# Reversible Non-Volatile Electronic Switching in a Near Room Temperature van
der Waals Ferromagnet

Han Wu  Department of Physics and Astronomy and Rice Center for Quantum
Materials, Rice University, Houston, TX, 77005 USA  Lei Chen  Department of
Physics and Astronomy and Rice Center for Quantum Materials, Rice University,
Houston, TX, 77005 USA  Paul Malinowski  Department of Physics, University of
Washington, Seattle, Washington 98195, USA  Jianwei Huang  Department of
Physics and Astronomy and Rice Center for Quantum Materials, Rice University,
Houston, TX, 77005 USA  Qinwen Deng  Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA 19104, USA  Kirsty Scott
Department of Physics, Yale University, New Haven, Connecticut 06520, USA
Energy Sciences Institute, Yale University, West Haven, Connecticut 06516, USA
Bo Gyu Jang  Theoretical Division and Center for Integrated Nanotechnologies,
Los Alamos National Laboratory, Los Alamos, NM, USA  Jacob P. C. Ruff  Cornell
High Energy Synchrotron Source, Cornell University, Ithaca, NY 14853, USA  Yu
He  Department of Applied Physics, Yale University, New Haven, Connecticut
06511, USA  Xiang Chen  Department of Physics, University of California,
Berkeley, Berkeley, California 94720, USA  Chaowei Hu  Department of Physics,
University of Washington, Seattle, Washington 98195, USA  Department of
Materials Science and Engineering, University of Washington, Seattle,
Washington 98195, USA  Ziqin Yue  Department of Physics and Astronomy and Rice
Center for Quantum Materials, Rice University, Houston, TX, 77005 USA  Ji Seop
Oh  Department of Physics, University of California, Berkeley, Berkeley,
California 94720, USA  Department of Physics and Astronomy and Rice Center for
Quantum Materials, Rice University, Houston, TX, 77005 USA  Xiaokun Teng
Department of Physics and Astronomy and Rice Center for Quantum Materials,
Rice University, Houston, TX, 77005 USA  Yucheng Guo  Department of Physics
and Astronomy and Rice Center for Quantum Materials, Rice University, Houston,
TX, 77005 USA  Mason Klemm  Department of Physics and Astronomy and Rice
Center for Quantum Materials, Rice University, Houston, TX, 77005 USA  Chuqiao
Shi  Department of Materials Science and NanoEngineering, Rice University,
Houston, TX, 77005, USA  Yue Shi  Department of Physics, University of
Washington, Seattle, Washington 98195, USA  Chandan Setty  Department of
Physics and Astronomy and Rice Center for Quantum Materials, Rice University,
Houston, TX, 77005 USA  Tyler Werner  Department of Applied Physics, Yale
University, New Haven, Connecticut 06511, USA  Makoto Hashimoto  Stanford
Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo
Park, California 94025, USA  Donghui Lu  Stanford Synchrotron Radiation
Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California
94025, USA  T. Yilmaz  National Synchrotron Light Source II, Brookhaven
National Lab, Upton, New York 11973, USA  Elio Vescovo  National Synchrotron
Light Source II, Brookhaven National Lab, Upton, New York 11973, USA  Sung-
Kwan Mo  Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Alexei Fedorov  Advanced Light Source, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Jonathan Denlinger
Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA  Yaofeng Xie  Department of Physics and Astronomy and Rice Center
for Quantum Materials, Rice University, Houston, TX, 77005 USA  Bin Gao
Department of Physics and Astronomy and Rice Center for Quantum Materials,
Rice University, Houston, TX, 77005 USA  Junichiro Kono  Department of
Materials Science and NanoEngineering, Rice University, Houston, TX, 77005,
USA  Departments of Electrical and Computer Engineering, Rice University,
Houston, TX, 77005, USA  Department of Physics and Astronomy and Rice Center
for Quantum Materials, Rice University, Houston, TX, 77005 USA  Pengcheng Dai
Department of Physics and Astronomy and Rice Center for Quantum Materials,
Rice University, Houston, TX, 77005 USA  Yimo Han  Department of Materials
Science and NanoEngineering, Rice University, Houston, TX, 77005, USA
Xiaodong Xu  Department of Physics, University of Washington, Seattle,
Washington 98195, USA  Department of Materials Science and Engineering,
University of Washington, Seattle, Washington 98195, USA  Robert J. Birgeneau
Department of Physics, University of California, Berkeley, Berkeley,
California 94720, USA  Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Department of Materials Science
and Engineering, University of California, Berkeley, USA  Jian-Xin Zhu
Theoretical Division and Center for Integrated Nanotechnologies, Los Alamos
National Laboratory, Los Alamos, NM, USA  Eduardo H. da Silva Neto  Department
of Physics, Yale University, New Haven, Connecticut 06520, USA Energy Sciences
Institute, Yale University, West Haven, Connecticut 06516, USA Liang Wu
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia,
PA 19104, USA  Jiun-Haw Chu  Department of Physics, University of Washington,
Seattle, Washington 98195, USA  Qimiao Si  Department of Physics and Astronomy
and Rice Center for Quantum Materials, Rice University, Houston, TX, 77005 USA
Ming Yi<EMAIL_ADDRESS>Department of Physics and Astronomy and Rice Center
for Quantum Materials, Rice University, Houston, TX, 77005 USA

##  I Abstract

The ability to reversibly toggle between two distinct states in a non-volatile
method is important for information storage applications. Such devices have
been realized for phase-change materials, which utilizes local heating methods
to toggle between a crystalline and an amorphous state with distinct
electrical properties Wuttig2007 ; Wang2021_phase ; Wuttig2017 ; Kolobov2004 ;
Poccia2011 . To expand such kind of switching between two topologically
distinct phases requires non-volatile switching between two crystalline phases
with distinct symmetries. Here we report the observation of reversible and
non-volatile switching between two stable and closely-related crystal
structures with remarkably distinct electronic structures in the near room
temperature van der Waals ferromagnet Fe5-δGeTe2. From a combination of
characterization techniques we show that the switching is enabled by the
ordering and disordering of an Fe site vacancy that results in distinct
crystalline symmetries of the two phases that can be controlled by a thermal
annealing and quenching method. Furthermore, from symmetry analysis as well as
first principle calculations, we provide understanding of the key distinction
in the observed electronic structures of the two phases: topological nodal
lines compatible with the preserved global inversion symmetry in the site-
disordered phase, and flat bands resulting from quantum destructive
interference on a bipartite crystaline lattice formed by the presence of the
site order as well as the lifting of the topological degeneracy due to the
broken inversion symmetry in the site-ordered phase. Our work not only reveals
a rich variety of quantum phases emergent in the metallic van der Waals
ferromagnets due to the presence of site ordering, but also demonstrates the
potential of these highly tunable two-dimensional magnets for memory and
spintronics applications.

##  II Main

Materials that can toggle between two states with distinct properties are
important for information storage technology. Phase-change materials, for
example, have been widely used for rewriteable optical data storage Wuttig2007
; Wang2021_phase ; Wuttig2017 ; Jiang2018 ; Wang2017 ; Duan2021 ;
Zhang2022_switch ; Liu2023 ; Kolobov2004 ; Poccia2011 ; Fratini2010 ;
Seyler2015 ; Eric2023 ; Liu2023_2 ; Du2021 ; Chiu2016 ; Tang2109 ; Zhang2019 ;
Vergniory2019 . The key advantage is that the two phases are controlled by a
non-volatile process, which is realized via a transient laser pulse that
locally heats and changes the crystal structure, either resulting in a
crystalline state or a quenched amorphous state. 2D van der Waals (vdW)
materials is another class of material family whose properties are highly
tunable, such as by electrostatic doping, optical illumination, or strain
2d_Burch2018 ; 2d_Cheng2019 ; 2d_Gong2017 ; 2d_Huang2017 ; 2d_Mak2019 ;
Neto2009 ; Deng2020 ; Xu2020 ; Tokura2019 ; Bernevig2022 ; Chu2010 ; Yi2019 ;
Mutch2019 ; Ricco2018 ; Cenker2022 ; Li2022 ; ZHENG2022 ; Ideue2019 ;
Liang2017 ; Xi2013 ; Nikola2022 . They are valued not only for their versatile
tunability but also the low dimensionality that allows exotic properties to
arise due to quantum confinement. The advent of the concept of topology adds
the potential to realize switching devices that go beyond resistive or optical
readouts. As topology is often distinguished by crystalline symmetries,
switching between two topologically distinct states can be realized via tuning
knobs that change symmetries Du2021 , most often achieved with a structural
transition. However, such tuning knobs typically modulate temperature,
electrostatic doping, strain, field, or pressure, all difficult to achieve in
a non-volatile method.

Here in this work, we demonstrate non-volatile reversible switching of two
closely related crystal structural phases in the vdW ferromagnet Fe5GeTe2 via
an annealing and quenching procedure. Fe5GeTe2 belongs to a class of Fe-based
metallic vdW ferromagnets that exhibits relatively high Curie temperatures
($T_{\mathrm{C}}$ = 275 K to 330 K in the bulk limit) Li2020_F5 ;
Li2021_F5ARPES ; Ly2021_F5ordering ; Wu2021_F5arpes ; Gao2020_F5domainwall ;
May2019_F5ACSNANO ; May2019_F5PRM ; Huang2022_F5ARPES ; Zhang2020_F5 ;
Zhang2022_F5dopedskymion ; Junho2022_F345 ; Ribeiro2022_F5MBE ;
Chenxiang2022_F5doped . Different from other widely studied 2D ferromagnets
such as CrI3 and Cr2X2Te6 (X = Ge, Si) 2d_Huang2017 ; 2d_Gong2017 , Fe5GeTe2
is air-stable and metallic, hence has been considered a top candidate for
spintronics applications 2d_Huang2017 ; 2d_Gong2017 ; 2d_Cheng2019 ;
May2019_F5PRM ; May2019_F5ACSNANO . The two phases share similar overall
crystal structure but differ only in the ordering or disordering of an Fe
vacancy site occupation that results in distinct crystalline symmetries.
Second harmonic generation (SHG) measurements show the site-disordered phase
to exhibit global inversion symmetry while the site-ordered phase breaks
inversion symmetry, with intensity differing by a factor of 30. Remarkably,
the electronic structures in the two phases are qualitatively distinct, as
observed by angle-resolved photoemission spectroscopy (ARPES). From a
combination of symmetry analysis and first principle calculations, we also
provide a understanding of the key features of the observed electronic
structure. In the site-disordered phase, we observe topological nodal lines
that are compatible with the preserved global inversion symmetry, while in the
site-ordered phase, we observe the lifting of the topological degeneracy due
to the broken inversion symmetry as well as flat bands that are compatible
with the quantum destructive interference of a bipartite crystalline lattice
formed by the site-order. Our work not only demonstrates the exciting
potential of using site order in the Fe-based 2D materials as a novel tuning
knob to engineer and control correlated topological phases, but also reveals
the potential of this class of 2D materials as a novel type of phase-change
materials for non-volatile spintronics, memory or non-linear optical
applications.

###  II.1 Reversible switching of two distinct electronic structures

Fe5GeTe2 belongs to a larger class of Fe-based metallic ferromagnets, FenGeTe2
(n=3 to 5) Deng2018 ; Junho2022_F345 ; May2019_F5ACSNANO ; May2019_F5PRM ;
wu2023spectral , and is known to have a unique partially occupied split site
May2019_F5PRM ; May2019_F5ACSNANO . The crystal lattice of Fe5GeTe2 is
rhombohedral (space group $R\overline{3}m$, No. 166) May2019_F5PRM ;
May2019_F5ACSNANO . The crystal structure consists of an ABC-stacking of the
vdW slabs (Fig. 1a). Each slab consists of Fe and Ge sites sandwiched between
layers of Te. In addition, each slab consists of three distinct Fe sites,
marked as Fe(1), Fe(2), and Fe(3) in Fig. 1a. While Fe(2) and Fe(3) sites are
fully occupied, Fe(1) sites are known to be split-sites where for each up-down
pair within a single slab, they are either occupied in the up or down site
May2019_F5PRM ; May2019_F5ACSNANO ; Ly2021_F5ordering . This choice of either
the up or down site for Fe(1) pushes the Ge sites to also occupy a split site,
where the site farther away from the occupied Fe(1) site is preferred. The
choice of either occupying the up or down Fe(1) sites can be uncorrelated
spatially or form an order depending on the rate at which the crystals are
formed from growth May2019_F5ACSNANO ; May2019_F5PRM . In particular, the
occupancy of the Fe(1) sites could form an up-down-down (UDD) or down-up-up
(DUU) pattern, resulting in a $\sqrt{3}\times\sqrt{3}$ superstructure
Ly2021_F5ordering . This ordered occupancy is favored when the crystals are
quenched from above a structural transition identified by previous literature
as $T_{\mathrm{HT}}$ = 550 K while the random distribution is favored with
slow cooling May2019_F5PRM ; May2019_F5ACSNANO . For simplicity, we refer to
the uncorrelated phase the site-disordered phase and the ordered phase the
site-ordered phase. The ordering of the Fe(1) sites plays a crucial role in
modifying the global symmetry of the crystal. In the site-disordered phase,
the global inversion symmetry is preserved. This can be seen in Fig. 1a, where
the inversion centers of each vdW slab is between the Ge split sites. In the
site-ordered phase, the inversion symmetry is broken by the Fe(1) sites
Ly2021_F5ordering . Such symmetry breaking has profound impact on the
electronic structure, and as we will demonstrate, is the key to the
tunability.

To probe such an effect, we carried out ARPES measurements on crystals that
were prepared in the two thermal methods. The measured Fermi surface (FS) of
the slow-cooled crystals (Fig. 1g) and the quenched crystals (Fig. 1h) under
the same measurement conditions are drastically different. In particular, the
quenched crystals exhibit small pockets at the K points of the BZ, which are
absent in the slow-cooled crystals. Instead, the slow-cooled crystals exhibit
additional large pockets centered at the $\Gamma$ point. As we will show in
detail in each of the two subsequent sections, the band dispersions leading to
these FSs are significantly different, belonging to distinct topologically
non-trivial phases. Before we discuss the electronic structure in depth, we
first demonstrate the reversible non-volatile switching of these two phases.
To confirm that it is the last thermal cooling step that dictates the
electronic phase, we performed the following test (see Fig. S8 in the SI).
First, we prepared the crystals by quenching them from above $T_{\mathrm{HT}}$
down to room temperature. Then we cut a crystal into halves and annealed a
half piece to the metastable phase above $T_{\mathrm{HT}}$ and slowly cooled
it back down to room temperature while leaving the other half untreated (Fig.
1b). The half pieces are then measured by ARPES. The electronic structure of
the two halves are observed to be distinct, with the original quenched half
identical to that shown in Fig. 1h, while the annealed and slow-cooled half
identical to that presented in Fig. 1g. We have also checked the reverse
process, which is to start with a crystal that was first formed via slow-
cooling to room temperature, cut it in half, and annealing one half to above
$T_{\mathrm{HT}}$ and then quenched in water (Fig. 1c). The subsequent ARPES
measurement on the two halves again show the contrasting electronic
structures, with the re-quenched half showing electronic structure identical
to that in Fig. 1h and the original slow-cooled half identical to Fig. 1g.
This procedure demonstrates that the key for the distinct electronic
structures is the cooling rate in the final thermal treatment from above
$T_{\mathrm{HT}}$. Hence we have demonstrated that there are two stable phases
with drastically distinct electronic structures that can be reversibly
switched in a non-volatile method.

###  II.2 Fe(1) site ordering as origin for distinct electronic phases

As reported, there are three types of possible variations of the Fe5-δGeTe2
single crystals: Fe deficiency ($\delta$), stacking faults, and the formation
of the $\sqrt{3}\times\sqrt{3}$ Fe(1) site order Ly2021_F5ordering ;
May2019_F5PRM ; May2019_F5ACSNANO . Since each pair of half crystals used
above originate from the same original piece, the above procedure also rules
out any difference in Fe deficiency as a potential cause for the difference in
the electronic structure. Furthermore, we can also rule out the vdW stacking
faults as a possible cause of the distinct electronic structure. From our
transmission electron microscopy (TEM) images on the two types of crystals
(see Fig. S1 in the SI), we do not observe any regular appearance of stacking
faults in either the slow-cooled or quenched crystals. Both crystals exhibit
ABC stacking, with occasional stacking faults between the vdW layers. Such
rare occurrence cannot constitute a qualitative electronic structure
distinction between the two types of crystals.

Therefore we are left with the appearance of the $\sqrt{3}\times\sqrt{3}$
Fe(1) site order as the likely cause of the dichotomy of the electronic
structure. First from single crystal x-ray diffraction (XRD) measurements,
while the diffraction peaks corresponding to the $\sqrt{3}\times\sqrt{3}$
order are observed in the two types of crystals, their intensity relative to
the Bragg peaks is reduced in the slow-cooled samples compared to those
measured on a quenched crystal (see Extended Data Fig. S3). This suggests that
while both site-disordered and site-ordered regions exist in the slow-cooled
crystals, the population of the site-ordered regions is smaller. To further
confirm this, we carried out STM measurements on both quenched and slow-cooled
crystals, revealing regions with a $\sqrt{3}\times\sqrt{3}$ superlattice with
both UDD and DUU ordering of Fe(1) occupation sites (Fig. 1e-f), consistent
with previous STM reports on Fe5-δGeTe2 Ly2021_F5ordering . While the field of
view (on the order of 1 $\mu$m2) of our STM measurements on quenched crystals
showed only $\sqrt{3}\times\sqrt{3}$ ordered regions, similar measurements on
slow-cooled crystals also showed disordered regions without the
$\sqrt{3}\times\sqrt{3}$ superlattices. Interestingly, in slow-cooled
crystals, these disordered regions dominate the field-of-view, surrounding
small domains of $\sqrt{3}\times\sqrt{3}$ superlattices (see SI Extended Data
Fig. S2), consistent with the XRD results. The existence of regions with
$\sqrt{3}\times\sqrt{3}$ order in the two types of crystals revealed by STM is
further confirmed by SHG measurement. We carried out polarization-dependent
SHG measurements at 5 K on the two types of crystals. The quenched crystals
reveal a 30 times stronger SHG signal compared to that of the slow-cooled
crystals (Fig. 1i,j), note that to observe the tiny SHG signal (less than 10.
c.p.s.) in the slow-cooled sample, a incident power of 4 mW with a 50 X
objective is needed, which is just below the damage threshold, requiring a
photon counter. As the SHG signal is contributed by the electric dipole (ED),
$I_{i}^{\text{ED}}(2\omega)\propto\left|{\Sigma}_{jk}\chi_{ijk}^{\text{ED}}E_{j}(\omega)E_{k}(\omega)\right|^{2}$,
where $k$ and $\omega$ are the wavevector and frequency of incident beam
respectively, $\chi$ is the nonlinear susceptibility tensor and $i,\ j,k,l$
are Cartesian coordinate indices, it is a sensitive probe of the presence of
inversion-symmetry-breaking. On one hand, for the quenched crystals, the clear
presence of inversion symmetry breaking is consistent with the formation of
the $\sqrt{3}\times\sqrt{3}$ order. On the other hand, in the slow-cooled
crystals dominated by regions with random Fe(1) site occupancy, the electric
dipole contribution to SHG would be forbidden due to the preserved global
inversion symmetry while only a smaller electric quadrupole (EQ) SHG
contribution following the three-fold rotational symmetry would be allowed,
$I_{i}^{\text{EQ}}(2\omega)\propto\left|{\Sigma}_{jkl}\chi_{ijkl}^{\text{EQ}}k_{j}E_{k}(\omega)E_{l}(\omega)\right|^{2}$
under normal incidence. The much enhanced SHG signal in the quenched crystals
is consistent with both the STM and XRD observations. Hence, we associate the
electronic structure measured on the quenched crystals to that of the Fe(1)
site-ordered phase and that measured on the slow-cooled crystals to that of
the Fe(1) site-disordered phase. As we will demonstrate subsequently in the
discussion section, the crystal symmetries for the site-disordered and site-
ordered phases are highly compatible with the topological band dispersions
that we observe from ARPES.

###  II.3 Nodal lines in the site-disordered phase

Next, we present in detail the key features in the measured electronic
structure of the site-disordered phase achieved from slow-cooling the
crystals. The FS in the ferromagnetically ordered state is shown in Fig. 2b,
consisting of several circular Fermi pockets centered at the BZ center and
elliptical pockets surrounding the $\bar{K}$\- $\bar{M}$\- $\bar{K^{\prime}}$
BZ boundaries, as highlighted by white dashed lines. To understand these
features, we measured the electronic dispersions along the high symmetry
direction of $\bar{M}-\bar{K}-\bar{\Gamma}-\bar{K}$ (Fig. 2e-f). First, we
observe a number of hole bands centered at the $\bar{\Gamma}$ point, giving
rise to the observed circular Fermi pockets. Interestingly, near the $\bar{K}$
point, we also observe a band crossing near -0.18 eV. The crossing can be
better visualized from the energy distribution curves (EDC) as well as second
energy derivatives of the raw spectra (Fig. 2g). The nature of the crossing
can be further demonstrated from a series of cuts of the measured band
dispersions close to the $\bar{K}$ point. In Fig. 2d, both horizontal cuts
(cuts 1 to 5) and vertical cuts (cuts 6 to 10) in the crossing region reveal
two bands that cross near the $\bar{K}$ point and become gapped away from
$\bar{K}$.

Having demonstrated the band crossing at the $\bar{K}$ points in the in-plane
direction, we also examine the dispersion along the out-of-plane direction
(kz) by varying the photon energy. As the inter-layer interactions in vdW
materials are quite weak, we do not observe strong variation along kz (see SI
Extended Data Fig. S6). For a range of photon energies that probes a range
much beyond that of a single BZ along kz, we always observe the crossing near
the $\bar{K}$ point (Fig. 2h), hence the in-plane nodal crossing takes the
form of nodal lines along the out-of-plane direction. Taking these findings
together, our ARPES data reveal the existence of nodal lines along the BZ
boundaries. As these nodal lines are observed in a ferromagnetic phase, the
time-reversal symmetry is broken and hence the spin degree of freedom is
quenched, giving rise to two-fold degenerate lines.

###  II.4 Flat bands in the site-ordered phase

Having presented the existence of the nodal lines in the site-disordered
phase, we now focus on the observed electronic structure of the site-ordered
phase. Figure 3 summarizes the measured electronic structure of quenched
crystals. Instead of the elliptical Fermi pockets surrounding the
$\bar{K}-\bar{M}$ direction resulting from the Dirac crossing at $\bar{K}$
points in the slow-cooled crystals, the quenched crystals exhibit circular
pockets at the $\bar{K}$ points. This distinction can be further seen from
dispersions measured along the high symmetry direction
$\bar{M}-\bar{K}-\bar{\Gamma}-\bar{K}$. In stark contrast to that measured for
the site-disordered phase (Fig. 2), the site-ordered crystals show electron
bands at the $\bar{K}$ points with clear band bottoms and no band crossings,
and hence the absence of the nodal lines observed in the site-disordered
crystals.

More interestingly, three flat bands are observed in the site-ordered crystals
that are not observed in the site-disordered crystals. We first illustrate
them along the $\bar{K}-\bar{\Gamma}-\bar{K}-\bar{M}$ direction, captured in
measurements under both linear horizontal (LH) and linear vertical (LV)
polarizations (Fig. 3a-b). The location of the flat bands can be identified as
peaks in the integrated EDCs from both polarizations, at $E_{F}$, -0.2 eV, and
-0.6 eV. Beyond the high symmetry direction, the flat bands are observed to
persist across a large region of the BZ. We illustrate this from five cuts
measured across the in-plane BZ (Fig. 3d-e). The flat band near $E_{F}$ could
be clearly seen along the $\bar{K}-\bar{M}-\bar{K}$ direction as shown on cut
1. When the $\bar{\Gamma}$ point is approached from cut2 to cut5, the flat
band near $E_{F}$ shifts to above $E_{F}$ and could no longer be observed. The
second and third flat bands located at -0.2 eV and -0.6 eV are flat throughout
the BZ except where they hybridize with the dispersive bands near the
$\bar{\Gamma}$ point. We note that this hybridization indicates that these
flat dispersions are intrinsic to the crystal and cannot be due to disorders
or impurities that would otherwise form momentum-independent states that do
not interact with intrinsic band structure. Furthermore, we carried out photon
energy-dependent measurements, where the flat bands are observed to persist
across $k_{z}$ (see SI), consistent with the 2D nature of the vdW materials.

###  II.5 Topology for the distinct electronic phases

The drastically distinct electronic structures of the two types of crystals,
with one exhibiting two-fold nodal lines and the other flat dispersions,
belong to distinct topological states. Here we show that they can be
understood from the symmetries dictated by the site-disordered or site-ordered
phases, respectively. We first discuss the case of the site-disordered phase
where we observe nodal lines at the K points. For a single vdW slab with 50%
occupation of the Fe(1) sites, the crystalline symmetry belongs to the
centrosymmetric space group $P\bar{3}1m$ (No. 164). Here, the crystal has both
two-fold rotational symmetry about the $y$ axis ($C_{2y}$) (Fig. 4b) and
three-fold rotational symmetry about the $z$ axis ($C_{3z}$) (Fig. 4c),
similar to the case of graphene. The momentum point $K$ ($K^{\prime}$) is
invariant under these two symmetry operations and allows the existence of a 2D
irreducible representation. In the ferromagnetic phase where time-reversal
symmetry is broken, the spin-polarized bands in the ferromagnetic state can be
regarded as spinless states and would cross at the K (K’) points, where the
two-fold degeneracy comes from the orbital degree of freedom (see SI for a
discussion of the orbitals), leading to a symmetry-enforced ferromagnetic
Dirac crossing. To demonstrate this, we built an effective tight binding model
considering the different Fe 3d orbitals and show that such a crossing is
indeed protected at the K (K’) point (see SI for a full discussion of the
tight-binding model). When we incorporate spin-orbit coupling (SOC), (see SI),
in general, the SOC can be expressed as
$H_{SO}=\lambda_{SO}{\bm{L}}\cdot{\bm{S}}$, where $\bm{L}$ and $\bm{S}$ are
the angular and spin momenta, respectively. In a ferromagnetic system,
${\bm{S}}\sim\langle{\bm{S}}\rangle$ plays the role of an effective Zeeman
splitting field in the orbital basis. Since the direction of the magnetic
moment is along the $z$-direction May2019_F5PRM ; May2019_F5ACSNANO , which is
parallel to the direction of the orbital angular momentum, the SOC would lift
the two-fold degeneracy at K and K’. The band structure calculation from the
tight binding model with SOC is illustrated in Fig. 4d. This is consistent
with our experimental observation of the crossing at K points except that the
gap due to SOC is not resolved in the experiment due to the energy resolution.
The appearance of this degeneracy at K points is similar to that reported in
the related Fe3GeTe2, where the topological nodal lines are theoretically
identified to give rise to a large anomalous Hall effect Kim2018 , but
difficult to resolve in the ARPES measured dispersions. Here in Fe5GeTe2, they
are clearly observed.

Having understood the single layer case, we now consider the bulk system of
the site-disordered phase. In a simple hypothetical AAA stacking scenario, the
hopping along the z direction extends the original 2D hexagonal BZ into a 3D
hexagonal prism and would extend the topological crossings at K and K’ to
nodal lines along the K-H direction. This is protected by a combination of
$C_{3z}$ and $PT$ symmetries. For the real ABC stacking of the layers, the BZ
changes from a hexagonal prism into the BZ of a rhombohedral space group (Fig.
4e). Due to the ABC stacking of the layers, the K-H direction is no longer a
high symmetry line of the BZ. Instead, the topological crossings at each
$k_{z}$ plane shift away from the K and K’ points, forming helical nodal lines
that wind around K-H, where the magnitude of the shift is proportional to the
strength of the interlayer hopping, similar to helical nodal lines reported in
other ABC-stacked materials including Fe3Sn2 Heikkila2011 ; Ye2018 ; Fang2022
. Here in Fe5GeTe2, due to the weak vdW interlayer coupling, the in-plane
deviation of the crossing from the K-H line is too small to be experimentally
resolved. Hence we cannot directly observe the winding but only observe nodal
lines near K-H.

In addition to tight-binding calculations, we also carried out density
functional theory (DFT) calculations to check for the symmetry-enforced
crossings (see SI Extended Data Fig. S4a-c). To demonstrate the importance of
the globally preserved inversion symmetry of the random Fe(1) occupation, we
carried out the following comparison. First, we calculated the band structure
for the Fe(1) sites all occupying the up sites (UUU). The inverted case is the
structure with Fe(1) sites all occupying the down sites (DDD). The average of
the two from directly overlapping the UUU and DDD band structures would give
an average stoichiometry of Fe5GeTe2. We note that while such structure does
not exist in the crystal, it mimics the site-disordered phase except it lacks
inversion symmetry. To directly compare this calculation with an inversion
symmetric structure, we also calculated the band structure of a crystal
structure with both up and down sites fully occupied, giving a stoichiometry
of Fe6GeTe2. By comparing calculations without and with SOC, only the
inversion symmetric Fe6GeTe2 shows band crossings at the K point that open up
a gap with the inclusion of SOC, demonstrating the symmetry-enforced nature of
the topological nodal lines. The UUU and DDD band structures do not exhibit
such kind of band crossing, confirming that the presence of global inversion
symmetry is consistent and also required for the observed topological nodal
lines.

Finally, we discuss the site-ordered phase. Consistent with the inversion
symmetry breaking observed by SHG, we no longer observe the topological
crossings at the K point. Such inversion symmetry breaking is consistent with
the $\sqrt{3}\times\sqrt{3}$ order caused by the Fe(1) site ordering.
Interestingly, for such DUU occupation order of the Fe(1) site (Fig. 4f), the
shortest bond occurs between the Fe(1) sublayer and the adjacent Fe(3)
sublayer May2019_F5PRM ; May2019_F5ACSNANO . The in-plane projection of these
two sublayers form a clover unit pattern, with the center being the missing
Fe(1) site, as shown in Fig. 4g-h. Considering only the nearest neighbor
hopping, $t_{1}$, which is between the red Fe(1) sites and yellow Fe(3) sites,
the lattice is manifested as a bipartite crystalline lattice (BCL), in which
the lattice is categorized into two sublattices with different numbers of
atoms (Fig. 4h). BCLs are predicted to be a generic platform to realize
destructive interference of the electronic wavefunction and further lead to
flat bands Regnault2022 ; Calugaru2022 , but have never been directly observed
in bulk materials.

To see this clearly, we consider the Hamiltonian for the single orbital clover
lattice with nearest neighbor hopping

$H(\bm{k})=\begin{pmatrix}\mathbf{0_{2\times 2}}&\mathcal{H}_{\bm{k}}\\\
\mathcal{H}_{\bm{k}}&\mathbf{0_{3\times 3}}\end{pmatrix},$ (1)

where $\mathcal{H}_{\bm{k}}$ is the hopping matrix between two sublattices
(see explicit form in Eq. LABEL:eq:fb_ham in methods). Given that
$\mathcal{H}_{\bm{k}}$ is a $3\times 2$ rectangular matrix, the Hamiltonian
contains at least $3-2=1$ zero modes for all $\bm{k}$. The band structure is
shown in Fig. 4j, after diagonalizing the Hamiltonian. The correspinding
localized wavefunction for the flat band is:

$\psi(k_{x},k_{y})=\begin{pmatrix}0&0&e^{\frac{k_{y}}{3}}-e^{-\frac{2}{3}k_{y}}&e^{-\frac{k_{x}}{2\sqrt{3}}-\frac{k_{y}}{6}}-e^{\frac{k_{x}}{\sqrt{3}}+\frac{k_{y}}{3}}&e^{\frac{k_{x}}{2\sqrt{3}}-\frac{k_{y}}{6}}-e^{-\frac{k_{x}}{\sqrt{3}}+\frac{k_{y}}{3}}\end{pmatrix},$
(2)

leading to a real space Wannier function as shown in Fig. 4h. The Wannier
amplitude is identically zero on the red sites because of the destructive
interference effect. When SOC is incorporated, the flat band gains dispersion
and also acquires a finite Chern number, and becomes topologically non-
trivial. The consideration for different orbital groups is also provided in
the methods. Such kind of destructive-interference induced flat bands have
been discussed in kagome Regnault2022 ; Calugaru2022 ; Yin2018 ; Kang2020 ;
Kang2020_2  and pyrochlore Regnault2022 ; Calugaru2022  lattices. Here in
quenched Fe5GeTe2, they are directly the result of the geometrically
frustrated lattice formed by the Fe(1) occupation site ordering enabled by the
quenching process.

We carried out DFT calculations to check for the BCL flat bands associated
with the clover lattice (see SI Extended Data Fig. S4d). To mimic the site-
disordered phase, we overlap the band structures for the UUU and DDD
structures and compare it to the band structure calculated with the site-
ordered phase with the $\sqrt{3}\times\sqrt{3}$ order. A direct comparison of
the two shows that no flat bands are observed in the UUU+DDD calculation but
flat bands are observed for the site-ordered phase. We can also unfold the
band structure of the $\sqrt{3}\times\sqrt{3}$ order back to the original
unfolded BZ to compare more directly with the observed dispersions, and find
reasonable agreement (see SI Extended Data Fig. S5). Projection of the density
of states unto the different Fe sites also show that the peaks corresponding
to the flat bands have large contributions from the Fe(1) and Fe(3) sites that
form the clover lattice. This demonstrates that the flat dispersions that we
observe only in the site-ordered phase and not in the site-disordered phase
are associated with the clover unit that only forms with the Fe(1) site order,
and is a direct result of the quantum destructive interference of the
bipartite crystalline lattice.

###  II.6 Discussion

Taking all experimental and theoretical evidence presented together, we have
demonstrated the reversible switching of two remarkably distinct electronic
structures ascribed to two closely-related crystalline phases via a non-
volatile thermal process in Fe5GeTe2. The capability is enabled by the Fe(1)
site ordering that changes the crystal symmetries leading to distinct
topological characteristics. On one hand, the random occupation of the Fe(1)
sites leads to global inversion symmetry that allows symmetry-enforced
topological nodal lines, which are observed to be lifted when the inversion-
symmetry breaking order forms. On the other hand, the formation of the site
order creates a bipartite crystalline lattice that localizes electronic states
to form flat bands, which are observed to be destroyed with the breaking of
the site order. Our findings indicate that the Fe5GeTe2 system is a rich
system for probing and understanding topology in the correlated regime. As
Fe5GeTe2 is known to exhibit high Curie temperature, it would be interesting
to compare the magnetic properties of the two phases, including manipulation
of the topological nodal lines in the site-disordered phase and the role of
the topological flat bands for magnetism in the site-ordered phase. Fe5GeTe2
is also an interesting system to probe from the order-disorder perspective. As
our STM results show that the slow-cooled samples exhibit domains of ordered
regions, it would be interesting to understand how the domains form and
propagate in the cooling process as a function of cooling rate, especially
given that Fe5GeTe2 behaves counterintuitively in that the ordered phase is
preferred via quenching. Such studies would benefit from the vast expertise
developed for probing and understanding order-disorder formation in other
quantum materials Wuttig2007 ; Wang2021_phase ; Wuttig2017 ; Jiang2018 ;
Wang2017 ; Duan2021 ; Zhang2022_switch ; Liu2023 ; Kolobov2004 ; Poccia2011 ;
Fratini2010 ; Seyler2015 ; Eric2023 ; Liu2023_2 . Finally, the non-volatile
switch our work exemplifies promises versatile settings to apply a novel
design principle, viz to utilize the cooperation of crystalline symmetry and
strong correlations to produce new correlated topological materials
ChenLei2022 .

Aside from fundamental physics, our work also indicates that Fe5GeTe2 has
great potential for applications. Skyrmions have recently been reported in
this class of Fe-based vdW ferromagnets F3_sky_Birch2022 ; F3_sky_Ding2020 ;
F3_sky_Yang2020 ; F5_sky_Brian2023 ; F5_sky_Fujita2022 ;
Schmitt2022_F5skyrmion . As skyrmions are stablized by Dzyaloshinsky–Moriya
interaction, which is only allowed when inversion symmetry is broken, there
has been debates on how to understand the appearance of skyrmions in these
seemingly centrosymmetric crystals. In the case of Fe3GeTe2, this has been
explained via random Fe deficiencies that on average occur asymmetrically in
the crystal Anirban2022 . In the case of (Fe,Co)5GeTe2, this is ascribed to
AA’ stacking of the vdW layers meisenheimer2022controlled . Here we show that
the two phases have clean distinction on inversion symmetry via the site-
ordering process, hence provides a platform to potentially control skyrmion
formation. The process by which we demonstrate the switching–heating and
cooling all above room temperature–is similar to that already commercially
used for phase-change materials such as Ge2Sb2Te5 Wuttig2007 ; Wuttig2017 ;
Kolobov2004 . Different from phase-change materials, we only need to surpass a
submelting temperature where the Fe(1) sites are mobilized instead of having
to achieve the melting and crystallization temperature. Techniques such as
local laser heating can be explored for spatial writing of the two phases
especially given that the overall crystal structures are compatible. This is
in contrast to some vacancy ordered materials such as KxFe2-ySe2 where the
metallic regions are structurally unstable and only appears as microstructures
amidst the insulating vacancy ordered phases Bao_2013 ; Ding2013 . The heating
and quenching process that we utilize is non-volatile and above room
temperature, which is advantageous compared from those controls that require
the presence of field, strain, pressure, or current. Nevertheless, modifying
the Fe(1) sites and their vacancies appears to have lower energy barrier than
re-crystallization, suggesting that electrical current, photo illumination or
other commonly utilized switching methodologies could also be explored for
this 2D vdW material.

Finally, the concept of using vacancy order-disorder to realize distinct
topological phases goes beyond Fe5GeTe2. Phase change via order-disorder has
been explored extensively for realizing switches based on electrical or
optical properties. Here we demonstrate the concept that vacancy order can be
utilized to change the crystalline symmetries of two otherwise energetically
similar ground states with dramatically distinct consequences on their
topological character. A large base of quantum materials are known to exhibit
vacancies or site disorder. The consideration of the symmetries of these
phases may open up new routes towards realizing exotic topological phases as
well as novel spintronics applications.

##  III Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  IV Acknowledgments

The authors acknowledge insightful discussions with Kai Sun, Luis Balicas, and
Alex Frano. This research used resources of the Advanced Light Source, the
Stanford Synchrotron Radiation Lightsource, and the National Synchrotron Light
Source-II, all U.S. Department Of Energy (DOE) Office of Science User
Facilities under contract Nos. DE-AC02-05CH11231, AC02-76SF00515 and No. DE-
SC0012704, respectively. Rice ARPES work is supported by the U.S. DOE grant
No. DE-SC0021421 and the Gordon and Betty Moore Foundation’s EPiQS Initiative
through grant no. GBMF9470. The theory work at Rice is primarily supported by
the U.S. DOE, BES, under Award No. DE-SC0018197 (L.C., symmetry analysis), by
the AFOSR under Grant No. FA9550-21-1-0356 (C.S., electronic structure
construction), and by the Robert A. Welch Foundation Grant No. C-1411 (Q.S.).
Work at Los Alamos was carried out under the auspices of the U.S. Department
of Energy (DOE) National Nuclear Security Administration (NNSA) under Contract
No. 89233218CNA000001, and was supported by LANL LDRD Program, UC Laboratory
Fees Research Program (Grant Number: FR-20-653926), and in part by the Center
for Integrated Nanotechnologies, a DOE BES user facility. The development of
the SHG photon counter is supported by the Army Research Office and was
accomplished under grant no. W911NF-19-1-0342. The sample exfoliation is based
upon work supported by the Air Force Office of Scientific Research under award
number FA9550-22-1-0449. Q.D. is supported by the NSF EPM program under grant
no. DMR-2213891. L.W. acknowledges the support by the Air Force Office of
Scientific Research under award no. FA9550-22-1-0410. TEM study is supported
by Welch Foundation (C-2065-20210327). The authors acknowledge the use of the
Electron Microscopy Center at Rice. The work at LBL and UC Berkeley was funded
by the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, Materials Sciences and Engineering Division under Contract No. DE-
AC02-05-CH11231 (Quantum Materials program KC2202). Research conducted at the
Center for High-Energy X-ray Sciences (CHEXS) is supported by the National
Science Foundation (BIO, ENG and MPS Directorates) under award DMR-1829070.
Materials synthesis at UW was supported as part of Programmable Quantum
Materials, an Energy Frontier Research Center funded by the U.S. Department of
Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DE-
SC0019443

##  V Author Contributions

The project was initiated and organized by M.Y. The single crystals were grown
by P.M., Y.S., X.C., Y.H, C.H, X.X and J.C. The ARPES measurements and
analyses were carried out by H.W., J.W.H., J.S.O., R.J.B. and M.Y. with the
help of D.H.L., M.H., S.-K.M., A.F., J.D., T.Y. and E.V. The tight binding
model and symmetry analyses were proposed and carried out by L.C., C.S. and
Q.S. The first principle calculations were carried out by B.G.J. and J.Z. The
SHG were carried out by Q.D. and L.W. The STM measurements were measured by
K.S. and E.dsN. The x-ray diffraction was done by J.R. The TEM were measured
by C.S. and Y.-M.H. The sample annealing and quenching process and
characterization were carried out by P.M., J.C., X.C., Y.X., B.G., X.T., M.K.,
H.W. and P.D. The manuscript was written by H.W. and M.Y. and contributed by
all the authors.

##  VI Competing Interests

The authors declare no competing interests.

## References

  * (1) Wuttig, M. & Yamada, N.  Phase-change materials for rewriteable data storage.  _Nat. Mater._ 6, 824–832 (2007).
  * (2) Wang, Y. _et al._ Electrical tuning of phase-change antennas and metasurfaces.  _Nat. Nanotechnol._ 16, 667–672 (2021).
  * (3) Wuttig, M., Bhaskaran, H. & Taubner, T.  Phase-change materials for non-volatile photonic applications.  _Nat. Photon._ 11, 465–476 (2017).
  * (4) Kolobov, A. V. _et al._ Understanding the phase-change mechanism of rewritable optical media.  _Nat. Mater._ 3, 703–708 (2004).
  * (5) Poccia, N. _et al._ Evolution and control of oxygen order in a cuprate superconductor.  _Nat. Mater._ 10, 733–736 (2011).
  * (6) Jiang, S., Shan, J. & Mak, K. F.  Electric-field switching of two-dimensional van der waals magnets.  _Nat. Mater._ 17, 406–410 (2018).
  * (7) Wang, Y. _et al._ Structural phase transition in monolayer MoTe2 driven by electrostatic doping.  _Nature_ 550, 487–491 (2017).
  * (8) Duan, S. _et al._ Optical manipulation of electronic dimensionality in a quantum material.  _Nature_ 595, 239–244 (2021).
  * (9) Zhang, P. _et al._ All-optical switching of magnetization in atomically thin CrI3.  _Nat. Mater._ 21, 1373–1378 (2022).
  * (10) Liu, G. _et al._ Electrical switching of ferro-rotational order in nanometre-thick 1T-TaS2 crystals.  _Nat. Nanotechnol._ (2023).
  * (11) Fratini, M. _et al._ Scale-free structural organization of oxygen interstitials in La2CuO4+y.  _Nature_ 466, 841–844 (2010).
  * (12) Seyler, K. L. _et al._ Electrical control of second-harmonic generation in a WSe2 monolayer transistor.  _Nat. Nanotechnol._ 10, 407–411 (2015).
  * (13) Anderson, E. _et al._ Programming correlated magnetic states with gate-controlled moire geometry.  _Science_ 0, eadg4268 (2023).
  * (14) Liu, D. _et al._ Lattice plainification advances highly effective snse crystalline thermoelectrics.  _Science_ 380, 841–846 (2023).
  * (15) Du, L. _et al._ Engineering symmetry breaking in 2d layered materials.  _Nat Rev Phys_ 3, 193–206 (2021).
  * (16) Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S.  Classification of topological quantum matter with symmetries.  _Rev. Mod. Phys._ 88, 035005 (2016).
  * (17) Tang, F., Po, H. C., Vishwanath, A. & Wan, X.  Comprehensive search for topological materials using symmetry indicators.  _Nature_ 566, 486–489 (2019).
  * (18) Zhang, T. _et al._ Catalogue of topological electronic materials.  _Nature_ 566, 475–479 (2019).
  * (19) Vergniory, M. G. _et al._ A complete catalogue of high-quality topological materials.  _Nature_ 566, 480–485 (2019).
  * (20) Burch, K. S., Mandrus, D. & Park, J.-G.  Magnetism in two-dimensional van der Waals materials.  _Nature_ 563, 47–52 (2018).
  * (21) Cheng, G. & Xiang, Z.  Two-dimensional magnetic crystals and emergent heterostructure devices.  _Science_ 363, eaav4450 (2019).
  * (22) Gong, C. _et al._ Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals.  _Nature_ 546, 265–269 (2017).
  * (23) Huang, B. _et al._ Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit.  _Nature_ 546, 270–273 (2017).
  * (24) Mak, K. F., Shan, J. & Ralph, D. C.  Probing and controlling magnetic states in 2d layered magnetic materials.  _Nat Rev Phys_ 1, 646–661 (2019).
  * (25) Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K.  The electronic properties of graphene.  _Rev. Mod. Phys._ 81, 109–162 (2009).
  * (26) Deng, Y. _et al._ Quantum anomalous hall effect in intrinsic magnetic topological insulator MnBi2Te4.  _Science_ 367, 895–900 (2020).
  * (27) Xu, Y. _et al._ High-throughput calculations of magnetic topological materials.  _Nature_ 586, 702–707 (2020).
  * (28) Tokura, Y., Yasuda, K. & Tsukazaki, A.  Magnetic topological insulators.  _Nat Rev Phys_ 1, 126–143 (2019).
  * (29) Bernevig, B. A., Felser, C. & Beidenkopf, H.  Progress and prospects in magnetic topological materials.  _Nature_ 603, 41–51 (2022).
  * (30) Chu, J.-H. _et al._ In-plane resistivity anisotropy in an underdoped iron arsenide superconductor.  _Science_ 329, 824–826 (2010).
  * (31) Yi, M. _et al._ Nematic energy scale and the missing electron pocket in FeSe.  _Phys. Rev. X_ 9, 041049 (2019).
  * (32) Mutch, J. _et al._ Evidence for a strain-tuned topological phase transition in ZrTe5.  _Sci. Adv._ 5, eaav9771 (2019).
  * (33) Riccò, S. _et al._ In situ strain tuning of the metal-insulator-transition of Ca2RuO4 in angle-resolved photoemission experiments.  _Nat. Commun._ 9, 4535 (2018).
  * (34) Cenker, J. _et al._ Reversible strain-induced magnetic phase transition in a van der Waals magnet.  _Nat. Nanotechnol._ 17, 256–261 (2022).
  * (35) Li, Y.-S. _et al._ Elastocaloric determination of the phase diagram of Sr2RuO4.  _Nature_ 607, 276–280 (2022).
  * (36) Zheng, L. _et al._ Emergent charge order and unconventional superconductivity in pressurized kagome superconductor CsV3Sb5.  _arXiv preprint arXiv:2209.07340_ (2022).
  * (37) Ideue, T. _et al._ Pressure-induced topological phase transition in noncentrosymmetric elemental tellurium.  _Proc. Natl. Acad. Sci. USA_ 116, 25530–25534 (2019).
  * (38) Liang, T. _et al._ A pressure-induced topological phase with large berry curvature in Pb1-xSnxTe.  _Sci. Adv._ 3, e1602510 (2017).
  * (39) Xi, X. _et al._ Signatures of a pressure-induced topological quantum phase transition in BiTeI.  _Phys. Rev. Lett._ 111, 155701 (2013).
  * (40) Maksimovic, N. _et al._ Evidence for a delocalization quantum phase transition without symmetry breaking in CeCoIn5.  _Science_ 375, 76–81 (2022).
  * (41) Li, Z. _et al._ Magnetic critical behavior of the van der Waals Fe5GeTe2 crystal with near room temperature ferromagnetism.  _Sci Rep_ 10, 15345 (2020).
  * (42) Li, Z. _et al._ Weak antilocalization effect up to 120K in the van der Waals crystal Fe5-xGeTe2 with near room temperature ferromagnetism.  _arXiv preprint arXiv:2109.02085_ (2021).
  * (43) Ly, T. T. _et al._ Direct observation of Fe$-$Ge ordering in Fe5-xGeTe2 crystals and resultant helimagnetism.  _Adv. Funct. Mater._ 31, 2009758 (2021).
  * (44) Wu, X. _et al._ Direct observation of competition between charge order and itinerant ferromagnetism in the van der Waals crystal Fe5-xGeTe2.  _Phys. Rev. B_ 104, 165101 (2021).
  * (45) Gao, Y. _et al._ Spontaneous (anti) meron chains in the domain walls of van der Waals ferromagnetic Fe5-xGeTe2.  _Adv. Mater._ 32, 2005228 (2020).
  * (46) May, A. F. _et al._ Ferromagnetism near room temperature in the cleavable van der Waals crystal Fe5GeTe2.  _ACS Nano_ 13, 4436–4442 (2019).
  * (47) May, A. F., Bridges, C. A. & McGuire, M. A.  Physical properties and thermal stability of Fe5-xGeTe2 single crystals.  _Phys. Rev. Mater._ 3, 104401 (2019).
  * (48) Huang, K. _et al._ Measurement of electronic structure in van der Waals ferromagnet Fe(5-x)GeTe2.  _Chinese Physics B_ 31, 057404 (2022).
  * (49) Zhang, H. _et al._ Itinerant ferromagnetism in van der Waals Fe5-xGeTe2 crystals above room temperature.  _Phys. Rev. B_ 102, 064417 (2020).
  * (50) Zhang, H. _et al._ Room-temperature skyrmion lattice in a layered magnet (Fe0.5Co0.5)5GeTe2.  _Sci. Adv._ 8, eabm7103 (2022).
  * (51) Seo, J. _et al._ Nearly room temperature ferromagnetism in a magnetic metal-rich van der Waals metal.  _Sci. Adv._ 6, eaay8912 (2020).
  * (52) Ribeiro, M. _et al._ Large-scale epitaxy of two-dimensional van der Waals room-temperature ferromagnet Fe5GeTe2.  _npj 2D Mater Appl_ 6, 10 (2022).
  * (53) Chen, X. _et al._ Pervasive beyond room-temperature ferromagnetism in a doped van der Waals magnet.  _Phys. Rev. Lett._ 128, 217203 (2022).
  * (54) Deng, Y. _et al._ Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2.  _Nature_ 563, 94–99 (2018).
  * (55) Wu, H. _et al._ Spectral evidence for local-moment ferromagnetism in van der waals metals Fe3GaTe2 and Fe3GeTe2.  _arXiv preprint arXiv:2307.00441_ (2023).  eprint 2307.00441.
  * (56) Kim, K. _et al._ Large anomalous hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal.  _Nat. Mater._ 17, 794–799 (2018).
  * (57) Heikkilä, T. T. & Volovik, G. E.  Dimensional crossover in topological matter: Evolution of the multiple Dirac point in the layered system to the flat band on the surface.  _JETP Letters_ 93, 59–65 (2011).
  * (58) Ye, L. _et al._ Massive Dirac fermions in a ferromagnetic kagome metal.  _Nature_ 555, 638–642 (2018).
  * (59) Fang, S. _et al._ Ferromagnetic helical nodal line and kane-mele spin-orbit coupling in kagome metal Fe3Sn2.  _Phys. Rev. B_ 105, 035107 (2022).
  * (60) Regnault, N. _et al._ Catalogue of flat-band stoichiometric materials.  _Nature_ 603, 824–828 (2022).
  * (61) Călugăru, D. _et al._ General construction and topological classification of crystalline flat bands.  _Nat. Phys._ 18, 185–189 (2022).
  * (62) Yin, J.-X. _et al._ Giant and anisotropic many-body spin–orbit tunability in a strongly correlated kagome magnet.  _Nature_ 562, 91–95 (2018).
  * (63) Kang, M. _et al._ Dirac fermions and flat bands in the ideal kagome metal FeSn.  _Nat. Mater._ 19, 163–169 (2020).
  * (64) Kang, M. _et al._ Topological flat bands in frustrated kagome lattice CoSn.  _Nat. Commun._ 11, 4004 (2020).
  * (65) Chen, L. _et al._ Topological semimetal driven by strong correlations and crystalline symmetry.  _Nat. Phys._ 18, 1341–1346 (2022).
  * (66) Birch, M. T. _et al._ History-dependent domain and skyrmion formation in 2d van der waals magnet Fe3GeTe2.  _Nat. Commun._ 13, 3035 (2022).
  * (67) Ding, B. _et al._ Observation of magnetic skyrmion bubbles in a van der waals ferromagnet Fe3GeTe2.  _Nano Letters_ 20, 868–873 (2020).
  * (68) Yang, M. _et al._ Creation of skyrmions in van der waals ferromagnet Fe3GeTe2 on (Co/Pd) superlattice.  _Sci. Adv._ 6, eabb5157 (2020).
  * (69) Casas, B. W. _et al._ Coexistence of merons with skyrmions in the centrosymmetric van der waals ferromagnet Fe5-xGeTe2.  _Adv. Mater._ 35, 2370118 (2023).
  * (70) Fujita, R. _et al._ Layer-dependent magnetic domains in atomically thin Fe5GeTe2.  _ACS Nano_ 16, 10545–10553 (2022).
  * (71) Schmitt, M. _et al._ Skyrmionic spin structures in layered Fe5GeTe2 up to room temperature.  _Commun Phys_ 5, 254 (2022).
  * (72) Chakraborty, A. _et al._ Magnetic skyrmions in a thickness tunable 2D ferromagnet from a defect driven Dzyaloshinskii–Moriya interaction.  _Adv. Mater._ 34, 2108637 (2022).
  * (73) Meisenheimer, P. _et al._ Controlled ordering of room-temperature magnetic skyrmions in a polar van der waals magnet.  _arXiv preprint arXiv:2210.02546_ (2022).
  * (74) Wei, B. _et al._ Superconductivity tuned by the iron vacancy order in KxFe2-ySe2.  _Chinese Physics Letters_ 30, 027402 (2013).
  * (75) Ding, X. _et al._ Influence of microstructure on superconductivity in KxFe2-ySe2 and evidence for a new parent phase K2Fe7Se8.  _Nat. Commun._ 4, 1897 (2013).

Figure 1: Phase-tuning in Fe5-δGeTe2. a, Crystal structure of Fe5GeTe2 with
atomic sites labeled. Fe(1) and Ge are modeled as split sites, marked by half-
filled circles to represent 50% occupancy. b-d, Schematic and procedures for
tuning global symmetry via sublattice ordering. The two quantum phases can be
switched by first annealing to above THT = 550K and either quench or slowly
cool to achieve the site-ordered or the site-disordered phase, respectively.
The inset in d shows the real steps of tuning phases. e-f, Schematic model and
STM topographic image of the $\sqrt{3}\times\sqrt{3}$ superstructure on Te
termination. The right panels are Fourier transforms of the respective
topographies with peaks corresponding to the lattice periodicity (white), a
$\sqrt{3}\times\sqrt{3}$ superstructure periodicity (blue), and the second
order of the $\sqrt{3}\times\sqrt{3}$ superstructure periodicity (pink). The
inset scale bars are 15 nm-1. The schematic lattice in e corresponds to a up-
down-down (UDD) ordering of the Fe(1) atoms while that in f corresponds to a
down-up-up (DUU) ordering of the Fe(1) sites. g-h, The Fermi surfaces of the
two phases measured at 15 K using 114 eV LV photons, achieved by slow-cooling
the crystal to room temperature from above THT (g) and quenching from above
THT (h), respectively. i-j, Polarization-resolved SHG intensity on slow-cooled
and quenched crystals measured at 5K. In both figures, the crossed and
parallel configurations correspond to E(2$\omega$)$\bot$ E($\omega$) and
E(2$\omega$) ||E($\omega$) respectively, while E(2$\omega$) and E($\omega$)
were simultaneously rotated in the crystal ab plane. c.p.s. stands for counts
per second. The dots are experimental data and the solid curves are the fits
by a six-fold sinusoidal function. Note that the radial axis for slow-cooled
crystal in the inset is enhanced by a factor of 10.  Figure 2: Topological
nodal lines in the slow-cooled site-disordered phase. a, Schematic showing
Dirac cones at K and K’ on a 2D projected BZ plane. b, Fermi surface mapping
with LV polarized photons. The red line outlines the 2D BZ boundary and the
white dashed line marks the electron pocket surrounding the BZ boundary. c,
Zoomed in view of the box in b. d, Dispersions measured across the Dirac node,
showing the crossing bands and the opening of a gap away from the Dirac node.
e-f, Dispersions measured along the
$\bar{M}-\bar{K}-\bar{\Gamma}-\bar{K^{\prime}}$ direction with LH and LV
polarization, respectively. The red arrows in f point to the Dirac crossings.
g, Energy distribution curves (EDCs) and the second energy derivative cut
within the momentum range marked by black arrow in f. h Photon energy
dependence of the cut near the $\bar{K}$ point. The red solid line shows the
Dirac nodal line along $k_{z}$. All data from b to g were taken with 132 eV
photons.  Figure 3: Flat bands in the quenched site-ordered phase. a,b,
measured energy-momentum dispersions along the
$\bar{M}-\bar{K}-\bar{\Gamma}-\bar{K^{\prime}}$ direction with LV and LH
polarized photons. The EDCs are integrated from a small range around $\bar{M}$
marked by the arrows in Fig. 3a, b, showing three peaks that correspond to the
location of the flat bands. c,d, Corresponding Fermi surfaces measured with LV
and LH polarizations, respectively. Data in c were measured under the same
geometry as Fig. 2b. e, Measured dispersions along a series of parallel cuts
within the first BZ as shown in d. All data from a to e were taken with 114 eV
photons.  Figure 4: Effect of the Fe(1) site ordering on inversion-symmetry
and quantum destructive interference. a, Illustration of one vdW slab of the
Fe(1) site-disordered phase, in which the Fe(1) sites are randomly distributed
such that global inversion symmetry is preserved. b-c, The schematic view of
the lattice showing the C2y and C3z symmetries in the Fe(1) site-disordered
phase, respectively. d, The tight-binding model showing Dirac crossings at K
(K’) protected by the above symmetries and the associated two-fold topological
crossing due to the ferromagnetic order and the gap by SOC. e, The helical
topological nodal lines induced by the ABC stacking of the vdW slabs that
results in winding of the nodal lines with opposite chiralities around
$\bar{K}$ and $\bar{K^{\prime}}$. f, Illustration of one vdW slab of the Fe(1)
ordered-site phase, in which the Fe(1) is DUU-ordered, forming a bipartite
crystalline lattice. g, The Fe(1) and the nearest neighbor Fe(3) sites form a
clover lattice as shown in the red dashed box in f. h, Single clover unit
showing the destructive interference of the hopping amplitude at the Fe(1) and
Fe(3) sites due to the alternating sign of the Wannier phase, leading to a
localization of the electronic wavefunction. The sites and wavefunction
amplitudes are labeled on the corresponding atoms. i, The original (blue) and
the $\sqrt{3}\times\sqrt{3}$ superstructure (red) BZ. j-k, Tight-binding model
for the clover lattice without and with spin orbital coupling (SOC),
respectively. With SOC, the flat band is gapped with a Chern number of 1, and
is hence topologically nontrivial.