# Hundness and band renormalization in the kagome antiferromagnets Mn${}_{3}X$

Yingying Cao  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Yuanji Xu  Institute for Applied Physics, University of
Science and Technology Beijing, Beijing 100083, China  Yi-feng Yang
<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong
523808, China

###### Abstract

The interplay of topological band structures and electronic correlations may
lead to novel exotic quantum phenomena with potential applications. First-
principles calculations are critical for guiding the experimental discoveries
and interpretations, but often fail if electronic correlations cannot be
properly treated. Here we show that this issue occurs also in the
antiferromagnetic kagome lattice Mn${}_{3}X$ ($X=$ Sn, Ge), which exhibit a
large anomalous Hall effect due to topological band structures with Weyl nodes
near the Fermi energy. Our systematic investigations reveal a crucial role of
the Hund’s rule coupling on three key aspects of their magnetic, electronic,
and topological properties: (1) the establishment of noncollinear
antiferromagnetic orders, (2) the weakly renormalized bands in excellent
agreement with ARPES, and (3) a sensitive tuning of the Weyl nodes beyond
previous expectations. Our work provides a basis for understanding the
topological properties of Mn${}_{3}X$ and challenges previous experimental
interpretations based on incorrect band structures.

Topological quantum materials have attracted intensive interest for their
importance on both fundamental physics and potential applications. Increasing
attentions have recently focused on correlated topological materials, in which
exotic many-body phenomena may emerge due to the interplay of nontrivial
topology and electronic correlations 1; 2. One notable example is Mn3Sn, which
adopts a kagome bilayer structure and exhibits a noncollinear triangular
antiferromagnetic (AFM) order in the kagome plane below 430 K3; 4. This
compound has a considerable spin-orbit coupling (SOC) that breaks the
effective time reversal symmetry and may cause nonzero Berry curvatures and
sizable anomalous Hall conductivities without external magnetic field 5. A
large anomalous Hall effect (AHE) has indeed been observed at room temperature
despite of the vanishingly small net magnetization 4. Resistivity and thermal
conductivity measurements reported an anomalous Lorenz number and excluded its
origin from inelastic scattering6. A large magneto-optical Kerr effect (MOKE)
was also observed, implying the presence of analogous symmetry requirement for
the AHE 7. Hence, Mn3Sn has been established to be the first AFM material with
intrinsic AHE.

First-principles calculations of electronic band structures played a key role
in guiding the above experimental discoveries and interpretations. Density
functional theory (DFT) studies of Mn3Sn predicted the existence of Weyl
points near the Fermi energy 8; 9; 10, which are singular points of Berry
curvatures (magnetic monopoles in momentum space) 11; 12 and may give a large
contribution to the AHE. Comparison with the angle-resolved photoemission
spectroscopy (ARPES) experiment 13 seems to support the predicted DFT band
structures, but requires a very large band renormalization factor of about
$5$, suggesting significant electronic correlations. However, the effects of
electronic correlations typically cannot be captured by DFT. In addition, the
Hund’s rule coupling is also not well treated but has been shown to be
important for the partially filled $3d$ shell 14; 15, especially in Mn-based
compounds 16. This raises the concern on the reliability of the DFT-predicted
band structures, which has not been paid much attention in previous studies.

Mn3Ge is an isostructural compound with the same noncollinear triangular AFM
order and a small residual in-plane magnetic moment below 380 K 17. It
exhibits similar giant AHE, anomalous Nernst effect, and MOKE 18; 19; 20,
suggesting similar electronic structure compared to Mn3Sn. However, comparison
between DFT calculations with ARPES measurements yields a small
renormalization factor of 1.18, in sharp contrast to that of Mn3Sn 21. This
large divergence between two similar compounds also calls for reinvestigation
of their correlated electronic structures. This issue becomes particularly
important considering that DFT predictions of the Weyl points have been widely
adopted in experimental interpretations, although their presence has not been
visualized in ARPES. Recent DFT+U calculations have even claimed possible
topological phase transition with the onsite Coulomb interaction 22.

In this work, we show that the Hund’s rule coupling is critical in resolving
the above controversy. We performed systematic correlated electronic structure
calculations by combining DFT 23 and the dynamical mean-field theory (DMFT)
24; 25; 26; 27; 28; 29, taking into account the onsite Hubbard repulsion, the
Hund’s rule coupling, and the SOC. Detailed comparison with ARPES data
uncovers a primary role of the Hund’s rule coupling on some key aspects of the
magnetic, electronic, and topological properties of both Mn3Sn and Mn3Ge. By
determining the Mn-ionic configurations, it helps to establish their spin
polarization and hence the AFM order and yield correct electronic band
structures in excellent agreement with ARPES data in many fine details. We
obtain a much smaller renormalization factor ($Z^{-1}\approx 1.6$) of Mn3Sn
than previously expected from DFT-ARPES comparison. The Weyl nodes are found
to depend sensitively on the Hund’s rule coupling and may evolve into line
nodes with varying parameters. A similar renormalization is also obtained in
Mn3Ge by comparing our calculations and latest ARPES measurements. These
reveal the potential importance of Hund physics for understanding their exotic
topological properties.

Figure 1: Crystal and magnetic properties. a Illustration of the crystal
structure of Mn3Sn with AFM1 magnetic configuration showing inverse symmetry
between two adjacent Mn3Sn layers 30; b Symmetry of the triangular
antiferromagnetic order for two spin configurations, AFM1 and AFM2, depending
on the applied magnetic field $B||x$ and $B||y$, respectively. Three mirror
planes are shown for $M_{z}T$ (magenta), $M_{x}$ (yellow), and $M_{x}T$
(green), where $M_{x}$ and $M_{z}$ are mirror operations and $T$ is the time
reversal operation. c The spin moment of a single Mn-ion in Mn3Sn for
different choices of $U$ and $J$ calculated using DFT+DMFT at 300 K.

Results

Structural and magnetic analyses. Mn${}_{3}X$ adopt a hexagonal structure with
the space group $P6_{3}/mmc$. The lattice constants are $a=5.67\,$Å and
$c=4.53\,$Å for Mn3Sn and $a=5.39\,$Å and $c=4.35\,$Å for Mn3Ge31; 32. The Mn-
ions form a kagome lattice with mixed triangles and hexagons in the
$ab$-plane, and the Sn(Ge)-ions are located at the center of the hexagons. As
shown in Fig. 1a for Mn3Sn, the adjacent layers stack along the $c$-axis with
an offset. Below the Néel temperature, a noncollinear AFM state is formed with
the Mn moments aligned in the $ab$-plane and forming an inverse triangular
spin structure 3; 33. As the temperature decreases below $200$ K, a helical or
incommensurate spin structure may appear in Mn3Sn depending sensitively on the
samples and suppress the triangular AFM until a spin glass is formed below
about $50$ K 34. For simplicity, we will only consider the inverse triangular
AFM phase, in which the inverse symmetry is preserved to transform the two
kagome layers to each other. As illustrated in Fig. 1b, there exist two
quenched triangular spin configurations, AFM1 and AFM2. While Mn3Ge is
reported to adopt AFM2 order as the ground state 33, the AFM configuration of
Mn3Sn switches between two orders depending on the direction of the applied
magnetic field 35. The coplanar AFM1 order has two additional symmetries: the
mirror symmetry $M_{x}$ and the combined symmetry $M_{z}T$. The mirror
reflection reverses the components of Berry curvatures parallel to the mirror
plane and the time reversal symmetry ($T$) reverses the Berry curvatures along
all directions. Since the Berry curvatures are not affected by the inverse
symmetry, the Hall conductivity given by their integration over the Brillouin
zone is only nonzero along $x$ direction. Similarly, the AFM2 has $M_{z}T$ and
$M_{x}T$ symmetries, with the only nonzero component of Hall conductivity
along $y$ direction. In real materials, $M_{x}$ and $M_{x}T$ can be broken due
to a tiny net moment in $ab$-plane 13, causing nonzero anomalous Hall
conductivity in both $x$ and $y$ directions. This was ignored in previous
first-principles calculations 10; 13. Our DFT+DMFT results in Supplementary
Fig. S3 confirm that it is indeed a small perturbation to the electronic band
structures. We will therefore focus on the ideal inverse triangular
antiferromagnetic orders here.

Effects of the Hund’s rule coupling in Mn3Sn. We only consider AFM1 for Mn3Sn,
whose electronic band structures have been measured by ARPES 13. Figure 1c
plots the calculated Mn spin moment as a function of the Hubbard $U$ and the
Hund’s rule coupling $J$. Such a value of the Hund’s rule coupling is
reasonable for Mn-ions and has been widely used in the literature 36; 37. The
calculations were carried out using DFT+DMFT at 300 K below
$T_{\text{N}}=430\,$K. For each value of $U$, the AFM order can only be
stabilized for sufficiently large $J$. The critical $J_{\text{c}}$ lies
roughly between 0.5 and 0.6 eV and varies slightly when $U$ increases from 3
to 6 eV. Its increase with decreasing $U$ indicates their compensating role
for local spin polarization. Without the Hund’s rule coupling, the moment no
longer exists and the triangular AFM order cannot be formed regardless of the
value of $U$, at least within our calculated parameter range. By contrast, the
size of the moment grows quickly as $J>J_{\text{c}}$, implying its sensitive
dependence on the Hund’s rule coupling. Thus, the Hund’s rule coupling plays a
key role in establishing the AFM order in Mn3Sn, which was not well noticed in
previous studies 38.

To understand what happens, we plot in Figs. 2a and 2b the projection of the
calculated DMFT states on the ionic configurations $(N,S_{z})$ of the Mn-$3d$
orbitals for a fixed $U=6\,$eV and different values of $J$ 39. Here $N$ is the
occupation number of electrons on Mn-$3d$ orbitals and $S_{z}$ is the local
$z$-component of the total spin. For $J=0$ shown in Fig. 2a, we find strong
fluctuations among the three valence states $N=4$, 5, 6 indicated by the
similar probabilities of DMFT state on these configurations. Meanwhile, the
spin also distributes equally along opposite directions, with the highest
probability in the $S_{z}=\pm 0.5$ low spin states. The system is in a
nonmagnetic phase despite of the large Hubbard $U=6\,$eV. By contrast, once
the Hund’s rule coupling $J=0.55\,$eV is switched on as shown in Fig. 2b, the
$N=5$, $S_{z}=-2.5$ state becomes dominant and the Mn-spins become polarized,
with most of the weight on negative $S_{z}$ albeit fluctuating strongly among
$S_{z}=-2.5$, -1.5, and -0.5. The total spin moment is about 2.6
$\mu_{\text{B}}$/Mn, smaller than the DFT prediction of 3.3 $\mu_{\text{B}}$.
We will show that this difference is crucial for yielding a spectral function
consistent with ARPES measurement 13. From experiments 3; 35, the estimated
value of magnetic moment varies from 1.78 to 3.0 $\mu_{B}$ depending on the
detection technologies, temperature and samples, whose implications await a
better understanding.

Figure 2: Effect of the Hund’s rule coupling on the Mn-ionic states and band
renormalization in Mn3Sn. a,b Probabilistic distribution of the DMFT state on
different Mn-ionic configurations with the electron occupancy $N=4$, 5, 6 with
($J=0.55\,$eV) and without the Hund’s rule coupling at $U=6\,$eV. The
horizontal axis denotes their respective energy difference with respect to the
configuration of largest probability. The insets show the distribution of
$N=5$ state on the local $z$-component of the total spin, $S_{z}$. c
Illustration of the fourth-order polynomial fitting (solid and dashed lines)
to the imaginary part of the orbital-averaged self-energy as a function of the
Matsubara frequency for $U=6\,$eV and $J=0.55\,$eV. d The renormalization
factor $Z$ estimated from the DFT+DMFT self-energies with varying $U$ and $J$.

The strength of electronic correlations may be seen straightforwardly from the
band renormalization. Figure 2c plots the orbital-averaged self-energies
$\Sigma(i\omega)$ in Matsubara frequency for different spin directions and
inequivalent Mn-ions. We find that they behave similarly on two inequivalent
ions, possibly due to their similar crystal field environments. The solid and
dashed lines give the fourth-order polynomial fitting 15, which we use to
estimate the renormalization factor
$Z^{-1}=1-\partial{\text{Im}}\Sigma(i\omega)/\partial\omega|_{\omega\to
0^{+}}$. For clarity, we take the average over all Mn-$3d$ orbitals in the
unit cell and plot the derived $Z$ as a function of $U$ and $J$ in Fig. 2d.
Interestingly, $Z$ increases gradually with increasing $J$ in the AFM phase
for $J>J_{c}$, but behaves oppositely in the paramagnetic phase. This trend
may come from the enhanced spin fluctuations near the critical point. At
$J_{c}$, we obtain a maximum band renormalization $Z^{-1}$, which is, however,
less than 1.8 and hence three times smaller than previous estimate from the
comparison between DFT and ARPES in Mn3Sn 13.

Figure 3: Comparison of ARPES 13 and DFT+DMFT band structures in Mn3Sn.
Intensity map of the theoretical and experimental spectral functions along:
a,b the high symmetry $\Gamma$-K-M-K-$\Gamma$ line (cyan arrows in the inset
of a); c the high symmetry M-K-$\Gamma$-K-M line (magenta arrows in the inset
of a); d-f the $k_{x}$ direction with $|\Delta k_{y}|=0.072\,(2\pi/a)$
slightly off the $\Gamma$-K-M-K-$\Gamma$ path; g the $k_{x}$ direction with
$\Delta k_{y}=-0.144\,(2\pi/a)$ from $\Gamma$-K-M-K-$\Gamma$; h the high
symmetry H-K-H line. The arrows of different colors highlight some
characteristic features of excellent agreement. i Comparison of the
theoretical and experimental Fermi surface mapping in the $k_{z}=0$ plane. To
mimic the low resolution of the ARPES data, we have intentionally chosen the
DFT+DMFT results at 300 K. The calculated Fermi surfaces are qualitative the
same at 60 K. The yellow line in the left one indicates the hexagonal
Brillouin zone. The red curves show the DFT-predicted Fermi surfaces
reproduced from Ref. 13.

Comparison with ARPES on Mn3Sn. The much smaller renormalization factor
questions the validity concerning the interpretation of ARPES data using the
artificially renormalized DFT band structures. In fact, as shown in
Supplementary Fig. S1, the renormalized DFT bands fail to conform to
experimental observations in many aspects on a qualitative level.
Surprisingly, we find that DFT+DMFT can actually give an excellent agreement
with the ARPES data for reasonable parameters. Figure 3 shows the intensity
map of the spectral function along different line cuts in the Brillouin zone.
The ARPES data were reproduced from Ref. 13 and the calculations were
performed for $U=6\,$eV and $J=0.55\,$eV at $T=60\,$K. As compared in Figs.
3a-3c, DFT+DMFT successfully captures several major features of the ARPES
along the high symmetry path M-K-$\Gamma$, where three bands labeled as
$\alpha,\beta$, and $\gamma$ cross the Fermi energy (red arrows) and a flat
band appears around M at about -0.05 eV. The latter exhibits a peak near the K
point (blue arrows), then bends down, and reaches about -0.3 eV at $\Gamma$.
By contrast, DFT gives a flat band close to the M point at about -0.25 eV
(Supplementary Fig. S1), which is far away from the Fermi energy and hence has
to be renormalized by a factor of 5 to explain the ARPES observation. But the
large artificial renormalization induces a lot of spectral weights between -3
and -1 eV near K-M (Supplementary Fig. S1), which are not present in
experiment 13.

Away from the high symmetry path by $\Delta k_{y}=-0.072$, two bands (red
arrows) were observed to cross the Fermi energy along $k_{x}$ in ARPES, as
plotted in Figs. 3e and 3f. This feature is well captured in DFT+DMFT as
marked by the arrows in Fig. 3d. The numerical results provide a better view
on the band evolution with $k_{y}$. Further away from the high symmetry path,
the conduction bands gradually move upwards and the number of bands crossing
the Fermi energy decreases. For $\Delta k_{y}=-0.144$ in Fig. 3g, only the
$\alpha$ band (red arrow) crosses the Fermi energy. The bands that intersect
around K start to open a gap, display a shoulder (blue arrows), and bulge
upward at $k_{x}=0$ (green arrow). All features are in good consistency with
experiment.

The dispersion along another high symmetry line K-H is also compared in Fig.
3h. The quantitative agreement with ARPES again supports our DFT+DMFT
calculations, while in DFT, the renormalization would cause an almost flat
band along K-H as shown in Supplementary Fig. S1, which is missing in ARPES.
Figure 3i compares the projected Fermi surfaces on $k_{z}=0$ plane. We find a
better agreement between ARPES and DFT+DMFT. In particular, both have larger
Fermi surfaces around some K points than those of DFT (red lines).

These band features depend sensitively on the Hund’s rule coupling as
discussed in detail in Supplementary Sec. B and Fig. S2. For a slightly larger
$J=0.6\,$eV, the flat band around M moves further away from the Fermi energy
to about -0.1 eV, and many other features also deviate from ARPES
observations. This highlights the importance of the Hund’s rule coupling via
tuning the spin states of the Mn-ions, and explains why DFT fails to yield the
correct band structures even with a large renormalization factor.

Figure 4: Electronic structures of Mn3Ge. a The spin moment of Mn-ions in
Mn3Ge as a function of the Hund’s rule coupling $J$ calculated using DFT+DMFT
for fixed $U=6\,$eV at 300 K. b The renormalization factor $Z$ extracted from
the DFT+DMFT self-energies with varying $J$ at $U=6\,$eV. c The spectral
function along the high symmetry line L-A-$\Gamma$-K-M-K-M’-A at $J=0.62\,$eV,
$U=6\,$eV, $T=300\,$K.

Correlation effects in Mn3Ge. To further support our calculations, we also
investigated the effects of the Hund’s rule coupling in Mn3Ge. Figure 4a shows
the variation of the Mn spin moment as a function of $J$ for a fixed $U=6\,$eV
from DFT+DMFT calculations for the triangular antiferromagnetic configuration.
Again, we find a critical $J_{c}\approx 0.58\,$eV, above which the spin moment
first grows rapidly and then gradually saturates to about 4
$\mu_{\text{B}}$/Mn at $J=1\,$eV. The slightly larger $J_{c}$ suggests that
the Mn ions are harder to be polarized in Mn3Ge than in Mn3Sn, consistent with
the smaller moment length estimated by spherical neutron polarimetry 3; 33.
The band renormalization factor $Z$ is shown in Fig. 4b and varies only
gradually from 0.5 to 0.8, indicating weak electronic correlations in Mn3Ge.
At $J=0.62\,$eV close to the observed spin moment 2.65 $\mu_{\text{B}}$/Mn 33,
our DFT+DMFT calculations also result in a small $Z^{-1}\approx 1.6$.

For comparison with ARPES, we plot in Fig. 4c the DFT+DMFT spectral function
along a high symmetry path. Flat bands are clearly seen around -0.05 eV along
the K-M path and bend steeply downwards halfway along K-$\Gamma$. Around
$\Gamma$, there are additional dispersive bands bending down towards K and A
below -0.38 eV. All these features are in good consistency with latest ARPES
measurements 21 without any artificial renormalization. The overall
consistency for both Mn3Ge and Mn3Sn validates our systematic DFT+DMFT
calculations for these two isostructural compounds.

Figure 5: Effect of the Hund’s rule coupling on the Weyl points in Mn3Sn. a
Illustration of the selected $\mathbf{k}$-paths marked by dashed lines of
different colors on the $k_{z}=0$ plane of the Brillouin zone. b The spectral
function along the high symmetry line L-A-$\Gamma$-K-M-K-M’-A, where
$\Gamma$-K-M-K-M’ is denoted by the red dashed line in a and the arrows mark
the band crossings or gaps. c Comparison of the spectral functions along
$\Gamma$-K-M (orange dashed line) and the line shifted by $\Delta
k_{y}=-0.072\,(2\pi/a)$ (green dashed line). Note that the orange $\Gamma$-K-M
line is different from the red one because of the lack of six-fold rotational
symmetry. d Comparison of the spectral function along K-M’-K (cyan dashed
line) and the line shifted by $\Delta k_{1}=-0.072\,(2\pi/a)$ (pink dashed
line). e,f The band maps on the $k_{z}=0$ plane at $E=9\,$meV for $J=0.6\,$eV
and $E=50\,$meV for $J=0.55\,$eV, illustrating the evolution of the band
crossings with the Hund’s rule coupling.

Tuning the Weyl points. Weyl points in Mn${}_{3}X$ were predicted to be
present near the Fermi energy in DFT and may contribute significantly to the
AHE 8; 9; 10. Their presence in Mn3Sn is supported by the observation of a
large anomalous Nernst effect beyond that in ferromagnetic metals 40 and the
chiral anomaly in magnetotransport41. To see if they might be affected, we
study in detail the DFT+DMFT band structures for $U=6\,$eV and $J=0.55\,$eV
along different paths in the Brillouin zone marked by dashed lines of
different colors in Fig. 5a. Figure 5b shows several crossing points of non-
degenerate linear bands along the high symmetry path L-A-$\Gamma$-K-M-K-M’-A
(red dashed line): two at about -30 meV between K and M marked by blue arrows,
two at about 13 meV marked by green arrows, and one at about 8 meV marked by
the purple arrow. These crossing points are candidate Weyl points to be
examined in more details. In addition, there are two small gaps (red arrows)
near K along the $\Gamma$-K and K-M’ paths. We first focus on the dispersion
near K and M and take the path along the $k_{x}$ direction. In Fig. 5c, it is
seen that among all three band crossings along $\Gamma$-K-M, only those around
the K point (red and blue arrows) become gapped when the path is moved
slightly by $\Delta k_{y}=-0.072$, while the one around M (green arrows)
remains unchanged. These indicate that those marked by red and blue arrows are
Weyl points, while the one marked by green arrow is a part of line nodes. Note
that the $\Gamma$-K path (orange dashed line) in Fig. 5c is different from the
$\Gamma$-K path (red dashed line) in Fig. 5b. Along the K-M’-K path in Fig.
5d, the two crossings marked by purple arrows remain unchanged away from the
high symmetry path and are also parts of line nodes. For clarity, we mark the
positions of the Weyl points by red and blue dots in Fig. 5a. Both are located
around K and at about -30 meV below the Fermi energy, in contrast to the DFT-
predicted Weyl points above the Fermi energy, which evolve into line nodes in
our calculations. Similar gap closure and the presence of nodes near the Fermi
energy are also found in Mn3Ge, as pointed out by colored arrows in Fig. 4c.

To understand the origin of this evolvement, we compare the band maps of Mn3Sn
on the $k_{z}=0$ plane for $J=0.6$ and 0.55 eV. For $J=0.6\,$eV shown in Fig.
5e, two bands cross at 9 meV along K-M-K (red line) but become gapped slightly
off this path (blue line). They correspond to the Weyl points around K in DFT
calculations and exist here for $J$ beyond 0.6 eV. As $J$ decreases, the top
of the $\beta$ band and the bottom of the $\alpha$ band are renormalized
towards the Fermi energy, and their dispersions start to deviate from
linearity near the nodes. For $J=0.55\,$eV, these nodes are pushed to about 50
meV but remain unaffected under a slight shift from the high symmetry path, so
they are no longer Weyl points (Fig. 5f). In the meanwhile, the bands around K
point at -0.1 eV that bend down at $J=0.6\,$eV are lifted towards the Fermi
energy and renormalized to bend upwards at $J=0.55\,$eV. They eventually
intersect with the $\beta$ and $\gamma$ bands around K and produce the Weyl
points below the Fermi energy in Fig. 5c. Thus, these Weyl points start to
show up only below $J\approx 0.6\,$eV and move closer to the Fermi energy as
$J$ further decreases. They are not present in DFT calculations as shown in
Supplementary Fig. S1. Their sensitive dependence on the Hund’s rule coupling
may be attributed to the rapid change of the Mn spin moment as $J$ exceeds its
critical value $J_{c}$, which in turn changes the band dispersion as well as
the relative band shift of different spin components (Supplementary Fig. S2).
The sensitive tuning with the Hund’s rule coupling implies that the Weyl nodes
in Mn${}_{3}X$ systems are accidental and not protected by symmetry 22.

Discussion
To summarize, we have performed systematic electronic band structure
calculations of the inverse triangular antiferromagnets Mn${}_{3}X$ ($X=$ Sn,
Ge) using DFT+DMFT and found very different results compared to previous DFT
calculations. The Hund’s rule coupling is seen to play a key role for the
magnetic, electronic, and topological properties. It determines the local spin
state of Mn-ions, consequently the triangular AFM order, and the relative
shift of the bands. A proper choice of the Hund’s rule coupling can yield
excellent agreement with the ARPES data in many details, which is not possible
in previous DFT calculations even with an artificial band renormalization. In
particular, the Mn-$d$ electrons are found to be weakly correlated in both
compounds, with a much smaller renormalization factor than previously
estimated in Mn3Sn.

The different band structures may have an influence on experimental
interpretations. So far, direct experimental evidences are still lacking for
the exact locations of the Weyl points, so that previous analyses were mostly
based on DFT band structures, which may need to be reconsidered. For example,
it has been argued that increasing Mn content may introduce more $d$
electrons, shift the Fermi energy upwards to the DFT Weyl points, and hence
enhance the anomalous Hall conductivity and longitudinal magnetoconductance
13. However, as shown in Supplementary Fig. S4 for our self-consistent
DFT+DMFT calculations of Mn3+δSn1-δ with more Mn-$d$ electrons, the band
contacts above the Fermi energy around M point are located at around 8.2 meV
in Mn3.03Sn2.97 and 7.4 meV in Mn3.06Sn0.94, which move less than 1 meV
towards the Fermi energy and cannot account for the $60\%$ reduction of the
longitudinal magnetoconductance. It is therefore important to reexamine all
previous experimental interpretations based on incorrect DFT band structures
for an improved understanding of the topological properties and their
experimental tuning. In addition, our results predict that the Weyl nodes in
Mn${}_{3}X$ family systems may be sensitively tuned by controlling the size of
the spin moment, possibly also by pressure 32; 42.

Very recently, the Ga-rich Mn3Ga was reported to have the highest Néel
temperature and largest AHE in the Mn${}_{3}X$ compounds 43. The DFT seems to
yield correct order of magnitude of the AHE, but its predicted monotonic
increase with Mn vacancy was not observed in experiments. A unified
explanation of the exotic properties in the whole Mn${}_{3}X$ family is
desired based on correlated electronic structure calculations with a proper
treatment of the Hund’s rule coupling.

Methods

DFT calculations
The first-principles DFT calculations were performed by using the full-
potential linearized augmented plane-wave method implemented in the WIEN2k
package23. The Muffin-tin radii ($R_{\text{M}T}$) were set to 2.5 a.u. for Mn
and Sn, and the cutoff parameter of the basis was chosen such that
$R_{\text{M}T}K_{\text{max}}=8.5$. We took 10000 ${\bf k}$-points in the whole
Brillouin zone and used the generalized-gradient approximation with the
Perdew-Burke-Ernzerhof (GGA-PBE) exchange-correlation potential 44.

DFT+DMFT calculations
Non-spin-polarized DFT calculations with SOC were first performed using the
full potential code WIEN2k 23. Cluster DMFT calculations were then carried out
for the noncollinear AFM order of Mn3Sn and Mn3Ge, but only the onsite self-
energies were considered to avoid the negative sign problem as performed in
Ref. 45. The directions of the spin polarization of local orbitals on each Mn-
ion were fixed according to experimental observation illustrated in Fig. 1a.
The magnitudes of the spin moments were determined self-consistently and found
to differ slightly on inequivalent Mn-ions. Only their averages are shown in
the figure for simplicity. Both the Hubbard $U$ and the Hund’s rule coupling
$J$ were added to all five Mn-3$d$ orbitals. The hybridization expansion
continuous-time quantum Monte Carlo method (CTQMC) was chosen as the DMFT
impurity solver 46, and the spectral functions were obtained using the maximum
entropy method for analytic continuation of the self-energies 47.

Acknowledgements

We thank Satoru Nakatsuji, Kenta Kuroda, and Takeshi Kondo for generously
sharing their ARPES data. This work was supported by the National Key Research
and Development Program of China (Grant No. 2022YFA1402203), the National
Natural Science Foundation of China (NSFC Grant No. 11974397), and the
Strategic Priority Research Program of the Chinese Academy of Sciences (Grant
No. XDB33010100).

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