# Spin-charge-lattice coupling across the charge density wave transition in a
Kagome lattice antiferromagnet

Xiaokun Teng  Department of Physics and Astronomy, Rice University, Houston,
Texas 77005, USA  David W. Tam  Laboratory for Neutron Scattering and Imaging,
Paul Scherrer Institut, 5232 Villigen, Switzerland  Lebing Chen  Department of
Physics and Astronomy, Rice University, Houston, Texas 77005, USA  Hengxin Tan
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
7610001, Israel  Yaofeng Xie  Department of Physics and Astronomy, Rice
University, Houston, Texas 77005, USA  Bin Gao  Department of Physics and
Astronomy, Rice University, Houston, Texas 77005, USA  Garrett E. Granroth
Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Alexandre Ivanov  Institut Laue-Langevin, 71 avenue des
Martyrs CS 20156, 38042 Grenoble Cedex 9, France  Philippe Bourges
Laboratoire L$\rm\acute{e}$on Brillouin, CEA-CNRS, Universit$\rm\acute{e}$
Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France  Binghai Yan
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
7610001, Israel  Ming Yi  Department of Physics and Astronomy, Rice
University, Houston, Texas 77005, USA  Pengcheng Dai<EMAIL_ADDRESS>Department
of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

###### Abstract

Understanding spin and lattice excitations in a metallic magnetic ordered
system form the basis to unveil the magnetic and lattice exchange couplings
and their interactions with itinerant electrons. Kagome lattice
antiferromagnet FeGe is interesting because it displays rare charge density
wave (CDW) deep inside the antiferromagnetic ordered phase that interacts with
the magnetic order. We use neutron scattering to study the evolution of spin
and lattice excitations across the CDW transition $T_{\rm CDW}$ in FeGe. While
spin excitations below $\sim$100 meV can be well described by spin waves of a
spin-1 Heisenberg Hamiltonian, spin excitations at higher energies are
centered around the Brillouin zone boundary and extend up to $\sim 180$ meV
consistent with quasiparticle excitations across spin-polarized electron-hole
Fermi surfaces. Furthermore, $c$-axis spin wave dispersion and Fe-Ge optical
phonon modes show a clear hardening below $T_{\rm CDW}$ due to spin-charge-
lattice coupling but with no evidence for a phonon Kohn anomaly. By comparing
our experimental results with density functional theory calculations in
absolute units, we conclude that FeGe is a Hund’s metal in the intermediate
correlated regime where magnetism has contributions from both itinerant and
localized electrons arising from spin polarized electronic bands near the
Fermi level.

In insulating magnets where unpaired electrons are localized on magnetic
atomic sites, interactions of local spin moments are governed by the
Heisenberg exchange couplings Heisenberg (1928). Magnons arising from spin
vibrations about their equilibrium positions should be characterized by
linearized spin wave theories that ignore all terms of order higher than
quadratic and interactions with lattice vibrations Holstein and Primakoff
(1940); Oguchi (1960). For a magnetic ordered material with more than one
magnetic ion per unit cell, we expect to observe acoustic and optical spin
waves, just like acoustic and optical phonon modes are expected for a
crystalline solid with more than one atom per unit cell Boothroyd (2020).For
example, spin waves in insulating kagome Chisnell  _et al._ (2015) and
honeycomb Chen  _et al._ (2018) lattice ferromagnet have well-defined acoustic
and optical modes as expected for a local moment Heisenberg magnet. A spin gap
between the acoustic and optical spin waves at the Dirac points can give rise
to protected topological magnon bands and edge modes Chisnell  _et al._
(2015); Chen  _et al._ (2018).

For metallic magnets, magnetic order can arise from either localized moments
similar to an insulating magnet or quasiparticle spin-flip excitations between
the valence (hole) and conduction (electron) bands at the Fermi level, dubbed
a spin density wave (SDW), due to electron-electron correlations Grüner
(1994). Spin excitations from these magnets can arise from vibrations of
localized moments Heisenberg (1928) or electron-hole Fermi surface
quasiparticle excitations of itinerant electrons Bloch (1929); Slater (1936);
Edmund C. Stoner (1947), respectively. Since SDW order can coexist and
intertwine with other orders such as charge density wave (CDW) Fawcett (1988)
and superconductivity Fradkin  _et al._ (2015); Dai (2015), a determination of
the interplay between magnetic and other intertwined orders forms the basis to
understand correlated electron materials.

Recently, kagome lattice magnet FeGe was found to have a CDW order deep inside
the magnetic ordered phase that couples with the ordered moment (Fig. 1a)
Ohoyama  _et al._ (1963); Bernhard  _et al._ (1984, 1988); Teng  _et al._
(2022); Yin  _et al._ (2022a); Teng  _et al._ (2023). With decreasing
temperature, FeGe first exhibits collinear $A$-type antiferromagnetic (AFM)
order at $T_{\rm N}\approx 400$ K (Fig. 1b), forms a $2\times 2\times 2$ CDW
order below $T_{\rm CDW}\approx 110$ K with an enhanced ordered moment, and
finally develops incommensurate AFM structure below $T_{\rm Canting}\approx
60$ K (Fig. 1c) Bernhard  _et al._ (1984, 1988); Teng  _et al._ (2022); Yin
_et al._ (2022a); Teng  _et al._ (2023); Miao  _et al._ (2023). In previous
angle resolved photoemission spectroscopy (ARPES) and inelastic neutron
scattering experiments, electron-boson interaction induced kink around 30 meV
seen in ARPES spectra was identified as electron and optical phonon coupling
Teng  _et al._ (2023). From temperature dependent low-energy spin wave
measurements, incommensurate spin fluctuations associated with incommensurate
AFM static order show a kink at $T_{\rm CDW}$ and survive up to $T_{\rm N}$
Chen  _et al._ (2024). Although these results suggest that incommensurate AFM
order is actually a SDW phase instead of the double cone AFM structure as
suggested originally Bernhard  _et al._ (1984, 1988), there is no
determination of the spin-charge-lattice coupling across $T_{\rm CDW}$ Teng
_et al._ (2023) and the microscopic origin of the magnetic order. Using first
principle calculations, it was predicted that the nearest-neighbor magnetic
exchange interactions are dominate and ferromagnetic (FM), incommensurate AFM
order is due to lattice distortion-induced Dzyaloshinskii-Moriya (DM)
interactions Zhou  _et al._ (2023). Furthermore, spin waves should be strongly
dispersive in the kagome plane with acoustic mode below about 100 meV and two
optical modes extending up to 250 meV Zhou  _et al._ (2023), similar to
optical modes in insulating kagome lattice magnet Chisnell  _et al._ (2015).

Here we use inelastic neutron scattering to study the evolution of spin and
lattice excitations of FeGe across $T_{\rm CDW}$ SI . While spin excitations
below $\sim$100 meV can be well described by a spin-1 ($S\approx 1$) local
moment Heisenberg Hamiltonian, spin excitations at higher energies are
centered around the Brillouin zone boundary and extend up to $\sim 180$ meV
(Figs. 1d-f), clearly different from the predictions of the first principle
calculations Zhou  _et al._ (2023) and the local moment picture. Instead, the
high energy spin waves are consistent with quasiparticle excitations between
spin-polarized electron-hole Fermi surfaces, similar to spin excitations in
metallic antiferromagnets FeSn Xie  _et al._ (2021); Do  _et al._ (2022); Tao
_et al._ (2023) and Fe0.89Co0.11Sn Xie  _et al._ (2022), van der Waals
metallic ferromagnet Fe2.72GeTe2 Bao  _et al._ (2022), weak itinerant
ferromagnet MnSi Chen  _et al._ (2020); Jin  _et al._ (2023), and heavy
Fermion CePd3 Goremychkin  _et al._ (2018). Furthermore, spin wave dispersions
and Fe-Ge optical phonon modes show a clear hardening below $T_{\rm CDW}$ due
to spin-charge-lattice coupling but with no evidence for phonon Kohn anomaly.
By comparing these results with density functional theory (DFT) calculations
in absolute units, we conclude that FeGe has an intimate coupling between
itinerant electrons and magnetism similar to iron pnictides Dai  _et al._
(2012), suggesting that it is a Hund’s metal with intermediate electronic
correlations Georges  _et al._ (2013); Stadler  _et al._ (2019). The strong
spin-charge-lattice coupling in FeGe is different from FeSn Xie  _et al._
(2021); Do  _et al._ (2022); Tao  _et al._ (2023) and all other kagome lattice
materials Yin  _et al._ (2022b), making this an unique system to investigate
intertwined orders in spin, charge, and lattice degrees of freedom.

Figure 1: (a) In-plane lattice structure of FeGe with red and grey denoting Fe
and Ge atoms respectively. Magnetic structure of FeGe at (b) $T>T_{\rm
Canting}$ and (c) $T<T_{\rm Canting}$. (d) Reciprocal space of FeGe. The thick
red line denotes the momentum path along the $[H,-0.5H,-1.5]$ direction. (e)
DFT (with no spin orbit coupling) calculation. The red and blue bands
represent spin up and down bands respectively. Thick arrows indicate particle-
hole (p-h) scattering. (f) Dynamic susceptibility calculated from p-h
scattering of the set of bands in (e), where the energy axis has been
renormalized by 1.7 Teng  _et al._ (2023).

We first examine energy ($E$) and momentum ($\bf Q$) dispersion of the in-
plane spin excitations of FeGe in the $A$-type AFM ordered phase at 120 K. At
low energy, spin-wave-like excitations stem from the $\Gamma$ point at the
zone center and gradually disperse to the zone boundary, first reaching the
$M$ points at $E=90\pm 10$ meV and then the $K$ points at $E=120\pm 10$ meV,
showing intensity modulation across zone boundaries (Figs. 1d and 2a). Figure
2b shows the spin excitation dispersion along the high-symmetry direction
$[H,-0.5H,-1.5]$ with $L=[-3,0]$ as defined in Fig. 1d taken with incident
neutron energy of $E_{i}=300$ meV. We take systematic constant energy cuts
from $E_{i}=45$, 100, and 300 meV data at temperatures across $T_{\rm CDW}$
and $T_{\rm Canting}$, and fit them with two Gaussians with a linear
background. After averaging between the left and right, we obtain the spin
wave dispersions at different temperatures, and find no significant change
change across $T_{\rm CDW}$ and $T_{\rm Canting}$ from 120 K to 8 K (Fig. 2c).

Figure 2: (a) Constant energy slices at $E=30\pm 10$, $50\pm 10$, $70\pm 10$,
$90\pm 10$, $110\pm 10$, $130\pm 10$, $150\pm 10$, $170\pm 10$ meV. The white
lines are Brillouin zone boundaries, and high symmetry points are labeled with
black points. (b) In-plane magnetic excitation along the $[H,-0.5H,-1.5]$
direction. The black solid lines and gray dashed lines indicate simulated
acoustic and optical spin wave branches, respectively, using a Heisenberg
model (SpinW) Toth and Lake (2015). Data in (a) and (b) is taken at 120 K. (c)
Spin wave dispersions extracted by Gaussian fitting with a linear background
of the symmetrized spectra along the $[H,-0.5H,-1.5]$ direction. The vertical
error bars indicate the energy integration range. Horizontal error bars are
from fitting. Integration along the orthogonal in-plane direction is [-0.1,
0.1]. (d) Integrated magnetic intensity in the first in-plane Brillouin zone
and averaged between $L=[-2.5,-0.5]$. The black line indicates the calculated
spin wave intensity in absolute units assuming $S$ = 1 in the SpinW + Horace
program Toth and Lake (2015). Red, green and blue symbols represent 120 K, 70
K and 8 K data respectively Ewings  _et al._ (2016). The red-dashed line is
the DFT estimated spin susceptibility from the two electronic bands in Figs.
1f and 1e.

To understand magnetic excitations using a local moment picture, we consider
the Heisenberg Hamiltonian:

$H_{0}=\sum_{<i,j>}J_{ij}S_{i}\cdot S_{j}+\sum_{i}D_{z}(S_{z_{i}})^{2}$

Here, $J_{ij}$ represents the magnetic exchange interaction between the $i$th
and $j$th Fe atoms, $S_{i}$ and $S_{j}$ are the local spins at the $i$th and
$j$th sites, respectively, and $D_{z}$ is the single-ion anisotropy. For in-
plane dispersions, we use $J_{1}$ and $J_{2}$ as the nearest and next-nearest-
neighbor couplings, respectively (Fig. 1a). The $c$-axis nearest and next-
nearest-neighbor couplings $J_{c1}$ and $J_{c2}$ (Fig. 1b) are determined in
Ref. Chen  _et al._ (2024). We can simulate spin wave dispersions using the
above Heisenberg Hamiltonian and compare with experiments.

Model | $J_{1}$ | $J_{2}$ | $J_{c1}$ | $J_{c2}$ |  $D_{z}$ (meV)
---|---|---|---|---|---
Heisenberg | $-16.4$ | $-7.2$ | $11.3$ | 0 | $-0.015$
First principle | $-41.97$ | $5.49$ | $8.44$ | $-2.04$ | 0
Table 1: The first row gives magnetic exchange coupling constants obtained
from the Heisenberg Hamiltonian simulation. The solid and dashed lines in Fig.
2b shows the resulting spin wave dispersions. The second row shows exchange
parameters predicted from the first principle calculations Zhou  _et al._
(2023).

In previous low-energy inelastic neutron scattering experiments, the
dispersion of spin excitations along the $L$ direction was found to have a
band top around $E\approx 24$ meV and a small $D_{z}\approx-0.015$ meV Chen
_et al._ (2024). To better fit the high-energy in-plane spin excitation data,
the anisotropic term was set loose in the fitting process due to the large $L$
integration range with $E_{i}=300$ meV. The solid black lines in Fig. 2b
represent the acoustic spin-wave branches that are in agreement with the data
below $\sim$100 meV with magnetic exchange couplings in Table 1. However, two
distinctions between the data and the simulation cannot be explained by the
Heisenberg model. First, the optical branches predicted by the Heisenberg
model and first principle calculations Zhou  _et al._ (2023) are absent in the
data (Fig. 2b). Second, when approaching the Brillouin zone boundary at $K$
(Dirac points) and $M$ points (Figs. 2b, 2c), the magnetic excitations
continue to rise and form a convex shape, instead of bending over to a concave
shape as predicted by the simulation. As a consequence, there is no spin gap
at the Dirac points and we do not expect spin excitations of FeGe to have
protected topological magnon bands and edge modes Chisnell  _et al._ (2015);
Chen  _et al._ (2018). Figure 2d compares the integrated local dynamic spin
susceptibility in the first Brillouin zone at 8 K, 70 K, and 120 K with
$L=[0.5,2.5]$, normalized by acoustic phonons near a nuclear Bragg peak to
absolute unit Xu  _et al._ (2013), with calculated results from Heisenberg
model assuming $S=1$. The susceptibility shows a broad peak around 100 meV,
and decays rapidly for energies above 100 meV (Fig. 2d). Although DFT
calculations of the hole-electron quasiparticle excitations (Figs. 1e and 1f)
correctly predicted the high energy spin excitations, the absolute magnetic
scattering estimated from the DFT is much smaller than the observation (Fig.
2d), similar to iron pnictides Wang  _et al._ (2013).

Figure 3: High energy magnetic excitations along (a) $[0,K]$ with
$[H,-0.5H]=[-0.1,0.1]$ and (b) $[H,-0.5H]$ with $K=[-0.1,0.1]$, $L=[-4,0]$
(r.l.u.) measured with $E_{i}=300$ meV. The dashed lines indicate the $M$
points in (a) and $K$ points in (b). Constant energy slices at (c) $E=130\pm
10$, (d) $150\pm 10$, (e) $170\pm 10$ meV and (f) $190\pm 10$ meV. The solid
black lines mark the Brillouin zone boundaries. Data is taken at 120 K.

Since spin excitations above 120 meV in FeGe are rod-like at both the $M$
(Fig. 3a) and the $K$ points (Fig. 3b), they are similar to metallic
antiferromagnets FeSn Xie  _et al._ (2021); Do  _et al._ (2022); Tao  _et al._
(2023); Xie  _et al._ (2022), ferromagnets Fe2.72GeTe2 Bao  _et al._ (2022),
MnSi Chen  _et al._ (2020); Jin  _et al._ (2023), and heavy Fermion CePd3
Goremychkin  _et al._ (2018). To further investigate the behavior of this rod-
like dispersion, we plot the constant energy slices in multiple Brillouin
zones (Figs. 3c-f). The magnetic excitations are concentrated at Brillouin
zone boundary (Figs. 3c-e), clearly different from that of MnSi where the rod-
like dispersions reside inside Brillouin zone boundaries Chen  _et al._
(2020); Jin  _et al._ (2023). Spin excitations gradually vanish around $E$ =
200 meV (Fig. 3f), different from expectations of a local moment Heisenberg
Hamiltonian and the first principle calculations Zhou  _et al._ (2023). These
results are similar to high-energy spin excitations of FeSn, which also has
$A$-type AFM but without CDW order Xie  _et al._ (2021); Do  _et al._ (2022).
The wave vector dependence of these excitations are clearly different from
cluster spin excitations associated with localized spins in insulating
frustrated pyrochlore Lee  _et al._ (2002) and triangular lattice
antiferromagnets Gao  _et al._ (2023), as well as in high energy spin
excitations in metallic kagome lattice ferrimagnet TbMn6Sn6 Riberolles  _et
al._ (2022, 2024).

Figure 4: Temperature dependent spin excitations along the $c$-axis ($L$
direction) at (a) 8 K, (b) 70 K and (c) 120 K measured with $E_{i}=45$ meV.
(d) Spin wave dispersions determined by fitting constant momentum cuts taken
from (a-c) with Lorentzian on a constant background. Red and blue arrows in
(a) and (d) mark $L=1.35$ and $L=1.25$, respectively. (e) Temperature
dependent spin wave energy at $L=1.35$ (blue dots) and 1.25 (red dots),
obtained from IN8. Grey and blue dashed lines mark $T_{\rm Canting}$ and
$T_{\rm CDW}$ respectively. Thick blue and red lines are guides to the eye.

Figures 4a-c show the out-of-plane spin wave excitations at different
temperatures across $T_{\rm CDW}$ and $T_{\rm Canting}$. The fitted spin wave
dispersions are shown in Fig. 4d. By extracting the band top $E_{top}$ at
$L=-1.25$ for each temperature, we find that spin waves harden by around 11.6%
from 120 K to 70 K across $T_{\rm CDW}$, but remain unchanged across $T_{\rm
Canting}$. The temperature dependence of the spin wave energy measured at
$L=1.25$ and $L=1.35$ near the zone boundary using a triple-axis spectrometer
shows a clear hardening of spin wave dispersion around 2 meV below $T_{\rm
CDW}$ (Fig. 4e). This hardening of spin waves coincides with an increase of
around 0.1 $\mu_{B}$/Fe in the static magnetic moment, from around 1.5
$\mu_{B}$/Fe to 1.6 $\mu_{B}$/Fe across $T_{\rm CDW}$ Teng  _et al._ (2022).
The behavior is not as prominent for the in-plane dispersion because of its
steeper dispersion SI . Therefore, the hardening of spin wave dispersion below
$T_{\rm CDW}$ arises from the CDW order induced moment increase and spin waves
from the $A$-type AFM order mostly conform to a local moment Hamiltonian below
about 100 meV.

Figure 5: (a) In-plane phonon dispersion at $L=[-3.1,-2.9]$ and 120 K. The
solid black lines are DFT calculations. Two separate optical phonon branches
are labeled as OP1 and OP2, corresponding to Fe-Ge $A_{2u}$ and Fe out-of-
plane vibrations, respectively. Phonon dispersions of (b) the OP1 and (c) OP2
at 50 K, 120 K, 200 K and 300 K. (d) Temperature dependence of the OP1 energy
shift throughout the Brillouin zone at 10 K, 50 K, 80 K, 120 K, 140 K, 160 K,
and 300 K. The phonon energy at the base temperature (10 K) is subtracted.
Temperature dependence of the OP1 phonon mode at (e) $\Gamma$, (f) $M$ points,
and the OP2 phonon mode at $M$ point. The thick colored lines in (b-d) and the
gray lines in (e-g) are guides to the eye. Temperature dependence of phonon
full width half maximum (FWHM) at the OP1 $\Gamma$ (h), OP1 $M$ (i) and OP2
$M$ (j).

After mapping out spin excitation evolution across $T_{\rm CDW}$ and $T_{\rm
Canting}$, we investigate temperature dependence of the in-plane phonon
spectra (Figs. 5a-5j) SI . In previous Raman and neutron Larmor diffraction
measurements, the crystal structure is found to change from hexagonal to
monoclinic with a small in-plane lattice distortion, but becomes more
symmetric below $T_{\rm CDW}$ Wu  _et al._ (2024). However, much is unclear on
the dynamic spin-lattice-charge interactions acrossing the CDW transition.
Figure 5a shows the overall phonon spectra where optical phonon 1 (OP1) and
optical phonon 2 (OP2) are marked. They agree well with the DFT calculated
spectra shown in solid black lines. For both OP1 and OP2, the entire phonon
dispersion hardens about 1 meV at all measured wave vectors on cooling from
300 K to 10 K (Figs. 5b-5d), distinct from previous work on acoustic phonon
mode where no phonon energy shift is observed at $M$ points Miao  _et al._
(2023). To further determine if the phonon hardening is coupled with CDW
order, we plot the temperature dependence of phonon energy at high symmetry
$\Gamma$ and $M$ points for OP1 and OP2 (Figs. 5e-5g). There is almost no
shift in phonon dispersion below around 80 K, and CDW order is clearly coupled
with phonon hardening at $M$ points for OP1 and OP2 modes (Figs. 5f and 5g).
Since OP1 is the optical $A_{2u}$ mode involving out of plane Fe-Ge
vibrational modes and OP2 is Fe out of plane vibrational mode, the results
suggest a strong coupling of these modes with the CDW order associated the Ge
$c$-axis dimerization Miao  _et al._ (2023). Note that the phonon full width
half maximum (FWHM) at different wave vectors consistently decreases with
decreasing temperature (Fig. 5h-j), corresponding to a larger phonon lifetime
at lower temperatures. For comparison, acoustic zone boundary phonon lifetime
was found to decrease at $A$ point from 400 K to 200 K but not at $M$ point
from inelastic X-ray scattering experiments Miao  _et al._ (2023).

In previous work on FeGe Miao  _et al._ (2023), the CDW transition is
suggested to arise from Ge $c$-axis dimerization different from the usual Kohn
anomaly in electron-phonon coupled CDW materials but similar to spin-lattice
coupling in FeSi Krannich  _et al._ (2015) and CuGeO3 Braden  _et al._ (1998);
Gros and Werner (1998). These results are consistent with first principle
calculations Wang (2023); Shao  _et al._ (2023); Zhang  _et al._ (2023) and
subsequent experiments Chen  _et al._ (2023); Zhao  _et al._ (2023),
suggesting that CDW order in FeGe arises from electron correlations instead of
the usual electron-phonon interaction Wu  _et al._ (2023). From our inelastic
neutron scattering experiments summarized in Figs. 2-5, we find that high
energy spin excitations of FeGe behave similarly to other itinerant magnets
Xie  _et al._ (2021); Do  _et al._ (2022); Tao  _et al._ (2023); Xie  _et al._
(2022); Bao  _et al._ (2022); Chen  _et al._ (2020); Jin  _et al._ (2023);
Goremychkin  _et al._ (2018). The lack of optical spin waves and rod-like spin
excitations confined to the Brillouin zone boundary for energies above 100 meV
(Fig. 3) are consistent with quasiparticle excitations from spin down to spin
up bands across the Fermi level as shown in our DFT calculations (Figs. 1e and
1f), indicating that itinerant electrons play an important role in determining
the high energy spin excitations. However, the magnitude of the magnetic
scattering determined from DFT is much smaller than that of the observation
(Figs. 1f and 2d). For comparison, we note that high energy spin excitations
in iron pnictides are consistent with a local moment picture while low-energy
spin excitations are from the nesting of electron and hole Fermi surfaces and
DFT calculated spectral weight is also much smaller than the observation Dai
_et al._ (2012); Wang  _et al._ (2013). Since the DFT calculated electronic
dispersions need to be renormalized by about 1.7 to account for those
determined from ARPES experiments Teng  _et al._ (2023) similar to the values
in iron pinctide superconductors Yin  _et al._ (2011), we conclude that FeGe
is a Hund’s metal in the intermediate electron correlation regime. Similarly,
while a pure local moment Heisenberg model with $S=1$ can account for
temperature dependent spin excitations above the anisotropy gap energy of
$\sim$1 meV Chen  _et al._ (2024), spin excitations above 100 meV appear to
have an itinerant origin (Fig. 2d). These results are consistent with recent
theoretical calculations indicating that FeGe is slightly more electron
correlated compared with FeSn Lin  _et al._ (2023), and the flatish electronic
bands near the Fermi level responsible for correlated properties of FeGe arise
from $(d_{xy},d_{x^{2}-y^{2}})$ and $(d_{xz},d_{yz})$ orbitals Jiang  _et al._
(2023). Therefore, FeGe is in the intermediate correlated regime where
magnetism has contributions from both itinerant and localized electrons, and
couples with CDW order to form a strong spin-charge-lattice coupled kagome
metal, suggesting that FeGe is a rare case where the energy scales of spin,
charge, lattice degrees of freedom are similar and their interactions give
rise to the observed exotic properties different from its sister compound FeSn
and all other known kagome lattice magnets Yin  _et al._ (2022b).

We are grateful to Andrea Piovano for helpful discussions. The neutron
scattering and single crystal synthesis work at Rice was supported by US NSF
DMR-2100741 and by the Robert A. Welch Foundation under grant no. C-1839,
respectively (P.D.). M.Y. acknowledges support by the U.S. DOE grant No.
DESC0021421 and the Robert A. Welch Foundation, Grant No. C-2175. A portion of
this research used resources at the Spallation Neutron Source, a DOE Office of
Science User Facility operated by Oak Ridge National Laboratory.

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