# Signatures of Weyl fermion annihilation
in a correlated kagome magnet

Ilya Belopolski111These authors contributed equally to this work.
<EMAIL_ADDRESS>Laboratory for Topological Quantum Matter and
Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA  RIKEN Center for Emergent Matter Science (CEMS), Wako,
Saitama 351-0198, Japan  Tyler A. Cochran∗ Laboratory for Topological Quantum
Matter and Spectroscopy (B7), Department of Physics, Princeton University,
Princeton, New Jersey 08544, USA  Xiaoxiong Liu∗ Department of Physics,
University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland  Zi-
Jia Cheng∗ Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Xian P. Yang  Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Zurab Guguchia  Laboratory for Topological Quantum Matter and Spectroscopy
(B7), Department of Physics, Princeton University, Princeton, New Jersey
08544, USA  Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute,
Villigen PSI, Switzerland  Stepan S. Tsirkin  Department of Physics,
University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland  Jia-
Xin Yin  Laboratory for Topological Quantum Matter and Spectroscopy (B7),
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Praveen Vir  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Straße 40, 01187 Dresden, Germany  Gohil S. Thakur  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany
Faculty of Chemistry and Food Chemistry, Technische Universitat, 01069
Dresden, Germany  Songtian S. Zhang  Laboratory for Topological Quantum Matter
and Spectroscopy (B7), Department of Physics, Princeton University, Princeton,
New Jersey 08544, USA  Junyi Zhang  Department of Physics, Princeton
University, Princeton, New Jersey 08544, USA  Konstantine Kaznatcheev
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  Guangming Cheng  Princeton Institute for Science and
Technology of Materials, Princeton University, Princeton, New Jersey, 08544,
USA  Guoqing Chang  Division of Physics and Applied Physics, School of
Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, 637371, Singapore  Daniel Multer  Laboratory for Topological
Quantum Matter and Spectroscopy (B7), Department of Physics, Princeton
University, Princeton, New Jersey 08544, USA  Nana Shumiya  Laboratory for
Topological Quantum Matter and Spectroscopy (B7), Department of Physics,
Princeton University, Princeton, New Jersey 08544, USA  Maksim Litskevich
Laboratory for Topological Quantum Matter and Spectroscopy (B7), Department of
Physics, Princeton University, Princeton, New Jersey 08544, USA  Elio Vescovo
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA  Timur K. Kim  Diamond Light Source, Didcot OX11 0DE, UK
Cephise Cacho  Diamond Light Source, Didcot OX11 0DE, UK  Nan Yao  Princeton
Institute for Science and Technology of Materials, Princeton University,
Princeton, New Jersey, 08544, USA  Claudia Felser  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany
Titus Neupert  Department of Physics, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland  M. Zahid Hasan
<EMAIL_ADDRESS>Laboratory for Topological Quantum Matter and
Spectroscopy (B7), Department of Physics, Princeton University, Princeton, New
Jersey 08544, USA  Princeton Institute for Science and Technology of
Materials, Princeton University, Princeton, New Jersey, 08544, USA  Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA

###### Abstract

The manipulation of topological states in quantum matter is an essential
pursuit of fundamental physics and next-generation quantum technology. Here we
report the magnetic manipulation of Weyl fermions in the kagome spin-orbit
semimetal Co3Sn2S2, observed by high-resolution photoemission spectroscopy. We
demonstrate the exchange collapse of spin-orbit-gapped ferromagnetic Weyl
loops into paramagnetic Dirac loops under suppression of the magnetic order.
We further observe that topological Fermi arcs disappear in the paramagnetic
phase, suggesting the annihilation of exchange-split Weyl points. Our findings
indicate that magnetic exchange collapse naturally drives Weyl fermion
annihilation, opening new opportunities for engineering topology under
correlated order parameters.

Quantum magnets exhibiting electronic topology are attracting considerable
interest for the magnetic manipulation of Weyl and Dirac quasiparticles, as
well as their topological surface states news_daSilvaNeto ;
RMPWeylDirac_Armitage ; ARCMP_me ; ReviewNatPhys_Fiete ; HighFold_NatRevMat ;
Review_KeimerMoore ; ReviewQuantumMaterials_Hsieh . To date, spectroscopic
signatures of electronic topological ground states have been observed in
several magnetic semimetals, comprising magnetic Weyl loops Co2MnGa_me ; Weyl
points Co3Sn2S2_HechangLei ; Co3Sn2S2_Enke ; Co3Sn2S2_Beidenkopf ;
Co3Sn2S2_YulinChen ; PrAlGe_DanDan ; and massive Dirac fermions
Fe3Sn2_Checkelsky . In parallel, the magnetic manipulation of Weyl and Dirac
fermions has been extensively explored in transport Transport_ARCMP ;
Na3Bi_XiongJun ; TaP_Chenglong ; Co2MnGa_Kaustuv ; Co2MnGa_Nakatsuji ;
Mn3Sn_Nakatsuji . However, direct spectroscopic observation of magnetic
control of topology remains challenging. Demonstrating coherent evolution of
topological quasiparticles under varying magnetic order, such as the
annihilation of Weyl points, offers the possibility to directly verify
fundamental notions of topological band theory RMPWeylDirac_Armitage ;
ARCMP_me ; TINI_BurkovBalents ; PyrochloreWeyl_Vishwanath . Furthermore, novel
transport and optical effects are enabled by tuning the relative energies of
Weyl loops and points WeylBrokenPT_Burkov ; ImbalancedLineNode_Zuyzin ;
Gyrotropic_Souza , controlling their positions relative to the Fermi level
OpticalWeyl_Qiong ; Burch_PhotovoltaicWeyl ; Heinonen_SpinChargeWeyl ;
Fe3Ga_Nakatsuji ; Co2MnGa_me  and switching on/off their topological surface
states PlasmonWeyl_Song ; PlasmonWeyl_Jafari .

We have investigated magnetic modulation of topological semimetallic states in
a range of materials by spectroscopy, including Fe3Sn2, Co2MnGa, PrAlGe,
Fe3GeTe2, TbMn6Sn6 and Co3Sn2S2 Fe3Sn2_Jiaxin ; Co2MnGa_me ; PrAlGe_DanDan ;
TbMn6Sn6_Jiaxin ; Co3Sn2S2_Jiaxin . Some of these materials exhibit high
magnetic transition temperatures $>600$ K, so that thermal broadening may
fundamentally overwhelm magnetic evolution of the Weyl or Dirac state
Co2MnGa_me ; Fe3Sn2_Checkelsky . Other systems, such as PrAlGe, exhibit low
transition temperatures of $\sim 10$ K, associated with only small magnetic
perturbations to the electronic structure PrAlGe_DanDan . Even in materials
such as Fe3GeTe2, with intermediate $T_{\rm C}=230$ K, the thermal evolution
appears to be dominated by a suppression of quasiparticle lifetime, without
significant coherent evolution of the dispersion Fe3GeTe2_Kim_Pohang ;
Fe3GeTe2_XinchunLai ; Fe3GeTe2_YulinChen . Using newly available high-quality
single crystals combined with state-of-the-art variable-temperature
photoemission spectroscopy, we have found that a large and previously
overlooked energy shift of a topological spin-orbit gapped Weyl loop occurs in
Co3Sn2S2 across $T_{\rm C}=176$ K Co3Sn2S2_QiunanXu ; Co3Sn2S2_Zurab ;
Co3Sn2S2_Takahashi ; Co3Sn2S2_XianggangQiu ; Co3Sn2S2_Rossi ;
Co3Sn2S2_temp_DFLiu . This shift takes place together with a magnetic exchange
gap collapse that suggests a ferromagnetic Weyl to paramagnetic Dirac loop
transition on raising temperature. This transition is further accompanied by
the removal of candidate topological Fermi arc surface states and the
annihilation of Weyl points.

Materials with inversion symmetry, mirror symmetry and ferromagnetism provide
a unique platform for a magnetic-topological phase transition. The
ferromagnetism produces singly-degenerate spin-split bands. In the limit of
weak spin-orbit coupling (SOC), mirror symmetry can then give rise to Weyl
loops on mirror planes of the bulk Brillouin zone Co2MnGa_me ;
WeylLoopSuperconductor_Nandkishore ; WeylLines_Kane ;
WeylDiracLoop_Nandkishore . A Weyl loop is a closed curve along which the
bands are everywhere two-fold degenerate; it is characterized by a $\pi$ Berry
phase topological invariant and a linear energy-momentum dispersion everywhere
along the loop. If the magnetic order is removed and no spin-splitting
remains, opposite-spin partner Weyl loops naturally collapse into a Dirac
loop, a closed curve along which the bands are everywhere four-fold degenerate
WeylDiracLoop_Nandkishore ; DiracLineNodes_WeiderKane ; CaAgX_Okamoto . Weyl
loops under SOC typically gap out, concentrating a loop of Berry curvature in
momentum space, leading to a giant anomalous Hall response Co3Sn2S2_Enke ;
Co2MnGa_me , large anomalous Nernst effect Co3Sn2S2_ZeroFieldNernst ;
Fe3Ga_Nakatsuji , large optical Hall conductivity Co3Sn2S2_Takahashi  and
other exotic response. Under SOC, Weyl loops may also leave behind some
discrete number of Weyl points. By contrast, Dirac points are generically
unstable under inversion and time-reversal symmetry Gapless_Murakami , so that
Dirac loops under SOC gap out fully. As a result, in this scenario upon
magnetic exchange collapse the Weyl points generically annihilate.

Co3Sn2S2 crystallizes in space group $R\bar{3}2/m$ (No. 166), with dihedral
point group $D_{3d}$, which includes inversion symmetry and three mirror
planes (Fig. 1a, S3). The system is ferromagnetic, with Curie temperature
$T_{\rm C}=176$ K Co3Sn2S2_Sobany ; Co3Sn2S2_Weihrich . Keeping in mind the
mirror symmetry and ferromagnetic order, we explore our Co3Sn2S2 samples by
ARPES at 20 K. Measuring with incident photon energy $h\nu=130$ eV, we observe
point-like electronic structures on $M_{y}$ (cyan arrows, Fig. 1d; the mirror
planes correspond to $\bar{\Gamma}-\bar{M}$). On cuts along $k_{y}$ through
the point-like features, we observe cone dispersions straddling the mirror
plane $M_{y}$ (Figs. 1e). On an energy-momentum cut along $k_{x}$, within the
mirror plane, we again observe cone-like dispersions (Fig. S4). The
observation of cone dispersions along both $k_{x}$ and $k_{y}$, coming
together at point Fermi surfaces, suggests a set of band crossings living in
the momentum-space mirror plane. To systematically understand the evolution of
the band crossings along the out-of-plane momentum-space direction $k_{z}$ we
acquire analogous datasets at a range of photon energies, from $h\nu=100$ to
$135$ eV (Fig. S5). We find that the cones persist in $h\nu$, with crossing
points consistently on the $M_{y}$ plane, but at varying $(k_{x},k_{z})$
coordinates (red diamonds, Fig. 1f). Taken together, these crossing points
appear to form an extended nodal electronic state encircling the L point of
the bulk Brillouin zone, suggesting the observation of a bulk loop node in
Co3Sn2S2. Since the system is ferromagnetic with generically singly-degenerate
bands, we interpret this loop node as a Weyl loop (Fig. 1c). To extract the
complete trajectory of the loop, we fit the ARPES locations of the cone
dispersions to a low-order polar Fourier decomposition around the L point of
the bulk Brillouin zone (blue loop, Fig. 1f; see Supplemental Material SM  for
fitting parameter values). In this way we extract the full momentum-space
trajectory of the Weyl loop from photoemission data alone.

Next we explore the evolution of the Weyl loop with temperature, focusing on
Cut (i). We systematically cycle the temperature of our samples from 20 K to
290 K and back to 20 K, moving across $T_{\rm C}=176$ K. On raising the
temperature, we observe a dramatic evolution of the Weyl cone on a large
energy scale of $\sim 0.1$ eV (Figs. 2a, b; S8), with the cone appearing to
recede above $E_{\textrm{F}}$. We next assemble the momentum distribution
curves (MDCs) of Cut (i) at $E_{\textrm{F}}$ for all temperatures (Fig. 2c).
Upon cycling the temperature, we observe a prominent and reversible evolution
of the Weyl cone across $T_{\rm C}$, consistent with a magnetic phase
transition. For further insight, we examine additional spectra on Cut (ii),
obtained during the course of the same measurement, and we consider a set of
deep bands $\sim 0.3$ eV below $E_{\textrm{F}}$, which are predominantly
formed from the same exchange-split Co $3d$ $a_{1g}$ and $e_{g}$ manifolds as
the Weyl loop (Figs. 2d; S11). At 20 K, these deep valence bands exhibit clear
splitting, consistent with the material’s ferromagnetic order. Upon raising
the temperature, the splitting appears to vanish and these deep bands collapse
together, suggesting a paramagnetic state with spin-degenerate bands. By
examining the evolution of the deep bands, we circumvent the limitations of
the photoemission $E_{\textrm{F}}$ cut-off and observe direct signatures of a
prominent magnetic exchange gap collapse across $T_{\rm C}$ in Co3Sn2S2.

To relate the Weyl loop temperature evolution to the magnetic exchange gap
collapse, we consider more carefully the interplay between topology and
ferromagnetism. In ab initio calculation, in the absence of SOC and in the
ferromagnetic state, the Weyl loop arises as a crossing of two spin-majority
bands, with a spin-minority partner Weyl loop above the Fermi level (schematic
blue and green loops, Figs. 2e; S6). In a non-magnetic ab initio calculation,
the exchange gap vanishes and these two Weyl loops coincide, forming a
spinless loop crossing—a Dirac loop (purple loop). Comparing the ab initio
calculations with ARPES, we find that the magnetic Weyl and non-magnetic Dirac
nodes exhibit overall agreement with the ferromagnetic and paramagnetic
spectra, respectively (magenta traces, Fig. 2b). Note that including SOC in
our ab initio results does not alter this interpretation, although the
expected gap appears in both loop nodes (blue traces). The observation that
the loop recedes above $E_{\textrm{F}}$ on increasing temperature is also
consistent with maintenance of charge balance in the spin-degenerate
electronic structure, further indicating a paramagnetic Dirac loop. We further
reduced the magnetic moment in our samples via nickel (Ni) doping, and again
observed a persistent loop node electronic structure despite suppression of
the ferromagnetism (Fig. S1). Taken together, our systematic ARPES spectra and
ab initio calculations suggest that we have observed the collapse of two
opposite-spin ferromagnetic Weyl loops into a paramagnetic Dirac loop.

To quantitatively characterize the Weyl loop collapse with temperature, we
perform a Lorentzian fit of energy distribution curves (EDCs) through the
extremum of the Weyl loop band (Cut(i), dotted line, Fig. 2a). The extracted
Weyl band extremum exhibits a clear evolution upward in $E_{\rm B}$ as the
temperature increases, 20 K $\rightarrow$ 250 K, consistent with exchange gap
collapse (Fig. 2d). We further extract the exchange gap $\Delta(T)$ on EDCs
through the deep bands (Cut (ii), dotted line, Fig. 2c) and compared the
resulting $\Delta(T)$ with the magnetization $M(T)$ as measured by a SQUID.
For $T<T_{\rm C}$ we find that the exchange splitting tracks $M(T)$. For
$T>T_{\rm C}$ we no longer observe an exchange splitting within our spectral
linewidth, consistent with the absence of magnetization. Remarkably, the
observed exchange gap and Weyl band shift are both $\sim 0.12$ eV, suggesting
a complete collapse of the opposite-spin partner Weyl loops across $T_{\rm C}$
and the formation of a spin-degenerate paramagnetic Dirac loop.

To further explore the paramagnetic Dirac loop we park our apparatus at 220 K,
well into the paramagnetic phase. At a range of $h\nu$ we observe
characteristic point-like iso-energy contours on $M_{y}$ and related mirror
planes (Figs. 3a, c; S7). Energy-momentum spectra through these point-like
contours further exhibit cone-like spectral weight straddling $M_{y}$,
indicative of Dirac loop cone dispersions above $E_{\textrm{F}}$ (Figs. 3b,
d). The presence of multiple cone features straddling $M_{y}$ at a range of
$h\nu$ again suggests an extended nodal electronic structure confined to the
mirror plane. Since we are in the paramagnetic phase with generically spin-
degenerate bands, we interpret these candidate band crossings as four-fold
degenerate, forming a Dirac loop. By analogy with our analysis in the
ferromagnetic phase, we again systematically collect the locations of all cone
features observed in the paramagnetic phase for $h\nu$ from 100 to 135 eV and
experimentally extract the full momentum-space trajectory of the Dirac loop
(red diamonds, Fig. 3e). Ab initio calculations of the Weyl and Dirac loops in
the ferromagnetic and non-magnetic states also agree with the experimentally-
observed trajectories (Fig. S2). A loop node electronic structure persisting
into the paramagnetic phase of Co3Sn2S2 again suggests the observation of a
ferromagnetic Weyl to paramagnetic Dirac loop collapse.

We next consider the fine structure of the Weyl loop collapse associated with
spin-orbit coupling (SOC). In Co3Sn2S2, ab initio calculations along with
ARPES and STM investigations suggest that each Weyl loop under SOC produces
two Weyl points above the Fermi level, with signatures of topological Fermi
arc surface states extending below the Fermi level. Co3Sn2S2_Enke ;
Co3Sn2S2_Beidenkopf ; Co3Sn2S2_QiunanXu ; Co3Sn2S2_HechangLei ; Co3Sn2S2_Zurab
; Co3Sn2S2_YulinChen ; Co3Sn2S2_Takahashi ; Co3Sn2S2_XianggangQiu . Our
observation of a Weyl to Dirac loop transition naturally motivates
investigation of Fermi arc and Weyl point annihilation across $T_{\rm C}$. At
20 K, we observe sharp arc-shaped states near the expected Weyl points,
consistent with topological Fermi arcs in ab initio calculation (Figs. 4a, c;
S9). At 220 K these Fermi arcs vanish, leaving behind bulk pockets broadly
consistent with the low temperature spectra (Figs. 4b, d). The disappearance
of the Fermi arcs above $T_{\rm C}$ provides evidence for the annihilation of
Weyl points in the paramagnetic phase. To further characterize this
annihilation, we consider the ab initio band structure under ferromagnetic
order on a momentum-space path connecting a pair of exchange-split Weyl points
(Fig. 4e). Upon exchange collapse, these partner Weyl points come together and
annihilate, opening a gap (Fig. 4f).

Our systematic variable-temperature ARPES experiments suggest that pairs of
ferromagnetic Weyl loops collapse into paramagnetic Dirac loops across $T_{\rm
C}$ in Co3Sn2S2 (Figs. 4g, h). Taken together with ab initio calculations, our
results additionally provide evidence for the annihilation of Fermi arcs and
Weyl points concomitant with this transition. Our findings suggest a general
mechanism for Weyl fermion annihilation, where the annihilation is driven by
magnetic exchange gap collapse and takes place predominantly along the energy
axis, rather than in momentum TaAs_DFT_ShinMing ; ARCMP_me ;
TINI_BurkovBalents . This novel mechanism should occur naturally in many
quantum magnets and motivates exploration of the rich topological evolution
associated with the onset of magnetic order. Such interplay between magnetism
and topology may also pave the way to magnetic design of correlated
topological states with exotic transport and optical response.

##  I Acknowledgments

The authors thank D. Lu and M. Hashimoto at Beamline 5-2 of the Stanford
Synchrotron Radiation Lightsource (SSRL) at the SLAC National Accelerator
Laboratory, CA, USA for support. The authors thank Diamond Light Source for
access to Beamline I05 (SI17924, SI19313). This research used Beamline 21-ID-1
(ESM-ARPES) of the National Synchrotron Light Source II, a U.S. Department of
Energy (DOE) Office of Science User Facility operated for the DOE Office of
Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. The
authors also acknowledge use of Princeton University’s Imaging and Analysis
Center, which is partially supported by the Princeton Center for Complex
Materials (PCCM), a National Science Foundation (NSF)-MRSEC program
(DMR-2011750). Use of the Stanford Synchrotron Radiation Lightsource (SSRL),
SLAC National Accelerator Laboratory, is supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.
DE-AC02-76SF00515. T. A. C. acknowledges the support of the National Science
Foundation Graduate Research Fellowship Program (DGE-1656466). T. N. and S. S.
T. acknowledge support from the European Union Horizon 2020 Research and
Innovation Program (ERC-StG-Neupert-757867-PARATOP). S. S. T. also
acknowledges support from the Swiss National Science Foundation (Grant No.
PP00P2-176877). X. L. acknowledges financial support from the China
Scholarship Council. G. C. would like to acknowledge the support of the
National Research Foundation, Singapore under its NRF Fellowship Award (NRF-
NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang
Technological University. M. Z. H. acknowledges visiting scientist support at
Berkeley Lab (LBNL) during the early phases of this work. Work at Princeton
University was supported by the Gordon and Betty Moore Foundation (Grants No.
GBMF4547 and No. GBMF9461; M. Z. H.). The ARPES and theoretical work were
supported by the United States Department of Energy (US DOE) under the Basic
Energy Sciences programme (Grant No. DOE/BES DE-FG-02-05ER46200; M. Z. H.). G.
S. T. thanks the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter – ct.qmat (EXC 2147) for postdoctoral funding. C.
F. acknowledges the DFG through SFB 1143 (project ID. 247310070) and the
WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum
Matter ct.qmat (EXC2147, project ID. 39085490).

## References

  * (1) E. H. da Silva Neto, “Weyl”ing away time-reversal symmetry. Science 365, 1248 (2019).
  * (2) N. P. Armitage, E. J. Mele, A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
  * (3) M. Z. Hasan, S.-Y. Xu, I. Belopolski, S.-M. Huang, Discovery of Weyl fermion semimetals and topological Fermi arc states. Ann. Rev. Cond. Matt. Phys. 8, 289 (2017).
  * (4) J. Maciejko, G. A. Fiete, Fractionalized topological insulators. Nat. Phys. 11, 385 (2015).
  * (5) M. Z. Hasan, et al., Weyl, Dirac and high-fold chiral fermions in topological quantum matter. Nat. Rev. Mat. (2021).
  * (6) B. Keimer, J. E. Moore, The physics of quantum materials. Nat. Phys. 13, 1045 (2017).
  * (7) D. N. Basov, R. D. Averitt, D. Hsieh, Towards properties on demand in quantum materials. Nat. Mat. 16, 1077 (2017).
  * (8) I. Belopolski, et al., Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet. Science 365, 1278 (2019).
  * (9) Q. Wang, et al., Large intrinsic anomalous Hall effect in half-metallic ferromagnet Co3Sn2S2 with magnetic Weyl fermions. Nat. Commun. 9, 3681 (2018).
  * (10) E. Liu, et al., Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal. Nat. Phys. 14, 1125 (2018).
  * (11) N. Morali, et al., Fermi-arc diversity on surface terminations of the magnetic Weyl semimetal Co3Sn2S2. Science 365, 1286 (2019).
  * (12) D. F. Liu, et al., Magnetic Weyl semimetal phase in a Kagomé crystal. Science 365, 1282 (2019).
  * (13) D. S. Sanchez, et al., Observation of Weyl fermions in a magnetic non-centrosymmetric crystal. Nat. Commun. 11, 3356 (2020).
  * (14) L. Ye, et al., Massive Dirac fermions in a ferromagnetic kagome metal. Nature 555, 638 (2018).
  * (15) J. Hu, S.-Y. Xu, N. Ni, Z. Mao, Transport of topological semimetals. Ann. Rev. Cond. Mat. Phys. 49, 207 (2019).
  * (16) J. Xiong, et al., Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413 (2015).
  * (17) C.-L. Zhang, et al., Magnetic-tunnelling-induced Weyl node annihilation in TaP. Nat. Phys. 13, 979 (2017).
  * (18) K. Manna, et al., From Colossal to Zero: Controlling the Anomalous Hall Effect in Magnetic Heusler Compounds via Berry Curvature Design. Phys. Rev. X 8, 041045 (2018).
  * (19) A. Sakai, et al., Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal. Nat. Phys. 14, 1119 (2018).
  * (20) K. Kuroda, et al., Evidence for magnetic Weyl fermions in a correlated metal. Nat. Mat. 16, 1090 (2017).
  * (21) A. A. Burkov, L. Balents, Weyl Semimetal in a Topological Insulator Multilayer. Phys. Rev. Lett. 107, 127205 (2011).
  * (22) X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
  * (23) A. A. Zyuzin, S. Wu, A. A. Burkov, Weyl semimetal with broken time reversal and inversion symmetries. Phys. Rev. B 85, 165110 (2012).
  * (24) S. F. Islam, A. A. Zyuzin, Collective modes in imbalanced nodal line semimetal (2021). arXiv:2108.13951.
  * (25) S. Zhong, J. E. Moore, I. Souza, Gyrotropic magnetic effect and the magnetic moment on the fermi surface. Phys. Rev. Lett. 116, 077201 (2016).
  * (26) Q. Ma, et al., Direct optical detection of Weyl fermion chirality in a topological semimetal. Nat. Phys. 13, 842 (2017).
  * (27) G. B. Osterhoudt, et al., Colossal mid-infrared bulk photovoltaic effect in a type-i weyl semimetal. Nat. Mat. 18, 471 (2019).
  * (28) S. S.-L. Zhang, A. A. Burkov, I. Martin, O. G. Heinonen, Spin-to-Charge Conversion in Magnetic Weyl Semimetals. Phys. Rev. Lett. 123, 187201 (2019).
  * (29) A. Sakai, et al., Iron-based binary ferromagnets for transverse thermoelectric conversion. Nature 581, 53 (2020).
  * (30) J. C. W. Song, M. S. Rudner, Fermi arc plasmons in Weyl semimetals. Phys. Rev. B 96, 205443 (2017).
  * (31) F. Adinehvand, Z. Faraei, T. Farajollahpour, S. A. Jafari, Sound of Fermi arcs: a linearly dispersing gapless surface plasmon mode in undoped Weyl semimetals. Phys. Rev. B 100, 195408 (2019).
  * (32) J.-X. Yin, et al., Giant and anisotropic many-body spin-orbit tunability in a strongly correlated kagome magnet. Nature (2018).
  * (33) J.-X. Yin, et al., Quantum-limit Chern topological magnetism in TbMn6Sn6. Nature 583, 533 (2020).
  * (34) J.-X. Yin, et al., Negative flat band magnetism in a spin-orbit-coupled correlated kagome magnet. Nat. Phys. 15, 443 (2019).
  * (35) K. Kim, et al., Large anomalous Hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal. Nat. Mat. 17, 794 (2018).
  * (36) Y. Zhang, et al., Emergence of Kondo lattice behavior in a van der Waals itinerant ferromagnet, Fe3GeTe2. Sci. Adv. 4, eaao6791 (2018).
  * (37) X. Xu, et al., Signature for non-Stoner ferromagnetism in the van der Waals ferromagnet $\mathrm{F}{\mathrm{e}}_{3}\mathrm{GeT}{\mathrm{e}}_{2}$. Phys. Rev. B 101, 201104 (2020).
  * (38) Q. Xu, et al., Topological surface Fermi arcs in the magnetic Weyl semimetal Co3Sn2S2. Phys. Rev. B 97, 235416 (2018).
  * (39) Z. Guguchia, et al., Tunable anomalous Hall conductivity through volume-wise magnetic competition in a topological kagome magnet. Nat. Commun. 11, 559 (2020).
  * (40) Y. Okamura, et al., Giant magneto-optical responses in magnetic Weyl semimetal Co3Sn2S2. Nat. Commun. 11, 4619 (2020).
  * (41) R. Yang, et al., Magnetization-Induced Band Shift in Ferromagnetic Weyl Semimetal ${\mathrm{Co}}_{3}{\mathrm{Sn}}_{2}{\mathrm{S}}_{2}$. Phys. Rev. Lett. 124, 077403 (2020).
  * (42) A. Rossi, et al., Electronic structure and topology across $T_{\rm C}$ in magnetic Weyl semimetal Co3Sn2S2 (2021). arXiv:2105.08265.
  * (43) D. F. Liu, et al., Topological Phase Transition in a Magnetic Weyl Semimetal (2021). arXiv:2106.03229.
  * (44) Y. Wang, R. Nandkishore, Topological surface superconductivity in doped Weyl loop materials. Phys. Rev. B 95, 060506 (2017).
  * (45) O. Stenull, C. L. Kane, T. C. Lubensky, Topological phonons and Weyl lines in three dimensions. Phys. Rev. Lett. 117, 068001 (2016).
  * (46) R. Nandkishore, Weyl and Dirac loop superconductors. Phys. Rev. B 93, 020506(R) (2016).
  * (47) Y. Kim, B. J. Wieder, C. L. Kane, A. M. Rappe, Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).
  * (48) A. Yamakage, Y. Yamakawa, Y. Tanaka, Y. Okamoto, Line-Node Dirac Semimetal and Topological Insulating Phase in Noncentrosymmetric Pnictides CaAg${X}$ (${X}=$ P, As). J. of the Phys. Soc. of Jpn. 85, 013708 (2016).
  * (49) S. N. Guin, et al., Zero-Field Nernst Effect in a Ferromagnetic Kagome-Lattice Weyl-Semimetal Co3Sn2S2. Adv. Mat. 31, 1806622 (2019).
  * (50) S. Murakami, Phase transition between the quantum spin Hall and insulator phases in 3D: emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).
  * (51) P. Vaqueiro, G. G. Sobany, A powder neutron diffraction study of the metallic ferromagnet Co3Sn2S2. Solid State Sci. 11, 513 (2009).
  * (52) W. Schnelle, et al., Ferromagnetic ordering and half-metallic state of Sn2Co3S2 with the shandite-type structure. Phys. Rev. B 88, 144404 (2013).
  * (53) See Supplemental Material at [url] for methods and additional systematics, including Refs. Co3Sn2S2_NCSS_Thakur ; Diamond_Hoesch ; DFT2 ; DFT_Efficiency_KressFurthmueller ; DFT4 ; Soluyanov_WannierTools ; Yates_Wannier90 ; Co3Sn2S2_oxidation_Claudia .
  * (54) S.-M. Huang, et al., A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015).
  * (55) G. S. Thakur, et al., Intrinsic Anomalous Hall Effect in Ni-Substituted Magnetic Weyl Semimetal Co3Sn2S2. Chem. Mat. 32, 1612 (2020).
  * (56) M. Hoesch, et al., A facility for the analysis of the electronic structures of solids and their surfaces by synchrotron radiation photoelectron spectroscopy. Rev. Sci. Inst. 88, 013106 (2017).
  * (57) G. Kresse, J. Furthmueller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
  * (58) G. Kresse, J. Furthmueller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comp. Mat. Sci. 6, 15 (1996).
  * (59) J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
  * (60) Q. S. Wu, S. Zhang, H.-F. Song, M. Troyer, A. A. Soluyanov, WannierTools: an open-source software package for novel topological materials. Comput. Phys. Commun. 224, 405 (2018).
  * (61) G. Pizzi, et al., Wannier90 as a community code: new features and applications. J. Phys.: Cond. Mat. 32, 165902 (2020).
  * (62) G. Li, et al., Surface states in bulk single crystal of topological semimetal Co3Sn2S2 toward water oxidation. Sci. Adv. 5, eaaw9867 (2019).

Figure 1:  Topological magnetic Weyl loop. (a) Primitive unit cell of
ferromagnetic Co3Sn2S2, with mirror symmetry. (b) Bulk and $(001)$ surface
Brillouin zones with bulk mirror plane ($M_{y}$, cyan) and several high-
symmetry points (red). (c) In the absence of spin-orbit coupling (SOC), the
combination of mirror symmetry and ferromagnetism generically gives rise to
Weyl loops, which live in the mirror planes of the bulk Brillouin zone. A Weyl
loop exhibits a ring of band crossings along a closed curve in momentum space
(blue loop) with a linear cone dispersion on any energy-momentum slice through
the loop. Under SOC, the Weyl loop typically gaps, possibly leaving behind
Weyl points. (d) ARPES Fermi surface at $T=20$ K and photon energy $h\nu=130$
eV, exhibiting multiple dot features (cyan arrows) on the mirror planes
($\bar{\Gamma}-\bar{M}$). (e) Cone dispersion at the Fermi level on an energy-
momentum spectrum through the dot feature (Cut i). (f) Collecting cone
dispersions for a range of $h\nu$ suggests a loop of band crossings (red
diamonds) living in $M_{y}$ and encircling the bulk L point (Fig. S4, S5).
Different $h\nu$ sample different out-of-plane $k_{z}$ momenta; representative
example shown for $130$ eV (dotted red curve). The crossing points can be fit
by a low-order polar coordinate Fourier decomposition around the L point (blue
curve SM ), mapping out the trajectory of the Weyl loop.

Figure 2:  Magnetic Weyl to Dirac loop collapse. (a) Cut (i) at 20 K and 220
K, with (b) corresponding ab initio calculation. Left: calculation in the
ferromagnetic state through the Weyl loop, without SOC (magenta) and with SOC
(blue). Right: calculation in the non-magnetic state through the Dirac loop.
(c) Momentum distribution curves (MDCs) of Cut (i) at the Fermi level for the
full temperature cycle, 20 K $\rightarrow$ 290 K $\rightarrow$ 20 K. (d) Cut
(ii), defined in Fig. 1d, exhibiting clear splittings in deeper energy bands
at 20 K (left), which collapse at 220 K (right). (e) Energy extremum of the
Weyl loop band, extracted from the temperature dependence on Cut (i), obtained
by Lorentzian fitting of energy distribution curves (EDCs, dotted lines in
(a)). Also, the magnetic exchange splitting as a function of temperature,
obtained from Cut (ii) by Lorentzian fitting to EDCs (dotted lines in (d)) and
compared with the magnetization $M(T)$. Cartoon: exchange gap collapse of two
opposite-spin Weyl loop partners (blue and green) into a single Dirac loop
(purple).

Figure 3:  Paramagnetic Dirac loop. (a) ARPES iso-energy contour slightly
above $E_{\textrm{F}}$, at $h\nu=110$ eV, acquired at 220 K, exhibiting point-
like features (cyan arrows) along $\bar{\Gamma}-\bar{M}$ (corresponding to
$M_{y}$ and the symmetry-related mirror planes). (b) Energy-momentum cut
through the point-like feature, exhibiting cone-like spectral weight (cut
location: white line in (a)). (c), (d) Analogous to (a), (b), at 125 eV. (e)
Locations of cones observed for all $h\nu$ (red diamonds, Fig. S7). Cones on
symmetry-related mirror planes are plotted all together in a single momentum-
space mirror plane $M_{y}$. Data points fit to a low-order polar coordinate
Fourier decomposition around L (purple curves), mapping out the trajectory of
the Dirac loop.

Figure 4:  Evidence for Fermi arc and Weyl point annihilation. (a) Fermi
surface acquired at $T=20$ K, $h\nu=130$ eV, exhibiting candidate Fermi arc
surface states (white arrow) near the Weyl point locations, as predicted by ab
initio calculations (blue, green circle; Fig. S9). Data symmetrized for
clarity. (b) Analogous Fermi surface at 220 K, with no signature of Fermi
arcs. (c) Energy-momentum cut through the candidate Fermi arcs at 20 K (green
dotted line, (a)). (d) Analogous cut at 220 K. (e) Calculation in the
ferromagnetic state, in the presence of SOC, slicing through a pair of
exchange-split Weyl points of opposite chirality. (f) Analogous calculation,
non-magnetic state. The two partner Weyl points annihilate, opening a gap of
$12$ meV. (g) Schematic of the ferromagnetic Weyl loop (non-SOC) and Weyl
point (SOC) configuration. (h) Schematic phase diagram: spin-orbit gapped
ferromagnetic Weyl loops collapse to a paramagnetic Dirac loop across $T_{\rm
C}$. Concurrently, the exchange-split Weyl points annihilate.