# X-ray Magnetic Circular Dichroism in Altermagnetic $\alpha$-MnTe

A. Hariki  Department of Physics and Electronics, Graduate School of
Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Nakaku, Sakai,
Osaka 599-8531, Japan  T. Yamaguchi  Department of Physics and Electronics,
Graduate School of Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho,
Nakaku, Sakai, Osaka 599-8531, Japan  D. Kriegner  Institute of Physics, Czech
Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6 Czech Republic  K. W.
Edmonds  School of Physics and Astronomy, University of Nottingham, Nottingham
NG7 2RD, United Kingdom  P. Wadley  School of Physics and Astronomy,
University of Nottingham, Nottingham NG7 2RD, United Kingdom  S. S. Dhesi
Diamond Light Source, Chilton OX11 0DE, United Kingdom  G. Springholz
Institute of Semiconductor and Solid State Physics, Johannes Kepler University
Linz, Altenbergerstr. 69, 4040 Linz, Austria  L. Šmejkal  Institut für Physik,
Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany  Institute of
Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6 Czech
Republic  K. Výborný  Institute of Physics, Czech Academy of Sciences,
Cukrovarnická 10, 162 00 Praha 6 Czech Republic  T. Jungwirth  Institute of
Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6 Czech
Republic  School of Physics and Astronomy, University of Nottingham,
Nottingham NG7 2RD, United Kingdom  J. Kuneš  Institute for Solid State
Physics, TU Wien, 1040 Vienna, Austria

###### Abstract

Altermagnetism is a recently identified magnetic symmetry class combining
characteristics of conventional collinear ferromagnets and antiferromagnets,
that were regarded as mutually exclusive, and enabling phenomena and
functionalities unparalleled in either of the two traditional elementary
magnetic classes. In this work we use symmetry and ab initio theory to explore
X-ray magnetic circular dichroism (XMCD) in the altermagnetic class. Our
results highlight the distinct phenomenology in altermagnets of this time-
reversal symmetry breaking response, and its potential utility for element-
specific spectroscopy and microscopy in altermagnets. As a representative
material for our XMCD study we choose $\alpha$-MnTe with the compensated
antiparallel magnetic order in which an anomalous Hall effect has been already
demonstrated both in theory and experiment. The predicted magnitude of XMCD
lies well within the resolution of existing experimental techniques.

Recent theoretical studies have identified magnetic crystals with
unconventional characteristics. On one hand, the crystal symmetries generate a
compensated antiparallel magnetic order. On the other hand, they enable time-
reversal ($\cal{T}$) symmetry breaking linear responses, such as the anomalous
Hall effect (AHE) Šmejkal  _et al._ (2022, 2022a, 2020); Samanta  _et al._
(2020); Naka  _et al._ (2020); Reichlová  _et al._ ; Hayami and Kusunose
(2021); Mazin  _et al._ (2021); Gonzalez Betancourt  _et al._ (2023); Naka
_et al._ (2022) or charge-spin conversion effects Šmejkal  _et al._ (2022a);
Naka  _et al._ (2019); González-Hernández  _et al._ (2021); Naka  _et al._
(2021); Ma  _et al._ (2021); Šmejkal  _et al._ (2022b, c), and strongly spin-
polarized electronic band structures Šmejkal  _et al._ (2022a, 2020); Ahn  _et
al._ (2019); Hayami  _et al._ (2019); Yuan  _et al._ (2020, 2021); Hayami  _et
al._ (2020); Reichlová  _et al._ ; Šmejkal  _et al._ (2022c); Mazin  _et al._
(2021); Liu  _et al._ (2022); Yang  _et al._ . Theoretical predictions have
gone beyond the previously established notion in which these
$\cal{T}$-symmetry breaking phenomena originate from a magnetization in
ferromagnets, or from a non-collinear order of spins on certain lattices of
magnetic atoms Šmejkal  _et al._ (2022). Ab initio calculations in RuO2 or
MnTe have identified large non-relativistic spin splittings in the band-
structure reaching an $\sim$ eV scale Šmejkal  _et al._ (2022a, 2020); Ahn
_et al._ (2019); Gonzalez Betancourt  _et al._ (2023). In contrast to
ferromagnets, and in line with the vanishing net magnetization in these
compensated collinear magnets, the sign of the spin splitting alternates
across the band structure. The alternating spin polarization in both real-
space crystal structure and momentum-space electronic structure suggested the
term altermagnetism Šmejkal  _et al._ (2022a, c).

Prompted by the theory predictions, the AHE has already been experimentally
observed and ascribed to the altermagnetic order in RuO2 or MnTe Feng  _et
al._ (2022); Gonzalez Betancourt  _et al._ (2023). The predictions of strong
spin currents, opening the prospect of robust writing and readout mechanisms
in stray-field-free ultra-fast memory devices Šmejkal  _et al._ (2022a);
González-Hernández  _et al._ (2021); Shao  _et al._ (2021); Šmejkal  _et al._
(2022b, c), have been supported by initial experiments in RuO2 Bose  _et al._
(2022); Bai  _et al._ (2022); Karube  _et al._ (2022).

Figure 1: Mn moments $\mathbf{m}_{A}$ and $\mathbf{m}_{B}$ in $\alpha$-MnTe
structure (with Te octahedra) for the studied Néel vector $\mathbf{L}$
orientations. The mirror plane $\mathcal{M}$ discussed in the text is marked
in blue. While in the right panel $\mathcal{M}$ is an element of the magnetic
symmetry groupk, in the left panel it is $\mathcal{MT}$ which leaves the
system invariant.

In this Letter we focus on the X-ray magnetic circular dichroism (XMCD) at the
Mn $L_{2,3}$ edge and demonstrate that, apart from the ferromagnetic or non-
collinear order Wimmer  _et al._ (2019); Kimata  _et al._ (2021), it can also
originate from the collinear altermagnetic order. Both AHE, the transverse
electric-current response to an applied electric bias, and XMCD, the
difference between absorption of left and right circularly polarized X-rays,
are given by the Hall vector ${\bf h}=(\sigma_{zy},\sigma_{xz},\sigma_{yx})$;
here $\sigma_{ij}=-\sigma_{ji}$ are the antisymmetric components of the
(frequency dependent) conductivity tensor Šmejkal  _et al._ (2022); Wimmer
_et al._ (2019). AHE and XMCD are, therefore, governed by the same symmetry
principles. Microscopically, they provide complementary information since XMCD
involves, besides the valence electronic states, also the atomic core states.

Before presenting our microscopic calculations of XMCD in $\alpha$-MnTe, we
summarize key symmetry considerations relevant to altermagnets. We start from
non-relativistic spin symmetries which consist of direct products of
transformations in decoupled real (orbital) space and spin space and classify
all crystals with collinear magnetic order into the following three mutually
exclusive symmetry types Šmejkal  _et al._ (2022a, c): First, a ferromagnetic
(ferrimagnetic) class has one spin lattice or opposite-spin sublattices not
connected by any symmetry transformation, i.e., there is no symmetry combining
a 180∘ spin-space rotation (${\cal C}_{2}^{S}$) with any real-space symmetry
transformation. Next, an antiferromagnetic class has opposite-spin sublattices
connected by a real-space translation or inversion Šmejkal  _et al._ (2017);
Elmers  _et al._ (2020). Finally, an altermagnetic class has opposite-spin
sublattices connected by a real-space rotation (proper or improper and
symmorphic or non-symmorphic), but not connected by a translation or
inversion. Unlike the ferromagnetic class with a non-relativistic
magnetization and spin-polarized bands that break $\cal{T}$-symmetry, and
unlike the antiferromagnetic class with non-relativistic zero net
magnetization and spin-unpolarized $\cal{T}$-invariant bands, altermagnets
have non-relativistic zero net magnetization combined with spin-polarized
bands that break $\cal{T}$-symmetry. The non-relativistic altermagnetic
symmetries thus suggest the presence of $\cal{T}$-symmetry breaking responses
analogous to ferromagnets. Simultaneously, the spin symmetries point to
qualitative difference in the phenomenology of these responses between
ferromagnets and the intrinsically anisotropic (even-parity $d$, $g$ or
$i$-wave) altermagnets Ahn  _et al._ (2019); Šmejkal  _et al._ (2022a, c).

Besides the above non-relativistic spin symmetries that distinguish the three
collinear magnetic classes, there are spin symmetries common to all collinear
magnets. First, it is an arbitrary rotation of the spin space around the
direction of ordered moments. Second, all collinear (co-planar) magnets share
a ${\cal C}_{2}^{S}\cal{T}$ non-relativistic spin symmetry combining the spin-
space inversion, i.e. the $\cal{T}$ transformation, with the ${\cal
C}_{2}^{S}$ spin-space rotation around an axis perpendicular to the ordered
moments. In the absence of the relativistic spin-orbit coupling (SOC), the
Hall vector is invariant under arbitrary spin-space rotations, including the
${\cal C}_{2}^{S}$ rotation. Since collinear magnets are invariant under the
above non-relativistic ${\cal C}_{2}^{S}\cal{T}$ spin-space symmetry, the Hall
vector must be also invariant under this symmetry. Combined with the
invariance under the spin-space rotation, the Hall vector is forced to be
$\cal{T}$-invariant, i.e., to vanish in the absence of SOC.

In relativistic physics, the real space and the spin space are coupled
eliminating pure spin rotations as well as the ${\cal C}_{2}^{S}\cal{T}$ spin-
space symmetry. In ferromagnets, the Hall vector is then always allowed in the
presence of SOC because both the magnetization and the Hall vector transform
as $\cal{T}$-odd axial vectors Šmejkal  _et al._ (2022c).

Figure 2: Mn $L_{2,3}$ XAS experimental spectrum (top, blue) and the LDA+DMFT
AIM calculation (bottom, blue). The XMCD predicted by the LDA+DMFT AIM for
${\bf L}\parallel[1\overline{1}00]$ (red). Here XMCD is given by $\sigma_{yx}$
(${\bf k}\parallel\mathbf{c}\perp{\bf L}$).

In altermagnets, the symmetry lowering by SOC also enables the presence of the
Hall vector, but not necessarily for all directions of the Néel vector. This
is because the Néel vector may or may not transform as a $\cal{T}$-odd axial
vector, depending on its crystal direction Šmejkal  _et al._ (2022c). In the
latter case, the relativistic symmetry lowering may not be sufficient to allow
for the $\cal{T}$-odd axial Hall vector Šmejkal  _et al._ (2022a, c). This
highlights the distinct phenomenology of AHE and XMCD in altermagnets compared
to ferromagnets.

We now proceed to the analysis of the XMCD in MnTe. A schematic crystal of
$\alpha$-MnTe with NiAs structure (crystallographic space group $P6_{3}/mmc$
#194 vil ) is shown in Fig. 1. The magnetic moments on Mn have a parallel
alignment within the $c$-planes and an antiparallel alignment between the
planes. The face-sharing Te octahedra surrounding the Mn atoms break the
translation or inversion symmetry connecting the opposite-spin sublattices,
but allow a non-symmorphic six-fold screw rotation ${\cal
C}_{6}^{R}t_{\nicefrac{{1}}{{2}}}$ to connect them Šmejkal  _et al._ (2022c);
Gonzalez Betancourt  _et al._ (2023) rendering $\alpha$-MnTe an altermagnet.

The X-ray absorption spectrum (XAS) and XMCD at the Mn $L_{2,3}$ edge are
calculated using a dynamical mean-field theory (DMFT) + Anderson impurity
model (AIM) approach of Refs. Hariki  _et al._ , 2017, 2018, 2020. Starting
with a density functional calculation using Wien2k Blaha  _et al._ we
construct a multi-band Hubbard model Kuneš  _et al._ (2010); Mostofi  _et al._
(2014) spanning the Mn $3d$ and Te $5p$ bands. The Coulomb interaction within
the Mn $d$-shell is parametrized by $U=$ 5.0 eV and $J=$ 0.86 eV, and the
double-counting correction $\mu_{\text{dc}}=$ 22.5 eV. DMFT with these
parameters reproduces well the valence-band both in experiment Sato  _et al._
(1999) and theory Faria Junior  _et al._ (2023) as well as Mn $2p$ core-level
x-ray photoemission spectra Yi  _et al._ (2010), see SM sm .

In Fig. 2 we compare the calculated XAS with the experiment. The measurements
were performed on an epitaxial $\alpha$-MnTe thin film grown on a InP (111)
substrate by molecular beam epitaxy as described in Ref. Kriegner  _et al._ ,
2017. Samples were transported from the growth chamber to beamline I06 at
Diamond Light Source in a vacuum suitcase. The XAS measurements we performed
using linearly polarized light propagating along the $c$-axis. The
polarization of the light was along the $[11\bar{2}0]$ direction and the
measurement was performed at 200 K. Total electron yield detection was used to
measure the absorption signal. All features seen in the experimental spectra
are correctly reproduced; the only apparent exception is the multiplet feature
on the low-energy side of the $L_{3}$ absorption edge, see SM sm  for
discussion.

The main quantitative prediction of this work is in Fig. 2 showing the XMCD
spectrum calculated for the Néel vector
$\mathbf{L}=\mathbf{m}_{A}-\mathbf{m}_{B}$ parallel to $[1\bar{1}00]$. Here
$\mathbf{m}_{A(B)}$ is the sublattice $A(B)$ magnetization, and $[1\bar{1}00]$
is one of the three equivalent magnetic easy-axes in the $c$-plane Kriegner
_et al._ (2017); Gonzalez Betancourt  _et al._ (2023). The relativistic
magnetic point group for this Néel vector direction is $m^{\prime}m^{\prime}m$
Gonzalez Betancourt  _et al._ (2023). The screw axis ${\cal
C}_{2}t_{\nicefrac{{1}}{{2}}}$ and the corresponding symmetry-allowed Hall
vector ${\bf h}=(0,0,\sigma_{yx})$ are along the crystal $c$-axis. In
contrast, for ${\bf L}$ parallel to an in-plane hard axis
$[2\bar{1}\bar{1}0]$, the corresponding magnetic point group $mmm$ implies
that all Hall vector components are zero, i.e., that a net XMCD signal is
excluded by symmetry. Note that experimentally, the Néel vector can be
reoriented within the $c$-plane by moderate magnetic fields Kriegner  _et al._
(2017).

Figure 3: The Mn site-resolved contributions to XMCD calculated by the
LDA+DMFT AIM for two Néel vectors $\mathbf{L}=$ $[1\bar{1}00]$ (left) and
$[11\bar{2}0]$ (right). The top and bottom panels correspond to
$\mathbf{k}\perp{\bf L}$ and $\mathbf{k}\parallel{\bf L}$, respectively. The
XMCD intensities are magnified by a factor indicated in panels.

Thanks to the localized nature of the core states, X-ray spectra are the sum
of contributions from distinct atomic sites, i.e., from the magnetic
sublattices $A$ and $B$ in the present case:
$\sigma_{\alpha\beta}=\sigma^{A}_{\alpha\beta}+\sigma^{B}_{\alpha\beta}$. For
Néel vector directions in the $c$-plane of $\alpha$-MnTe, the sublattices $A$
and $B$ are connected by the relativistic ${\cal
C}_{2}t_{\nicefrac{{1}}{{2}}}$ symmetry, which leads to
$\sigma^{A}_{yx}=\sigma^{B}_{yx}$, $\sigma^{A}_{zy}=-\sigma^{B}_{zy}$ and
$\sigma^{A}_{xz}=-\sigma^{B}_{xz}$.

The calculated site-resolved contributions to XMCD for $\mathbf{L}$ along
$[1\bar{1}00]$ and $[11\bar{2}0]$ are shown in Fig. 3. We associate the
$x$-axis of the conductivity tensor with the direction of ${\bf L}$. For both
Néel vector directions, the two sublattices taken separately yield large local
XMCD signals from $\sigma^{A(B)}_{zy}$ (i.e. for X-ray propagation vector
$\mathbf{k}\parallel\mathbf{L}$), which cancel each other exactly when summed
over. For XMCD given by $\sigma_{yx}$ (X-ray propagation vector
$\mathbf{k}\perp\mathbf{L}$), the identical contributions $\sigma^{A(B)}_{yx}$
of the two sublattices are finite for $\mathbf{L}\parallel[1\bar{1}00]$, and
add up constructively. For $\mathbf{L}\parallel[11\bar{2}0]$,
$\sigma^{A(B)}_{yx}=0$ by symmetry.

The Onsager relations for the conductivity Šmejkal  _et al._ (2022), and
correspondingly for XMCD, imply that XMCD(L,m)=$-$XMCD($-$L,$-$m). For
$\mathbf{L}\parallel[1\bar{1}00]$ in $\alpha$-MnTe, a net magnetization ${\bf
m}=\mathbf{m}_{A}+\mathbf{m}_{B}$ due to canting of the sublattice moments
towards the $c$-axis is allowed by the relativistic symmetry. It was estimated
to be extremely small $\leq 2\times 10^{-4}$ $\mu$B per Mn from ab initio
calculations, and remained experimentally undetectable in the thin-film MnTe
epilayers Gonzalez Betancourt  _et al._ (2023). This implies a nearly precise
relationship, XMCD(L)=$-$XMCD($-$L).

Figure 4: The single-site XMCD for $\mathbf{k}\parallel{\bf L}$ (left) and
$\mathbf{k}\perp{\bf L}$ (right) for models (i)-(iv) described in the text.
The XMCD intensities are magnified by a factor indicated in panels. Note that
here the Mn atomic model is employed.

While XMCD and AHE follow the same symmetry rules, they may originate from
different terms in the Hamiltonian and thus have different magnitudes or scale
differently upon changing the material parameters. An example of such a
behavior is different magnitudes of MCD in optical and x-ray regimes related
to their origin in valence-band and core-level SOC, respectively.

For $\mathbf{L}\parallel[1\bar{1}00]$ in $\alpha$-MnTe, and the corresponding
net XMCD given by $\sigma_{yx}$, it is instructive to analyze the single-site
XMCD contributions given by $\sigma^{A(B)}_{yx}$ ($\mathbf{k}\perp\mathbf{L}$)
and $\sigma^{A(B)}_{zy}$ ($\mathbf{k}\parallel\mathbf{L}$), although the sum
over the two sublattices of the latter vanishes as discussed above. In Fig. 4
we consider four different settings: i) full Hamiltonian, ii) no valence $3d$
SOC, iii) no core-valence interaction beyond monopole (e.g. no spin exchange),
iv) a combination of (ii) and (iii).

For $\mathbf{k}\parallel\mathbf{L}$, the valence $3d$ SOC plays a negligible
role. The core-valence multipole interaction changes the shape of the single-
site XMCD spectra, but does not change the magnitude of the effect. For
$\mathbf{k}\perp\mathbf{L}$, we find a moderate difference between (i) and
(ii). The single-site XMCD signal is substantially suppressed in (iii) and
vanished completely in (iv). This shows that the role of the valence $3d$ SOC
is marginal, while the core-valence multipole interaction is crucial for XMCD
in $\alpha$-MnTe.

The order of magnitude difference between the single-site XMCD for
$\mathbf{k}\parallel\mathbf{L}$ and $\mathbf{k}\perp\mathbf{L}$, as well as
their qualitatively different behavior in case (iv), suggest a different
microscopic origin. The single-site XMCD for $\mathbf{k}\parallel\mathbf{L}$
corresponds to the conventional XMCD known from ferromagnets, which arises
from the spin polarization of the valence states and SOC in the core states
while the role of valence $3d$ SOC and core-valence interaction is marginal
111The core-valence interaction changes the shape of XMCD, primarily to due
change of XAS..

To microscopically understand the XMCD for $\mathbf{k}\perp\mathbf{L}$ in
$\alpha$-MnTe, we have to answer the question: What is the symmetry origin of
vanishing of XMCD in case (iv)? We use the atomic model for sake of
simplicity, nevertheless, the same symmetry arguments apply to the full
Hamiltonian of $\alpha$-MnTe. The terms corresponding to case (iv) are
included in

$\begin{split}\hat{H}^{0}_{\text{at}}&=\epsilon_{c}\sum_{m,\sigma}\hat{p}^{\dagger}_{m\sigma}\hat{p}^{\phantom{\dagger}}_{m\sigma}+\\!\\!\sum_{m,m,\sigma,\sigma^{\prime}}\\!h^{\text{(2p)}}_{m\sigma,m^{\prime}\\!\sigma^{\prime}}\hat{p}^{\dagger}_{m\sigma}\hat{p}^{\phantom{\dagger}}_{m^{\prime}\\!\sigma^{\prime}}\\\
&+\sum_{m,m^{\prime},\sigma}h^{(\text{CF})}_{mm^{\prime}}\hat{d}^{\dagger}_{m\sigma}\hat{d}^{\phantom{\dagger}}_{m^{\prime}\\!\sigma}+b\sum_{m,\sigma}\hat{d}^{\dagger}_{m\sigma}\hat{d}^{\phantom{\dagger}}_{m\,-\\!\sigma}\\\
&+U_{pd}\hat{n}_{p}\hat{n}_{d}+\sum_{\begin{subarray}{c}i,j,k,l\\\
\sigma,\sigma^{\prime}\end{subarray}}u^{\sigma\sigma^{\prime}}_{ijkl}\hat{d}^{\dagger}_{i\sigma}\hat{d}^{\dagger}_{j\sigma^{\prime}}\hat{d}^{\phantom{\dagger}}_{k\sigma^{\prime}}\hat{d}^{\phantom{\dagger}}_{l\sigma}.\end{split}$
(1)

The remaining terms to complete the full atomic-model are

$\begin{split}\hat{H}^{1}_{\text{at}}=&\\!\sum_{\begin{subarray}{c}i,j,k,l\\\
\sigma,\sigma^{\prime}\end{subarray}}w^{\sigma\sigma^{\prime}}_{ijkl}\hat{d}^{\dagger}_{i\sigma}\hat{p}^{\dagger}_{j\sigma^{\prime}}\hat{d}^{\phantom{\dagger}}_{k\sigma^{\prime}}\hat{p}^{\phantom{\dagger}}_{l\sigma}+\\!\\!\sum_{\begin{subarray}{c}m,m^{\prime}\\\
\sigma,\sigma^{\prime}\end{subarray}}h^{\text{(3d)}}_{m\sigma,m^{\prime}\\!\sigma^{\prime}}\hat{d}^{\dagger}_{m\sigma}\hat{d}^{\phantom{\dagger}}_{m^{\prime}\\!\sigma^{\prime}}.\\\
\end{split}$

Here, the operators $\hat{p}^{\dagger}_{m\sigma}$ and
$\hat{d}^{\dagger}_{m\sigma}$, respectively, create an electron in a core
($2p$) and valence ($3d$) orbital with angular momentum projection $m$ and
spin projection $\sigma=\pm 1$. The magnetic order is represented by a Weiss
field $b$ 222In AIM the Weiss field is represented by a spin-dependent bath.,
chosen to point along the $x$-axis, without loss of generality.

The XMCD signal for $\mathbf{k}\perp\mathbf{L}$ (which is related to the
antisymmetric part of $\sigma_{xy}$) is obtained by the Fermi’s golden rule

$\begin{split}&F_{\pm}(\omega)=\sum_{f,i}\left|\expectationvalue{f|\hat{T}_{\pm}|i}\right|^{2}\delta\left(\omega-
E_{fi}\right)\\\
&\hat{T}_{\pm}\equiv\sum_{\sigma}\hat{T}^{\sigma}_{\pm}=\sum_{m,\sigma}\Gamma^{\phantom{n}}_{\pm
m}\hat{d}^{\dagger}_{m\pm
1\sigma}\hat{p}^{\phantom{\dagger}}_{m\sigma}+H.c.\end{split}$ (2)

where $E_{fi}=E_{f}-E_{i}$ is the excitation energy from the state $|i\rangle$
to state $|f\rangle$. The dipole operators $\hat{T}_{\pm}$, with real
coefficients $\Gamma_{m}$, describe absorption of circularly polarized light
propagating along the $c$-axis. The square of the matrix element in (2) has
the form

$\left|\expectationvalue{\hat{T}_{\pm}}\right|^{2}\\!=\\!\sum_{\sigma}\left|\expectationvalue{\hat{T}^{\sigma}_{\pm}}\right|^{2}\\!+\sum_{\sigma}\expectationvalue{\hat{T}^{\sigma}_{\pm}}\overline{\expectationvalue{\hat{T}^{-\sigma}_{\pm}}}.$
(3)

First, we use the local ${\cal C}_{3}$ rotation symmetry about the $c$-axis to
show that the second term in (3) does not contribute to $F_{\pm}(\omega)$.
Consider the $\mathcal{C}_{3}$ transformation acting only on the valence
orbital indices (valence SOC neglected), and on both the core orbital and spin
indices (core SOC included)
333${\mathcal{C}_{3}^{\phantom{1}}\hat{p}_{m\sigma}\mathcal{C}^{-1}_{3}=e^{-i(2m+\sigma)\tfrac{\pi}{3}}\hat{p}_{m\sigma}}$,
${\mathcal{C}_{3}^{\phantom{1}}\hat{d}_{m\sigma}\mathcal{C}^{-1}_{3}=e^{-im\tfrac{2\pi}{3}}\hat{d}_{m\sigma}}$
. This operation commutes with $\hat{H}^{0}_{\text{at}}$ and thus

$F_{\pm}(\omega)=\tfrac{1}{3}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{g\in\\{I,\mathcal{C}_{3},\mathcal{C}^{2}_{3}\\}}\sum_{f,i}\left|\expectationvalue{f|g\hat{T}_{\pm}g^{-1}|i}\right|^{2}\\!\delta\left(\omega\\!-\\!E_{fi}\right).$
(4)

The transformation of the dipole operator

$\mathcal{C}^{\phantom{1}}_{3}\hat{T}^{\sigma}_{\pm}\mathcal{C}^{-1}_{3}=\varepsilon\hat{T}_{\pm}^{\sigma}e^{-i\sigma\tfrac{\pi}{3}},$
(5)

introduces a spin-dependent phase shift ($\varepsilon$ is an overall phase
factor) arising from the phase difference between the valence spins (not
rotated) and the core spins (rotated). As a result the crossed spin term in
(3) drops out upon the summation in (4).

Next, we consider the role of $\mathcal{T}$. The $\mathcal{T}$-symmetry is
broken by the presence of the Weiss field $b$. However, since the valence spin
is coupled neither to the valence orbitals, nor to the core spin or orbitals,
the transformation
$\mathcal{T}^{\prime}\equiv\mathcal{T}\mathcal{C}_{2}^{S,3d}$
444${\mathcal{T}^{\prime}\hat{d}_{m\sigma}\mathcal{T}^{\prime-1}=(-1)^{m}\hat{d}_{-\\!m-\\!\sigma}}$,
${\mathcal{T}^{\prime}\hat{p}_{m\sigma}\mathcal{T}^{\prime-1}=(-1)^{m+\tfrac{\sigma-1}{2}}\hat{p}_{-\\!m-\\!\sigma}}.$
($\mathcal{T}$ combined with a $\mathcal{C}_{2}^{S,3d}$ rotation of the
valence spin) is an anti-unitary symmetry, see SM sm  for details, which
transforms the dipole operators as

$\mathcal{T}^{\prime}\hat{T}^{\sigma}_{\pm}\mathcal{T}^{\prime-1}=(-1)^{\tfrac{\sigma-1}{2}}\hat{T}^{\sigma}_{\mp}.$
(6)

As a result we can replace

$\expectationvalue{f|\hat{T}^{\sigma}_{+}|i}\rightarrow(-1)^{\tfrac{\sigma-1}{2}}\overline{\expectationvalue{f|\hat{T}^{\sigma}_{-}|i}}$
(7)

in the sum over eigenstates (2), which together with vanishing spin-crossed
terms in (7), leads to $F_{+}(\omega)=F_{-}(\omega)$, and thus zero XMCD.

Turning on either of the terms in $\hat{H}^{1}_{\text{at}}$ eliminates the
above symmetries, and the arguments for zero XMCD break down. However, for
$\mathbf{L}\parallel[11\bar{2}0]$, XMCD for the full Hamiltonian
$\hat{H}^{0}_{\text{at}}+\hat{H}^{1}_{\text{at}}$ still vanishes. Because of
the presence of a mirror plane $\mathcal{M}$ perpendicular to $\mathbf{L}$,
see Fig. 1, the Hamiltonian is invariant under $m\rightarrow-m$ and
$\sigma\rightarrow-\sigma$ transformations for both valence and core orbitals.
The $\mathcal{M}\hat{T}_{+}\mathcal{M}^{-1}=\hat{T}_{-}$ then implies
vanishing of XMCD for $\mathbf{L}\parallel[11\bar{2}0]$.

Although both AHE and XMCD are given by the anti-symmetric components of
(frequency-dependent) $\sigma_{\alpha\beta}$, they arise from different terms
in the Hamiltonian. AHE originates from SOC in the valence orbitals, i.e., the
same interaction responsible for example for magneto-crystalline anisotropy.
The electron-electron interactions tend to play a minor role for AHE Li and
Levchenko (2020), beyond establishing the magnetic order. XMCD in
$\alpha$-MnTe, on the other hand, shows little sensitivity to valence SOC, but
arises from a combination of core SOC and core-valence exchange interaction,
which affect the excited state containing a core hole. Only a minor
modification of the XMCD signal is observed when the relativistic effects in
the valence orbitals are completely neglected.

In conclusion, we have calculated XAS and XMCD at the Mn $L_{2,3}$ edge in a
prototypical altermagnet $\alpha$-MnTe. We have measured XAS and found a good
agreement with the numerical results, establishing the reliability of the
theoretical approach. The magnitude of calculated XMCD is well within the
resolution of present-day instruments. The effect follows the same symmetry
rules as established for AHE. In $\alpha$-MnTe, it is present for the
circularly polarized X-rays propagating along the $c$-axis and the Néel vector
in the plane perpendicular to the $c$-axis. Within the plane, the effect
vanishes for $\mathbf{L}\parallel[2\bar{1}\bar{1}0]$ and the other two
equivalent in-plane axes. Unlike AHE, SOC in the valence orbitals plays only a
marginal role for XMCD in $\alpha$-MnTe.

###### Acknowledgements.

We thank Jakub Železný, Satya Prakash Bommanaboyena and Anna Kauch for
discussions and critical reading of the manuscript. This work was supported by
JSPS KAKENHI Grant Numbers 21K13884, 21H01003, 23K03324, 23H03816, 23H03817
(A.H.), projects P30960-N27, I 4493-N (G.S.) and QUAST-FOR5249 project I
5868-N (J.K.) of the Austrian Science Fund (FWF), the Czech Academy of
Sciences, project LQ100102201 (D.K.) and project 22-22000M of the Czech
Science Foundation (GACR), and Johannes Gutenberg University Grant TopDyn
(L.Š).

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  * Note (1) The core-valence interaction changes the shape of XMCD, primarily to due change of XAS.
  * Note (2) In AIM the Weiss field is represented by a spin-dependent bath.
  * Note (3) ${\mathcal{C}_{3}^{\phantom{1}}\hat{p}_{m\sigma}\mathcal{C}^{-1}_{3}=e^{-i(2m+\sigma)\tfrac{\pi}{3}}\hat{p}_{m\sigma}}$, ${\mathcal{C}_{3}^{\phantom{1}}\hat{d}_{m\sigma}\mathcal{C}^{-1}_{3}=e^{-im\tfrac{2\pi}{3}}\hat{d}_{m\sigma}}$.
  * Note (4) ${\mathcal{T}^{\prime}\hat{d}_{m\sigma}\mathcal{T}^{\prime-1}=(-1)^{m}\hat{d}_{-\\!m-\\!\sigma}}$, ${\mathcal{T}^{\prime}\hat{p}_{m\sigma}\mathcal{T}^{\prime-1}=(-1)^{m+\tfrac{\sigma-1}{2}}\hat{p}_{-\\!m-\\!\sigma}}.$.
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