delta K_{xx}$ is generated at the first optical resonance of $M \parallel a$, indicated by the left arrow in Fig. 4(b). Moreover, the largest quantum metric enhancement appears at the redshift of the band edge, or the spin-orbit resonance studied in this work (marked by the second arrow). Therefore, the critical quantum geometry enhancements by the magnetization reorientation appear at low frequencies, directly related to the nodal line reconstruction.

Through the Souza-Willkins-Martin sum rule [35], the quantum weight is related to the electron localization length, which encodes the delocalization of the ground-state wavefunctions. That is, the stronger the quantum metric is, the stronger the wavefunctions overlap, the stronger the hopping amplitude becomes, the more metallic the states are, and the smaller the gap becomes. This explains why quantum geometry enhancement leads to a shrinking optical gap and redshift of the peak. We provide more details in Supplemental Note 1, including the applicability of the quantum weight theory to $Co_3Sn_2S_2$.

(Eq. 1) as well as the comparison of hopping amplitudes near the band inversion of the nodal loop to support the observed quantum geometry enhancement in $Co_3Sn_2S_2$. Overall, we have demonstrated the experimental enhancement of quantum geometry in Weyl or Dirac semimetals with nodal crossings by low-frequency magneto-optical spectroscopy [14].

In summary, we presented a detailed magneto-optical study to monitor the reconstruction of the topological band structure in a prominent kagome magnet upon manipulation of the magnetic order by external fields. We find that in $Co_3Sn_2S_2$, a low-energy resonance associated with a transition between bands of the nodal lines is highly sensitive to in-plane fields, namely it shows a red shift as the magnetization cants into the ab plane. Our material-specific theory is able to reproduce this shift and shows that it originates from the reduction of the SOC-induced gap upon the rotation of the magnetization to the kagome plane. In particular, a complete degenerate nodal line emerges for $M \parallel a$. Compared to quantum oscillation or STM measurements, this direct access to the interband response highlights the unique capabilities of optical spectroscopy to monitor changes in the band structure and its quantum geometry. As the dimension of the degenerate manifolds increases from Weyl points for $M \perp a$ to loops for $M \parallel a$, we expect an enhancement of the Hall response and reconstruction of the Hall spectral weight at finite frequencies. Following the changes in the topology of the bulk band structure, surface states may also be restructured. Since the nodal loop stays intact for $M \parallel a$, we expect the formation of a drumhead surface state spanning the projection of NL1 on the surface, instead of the Fermi arcs present between the isolated Weyl points, with possible implications for e.g. surface catalysis of hydrogen evolution [36, 37]. The magnetization reorientation, therefore, provides an exciting pathway to study these exotic surface states, also applicable to other materials.

# IV. METHODS

## A. Experimental Details

We carried out magneto-reflectance experiments on the same ab cut crystal which was used in a previous magneto-optical study [19]. Relative magneto-reflectance spectra, $[R(B) - R(0)]/R(0)$ were collected in Voigt configuration with the field aligned perpendicular to the in-plane $a$ crystal axis, where $R(0)$ is the zero field and $R(B)$ is the finite field reflectivity spectrum. The light of a globar was analyzed by a Bruker Vertex 80v Fourier-transform spectrometer in the energy range between 12 and 500 meV and guided to the sample position by a light-pipe. The sample was cooled to 4 K and placed in a superconducting coil (SC) for measurements up to 16 T and in a resistive magnet for high field measurements (HF) up to 34 T. The reflected light was collected and guided onto an external bolometer after passing a silicon beamsplitter. Due to the different optical path lengths in the SC and HF experiments, the raw signal was smaller in the high-field data in the overlapping field range. In order to compare the two datasets, the HF data was scaled with the corresponding SC data set at 5 and 10
T, whereas for 15 T and higher fields, the 15 T SC data was used for scaling. Therefore, the presented data for high fields shows a lower limit for the actual magnitude of the magnetic field-induced changes. To calculate the optical conductivity in field, we multiply the smoothed magneto-reflectance spectra with the 10 K zero-field reflectivity spectrum from Ref. 19 and perform a Kramers Kronig analysis with the same parameters as in the earlier work (see Fig. S2).

## B. DFT calculations

The calculations of the electronic structures are conducted using DFT as implemented in the Vienna ab initio simulation package (VASP) [38, 39]. All the calculations are performed using the projector-augmented wave (PAW) [40] pseudopotential with the generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE) [41, 42]. A plane-wave cutoff value of 400 eV and a Γ-centered 16 × 16 × 16 k-mesh is used to describe the electronic structure. The valence orbital set is 4s²3d⁷ for Co, 5s²5p² for Sn, and 3s²3p⁴ for S, respectively. For the crystal structure, we use the experimental lattice parameters, a = 5.379 Å and α = 59.8658° [43]. Based on the DFT electronic band structures, we construct the Wannier functions using Co d, Sn s, p, and S s, p orbitals. Finally, the optical properties are evaluated using the Kubo-Greenwood formula under the Wannier interpolation [44]. A fine mesh of 200 × 200 × 200 k points is applied during the integration with good convergence.

The optical conductivity given in Fig. 2(c) and the deduced spectral weight of certain photon energies plotted in Fig. 3 can be calculated through the Brillouin zone integration of the transition matrix elements:

$$ \begin{aligned} \mathrm{Re} \, \sigma_{\alpha\beta}(\hbar\omega) &= \frac{-\pi\omega e^2}{\hbar} \int_{\mathrm{BZ}} \sum_{nm} f_{mn}^{\mathrm{FD}} A_{nm,\alpha}(\mathbf{k}) A_{mn,\beta}(\mathbf{k}) \delta(\omega_{mn} - \omega) \\ &= \frac{-\pi e^2}{\hbar} \int_{\mathrm{BZ}} H_{\alpha\beta}(\hbar\omega; \mathbf{k}) \\ &= \frac{-\pi e^2}{\hbar} \int_{\mathrm{BZ}} \omega g_{\alpha\beta}(\hbar\omega; \mathbf{k}) \end{aligned} \quad (2) $$

α and β are the indices in Cartesian coordinates, ∫BZ is the Brillouin zone integration ∫d³k/(2π)³, fFD is the Fermi-Dirac distribution of band n, fFD = fFD - fFD. $\hbar\omega_{mn} = \hbar\omega_m - \hbar\omega_n$ is the energy difference between bands m and n. Finally,

$A_{nm,\alpha} = \langle u_{n,k} | i\nabla_{k\alpha} | u_{m,k} \rangle$ is the Berry connection, $H_{\alpha\beta}(\hbar\omega)$ is the spectral weight at certain transition energy $\omega$, and $g_{\alpha\beta}(\hbar\omega)$ is the quantum metric [15, 34]. They are all functions of the crystal momentum $\mathbf{k}$. Due to the fixed proportionality between $H_{\alpha\beta}(\hbar\omega_{mn})$ and $g_{mn;\alpha\beta}(\hbar\omega)$ at a fixed photon energy, the maps in Figs. 3(d)-(f) and S4(d)-(f) can be equivalently interpreted as maps of the quantum metric.

# V. DECLARATIONS

## A. Data availability

The datasets generated during and/or analyzed during this study are available on Zenodo 10.5281/zenodo.15354907.

## B. Acknowledgements

This work was supported by the Hungarian National Research, Development and Innovation Office NKFIH Grants No. FK 135003 and by the Ministry of Innovation and Technology and the National Research, Development and Innovation Office within the Quantum Information National Laboratory of Hungary, by the National Science and Technology Council in Taiwan, by the IPA program of RIKEN, Japan and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 360 – 492547816. We acknowledge the support of the LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL).

## C. Author Contributions

M.A.K., Y.T., T.W. and H.N. synthesized and characterized the crystals; F.S. measured the zero-field reflectivity; F.S., F.L., L.B.P., I.M., M.O. and S.B. performed the magnetoreflectance experiments. M.C.J., G.Y.G. and R.A. performed the ab initio calculations; F.S., M.C.J. and S.B. wrote the paper; I.K. and S.B. planned and coordinated the project. All authors contributed to the discussion and interpretation of the experimental and theoretical results and to the completion of the paper.

Correspondence to Felix Schilberth or Sándor Bordács on the magnetoreflectance experiments or to Ming-Chun Jiang for the DFT calculations.

## D. Competing interests

The authors declare no competing interests.

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<figure><img src="image_2.png" /><figcaption>FIG. 1. Symmetries and topology for out-of- and in-plane magnetic field in Co₃Sn₂S₂. The first row (a, d, g) shows a real-space representation of the magnetization for **M** ⊥ **a**, **M** || **c** and **M** || **a**. The second and third rows show a top and 3D view of the corresponding BZ for the respective configuration. (a–c) Inequivalent nodal loops (NL1/2) and emerging Weyl points (WP) for **M** ⊥ **a**. Those have the same color as their corresponding loop or are dark blue if located at other *k* points. (d–f) Three equivalent nodal loops related by the C₃ rotational symmetry for out-of-plane magnetization. (g–i) Mirror plane protected nodal loops for **M** || **a**.</figcaption></figure>

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<figure><img src="image_3.png" /></figure>

FIG. 2. Magneto-reflectance data measured for fields up to 34 T. (a, d) Raw magneto-reflectance data $[(R(B) - R(0))]/R(0)$ for $H \perp a$ and $H \parallel a$, respectively (b, e) Optical conductivity in field calculated from the zero-field reflectivity of Ref. 19 and the magneto-reflectance data for both in-plane field orientations. Insets: evolution of the 29 meV peak as a function of the field for the merged datasets from HF and SC experiments. The arrows, indicating the peak position for selected fields, highlight the stronger redshift for $M \parallel a$ (5,15,34 T in (b) and 1,5,10,16 T in (e)). (c, f) Comparison of the zero-field and high-field conductivities with *ab initio* calculations for both field directions.
<figure><img src="image_4.png" /><figcaption>FIG. 3. Evolution of the nodal loop upon reorienting the magnetization to $M \parallel a$. (a-c) gap of the nodal lines for out-of-plane and NL1 and NL2 in-plane loops, respectively. Points C and D are equivalent to A and B but lie on the BZ planes that contain NL2 instead of NL1. (d-f) Optical weight $H_{xx}(\omega, k)$ distribution at the peak energies (42.0-45.6 meV in (d), 24.4-30.8 meV in (e) and (f)) on the high symmetry planes containing the nodal loops from (a-c). (g) band structure along the triangles in the right column. The high symmetry points are shown above and in Fig. 1(f).</figcaption></figure>
<figure><img src="image_5.png" /></figure>

FIG. 4. (a) Optical conductivity $\sigma_{xx}$ and the cutoff-frequency dependent quantum weight $K_{xx}$ of Co$_3$Sn$_2$S$_2$ under the in-plane ($M \parallel c$) and out-of-plane ($M \parallel a$) magnetic configurations. (b) Difference of the quantum weight $\Delta K_{xx}$ between in- and out-of-plane magnetization, which peaks exactly at the energy of the nodal line resonance, highlighting its critical origin by comparison with the optical conductivity (shown in the same line style as in panel (a)). The dotted horizontal line denotes the saturation value, visible more clearly in the inset which shows the enhancement on a broad energy range. The gray shading denotes the energy scale of the main panel.
# Supplementary Material: Generation of a nodal line and Weyl points by magnetization reorientation in Co₃Sn₂S₂

F. Schilberth,¹,² M.-C. Jiang,³,⁴ F. Le Mardelé,⁵ L. B. Papp,² I. Mohelsky,⁵ M. A. Kassem,⁶,⁷ Y. Tabata,⁶

T. Waki,⁶ H. Nakamura,⁶ G.-Y. Guo,³,⁸ M. Orlita,⁵,⁹ R. Arita,⁴,¹⁰ I. Kézsmárki,¹ and S. Bordács¹,²,¹¹

¹Experimentalphysik V. Center for Electronic Correlations and Magnetism,
Institute for Physics, Augsburg University, D-86135 Augsburg, Germany

²Department of Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary

³Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan

⁴RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako 351-0198, Japan

⁵Laboratoire National des Champs Magnétiques Intenses, LNCMI-EMFL,
CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse,
Univ. Toulouse 3, INSA-T, Grenoble and Toulouse, France

⁶Department of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan

⁷Department of Physics, Faculty of Science, Assiut University, 71516 Assiut, Egypt

⁸Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan

⁹Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, Prague, 121 16, Czech Republic

¹⁰Department of Physics, University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo, 113-0033, Japan

¹¹HUN-REN-BME Condensed Matter Physics Research Group,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary

(Dated: May 16, 2025)

## Supplemental Note 1: Quantum Geometry Enhancement by Magnetic Reorientation in Co₃Sn₂S₂

We note that the quantum weight was generally defined for insulators with a finite gap [1], since the quantum weight would diverge in a metal due to the presence of $\omega$ in the denominator. Although Co₃Sn₂S₂ is metallic, spin-orbit coupling lifts the degeneracy of the nodal line, providing an optical gap. The reduction of this spin-orbit-induced gap is a manifestation of the energy-resolved quantum metric. Since quantum geometry is a local effect, we are probing the quantum weight of the gapped nodal line.

To further support this argument, we refer to Fig. S5 to recall that the band inversion responsible for the nodal line in Co₃Sn₂S₂ arises from the hybridization between Co out-of-plane orbitals ($d_{z^2,xz,yz}$) and the Co in-plane orbitals ($d_{xy,x^2-y^2}$). Thus, we investigate the hopping amplitudes between them. For $M \parallel c$, the sum of the absolute value of hopping amplitudes between $d_{z^2,xz,yz}$ and $d_{xy,x^2-y^2}$ is 426 meV, while for the case of $M \parallel a$ it is 438 meV. This enhancement indicates that the reorientation of magnetic moments indeed strengthens the inter-orbital hybridization responsible for the gapped nodal line.

<footnote>[1] Y. Onishi and L. Fu, Fundamental bound on topological gap, Physical Review X 14, 011052 (2024).</footnote>

<footnote>[2] F. Schilberth, M.-C. Jiang, S. Minami, M. A. Kassem, F. Mayr, T. Koretsune, Y. Tabata, T. Waki, H. Nakamura, G.-Y. Guo, R. Arita, I. Kézsmárki, and S. Bordács, Nodal-line resonance generating the giant anomalous Hall effect of Co₃Sn₂S₂, Physical Review B 107, 214441 (2023).</footnote>
<figure><img src="image_6.png" /><figcaption>FIG. S1. (a,b) Magnitude of the peak at 29 meV for **H** ⊥ **a** and 25 meV for **H** || **a** as function of field, respectively</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. S2. Field dependent reflectivity spectra generated by multiplying the magnetoreflectance data onto the 10 K spectrum of Ref. 2. Panels (a)&(b) show the spectra for **H** ⊥ **a** and **H** || **a** as function of field, respectively</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. S3. Comparison of the calculated in plane and out of plane optical conductivity spectra</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIG. S4. Evolution of the nodal loop upon reorienting the magnetization to $M \perp a$ (a-c) gap of the nodal lines for out-of-plane and NL1 and NL2 in-plane loops, respectively. For (a), the average gap is larger than for (b) and (c). Points C and D are equivalent to A and B but lie on the BZ planes that contain NL2 instead of NL1. (d-f) Optical weight $H_{xx}(\omega, k)$ at the peak energies (42.0-45.6 meV in (d), 28.8-33.6 meV in (e) and (f)) distributed on the high symmetry planes of the BZ containing the nodal loops from (a-c). We obtain qualitatively similar distributions despite the different energy ranges. (g) band structure along the triangles in the right column. The high symmetry points are shown above and in Fig. 1(d). For in-plane magnetization, the SOC gap along the nodal line is clearly smaller than for out-of-plane.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>FIG. S5. Projected band structure of Co₃Sn₂S₂ at low energy and without the spin-orbit coupling.</figcaption></figure>