# A tensorial approach to ‘altermagnetism’

Paolo G. Radaelli  Clarendon Laboratory, Department of Physics, University of
Oxford, Oxford, OX1 3PU, United Kingdom<EMAIL_ADDRESS>
###### Abstract

I present a tensorial approach to the description of
$\mathbf{k}/-\mathbf{k}$-symmetric, non-relativistic splitting of electronic
bands in magnetic materials, which was recently given the name of
‘altermagnetism’. I demonstrate that tensors provide a general framework to
discuss magnetic symmetry using both spin groups and magnetic point groups,
which have often been contrasted in recent literature. I also provide a
natural classification of altermagnets in terms of the lowest-order tensorial
forms that are permitted in each of the 69 altermagnetic point groups. This
approach clarifies the connection between altermagnetism and well-known bulk
properties, establishing that the vast majority of altermagnetic materials
must also be piezomagnetic and MOKE-active, and provides a rational criterion
to search for potential altermagnets among known materials and to test them
when the magnetic structure is unknown or ambiguous.

##  I Introduction

In the past two decades, there has been a resurgence of interest in compounds
having electronic bands with lifted spin degeneracy, partly motivated by the
requirement of new materials for spintronics. In addition to spin polarisation
in ferromagnets, it is well known that spin degeneracy can be lifted even in
non-magnetic materials by the famous Rashba-Dresselhaus (R-D)
effect,Dresselhaus (1955); Rashba and Sheka (1959) which requires spin-orbit
coupling (SOC) and is therefore largest in the presence of heavy elements. The
R-D effect also requires the absence of inversion symmetry, Manchon  _et al._
(2015); Bihlmayer  _et al._ (2015) due either to the bulk crystal structure
being acentric or to symmetry breaking at interfaces.

In their 2020 pioneering paper, Yuan et al. Yuan  _et al._ (2020a) came to the
surprising realisation that spin degeneracy can also be lifted in some fully
compensated antiferromagnets (AFM) due to the interaction between electron
spins and the ‘effective Zeeman field’ (largely of magnetic exchange origin)
produced by ordered magnetic moments. This effect is clearly distinct from the
R-D effect (most notably, it is $\mathbf{k}/-\mathbf{k}$ symmetric) and,
crucially, does not require SOC, opening the possibility to observe large spin
splitting even in light-element compounds. In fact, using Density Functional
Theory (DFT), Yuan et al. demonstrated spin splitting at ‘atomic-like energy
scales’ in the light-element insulator MnF2,Yuan  _et al._ (2020a) which is a
fully compensated collinear AFM. Although insulators provide excellent proofs
of principle, applications of AFM-induced spin splitting for spintronics would
be greatly facilitated by the discovery of metallic systems displaying this
effect. Using DFT calculations, Šmejkal et al. demonstrated large spin
splitting in the absence of SOC in several metallic/semiconducting candidates,
including RuO2, CrSb and MnTe,Šmejkal  _et al._ (2022a, b) and coined the term
‘altermagnetism’ to mark the distinction between these materials, ordinary
spin-split ferromagnets and non-spin-split AFMs. This terminology has been
widely used in subsequent literature, which includes both theoreticalMcClarty
and Rau (2024); Cheong and Huang (2024) and experimentalBai  _et al._ (2023);
Reimers  _et al._ (2024); Osumi  _et al._ (2024); Krempaský  _et al._ (2024)
contributions. Although different authors employ somewhat different
definitions of altermagnetism, 111 Cheong et. al. define ‘Type-III
altermagnets’ as systems with symmetries including time reversal is a distinct
operator. For reasons explained at length in this paper, this can only give
rise to $\mathbf{k}/-\mathbf{k}$ _anti_ symmetric textures of the R-D type,
though magnetism can play a very important role in them. Here, I am not
considering Type-III altermagnetism and, more generally magnetism-induced R-D
type textures, though I am including a brief discussion of these effects in
sec. XIII . for the purpose of this paper ‘altermagnetic textures’ are defined
as being $\mathbf{k}/-\mathbf{k}$ symmetric and time-reversal odd (switching
sign between time-reversed magnetic domains).

Šmejkal’s emphasis was very much on _collinear_ metallic antiferromagnets, for
which spin-splitting effects should be stronger, DFT is much easier to
perform, and which may have advantages for certain spintronic applications.
However, more recently, Cheong and Huang showed that altermagnetism can also
be expected in many non-collinear magnets, which may present distinct
practical advantages and should not be excluded a priori.Cheong and Huang
(2024).

From the very beginning, symmetry analysis has been recognised as an essential
tool of this research fields, since the guiding principle to select
potentially altermagnetic materials must lie in their underlying symmetry
properties. Yuan et al. Yuan  _et al._ (2020a) proposed an approach based on
magnetic space groups (MSG), while Šmejkal et al. proposed the so-called spin
groups (SGs) Litvin (1977); Šmejkal  _et al._ (2022a); Liu  _et al._ (2022),
which allow for extra symmetries to be considered when spins and the lattice
are approximately decoupled (see below). Cheong propose a classification based
on magnetic point groups (MPGs) Cheong and Huang (2024), while Liu et al.,Liu
_et al._ (2023) employ little co-groups at specific points of the Brillouin
zone (BZ), and propose a classification that also includes higher-order
effects (quartic altermagnetism). 222 I specifically highlight the approaches
adopted by Cheong Cheong and Huang (2024) and Liu, Liu  _et al._ (2023)
because they are the closest to the one I will describe herein, though neither
makes an explicit connection with Cartesian/sperical tensors. Moreover, the
MPG classifications proposed by Cheong and Liu largely overlap with the one
proposed in the present paper.

The interest in altermagnetism in non-collinear magnetic structuresCheong and
Huang (2024) and in ‘weak-altermagnetic’ effects Krempaský  _et al._ (2024)
require an extension of the SG approach as employed in ref. Šmejkal  _et al._
, 2022a. 333Ref. Šmejkal  _et al._ , 2022a employs ‘binary’ SGs, i.e., SG with
2-element groups acting on spins (see below). Binary SG cannot be used for
non-collinear structures. Certain non-collinear structures can be described
either with more complex SGs Litvin (1977) or with so-called multi-colour
groups,Harker (1981), but, to my knowledge, a complete theory for their use
has not been developed thus far. See Appendix C for more details. MPGs are a
flexible tool to deal with these cases, especially since there is no reason to
expect that altermagnetic splitting should always be weak for non-collinear
structures. Intuitively, a classification based on MPGs should be entirely
adequate, since spin splitting ultimately results in anomalous _macroscopic_
properties (e.g., the anomalous spin Hall effect) that are subjected to MPG
symmetry constraints via the Neumann’s principle.444The Neumann-Minnigerode-
Curie Principle (NMC Principle) enables one to derive the selection rules for
the physical properties from the symmetry of the crystal in question. For
macroscopic properties, the symmetry in question is precisely the MPG of the
crystal. One should also remark that, unlike the widely used MPGs, magnetic
_space_ groups have been almost entirely superseded among magnetic structures
specialists by irreducible representations analysis ‘à la Bertaut’,Bertaut
(1968) with additional symmetries in spin spaces being dealt with using
Izyumov’s exchange multiplet theory Izyumov (1980) However, one should not
lose sight of the advantages of SGs when it comes to pseudo-symmetric
structures, i.e., when the crystal symmetry is only ‘slightly’ broken upon
magnetic ordering.555Perhaps the best example of this is ferromagnetic
ordering in cubic magnetic metals such as Fe or Ni. Strictly speaking, Fe
becomes rhombohedral below the Curie temperature, but the deviation from cubic
symmetry due to magneto-striction is extremely small ($\approx 10^{-6}$). It
seems fair to state that, in previous work, the link between MPGs, SGs and the
complexity of spin textures in reciprocal space has not been entirely
clarified.

Here, I propose an approach based the complete expansion of altermagnetic spin
textures in momentum space in terms of Cartesian and spherical tensors. This
approach can be equally applied to vectorial textures (described by MPGs) and
scalar textures (described by binary SGs, referred to simply as as SGs in the
remainder), though my emphasis will be on the former. I demonstrate that, to
any given order, the expansion of vectorial magnetic textures is described by
a single Cartesian tensor of odd rank, while R-D
($\mathbf{k}/-\mathbf{k}$-antisymmetric) vectorial magnetic textures are
described by even-ranked tensors. I also propose a natural classification
based on the lowest-order tensor forms allowed in a particular set of MPGs
(class), and show that the vast majority of altermagnetic point groups (66 out
of 69) display quadratic (rank-3) altermagnetism, with the remaining three
groups allowing quartic (rank-5) altermagnetism. For the case of collinear AFM
structures, I demonstrate that, in most cases, SG scalar textures correspond
to the component of the MPG vector magnetic textures along the direction of
the staggered magnetisation, while the MPG approach also produces a pattern of
‘weak altermagnetic’ textures in other directions. One notable exception is
represented by cubic groups, since the pseudo-cubic symmetry introduces
constrains between texture components that would otherwise be independent. For
these cubic groups, I provide a conversion table between SG and MPG textures
for various direction of the magnetic moments, so that one could remain
entirely within the MPG framework, generally the more familiar to the
magnetism community.

One important practical implication of the tensor approach is the connection
with macroscopic properties: materials displaying _quadratic_ altermagnetism
(66 out of 69 MPGs) must also allow the piezomagnetic effect and the magneto-
optical Kerr effect (MOKE) — two easily-testable phenomena that have been
known for several decades Dzialoshinskii (1957); Kahn  _et al._ (1969), and
for which extensive materials databases are available (for example, ref.
Gallego  _et al._ , 2016). Testing for the presence of these effects might
also facilitate the screening of candidate altermagnetic materials before more
complex experiments are performed to to measure AFM-induced spin splitting of
electronic bands directly.

This paper is organised as follows: in secs. II, III and IV, I present a
general tensorial treatment of spin textures in momentum space, which can be
applied to both altermagnetic and R-D-type textures. In sec. IV, I also
outline how a parallel treatment for can be performed for _vector_ and
_scalar_ textures, described by MPGs and binary SGs respectively, and show
that dealing with the latter is greatly facilitated if one employs time
reversal (via Shubnikov groups) instead of 2-fold rotations. In secs. V, VI
and VII, I establish which altermagnetic tensor forms are allowed by symmetry
and propose a classification of MPGs based on the lowest-rank altermagnetic
tensors. Additional symmetry constraints at special points in reciprocal space
are discusses in sec. VIII. Up to this point, no distinction is made between
strong and weak effects, in keeping with the classical Neumann approach. In
secs. IX and X, I discuss the special case of collinear structures and explain
how the dominant components of the spin textures, which are expected to be
parallel to the Néel vector, can be extracted from the tensors. In sec. XI, I
explain the necessity of employing SGs (which I do via tensorial analysis) for
systems that are crystallographically cubic and in which symmetry is only
broken upon magnetic ordering, whilst also discussing the application of spin
groups to cases in which the magnetic symmetry is very low. Numerous examples
in different symmetries are presented in sec. XII, including graphical
depictions of several types of spin textures. The paper is concluded by a
summary and general discussion (sec. XIII).

##  II The tensorial description

The starting point to develop a tensorial description of altermagnetism and,
more generally, of spin splitting of electronic bands, is the expansion in
Cartesian tensors of the reciprocal-space spin textures, i.e., the vector
field of spin states in momentum space. This semiclassical vector field, which
I will denote as $\mathbf{B}^{eff}(\mathbf{k},m)$, determines both the
direction of the spin quantisation axis and the strength of the spin
splitting, and depends on both on the wavevector $\mathbf{k}$ and on the band
index $m$. 666More rigorously, within the context of DFT, for a given
wavevector $\mathbf{k}$ and band index $n$ the spin texture is defined by the
vector field
$\mathbf{s}_{n\mathbf{k}}=\bra{\Psi_{n\mathbf{k}}}\bm{\sigma}\ket{\Psi_{n\mathbf{k}}}$,
where $\bm{\sigma}$ is a vector of Pauli matrices and the integral implied by
the $\bra{}$ and $\ket{}$ is over the real-space unit cell.

$B_{i}^{eff}(\mathbf{k},m)=\sum_{l=n}^{n+1}T^{(l)}_{i,\alpha\beta\gamma\dots}k_{\alpha}k_{\beta}k_{\gamma}\dots$
(1)

The Cartesian tensors $T^{(l)}_{i,\alpha\beta\gamma}$ depend both on
$k=|\mathbf{k}|$ and on the band index $m$. In eq. 1, repeated indices are
implicitly summed, and the tensor if fully symmetric over the Greek indices.

The expression in eq. 1 is directly derived from to the expansion of the spin
texture into spherical tensors, and is therefore _complete_ on each spherical
shell in momentum space (or sections thereof near the BZ boundary). In fact,
one can show (see Appendix A.1) that the the sum of two Cartesian tensors of
ranks $n$ and $n+1$ is identical to the expansion of the vector field
$B_{i}^{eff}$ onto a spherical basis up to $\Gamma^{1}\otimes\Gamma^{n}$,
where $\Gamma^{n}$ is the irreducible representation (irrep) of the group
$SO(3)$ with $L=n$. Note that higher-rank Cartesian tensors need not be
necessarily small — in fact, reproducing the spin texture, especially near the
BZ boundary, may require a high-order expansion. Nevertheless, as I will show
in the remainder, a very natural classification can be obtained by considering
only low-rank Cartesian tensors.

In the presence of rotational symmetry, the tensor forms in eq. 1 are
constrained by the requirement that the spin textures have at least the
symmetry of the MPG of the crystal, which implies that the tensors themselves
must be totally symmetric by all the MPG operations — in other words, the
tensors must transform like the totally symmetric irrep of the MPG (see
Appendix A.2).

Hence, for a given tensor rank, tensors at all values of $k$ have the same
general form (imposed by symmetry) and their $k$ dependence is described in
terms of a reduced number of scalar functions. Additional constraints will
occur at the $\Gamma$ point (zone centre), at band crossing points and at
special points at the boundary of the BZ with non-trivial little-group
symmetry.

When considering inversion (parity) and time reversal symmetries in addition
to proper rotations, in the expansion in eq. 1 one can immediately distinguish
between _even-rank tensors_ , which are time reversal even and parity odd, and
_odd-rank tensors_ , which are time-reversal odd and parity even. Once again,
it is important to emphasise that only two Cartesian tensor are required for a
truncated spherical tensor expansions, and that all effects allowed by lower-
rank tensor are _automatically included_ in higher-rank tensors of the same
parity. For example, the rank-2 tensor is at linear order in $\mathbf{k}$ and
corresponds to the usual Rashba-Dresselhaus (R-D) effect, while the rank-4
tensor includes both the linear and cubic R-D tensor etc. In keeping with
recent literature, the rank-3 tensor can be called ‘quadratic altermagnetic
tensor’ and includes the ordinary ferromagnetic spin splitting (rank 1), while
the next-highest order that is $\mathbf{k}/-\mathbf{k}$ symmetric is a rank-5
tensor, which includes both quadratic and quartic altermagnetism as well as
ferromagnetic spin splitting.

A completely parallel treatment to the one I just outlined can be performed
for _scalar_ parity even and time-reversal-odd textures in real/reciprocal
space, which are used to describe collinear magnetic structures within the SG
framework. This becomes completely transparent if one casts binary SGs in the
slightly different language of time reversal rather than two-fold rotations —
see Appendix C for a complete discussion). The corresponding tensors are
totally symmetric and their rank is lowered by one with respect tho the
corresponding vectorial textures.

##  III Conditions for altermagnetism and the Rashba-Dresselhaus effect

Altermagnetism thus defined and for all odd ranks requires the following to be
satisfied:

  1. 1.

Time reversal symmetry must be broken — in other words, the effect is only
allowed in ferromagnets (FM) and some antiferromagnets (AFM).

  2. 2.

For a given point $\mathbf{k}$ in the BZ, the spin splitting must reverse in
the time-reversed FM or AFM domain.

  3. 3.

Spin splitting must be _symmetric_ between $\mathbf{k}$ and $-\mathbf{k}$

These conditions were thoroughly discussed in ref. Yuan  _et al._ , 2020a, and
exactly mirror those for the R-D effect at any even-rank tensor:

  1. 1.

Inversion symmetry must be broken.

  2. 2.

If the material is magnetic, for a given point $\mathbf{k}$ in the BZ, the
spin splitting must be the same in the time-reversed FM or AFM domains.

  3. 3.

Spin splitting must be _anti-symmetric_ between $\mathbf{k}$ and $-\mathbf{k}$

Note that spin splitting is not possible at any order in the presence of $PT$
symmetry (i.e., if the product of inversion and time reversal, $\theta I$, is
a symmetry operator)— a well-known result that can be obtained from simple
symmetry considerations. Tang  _et al._ (2016) Nevertheless, altermagnetism is
_not_ incompatible with the magneto-electric effect, tough the latter is often
associated with $PT$ symmetry conservation. 777An example of this is MPG
$3m^{\prime}$, which is admissible (i.e., compatible with ferromagnetism),
polar, magneto-electric and altermagnetic.

##  IV Jahn symbols and the connection with macroscopic properties

Further progress can be made by defining the so-called Jahn symbol for these
tensors,Jahn (1949) which for a tensor of rank $n$ in eq. 1 is generically:

$ea^{n}V[V^{n-1}]$ (2)

where $e$ indicates that all these are pseudo-tensors (with opposite parity to
that of ordinary tensor, which are parity-even for even ranks and parity-odd
for odd ranks), $a$ defines the time reversal properties ($a^{n}$ is time-
reversal-even when $n$ is even, odd when $n$ is odd) and the square brackets
indicate symmetrisation on all indices, since these are contracted with the
indices of $\mathbf{k}$.

The advantage of this notation is that one can sometimes connect these tensors
with tensors defining apparently unrelated _macroscopic_ properties that have
the same Jahn symbol. One can then mine the often extensive materials
databases for these known effects and extract candidate materials where the
‘novel’ effect is allowed by symmetry. For example, The Jahn symbol for the
linear R-D effect is $eV^{2}$, which is the same as the magnetotoroidic
tensor, while the Jahn symbol for the cubic R-D effect is $eV[V^{3}]$, which
does not have an obvious macroscopic counterpart. Most relevant for this
paper, the quadratic altermagnetic tensor $eaV[V^{2}]$ has the same Jahn
symbol as the tensors describing the piezomagnetic effect and the MOKE effect.
Piezomagnetism has been known since the 1950s, Dzialoshinskii (1957); Borovik-
Romanov (1960) and a symmetry classification can be found in the famous book
by Robert Birss Birss (1966), first published in 1964, which also lists
several materials now discussed in the context of altermagnetism. Interest in
MOKE activity in antiferromagnets and weak ferromagnets is almost as old,
dating back to the 1960’s, Kahn  _et al._ (1969); Smolenskiĭ  _et al._ (1975);
Zubov  _et al._ (1988); Zenkov  _et al._ (1989); Eremenko  _et al._ (1992),
and has recently experienced a resurgence. Higo  _et al._ (2018); Kang  _et
al._ (2022)

For _scalar_ altermagnetic textures, the corresponding Jahn symbols to those
in eq. 2 are $a[V^{n}]$, where $n$ is even. So, for example, the tensor for
scalar textures corresponding to quadratic altermagnetism has Jahn symbol
$a[V^{2}]$, one rank lower compared to the corresponding vectorial texture
$eaV[V^{2}]$ but with the same parity/time reversal properties.

##  V Tensor forms for quadratic altermagnetism

The allowed tensor forms are restricted by the MPG symmetry of the crystal, so
that the tensor itself is completely invariant by any of the MPG operations.
The number of free parameters in each tensor form corresponds to the number of
times the totally symmetric irrep is contained in the tensor representation.
In general a particular tensor form is shared by several MPGs, which can thus
be grouped into ‘classes’. For each MPG, the determination of the number of
free parameters via character decomposition and the construction of the
appropriate tensor form by projection is greatly facilitated by employing the
MTENSOR tool provided by the Bilbao Crystallographic Server. Gallego  _et al._
(2019) Tensors up to rank 6 were calculated using MTENSOR. For tensors above
rank 6, polynomial forms were obtained by the standard projection
(symmetrisation) method (see for example sec. XII.5).

Table LABEL:Table_1 lists the 17 unique symmetry-adapted quadratic
altermagnetic tensor forms, together with a set of the MPGs (class) that share
that tensor form, the number of free parameters and a simplified spherical
tensor decomposition (further discussed in section VI).888 I employed the
conventions adopted by the program MTENSOR of the Bilbao Crystallographic
Server (which are somewhat different from those in ref. Opechowski and
Guccione, 1965), except for $\bar{6}^{\prime}m^{\prime}2$, which has been
converted to $\bar{6}^{\prime}2m^{\prime}$ to be included in Class XIV. Note
that the textures of Classes X and XI differ by a $45^{\circ}$ rotation, and
are therefore equivalent at the MPG level, only distinguished by the
orientation of the texture with respect to the crystal axes — see further
discussion in section XII. In the interest of practical use and to follow
conventions, I kept these two classes separate. Straightforward axes
transformations may be required for particular MPGs I employed the usual
convention, which applies for example to the piezoelectric tensor, in which a
$3\times 6$ matrix is contracted with the array

$\left[k_{x}^{2},k_{y}^{2},k_{z}^{2},k_{y}k_{z},k_{x}k_{z},k_{x}k_{y}\right]$
(3)

to yield the three-component $\mathbf{B}^{eff}$, which depends on one or more
scalar functions $\Lambda_{ij}(k)$ that are constant on each spherical shell
in momentum space (for a note on axes conventions, see 999I adopted the
standard conventions for $x$, $y$, and $z$, which are related to the direction
of the symmetry directions. For example, for point group $3m1$, $z$ is
parallel to the 3-fold axis, $x$ is perpendicular to the mirror plane and $y$
completes the set. In the setting $31m$ of the same point group, $x$ and $y$
are interchanged. Examples on how to deal with conventions issues are provided
in section XII.). Examples of this procedure and on how to impose constraints
at special points of the BZ are provided in section XII.

Table LABEL:Table_1 also includes the single class of altermagnetic MPGs that
do not allow quadratic altermagnetism (Class XVIII), together with the
expression for their spin texture, described by a rank-5 spherical magnetic
hexadecapole (see discussion in section VII).

Among the 17 quadratic altermagnetic classes, eight classes (marked in bold
and with a (§) in Table LABEL:Table_1) allow a magnetic dipolar component,
i.e., they allow a net magnetic moment and uncompensated spin splitting,
though the latter need not arise primarily from and be parallel to the net
moment. The 31 magnetic groups in these classes are defined in ref. Cheong and
Huang, 2024 as ‘Type-I altermagnetic’, and coincide with the _admissible_
point groups, i.e., the ones allowing a net ferromagnetic moment. Out of the
remaining ten classes, 101010These classes are named ‘Type-II altermagnetic’
in ref. Cheong and Huang, 2024. six classes (VIII, XII, XIV, XV, XVII and
XVIII, marked in Table LABEL:Table_1 in italic and with an asterisk ∗) do not
allow the spin quantisation axis to be along an allowed collinear
antiferromagnetic direction, while the other four classes (V, IX, X and XI,
marked in Table LABEL:Table_1 with a dagger $\dagger$), do allow it. This
aspect will be discussed further in section IX.

##  VI Spherical tensor decomposition

In addition to the Cartesian tensor form, it is often useful to decompose
tensors onto a symmetry-adapted spherical basis, which makes the symmetry
constraints more transparent. Moreover, the spherical (multipole)
decomposition creates a bridge between MPG-based approaches and theories based
on multipoles, such as the Cluster Multipole Theory Suzuki  _et al._ (2017)
and the Landau approach to ferro-multipolar ordering proposed by P.
McClarty.McClarty and Rau (2024), 111111The connection with the latter is most
obvious, because ferro-ordering of a given multipolar form is allowed if and
only if this form is totally symmetric by the MPG operations. However, I
stress that here the symmetry-adapted spherical tensor forms were obtained by
direct projection onto the totally symmetric irrep of each MPG, rather than by
analysing the effect of each symmetry operators as it is done in ref. Suzuki
_et al._ , 2017.

As an example, I recall that the rank-2 R-D tensor can be decomposed into a
pseudoscalar ($L=0$), an ordinary polar vector ($L=1$) and a pseudo-
quadrupolar traceless tensor ($L=2$), all these being time-reversal even.
Consequently, the linear R-D effect is allowed in _chiral_ point groups (those
that allow a pseudoscalar), _polar_ point groups (which allow an ordinary
vector) and a few other non-centrosymmetric groups that only allow the pseudo-
quadrupolar traceless tensor (e.g., $\mkern
1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mum2$). When either of
these is allowed, these Cartesian tensors can be written as a linear
combination of symmetry-adapted spherical basis tensors.

In the case of quadratic altermagnetism, one can perform an entirely analogous
decomposition: let $\Gamma^{L}$ be the SO(3) irrep with dimensionality $2L+1$.
Then

$\displaystyle\left[V^{2}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{0}+\Gamma^{2}$ $\displaystyle V\left[V^{2}\right]$
$\displaystyle=$
$\displaystyle\Gamma^{1}\otimes\left(\Gamma^{0}+\Gamma^{2}\right)=2\Gamma^{1}+\Gamma^{2}+\Gamma^{3}$
(4)

Taking into account parity and time reversal, one concludes that the
altermagnetic tensor $eaV\left[V^{2}\right]$ decomposes into two magnetic
dipoles, a magnetic quadrupole and a magnetic octupole. Since the dipole irrep
occurs twice, there is some arbitrariness in the decompsition of the dipolar
field, but the natural choice is for one of the two components to be _parallel
to the dipole vector_ ($\tensorsym{D}^{I}\\!$, which is also allowed at the
lowest rank-1 order and represent the ordinary ferromagnetic uncompensated
spin splitting), while the other is _parallel to the wavevector_ $\mathbf{k}$
($\tensorsym{D}^{I\\!\\!I}$). The quadrupole terms yields a $\mathbf{B}^{eff}$
that is _perpendicular_ to $\mathbf{k}$, while octupolar terms always connect
different directions of $\mathbf{B}^{eff}$ .

With this choice and in the absence of any symmetry we can define the four
Cartesian tensors as:

$\displaystyle\tensorsym{D}^{I}$ $\displaystyle=$
$\displaystyle\delta_{ij}\delta_{\alpha\beta}v_{j}$ (8) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}v_{x}&v_{x}&v_{x}&0&0&0\\\
v_{y}&v_{y}&v_{y}&0&0&0\\\ v_{z}&v_{z}&v_{z}&0&0&0\\\ \end{array}\right)$

$\displaystyle\tensorsym{D}^{I\\!\\!I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta_{i\alpha}\delta_{\beta j}u_{j}$ (12)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}u_{x}&0&0&0&u_{z}&u_{y}\\\
0&u_{y}&0&u_{z}&0&u_{x}\\\ 0&0&u_{z}&u_{y}&u_{x}&0\\\ \end{array}\right)$

where $\mathbf{v}$ and $\mathbf{u}$ are the two dipolar components. The
quadrupolar tensor can be constructed from a traceless symmetric matrix
$S_{jk}$ as

$\displaystyle\tensorsym{Q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\epsilon_{i\alpha
k}\delta_{j\beta}+\epsilon_{i\beta k}\delta_{j\alpha}\right)S_{jk}$ (16)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&S_{23}&-S_{23}&-S_{11}-2S_{22}&-S_{12}&S_{13}\\\
-S_{13}&0&S_{13}&S_{12}&2S_{11}+S_{22}&-S_{23}\\\
S_{12}&-S_{12}&0&-S_{13}&S_{23}&S_{22}-S_{11}\\\ \end{array}\right)$

while the octupolar tensor $\tensorsym{O}$, defined as a linear combination of
$L=3$ tesseral harmonics, is:

$\tensorsym{O}=\left(\begin{array}[]{cccccc}-O_{122}-O_{133}&O_{122}&O_{133}&O_{123}&2O_{311}&2O_{211}\\\
O_{211}&-O_{211}-O_{223}&O_{223}&2O_{322}&O_{123}&2O_{122}\\\
O_{311}&O_{322}&-O_{311}-O_{322}&2O_{223}&2O_{133}&O_{123}\\\
\end{array}\right)$ (17)

In Appendix B, Table LABEL:Table_2, a symmetry-adapted spherical basis is
presented, comprising nine unique forms (two dipoles, three quadrupoles and
four octupoles) plus forms obtained by axis permutation. In each of the 17
quadratic altermagnetic classes, the unique tensor form can be expressed as a
linear combination of one or more symmetry-adapted spherical basis tensors, as
shown in Table LABEL:Table_3.

##  VII Quartic altermagnetism

Having classified all the MPGs that allow rank-3 (quadratic) altermagnetism,
one may wonder whether any higher-order altermagnetic point group has been
left out — in other words, are there symmetries where piezomagnetism and MOKE
activity are forbidden but higher-order altermagnetism is allowed? Having
excluded all paramagnetic point groups (where altermagnetism is not allowed at
any order) and groups that preserve $PT$ symmetry (where spin splitting is
entirely forbidden), only three MPGs are left: $432$, $\mkern
1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu3m$, and $m\mkern
1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mum$, which have been
included in Table LABEL:Table_1 as Class XVIII. All these MPGs allow quartic
(but not quadratic) altermagnetism and share the same one-parameter rank-5
tensor form, such that:

$B^{eff}_{i}=\Lambda(k)\epsilon_{ijl}(k_{j}^{3}k_{l}-k_{l}^{3}k_{j})$ (18)

The corresponding tensor is a pure $L=4$ (magnetic hexadecapole) spherical
tensor, since the $L=5$ spherical representation does not contain the totally
symmetric irrep of these point groups.

Including quartic altermagnetism will also affect the the classification of
the MPGs, since some of the classes will split on the basis of their rank-5
tensor forms. For example, the tetragonal and hexagonal groups in Classes
XIII, XV and XVI will split because they have different rank-5 tensor forms,
and Class XVIII will also split into two subclasses (this is indicated with
the letters (a) and (b) in Table LABEL:Table_1). However, _no further
splitting will occur at any higher tensor order_ , since the MPGs in each
class/subclass only differ by the proper/improper nature of some rotations,
which has no bearing on any parity-even tensor. Therefore, the classification
of altermagnetic groups in Table LABEL:Table_1 can be considered as complete.

##  VIII Symmetry in reciprocal space

Spin textures constructed with the tensorial method I just described have the
full symmetry of the MPG of the crystal, which, in turn, means that the field
at a given point in the _interior_ of the BZ will be locally symmetric by the
little co-group of that point, without any need for further symmetrisation.
Liu  _et al._ (2023) However, care must be taken at the zone centre and at
special points at the BZ boundary, which have additional symmetries due to the
fact that points related to them by some symmetry are also related by a
reciprocal lattice vector. Unlike the case of the R-D tensor and more
generally of even-ranked tensors, there is no general requirement that the
spin texture be zero at Kramers points (i.e., points for which $2\mathbf{k}$
is a reciprocal lattice vector). In general, spin splitting is allowed
everywhere if the dipolar term $\tensorsym{D}^{I\\!}$ is allowed, as is the
case for ordinary ferromagnets. Whenever dipolar terms are _not_ allowed, spin
splitting must vanish at the zone centre and at zone-boundary points with the
same little-group symmetry. In other cases, a reduced tensor may be obtained
by symmetrisation over the index that is not contracted with $\mathbf{k}$.
Some examples are given in section XII.

##  IX Collinearity

Collinearity is not strongly constrained by MPG symmetry, only being forbidden
in cubic MPGs. In all other cases, collinear ferromagnetic or
antiferromagnetic structures are allowed provided the following conditions are
met:

  1. 1.

The magnetic moment is either along the high-order rotation axis (hexagonal,
trigonal, tetragonal), along one of the 2-fold axes (orthorhombic) or either
parallel of perpendicular to the unique 2-fold axis (monoclinic). No special
condition exists for triclinic groups.

  2. 2.

The point-group symmetry of the magnetic site is _admissible_ , with the
admissible direction coinciding with one of the directions specified in point
1. above. This includes magnetic sites in a general position, since point
group $1$ is of course admissible.

It follows that collinear magnetic structures are allowed by symmetry in all
altermagnetic groups with the exception of the cubic ones (Classes XVII and
XVIII). However, as anticipated in section V, one must draw an important
distinction between classes that do not allow the spin quantisation axis to be
along a permitted collinear direction at the rank-3 tensor level (classes
VIII, XII, XIV, XV, for which I will use the adjective ‘weak-collinear’ and of
course XVII and XVIII, which do not admit collinearity) and those that do (the
‘strong-collinear’ classes V, IX, X and XI). In the latter case, spin
splitting in collinear structures could arise from a very similar effect as
for ordinary ferromagnets, in that spins travelling along certain
crystallographic directions would experience a net effective Zeeman field
originating from the ordered magnetic moments, the quantisation axis switching
sign depending on the wavevector direction. It is iimportant to distinguish
between true ferromagnets, where the magnetisation is uniform, and strong-
collinear ferri-antiferromagnets, where the internal magnetic field
distribution can never be entirely collinear, though one expects the effective
fields (mostly of exchange origin) to be strongest in the direction parallel
or antiparallel to the Néel vector. It is noteworthy that all MPGs allowing
ferromagnetism can trivially be strong-collinear in the ferromagnetic
direction, though some can also be strong-collinear in other directions as
well (see example XII.4).

By contrast, in the weak-collinear classes (VIII, XII, XIV, XV, XVII and
XVIII), the spin quantisation axis produced by the rank-3 tensor is not along
one of the allowed collinear antiferromagnetic directions. However, rank-3
spin splitting in these classes can still be very large for non-collinear
magnetic structures. All weak-collinear classes with the exception of Classes
XVII and XVIII (which cannot support collinear structures) become strong-
collinear at a higher tensor order — see sections X.

Examples of a strong-collinear and a weak-collinear class are given in section
XII with further discussed in section XIII.

##  X Strong-collinear higher-order altermagnetism

Strong-collinear altermagnetism will appear for all classes (except Classes
XVII and XVIII) even when it is forbidden by symmetry at the rank-3
(quadratic) level. Indeed, in their treatment employing SGs, Šmejkalet al.
Šmejkal  _et al._ (2022a) list a number of examples that require higher-order
tensors, so one needs to show that analogous results are obtained with MPGs.
Classes VIII, XII, XIV and XVb allow strong-collinear quartic (rank-5)
altermagnetism, while Class XVa only allows it at the next order (rank-7). The
expressions for the full tensors are naturally rather complex, but it is easy
to write the $\mathbf{B}^{eff}$ along the collinear direction $z$ (see example
in section XII.3 for more detail). The appropriate polynomials are reported in
Table 1. As shown in sec. XII, there is a close correspondence between these
forms and those listed in Šmejkalet al. Šmejkal  _et al._ (2022a), fig. 2,
demonstrating the equivalence between the MPG and SG approaches when it comes
to strong-collinear altermagnetism.

##  XI Cubic symmetry and the connection with spin groups

In several important cases (both collinear and non-collinear), the crystal
symmetry is only broken by the direction of the magnetic moments. A classic
case is that of hematite (Fe2O3),Morin (1959) wherethe 3-fold symmetry is
broken at room temperature but remains unbroken below the so-called Morin
transition ($\approx$260K). Since the magneto-elastic interaction is often
small, one needs to take into account the effect of the approximate crystal
symmetry (‘pseudo-symmetry’) on the spin textures. In the case of collinear
structures, this can be done effectively using SGs — a treatment that has been
presented in previous literature. Šmejkal  _et al._ (2022a, b) One needs to
establish to establish how the approach based on MPG needs to be modified to
take pseudo-symmetry into account.

For most collinear structures, the component of the effective field
$\mathbf{B}^{eff}$ along the staggered magnetisation can be readily identified
from the MPG treatment even in the presence of pseudo-symmetries (see sec.
XII.4 for a complete example). In the rare cases in which magnetic moments are
along a general direction, it is convenient to rotate them into a high-
symmetry direction and re-establish the MPG and the corresponding spin
texture. When the magneto-elastic interaction is weak, such high-symmetry
phase is almost always physical and can usually be reached by applying a small
magnetic field.

However, this approach is not viable in cubic symmetry, because, for collinear
phases, crystal symmetry can never be fully restored by any direction of the
staggered magnetisation, and it is precisely in these cases that SGs are most
useful. Lowest-order polynomial forms for all cubic SGs have been obtained
using scalar-field tensors, as discussed in sec. IV, and are listed in 2
together with the corresponding MPGs for magnetic moments along the three
cubic symmetry directions $[001]$, $[110]$ and $[111]$. When comparing these
forms with the corresponding strong-collinear forms of the same MPGs, one
notices that the former can be obtained from the latter by fixing some of the
parameters to have specific values. In other words, the pseudo-symmetry
manifests itself by establishing a link between effective-field parameters
that would otherwise be independent. This becomes clear when considering
specific cubic symmetries.

The cubic SGs $23$, $m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern
1.0mu$, $432$ $\mkern 1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu3m$
and $m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mum$ are all
ferro- (or ferri-) magnetic, and the lowest-order cubic polynomial form is
completely isotropic. The corresponding MPGs all admit a dipole component,
which is, however, not necessarily isotropic. For example, the polynomial form
for MPG $42^{\prime}2^{\prime}$ is
$\Lambda_{1}(k)(k_{x}^{2}+k_{y}^{2})+\Lambda_{2}(k)k_{z}^{2}$. Therefore, the
pseudo-cubic symmetry had the result of imposing
$\Lambda_{1}(k)=\Lambda_{2}(k)\;\forall k$, as stated.

The case of the three remaining cubic groups $\tilde{4}3\tilde{2}$,
$\tilde{{\mkern 1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern
1.0mu}}3\tilde{m}$ and $m\mkern
1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mu\tilde{m}$ is
considerably more interesting, and is discussed in more detail in sec. XII.5.
The lowest-order cubic polynomial form is of rank 7 (rank 6 for the
corresponding SGs), while the corresponding MPGs admit lower-order polynomial
forms (see tab. LABEL:Table_1 and 1). As always, these lower-order MPG forms
are contained in the rank-7 forms — for example, the trigonally-symmetric
(Class VIII) rank-5 form $\Lambda(k)(k_{x}^{2}-3k_{y}^{2})k_{x}k_{z}$ has
corresponding rank-7 components
$(k_{x}^{2}-3k_{y}^{2})k_{x}k_{z}\left(\Lambda_{1}(k)(k_{x}^{2}+k_{y}^{2})+\Lambda_{2}(k)k_{z}^{2}\right)$,
since $(k_{x}^{2}+k_{y}^{2})$ and $k_{z}^{2}$ are totally symmetric in
trigonal symmetry. However, in the presence of cubic pseudo-symmetry, these
components are linked together and with other rank-7 components, yielding the
pseudo-cubic rank-7 form displayed in tab. 2. All other trigonal components
only arise from the symmetry breaking due to the magnetic moment direction,
and are expected to be small for weak magneto-elastic interactions.

##  XII Examples

Here, I provide a set of examples to illustrate the practical use of this
method and in particular of Table LABEL:Table_1, and to highlight some of the
issues arising from different crystallographic conventions. All the examples
and references were generated using the program MAGNDATA from the Bilbao
Crystallographic Server. Gallego  _et al._ (2016) The figures display the
‘texture pattern’ as a normalised $\mathbf{B}^{eff}$ vector field, where the
normalisation function is $1/(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})^{(n-1)/2}$ where
$n$ is the rank of the tensor. These ‘texture patterns’ correspond to the
gnomonic projection of the textures, i.e., a projection from the centre of the
unit sphere to a plane tangent to it. This projection is most convenient to
display the symmetry (it preserves angles at the centre) and the details of
the textures. In these figures, $k_{x}$ and $k_{y}$ are in arbitrary
dimensionless units, and the conversion to spherical coordinates is
$\tan\theta=\sqrt{k_{x}^{2}+k_{y}^{2}}$, $\tan\phi=k_{y}/k_{x}$. There is a
close correspondence between these texture patterns and those displayed
schematically in fig. 2 of Šmejkal et al.Šmejkal  _et al._ (2022a) (more
details in the figure legends). An example of texture plotted on the surface
of a sphere in momentum space is shown in sec. XII.5.

###  XII.1 Class XIV (weak-collinear)

Class XIV comprises five hexagonal MPGs:
$\frac{6^{\prime}}{m^{\prime}}mm^{\prime}$, $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu^{\prime}2m^{\prime}$
(TmAgGe, ref. Baran  _et al._ , 2009), $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu^{\prime}m2^{\prime}$,
$6^{\prime}mm^{\prime}$ (HoMnO3, MSG $P6^{\prime}_{3}cm^{\prime}$, ref. Brown
and Chatterji, 2006) and $6^{\prime}22^{\prime}$. The expression for
$\mathbf{B}^{eff}(\mathbf{k})$ for this class at the rank-3 tensor level is:

$\displaystyle B_{x}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{11}(k)(k_{x}^{2}-k_{y}^{2})$ $\displaystyle
B_{y}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{11}(k)(-2k_{x}k_{y})$ $\displaystyle
B_{z}^{eff}(\mathbf{k})$ $\displaystyle=$ $\displaystyle 0$ (19)

the spin texture being parametrised by a _single_ scalar function of $k$,
$\Lambda_{11}(k)$. The spin quantisation axis is in the $xy$ plane and
therefore always perpendicular to the high-symmetry direction, i.e., the
allowed direction for the Néel vector. The spin texture is parallel to
$\mathbf{k}$ along the six directions $\Gamma-K$ and $\Gamma-K^{\prime}$,
while it is perpendicular to $\mathbf{k}$ along the $\Gamma-M$ lines (see ref.
Bradley and Cracknell, 2010 for the BZ point notation). Since the three $K$
points and the three $K^{\prime}$ points are equivalent, $\Lambda_{14}(k)$
must be $=0$ both at the $\Gamma$ point and at $k=|K|$. There is no additional
constraint at points $M$.

Class XIV is also useful to illustrate how to deal with different axes
orientations, which is essential to employ Table LABEL:Table_1 correctly. As a
preamble, one should remark that, in the presence of a crystal lattice, the
orientation of the point group directions is not arbitrary, but it related to
that of the crystal axes. So, for example, symbols $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu2m$ and $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mum2$ refer to the same
point group, being related by a 90∘ rotation of the in-plane axes. However,
$P\mkern 1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu2m$ and $P\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mum2$ are distinct space
groups, since the in-plane directions are now linked to the crystal axes.

The symbols, $6^{\prime}2^{\prime}2$, $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu^{\prime}m^{\prime}2$,
$\mkern 1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern
1.0mu^{\prime}2^{\prime}m$ (e.g., ThMn2, ref. Deportes  _et al._ , 1987 and
CsFeCl3, ref. Hayashida  _et al._ , 2018, both with MSG, $P\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu^{\prime}2^{\prime}m$,)
$6^{\prime}m^{\prime}m$ (e.g., YbMnO3, MSG $P6^{\prime}_{3}c^{\prime}m$, ref.
Fabrèges  _et al._ , 2008 ), and $\frac{6^{\prime}}{m^{\prime}}m^{\prime}m$
(e.g., CrSb, MSG $P\frac{6^{\prime}_{3}}{m^{\prime}}m^{\prime}c$, ref. Yuan
_et al._ , 2020b) refer to the _same_ point groups ($6^{\prime}22^{\prime}$,
$\mkern 1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern
1.0mu^{\prime}2m^{\prime}$, $\mkern
1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern 1.0mu^{\prime}m2^{\prime}$,
$6^{\prime}mm^{\prime}$, and $\frac{6^{\prime}}{m^{\prime}}mm^{\prime}$,
respectively), with the in-plane axes rotated by 90∘. Therefore, when dealing
with MSGs such as $P\mkern 1.0mu\overline{\mkern-1.0mu6\mkern-1.0mu}\mkern
1.0mu^{\prime}2^{\prime}m$ or $P\frac{6^{\prime}_{3}}{m^{\prime}}m^{\prime}c$,
one should employ the _rotated form of the tensor_ : 121212I did not create a
separate class for these symbols, since this tensor is obviously related to
that in Table LABEL:Table_1 and in order to avoid proliferation of classes.
However, the reader should be advised that simple transformations such as the
one in eq. 20 may be necessary, not only in this case but also to deal with
non-standard conventions (e.g., $2^{\prime}2^{\prime}2$ vs
$22^{\prime}2^{\prime}$, etc. in Class IV).

$\left(\begin{array}[]{cccccc}0&0&0&0&0&-2\Lambda_{22}\\\
-\Lambda_{22}&\Lambda_{22}&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{array}\right)$ (20)

The expression for $\mathbf{B}^{eff}(\mathbf{k})$ for this set being:

$\displaystyle B_{x}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{11}(k)(-2k_{x}k_{y})$ $\displaystyle
B_{y}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{11}(k)(k_{y}^{2}-k_{x}^{2})$ $\displaystyle
B_{z}^{eff}(\mathbf{k})$ $\displaystyle=$ $\displaystyle 0$ (21)

Among the compounds listed above, TmAgGe, HoMnO3, YbMnO3, CsFeCl3 and ThMn2
are reported to have non-collinear magnetic structures, while SbCr is
reportedly collinear.

Table 1:  Polynomial forms of the effective field component $z$, which in the listed classes is along the collinear staggered magnetisation (strong-collinear altermagnetism). Classes listed here are weak-collinear altermagnetic at the rank-3 (quadratic) level (see tab. LABEL:Table_1). Class | Rank | $B_{z}^{eff}(\mathbf{k})$
---|---|---
VIII | 5 | $\Lambda(k)(k_{x}^{2}-3k_{y}^{2})k_{x}k_{z}$
XII | 5 | $\left(\Lambda_{1}(k)(3k_{x}^{2}k_{y}-k_{y}^{3})-\Lambda_{2}(k)(3k_{x}k_{y}^{2}-k_{x}^{3})\right)k_{z}$
XIV | 5 | $\Lambda(k)(k_{x}^{3}-3k_{y}^{2}k_{x})k_{z}$
XVa | 7 | $\Lambda(k)(3k_{x}^{4}-10k_{x}^{2}k_{y}^{2}+3k_{y}^{4})k_{x}k_{y}$
XVb | 5 | $\Lambda(k)(k_{x}^{3}k_{y}-k_{y}^{3}k_{x})$

As discussed in sec. X, Class XIV is strong-collinear at the rank-5 (quartic)
level, the effective field along the $z$ axis (the allowed direction for the
collinear staggered magnetisation) being (see tab. 1):

$B^{eff}_{z}(\mathbf{k})=\Lambda(k)(k_{x}^{3}-3k_{y}^{2}k_{x})k_{z}$ (22)

Fig. 1 displays the corresponding texture patterns.

Figure 1:  In-plane, rank-3 (a) and out-of-plane, rank 5 (b) components of the
‘texture pattern’ (normalised $\mathbf{B}^{eff}$ vector field, corresponding
to the gnomonic projection of the spin texture — see text), for Class XIV
(e.g., MPG $6^{\prime}22^{\prime}$). The normalisation functions are
$1/(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})^{(n-1)/2}$, $n$ being the rank of the
tensor. Contours were plotted for $k_{z}=1$ and are symmetric/antisymmetric,
respectively for panels (a)/(b), by exchange $k_{z}\rightarrow-k_{z}$. Panel
(b) to be compared with fig. 2, sub-panel B4 of ref. Šmejkal  _et al._ ,
2022a.

###  XII.2 Classes X and XI (strong-collinear)

Classes X and XI are closely related, Note (4) because their tensor forms are
related by a $45^{\circ}$ rotation. However, these forms look sufficiently
different to make distinct classes useful in practice. The difference between
the two classes stems from the fact that in Class X, operators along the the
primary ($[001]$) and tertiary ($[110]$) symmetry directions are primed while,
for Class XI, operators along the the primary ($[001]$) and secondary
($[100]$) symmetry directions are primed.

The tensor form for Class X allows spin splitting along all three axes.
However, setting $\Lambda_{14}\approx 0$, one finds that the spin quantisation
axis is predominantly parallel or antiparallel to the $z$ axis, i.e., the
allowed Néel vector direction for collinear antiferromagnetism. The maximum
effective Zeeman field is for wavevectors in the $(110)$ and $(1\bar{1}0)$
directions, and the sign of the field switches between these two directions.

Class X comprises the MPGs $\mkern
1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu^{\prime}2m^{\prime}$
(e.g., Pb2MnO4, ref. Kimber and Attfield, 2007) and $4^{\prime}22^{\prime}$
(Er2Ge2O7, ref. Taddei  _et al._ , 2019), but also those corresponding to the
non-standard setting $\mkern 1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern
1.0mu^{\prime}2^{\prime}m$, $4^{\prime}mm^{\prime}$ and
$\frac{4}{m}^{\prime}mm^{\prime}$ (e.g., LiFe2F6, ref. Shachar  _et al._ ,
1972, and RuO2, ref. Berlijn  _et al._ , 2017, both with MSG
$P\frac{4^{\prime}_{2}}{m}nm^{\prime}$) . The characteristic spherical tensor
expansion for this class is:

$A(k)\,\tensorsym{Q}^{III}+B(k)\,\tensorsym{O}^{I}$ (23)

where $\tensorsym{Q}^{III}$ and $\tensorsym{O}^{I}$ are fixed tensors defined
in Table LABEL:Table_2. The spin texture is:

$\displaystyle B_{x}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{14}(k)k_{y}k_{z}$ $\displaystyle
B_{y}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{14}(k)k_{x}k_{z}$ $\displaystyle
B_{z}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{36}(k)k_{x}k_{y}$ (24)

In the primitive tetragonal cell (adopted by Pb2MnO4, Er2Ge2O7, LiFe2F6 and
RuO2), the four $M$ points have reciprocal-space coordinates
$(\frac{1}{2},\frac{1}{2},0)$, $(-\frac{1}{2},\frac{1}{2},0)$,
$(\frac{1}{2},-\frac{1}{2},0)$ and $(-\frac{1}{2},-\frac{1}{2},0)$ and are all
related by reciprocal lattice vectors. However, according to eq. XII.2,
$(\frac{1}{2},\frac{1}{2},0)/(\frac{1}{2},-\frac{1}{2},0)$ and
$(-\frac{1}{2},\frac{1}{2},0)/(\frac{1}{2},-\frac{1}{2},0)$ must have opposite
spin textures, which means that $\Lambda_{36}(k)=0$ for $k=k_{M}$. The in-
plane and out-of-plane spin-texture patterns, both of rank 3, are displayed in
fig. 2 .

Class XI comprises the standard-setting MPGs $\mkern
1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu^{\prime}2^{\prime}m$,
$4^{\prime}m^{\prime}m$, $\frac{4^{\prime}}{m}m^{\prime}m$ (e.g., the
pyrochlores Er2Ti2O7 and Er2Ru2O7, with MSG
$I\frac{4^{\prime}_{1}}{a}m^{\prime}d$, refs. Taira  _et al._ (2003); Poole
_et al._ (2007)), and also the non-standard-settings $\mkern
1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu^{\prime}2^{\prime}m$ and
$4^{\prime}2^{\prime}2$. The characteristic spherical tensor expansion for
this class is:

$A(k)\,\tensorsym{Q}_{z}^{I}+B(k)\,\tensorsym{O}_{z}^{II}$ (25)

where $\tensorsym{Q}_{z}^{I}$ and $\tensorsym{O}_{z}^{II}$ are fixed tensors
defined in Table LABEL:Table_2. The spin texture is rotated by 45∘ with
respect to Class X:

$\displaystyle B_{x}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{15}(k)k_{x}k_{z}$ $\displaystyle
B_{y}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\Lambda_{15}(k)k_{y}k_{z}$ $\displaystyle
B_{z}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{13}(k)(k_{x}^{2}-k_{y}^{2})$ (26)

Among the compounds listed above, Pb2MnO4 Er2Ge2O7, Er2Ti2O7 are reported to
have non-collinear magnetic structures, while LiFe2F6, Er2Ru2O7, and RuO2 are
reportedly collinear (but for RuO2, also see ref. Keßler  _et al._ , 2024).

Figure 2:  In-plane (a) and out-of-plane (b) components (both of rank 3) of
the texture pattern for Class X (e.g., MPG $4^{\prime}22^{\prime}$). The
normalisation function is $1/(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})$. Contours were
plotted for $k_{z}=1$ and are antisymmetric/symmetric, respectively for panels
(a)/(b), by exchange $k_{z}\rightarrow-k_{z}$.

###  XII.3 Class XV: higher-order strong-collinear altermagnetism

Class XV is weak-collinear at the rank-3 level, with

$\displaystyle B_{x}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda_{14}(k)k_{y}k_{z}$ $\displaystyle
B_{y}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-\Lambda_{14}(k)k_{x}k_{z}$ $\displaystyle
B_{z}^{eff}(\mathbf{k})$ $\displaystyle=$ $\displaystyle 0$ (27)

Subclasses XVa and XVb become strong-collinear at the rank-7 and rank-5 level,
respectively, the corresponding $B_{z}^{eff}$ components being listed in tab.
1. Figs. 3 and 4 display the corresponding texture patterns.

Figure 3:  In-plane, rank-3 (a) and out-of-plane, rank-7 (b) components of the
texture pattern for Class XVa (e.g., MPG $622$). The normalisation function is
as in previous figures. Contours were plotted for $k_{z}=1$ and are
antisymmetric/symmetric, respectively for panels (a)/(b), by exchange
$k_{z}\rightarrow-k_{z}$. Panel (b) to be compared with fig. 2, sub-panel P6
of ref. Šmejkal  _et al._ , 2022a. Figure 4:  In-plane, rank-3 (a) and out-of-
plane, rank-5 (b) components of the texture pattern for Class XVb (e.g., MPG
$422$). The normalisation function is as in previous figures. Contours were
plotted for $k_{z}=1$ and are antisymmetric/symmetric, respectively for panels
(a)/(b), by exchange $k_{z}\rightarrow-k_{z}$. Panel (b) to be compared with
fig. 2, sub-panel P4 of ref. Šmejkal  _et al._ , 2022a.

###  XII.4 Connection with spin groups

The classification in terms of MPGs is completely general even when a magnetic
space group cannot be defined, and is more flexible than the (binary) SG
approach employed in Refs. Šmejkal  _et al._ , 2022a; Liu  _et al._ , 2022
when dealing with non-collinear structures. I have shown several cases in
which the texture patterns generated along the collinear AFM direction using
the tensor approach are identical to those obtained using SGs. Nevertheless,
SGs are useful to show the connection between the spin textures of magnetic
structures related by spin rotations, and may also produce approximate
relations between tensor parameters that are unrelated at the MPG level via
the occurence of pseudo-symmetries (see also sec. XII.5). The relation between
spin textures on either sides of a spin-flip transition will be shown here by
way of an example.

Let us consider the magnetic structure of CoF2 (space group
$P\frac{4_{2}}{m}nm$), a well known piezomagnetic material. Dzialoshinskii
(1957); Borovik-Romanov (1960) In zero applied magnetic field, CoF2 possesses
a fully compensated collinear structure with spins along the $c$-axis. In this
compound, Co2+ (3d7) is strongly anisotropic, and a spin flop transition was
reported at a magnetic field of 7 T, Wu  _et al._ (2016) which is typical of
many 3d transition metal compounds. 131313Half-filled-shell ions such as Fe3+
have smaller anisotropies, but even in such cases spin-lattice coupling and
the Dzyaloshinskii-Moriya interaction cannot be neglected. The MSG in the
zero-field phase is $P\frac{4^{\prime}_{2}}{m}nm^{\prime}$, and the MPG is
$\frac{4^{\prime}}{m}mm^{\prime}$ (Class X). In the high-field phase and
assuming magnetic moments along the $[100]$ or $[110]$ directions, the MPG is
$m^{\prime}mm^{\prime}$, where the first $m^{\prime}$ is perpendicular to the
Néel vector direction (here chosen to be the $x$ axis), while the second
$m^{\prime}$ is perpendicular to the $z$ axis. This MPG is a member of Class
IV, with $y$ and $z$ exchanged. Using the conventions indicated here above, we
obtain:

$\displaystyle\tensorsym{T}_{S\parallel c}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&0&0&\Lambda_{14}&0&0\\\
0&0&0&0&\Lambda_{14}&0\\\ 0&0&0&0&0&\Lambda_{36}\\\ \end{array}\right)$ (31)
$\displaystyle\tensorsym{T}_{S\parallel a}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}0&0&0&0&0&\Lambda_{15}\\\
\Lambda_{31}&\Lambda_{33}&\Lambda_{32}&\Lambda_{24}&0&0\\\ 0&0&0&0&0&0\\\
\end{array}\right)$ (35)

where, for $\tensorsym{T}_{S\parallel a}$, I have exchanged $y$ and $z$ with
respect to the standard setting in Table LABEL:Table_1. The zero-field
magnetic structure $\tensorsym{T}_{S\parallel c}$ is strong-collinear for the
collinear AFM direction along the $z$ axis, via the tensor element
$\Lambda_{36}$. When $\Lambda_{14}$ is set to zero, $\mathbf{B}^{eff}$ is also
parallel to the $z$ axis and has the form $\Lambda_{36}k_{x}k_{y}$, being
maximum when $\mathbf{k}$ is in the $xy/x\bar{y}$ directions. The high-field
magnetic structure would allow weak FM, as indicated by the presence of dipole
terms in Class IV. However, $\tensorsym{T}_{S\parallel a}$ is also strong-
collinear in the $x$ direction via the $\Lambda_{15}$ element. When all other
elements are set to zero, $\mathbf{B}^{eff}$ is along the $x$ direction (the
direction of the AFM moments) and has the form $\Lambda_{24}k_{y}k_{z}$, being
maximum when $\mathbf{k}$ is in the $yz/y\bar{z}$ directions. This is exactly
what one would expect based on the SG approach (see below), i.e., spin
splitting along the AFM spin direction and with a reciprocal space pattern
that does not depend on the spin direction. The MPG approach correctly
describes this and also includes other tensor elements that are allowed by
symmetry, which are likely to be small for collinear structures, but may well
be large for non-collinear structures.

Regardless of the spin orientation, the spin space group is
$P\frac{\tilde{4}_{2}}{m}\tilde{n}m$, where the “$\tilde{\;\;}$” symbol
indicates spin flip in the SG sense, i.e., with the space-group operators only
acting on atoms and not on spins. The (point) SG is
$\frac{\tilde{4}}{m}\tilde{m}m$, and the scalar texture tensor, generated
using MTENSOR is:

$\tensorsym{T}_{SG}=\left(\begin{array}[]{ccc}0&\Lambda_{12}&0\\\
\Lambda_{12}&0&0\\\ 0&0&0\\\ \end{array}\right)$ (36)

The corresponding texture has the form $\Lambda_{12}k_{x}k_{y}$, which is
identical to the previous ones with an appropriate exchange of axes.

Clearly, this complete correspondence does not always hold when the magnetic
moment are along a low-symmetry direction. An extreme case occur when the
magnetic moments are along a completely generic direction $[x,y,z]$, since in
this case the MPG analysis yields six independent components along $[x,y,z]$.
However, these situations are extremely rare, and can be usually resolved
within the MPG framework by rotating the magnetic moment along a high-symmetry
direction (but see here below for the case of cubic groups).

###  XII.5 Strong-collinear altermagnetism in cubic groups

The relation between the SG and MPG approach becomes even clearer when
considering the case of the cubic groups. Cubic MPGs do not admit collinear
structures, since the direction of the magnetic moments always breaks the
cubic symmetry. However, cubic symmetry can still hold in an _approximate_
sense if the spin-lattice interaction is small. Highlighting this approximate
symmetries is one of the main advantages of SGs over MPGs. As shown for other
symmetries in sec. XII.4, it is often possible to choose a high-symmetry spin
direction where no space operator is lost, but this is clearly not possible
for cubic groups. It is therefore useful to work our a cubic example in
detail, so as to understand exactly how the two approaches are related.

Let us consider AFM ordering on a crystal with point group $m\mkern
1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mum$, such as the
pyrochlore Er2Ru2O7 (ref. Taira  _et al._ , 2003), in which cubic symmetry is
broken by a collinear AFM structure. The magnetic moments are along the $z$
axis, and the MPG is $\frac{4}{m}^{\prime}m^{\prime}m$ (Class XI). The
corresponding SG is $m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern
1.0mu\tilde{m}$ (full symbol $\frac{\tilde{4}}{m}\mkern
1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern
1.0mu\frac{\tilde{2}}{\tilde{m}}$) _regardless_ of the direction of the
magnetic moments, where, once again, SG space operators do not change the
direction of the spins and the “$\tilde{\;\;}$” symbol indicates spin flip. At
the rank-7 level, the effective field components for the two cases are:

$\displaystyle B^{SG}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\Lambda(k)(k_{x}^{2}-k_{y}^{2})(k_{x}^{2}-k_{z}^{2})(k_{y}^{2}-k_{z}^{2})$
$\displaystyle B_{z}^{MPG}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
B^{SG}(\mathbf{k})+\left(k_{x}^{2}-k_{y}^{2})(\Lambda_{1}(k)k_{x}^{2}k_{y}^{2}+\Lambda_{2}(k)(k_{x}^{2}+k_{y}^{2})^{2}+\Lambda_{3}(k)k_{z}^{4}\right)$
(37)

Corresponding MPG expressions for magnetic moment in other high-symmetry
directions and in the appropriate cooordinate systems are reported in tab. 2.
For the tetragonal case, is clear that $B^{SG}(\mathbf{k})$ is the symmetrised
version of $B_{z}^{MPG}(\mathbf{k})$, obtained by setting
$\Lambda_{1}(k)=\Lambda_{2}(k)=\Lambda_{3}(k)=0$. The additional components
would emerge as a consequence of symmetry breaking, and can be reasonably
expected to be small for a collinear structure where symmetry is only broken
by the spin system and spin-lattice interaction is weak. Fig. 5 shows the
texture pattern for $B^{SG}(\mathbf{k})$ (i.e., in the absence of the
tetragonal terms). Fig. 5 also shows that the scalar texture pattern is
_completely symmetric_ by the SG cubic symmetry operators, while the generic
tetragonal, trigonal or orthorhombic textures for the corresponding MPGs would
only have those specific symmetries.

Figure 5:  Views along the $[001]$ (a) and $[111]$ (b) directions of the
scalar texture pattern (i.e., the texture component along the direction of the
magnetic moments) for MPG $\frac{4}{m}^{\prime}m^{\prime}m$ (symmetrised to
cubic) or spin group $\frac{\tilde{4}}{m}m\tilde{m}$. The normalisation
function is as in previous figures. Contours were plotted for $k_{z}=1$ and
are both symmetric by exchange $k_{z}\rightarrow-k_{z}$. Panel (a) to be
compared with fig. 2, sub-panel B6 of ref. Šmejkal  _et al._ , 2022a.

For further clarity, views along different directions of these textures
plotted on the surface of a sphere in momentum space are displayed in fig. 6.
As remarked earlier, textures for different values of $k=|\mathbf{k}|$ only
differ by the multiplicative function $\Lambda(k)$ (see first line of eq.
XII.5).

Figure 6:  Views along the $[001]$ (a) and $[111]$ (b) and $[110]$ (c) directions of the scalar spin texture in fig. 5 plotted on the surface of a sphere in momentum space. Blue/red colours correspond to negative/positive values. Table 2:  Polynomial forms of the effective field components for cubic groups and different high-symmetry magnetic moment directions. The corresponding spin groups are indicated in the first column. For each moment direction, the corresponding MPG class is indicated together with one MPG example (to establish the correct setting). The symbol $..2$ means that the 2-fold axis is along the $z$ direction (see text). Spin groups |  | $S\parallel[001]$ | $S\parallel[110]$ | $S\parallel[111]$
---|---|---|---|---
$23$,$m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mu$ | Class |  II ($..2$) |  I ($1$) |  VIII ($3$)
Form | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$ | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$ | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$
$432$, $\mkern 1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu3m$ | Class |  XIVb ($42^{\prime}2^{\prime}$) |  IV ($2^{\prime}2^{\prime}2$) |  VII ($32^{\prime}$)
$m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mum$ | Form | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$ | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$ | $\Lambda(k)\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right)$
$\tilde{4}3\tilde{2}$ | Class |  XI ($\frac{4}{m}^{\prime}m^{\prime}m$) |  IV ($2^{\prime}2^{\prime}2$) |  VIII ($32$)
$\tilde{{\mkern 1.0mu\overline{\mkern-1.0mu4\mkern-1.0mu}\mkern 1.0mu}}3\tilde{m}$,$m\mkern 1.0mu\overline{\mkern-1.0mu3\mkern-1.0mu}\mkern 1.0mu\tilde{m}$ | Form | $\begin{array}[]{c}\Lambda(k)(k_{x}^{2}-k_{y}^{2})\cdot\\\ (k_{x}^{2}-k_{z}^{2})(k_{y}^{2}-k_{z}^{2})\end{array}$ | $\begin{array}[]{c}\Lambda(k)(-2k_{x}^{2}+(k_{y}-k_{z})^{2})\cdot\\\ \left(-2k_{x}^{2}+(k_{y}+k_{z})^{2}\right)k_{y}k_{z}\end{array}$ | $\begin{array}[]{c}\Lambda(k)(k_{x}^{2}-3k_{y}^{2})k_{x}k_{z}\cdot\\\ \left[6(k_{x}^{2}+k_{y}^{2})-16k_{z}^{2}-\sqrt{2}(k_{y}^{2}-3k_{x}^{2})\right]\end{array}$

##  XIII Discussion

In this paper, I have outlined a natural symmetry classification of all
$\mathbf{k}/-\mathbf{k}$-symmetric altermagnets, based on the lowest-rank
altermagnetic tensor forms allowed in each magnetic point group (MPG). I have
also tabulated all the higher-rank tensor forms required to describe ‘strong-
collinear’ altermagnetic textures, i.e., those that produce spin splitting
with quantisation axis parallel to the Néel vector. This include the case of
cubic groups, where the spin-group formalism (thoroughly developed by other
groups)Šmejkal  _et al._ (2022a) is most useful.

This classification has important practical consequences, particularly when
one considers the tensorial connection between altermagnetism (which describes
particular form of reciprocal-space spin textures) and well-known macroscopic
properties such as piezomagnetism and the MOKE effect. For example, the Bilbao
database MAGNDATA Gallego  _et al._ (2016) lists 139 known materials that are
MOKE active and do not allow ferromagnetism. The point groups of these
materials are the same as those in classes V, VIII, IX, X, XI, XII, XIV, XV
and XVII in which purely antiferromagnetic quadratic altermagnetism is
allowed. In fact, most materials so far discussed in the context of
altermagnetims (and many more) are well-known piezomagnets/MOKE-active AFM.
Moreover, while a positive identification of altermagnetism as distinct from
time-even effects such as the surface R-D effect is rather difficult,
detecting MOKE activity is rather straightforward and would enable screening
of candidate materials for which the magnetic structural determination may be
ambiguous. Keßler  _et al._ (2024)

One may also observe that altermagnetism is allowed in 69 MPGs, of which 66
allow quadratic altermagnetism and 31 allow ferromagnetism. Considering that
32 MPGs are paramagnetic (grey), out of a total of 122 MPGs only 21 groups are
black&white and do not allow altermagnetism, which is therefore hardly a rare
phenomenon.

Although the purpose of this paper is mainly to discuss altermagnetic
_symmetry_ , a few words about ‘mechanisms’ do not seem inappropriate,
particularly in connection with the role of the spin-orbit coupling (SOC). As
already discussed in ref. Yuan  _et al._ , 2020a, altermagnetic splitting is
caused by the time-reversal-odd effective Zeeman fields (mostly of exchange
origin) generated by real magnetic moments in the unit cell . Since the SO
interaction is time-reversal even (its sources being the gradients of the
electrical potential), at first sight it would seem that the SO interaction
could not be responsible for $\mathbf{k}/-\mathbf{k}$-symmetric
altermagnetism.

However, this in not entirely correct, since the SOC can affect both the
orientation of the localised magnetic moments and the spin density
distribution, most famously by inducing spin canting via the Dzyaloshinskii-
Moriya interaction. It is therefore not implausible that the SOC could be
responsible for ‘weak altermagnetism’, Krempaský  _et al._ (2024) particularly
when spin splitting along a unique high-symmetry direction is not allowed at
the lowest order (Classes VIII, XII, XIV, XV, XVII and XVIII). Interestingly,
one must also expect to observe R-D-like, $\mathbf{k}/-\mathbf{k}$-_anti_
symmetric splitting that is of pure magnetic origin, because magnetism
ordering can break inversion symmetry. The two best-known cases of this
phenomenon, which Cheong et al. describes as ‘type-III altermagnetism’,Cheong
and Huang (2024) are helical magnetic structures and type-II multiferroicity,
both of which can originate in centrosymmetric crystal due to competing
interactions.141414Altermagnetism as defined in this paper is not allowed in
paramagnetic (grey) point groups. However, many antiferromagnets (especially
incommensurate antiferromagnets) possess paramagnetic MPGs, because in there
MSG time reversal is equivalent to a (potentially incommensurate) translation
U — in other words, $U\theta$ is a symmetry operator. These systems are
candidates for ‘Type III altermagnetism’. A systematic treatment of these
system similar to the one presented here should be possible since, even in the
most complex cases, one can always define a MPG. Radaelli and Chapon (2007) In
both cases, the relevant properties (magnetic helicity and magnetic polarity,
respectively) are _time reversal even_ , so this materials do not violate the
general rule that the spin splitting should be the same in time-reversed
domains.

##  Appendix A Tensor equalities and invariance

###  A.1 Equivalence of Cartesian and Spherical tensor decompositions

In general, Cartesian tensors like those in eq. 1 are members of the tensor
product space
$\Gamma^{V}\otimes\left[\Gamma^{V}\otimes\Gamma^{V}\dots\right]$, where the
square bracket indicates index symmetrisation while $\Gamma^{V}$ is the
representation of ordinary vectors, which coincides with the $L=1$ irrep of
$SO(3)$ ($\Gamma^{1}$) if one considers only proper rotations. In turn,
$\left[\Gamma^{V}\otimes\Gamma^{V}\dots\right]$ can be decomposed in $SO(3)$
irreps as follows:

$\displaystyle\left[\Gamma^{V}\otimes\Gamma^{V}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{2}+\Gamma^{0}$
$\displaystyle\left[\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\right]$
$\displaystyle=$ $\displaystyle\Gamma^{3}+\Gamma^{1}$
$\displaystyle\left[\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\right]$
$\displaystyle=$ $\displaystyle\Gamma^{4}+\Gamma^{2}+\Gamma^{0}$
$\displaystyle\left[\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\otimes\Gamma^{V}\right]$
$\displaystyle=$ $\displaystyle\Gamma^{5}+\Gamma^{3}+\Gamma^{1}$
$\displaystyle\dots$ (38)

where $\Gamma^{2},\Gamma^{3}$, etc. are the irreps for $L=2,3,\dots$. Taking
once again the tensor product with $\Gamma^{V}$

$\displaystyle\Gamma^{V}\otimes\left[(\Gamma^{V})^{2}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{3}+\Gamma^{2}+2\Gamma^{1}$
$\displaystyle\Gamma^{V}\otimes\left[(\Gamma^{V})^{3}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{4}+\Gamma^{3}+2\Gamma^{2}+\Gamma^{1}+\Gamma^{0}$
$\displaystyle\Gamma^{V}\otimes\left[(\Gamma^{V})^{4}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{5}+\Gamma^{4}+2\Gamma^{3}+\Gamma^{2}+2\Gamma^{1}$
$\displaystyle\Gamma^{V}\otimes\left[(\Gamma^{V})^{5}\right]$ $\displaystyle=$
$\displaystyle\Gamma^{6}+\Gamma^{5}+2\Gamma^{4}+\Gamma^{3}+2\Gamma^{2}+\Gamma^{1}+\Gamma^{0}$
$\displaystyle\dots$ (39)

where I have used the shorthand $(\Gamma^{V})^{2}=\Gamma^{V}\otimes\Gamma^{V}$
etc. The spherical tensor decomposition is the familiar one:

$\displaystyle\Gamma^{1}\otimes\Gamma^{0}$ $\displaystyle=$
$\displaystyle\Gamma^{1}$ $\displaystyle\Gamma^{1}\otimes\Gamma^{1}$
$\displaystyle=$ $\displaystyle\Gamma^{2}+\Gamma^{1}+\Gamma^{0}$
$\displaystyle\Gamma^{1}\otimes\Gamma^{2}$ $\displaystyle=$
$\displaystyle\Gamma^{3}+\Gamma^{2}+\Gamma^{1}$
$\displaystyle\Gamma^{1}\otimes\Gamma^{3}$ $\displaystyle=$
$\displaystyle\Gamma^{4}+\Gamma^{3}+\Gamma^{2}$
$\displaystyle\Gamma^{1}\otimes\Gamma^{4}$ $\displaystyle=$
$\displaystyle\Gamma^{5}+\Gamma^{4}+\Gamma^{3}$
$\displaystyle\Gamma^{1}\otimes\Gamma^{5}$ $\displaystyle=$
$\displaystyle\Gamma^{6}+\Gamma^{5}+\Gamma^{4}$ $\displaystyle\dots$ (40)

So, for example, the sum of all the spherical harmonics terms is eq. A.1 is:

$\displaystyle\sum_{n=0}^{5}\Gamma^{1}\otimes\Gamma^{n}$ $\displaystyle=$
$\displaystyle\Gamma^{0}+3\Gamma^{1}+3\Gamma^{2}+3\Gamma^{3}+3\Gamma^{4}+2\Gamma^{5}+\Gamma^{6}$
(41) $\displaystyle=$
$\displaystyle\Gamma^{V}\otimes\left[(\Gamma^{V})^{4}\right]+\Gamma^{V}\otimes\left[(\Gamma^{V})^{5}\right]$

We can therefore conclude that, if one considers only proper rotations (i.e.,
elements of the continuous group $SO(3)$), the decomposition into spherical
tensors of of a vector field on a spherical shell up to order $n$ is
equivalent to the sum of two Cartesian tensors of ranks $n$ and $n+1$.
Including parity and time reversal, it is found that the Cartesian tensors of
odd rank are parity-even and time-reversal odd, while the Cartesian tensors of
even rank are parity-odd and time-reversal even.

###  A.2 Tensor invariance

Here, I show that the symmetry of the global point-group symmetry of spin
texture generated by a certain Cartesian tensor requires that tensor to be
totally symmetric by the symmetry operations of that point group. In general,
given a vector field $\mathbf{V}(\mathbf{k})$ and a proper rotation
$\matrixsym{R}$ the transformed vector field is:

$\tilde{\mathbf{V}}(\mathbf{k})=\matrixsym{R}\cdot\mathbf{V}(\matrixsym{R}^{-1}\cdot\mathbf{k})$
(42)

Particularising to the case of $\mathbf{B}^{eff}(\mathbf{k})$ we obtain for
_proper_ rotations:

$\displaystyle\tilde{\mathbf{B}}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle
R_{ij}T^{(n)}_{j,\alpha\beta\gamma\dots}R^{-1}_{\alpha\rho}R^{-1}_{\beta\sigma}R^{-1}_{\gamma\tau}k_{\rho}k_{\sigma}k_{\tau}\dots$
$\displaystyle=$ $\displaystyle
R_{ij}R_{\rho\alpha}R_{\sigma\beta}R_{\tau\gamma}T^{(n)}_{j,\alpha\beta\gamma\dots}k_{\rho}k_{\sigma}k_{\tau}\dots$

After some bookeeping can generalise this to proper/improper rotations that
may include time reversal by considering that $\mathbf{k}$ is time-reversal-
odd and parity-odd, as:

$\displaystyle\tilde{\mathbf{B}}^{eff}(\mathbf{k})$ $\displaystyle=$
$\displaystyle(-1)^{(n+1)p+nt}$ $\displaystyle
R_{ij}R_{\rho\alpha}R_{\sigma\beta}R_{\tau\gamma}\dots
T^{(n)}_{j,\alpha\beta\gamma\dots}k_{\rho}k_{\sigma}k_{\tau}\dots$

If we now require that, $\forall\mathbf{k}$,
$\tilde{\mathbf{B}}^{eff}(\mathbf{k})=\mathbf{B}^{eff}(\mathbf{k})$ , we
conclude:

$\displaystyle(-1)^{(n+1)p+nt}R_{ij}R_{\rho\alpha}R_{\sigma\beta}R_{\tau\gamma}\dots
T^{(n)}_{j,\alpha\beta\gamma\dots}$ $\displaystyle=$ $\displaystyle
T^{(n)}_{i,\rho\sigma\tau\dots}$

which is precisely the definition of a tensor that is totally symmetric by a
generalised rotation $\matrixsym{R}$.

##  Appendix B Spherical basis tensors

In this appendix, I present a decomposition of the 17 unique tensor forms for
quadratic altermagnetism into a set of nine simple spherical basis tensors
plus those obtained by axis permutations (Table LABEL:Table_2). Each of the 17
tensor forms can be obtained as a linear combination of the spherical basis
tensors (Table LABEL:Table_3). For example, for Class V, we have:

$\displaystyle\left(\begin{array}[]{cccccc}0&0&0&\Lambda_{14}&0&0\\\
0&0&0&0&\Lambda_{25}&0\\\ 0&0&0&0&0&\Lambda_{36}\\\ \end{array}\right)$
$\displaystyle=$
$\displaystyle\Lambda_{1}Q^{I\\!I}+\Lambda_{2}Q^{I\\!I\\!I}+\Lambda_{3}O^{I}=$
(49) $\displaystyle=$
$\displaystyle\Lambda_{1}\left(\begin{array}[]{cccccc}0&0&0&-1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&0\\\
\end{array}\right)+\Lambda_{2}\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&-2\\\
\end{array}\right)+\Lambda_{3}\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ \end{array}\right)$ (59)

with

$\displaystyle\Lambda_{14}$ $\displaystyle=$
$\displaystyle-\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$
$\displaystyle\Lambda_{25}$ $\displaystyle=$
$\displaystyle\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$ $\displaystyle\Lambda_{36}$
$\displaystyle=$ $\displaystyle-2\Lambda_{2}+\Lambda_{3}$

##  Appendix C Binary spin groups in the time-reversal language

Spin groups (SG) are generally defined as the outer direct product of a point
(or space) group and a spin rotation group. Litvin (1977) The simplest non-
trivial SGs are _binary_ SG, in which the group acting on spins is a 2-element
set. To classify collinear structures Šmejkal et al. adopt the two-element
group $\\{1,2_{z}\\}$, where $1$ is the identity and $2_{z}$ is a 2-fold
rotation perpendicular to the Néel vector. Another choice is to use the two-
element group $\\{1,1^{\prime}\\}$, where $1^{\prime}$ is the time reversal
operator. With either choice, binary SG coincide with the the celebrated
Shubnikov groups. Opechowski and Guccione (1965) The only difference between
binary SG with the time-reversal operator and the MPG treatment (which also
employs Shubnikov point groups) is that binary SG are made to act on _scalar,
time-reversal-odd textures_ both in real and in reciprocal space (space
operators have no effect on scalars), while MPGs act on _axial vector, time-
reversal-odd textures_. For binary SGs, I will adopt the time-reversal
operator, indicated with the symbol “$\tilde{\;\;}$” to distinguish it from
the MPG equivalent, indicated with a prime (’) This choice enables one to
recast the action of binary SGs onto the familiar language of parity, time
reversal etc., and to derive a parallel tensorial treatment to that of MPGs.

One should remark that, for a given collinear structure, there is no simple
correspondence between its MPGs and binary SG, with the latter generally
having more symmetry operators than the former. Moreover, converting SG into
MPGs and vice versa require knowledge of the direction of the Néel vector. For
example, for a tetragonal structure with binary SG
$\frac{\tilde{4}}{m}\tilde{m}m$, the MPG is $\frac{4^{\prime}}{m}mm^{\prime}$
for magnetic moments along the $z$ axis and $m^{\prime}mm^{\prime}$ or
$mm^{\prime}m^{\prime}$ for Néel vector along $x$ and $y$, respectively.
Several examples of these conversions are given in sec. XII.

##  Appendix D Construction of gnomonic projections from band dispersions

Here, I explain how to construct gnomonic projections of spin textures,
similar to those displayed in the figures of sec. XII, starting from spin-
polarised band dispersions such as those obtained from DFT calculations. One
starts by calculating a grid of points at constant $k=|\mathbf{k}|$ for a
particular band. It is most convenient to calculate points at constant
$\theta$ and equal intervals $\Delta\phi$, where $k$, $\theta$ and $\phi$ are
spherical coordinates in momentum space. To reproduce the figures of sec. XII,
one plots the spin polarisation for points at constant $\theta$ on a circle of
radius $\tan\theta$ and with the original values of $\phi$. Figures
constructed from points calculated by DFT will include the pre-factors
$\Lambda_{i}(k)$, which depend on both $k$ and the band index. Moreover, one
may require higher-rank tensors than those discussed in the paper to reproduce
DFT data accurately.

###### Acknowledgements.

I acknowledge discussions with A. Stroppa (CNR-SPIN), Roger D. Johnson
(University College London), Dmitry Khalyavin (STFC UKRI) and Gautam Gurung
(Trinity College, Oxford).

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  * Note (1) Cheong et. al. define ‘Type-III altermagnets’ as systems with symmetries including time reversal is a distinct operator. For reasons explained at length in this paper, this can only give rise to $\mathbf{k}/-\mathbf{k}$ _anti_ symmetric textures of the R-D type, though magnetism can play a very important role in them. Here, I am not considering Type-III altermagnetism and, more generally magnetism-induced R-D type textures, though I am including a brief discussion of these effects in sec. XIII .
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  * Note (2) I specifically highlight the approaches adopted by Cheong Cheong and Huang (2024) and Liu, Liu  _et al._ (2023) because they are the closest to the one I will describe herein, though neither makes an explicit connection with Cartesian/sperical tensors. Moreover, the MPG classifications proposed by Cheong and Liu largely overlap with the one proposed in the present paper.
  * Note (3) Ref. Smejkal2022 employs ‘binary’ SGs, i.e., SG with 2-element groups acting on spins (see below). Binary SG cannot be used for non-collinear structures. Certain non-collinear structures can be described either with more complex SGs Litvin (1977) or with so-called multi-colour groups,Harker (1981), but, to my knowledge, a complete theory for their use has not been developed thus far. See Appendix C for more details.
  * Note (4) The Neumann-Minnigerode-Curie Principle (NMC Principle) enables one to derive the selection rules for the physical properties from the symmetry of the crystal in question. For macroscopic properties, the symmetry in question is precisely the MPG of the crystal. One should also remark that, unlike the widely used MPGs, magnetic _space_ groups have been almost entirely superseded among magnetic structures specialists by irreducible representations analysis ‘à la Bertaut’,Bertaut (1968) with additional symmetries in spin spaces being dealt with using Izyumov’s exchange multiplet theory Izyumov (1980).
  * Note (5) Perhaps the best example of this is ferromagnetic ordering in cubic magnetic metals such as Fe or Ni. Strictly speaking, Fe becomes rhombohedral below the Curie temperature, but the deviation from cubic symmetry due to magneto-striction is extremely small ($\approx 10^{-6}$).
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  * Note (8) I employed the conventions adopted by the program MTENSOR of the Bilbao Crystallographic Server (which are somewhat different from those in ref. Opechowski1965), except for $\bar{6}^{\prime}m^{\prime}2$, which has been converted to $\bar{6}^{\prime}2m^{\prime}$ to be included in Class XIV. Note that the textures of Classes X and XI differ by a $45^{\circ}$ rotation, and are therefore equivalent at the MPG level, only distinguished by the orientation of the texture with respect to the crystal axes — see further discussion in section XII. In the interest of practical use and to follow conventions, I kept these two classes separate. Straightforward axes transformations may be required for particular MPGs.
  * Note (9) I adopted the standard conventions for $x$, $y$, and $z$, which are related to the direction of the symmetry directions. For example, for point group $3m1$, $z$ is parallel to the 3-fold axis, $x$ is perpendicular to the mirror plane and $y$ completes the set. In the setting $31m$ of the same point group, $x$ and $y$ are interchanged. Examples on how to deal with conventions issues are provided in section XII.
  * Note (10) These classes are named ‘Type-II altermagnetic’ in ref. Cheong2024.
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  * Note (13) Half-filled-shell ions such as Fe3+ have smaller anisotropies, but even in such cases spin-lattice coupling and the Dzyaloshinskii-Moriya interaction cannot be neglected.
  * Note (14) Altermagnetism as defined in this paper is not allowed in paramagnetic (grey) point groups. However, many antiferromagnets (especially incommensurate antiferromagnets) possess paramagnetic MPGs, because in there MSG time reversal is equivalent to a (potentially incommensurate) translation U — in other words, $U\theta$ is a symmetry operator. These systems are candidates for ‘Type III altermagnetism’. A systematic treatment of these system similar to the one presented here should be possible since, even in the most complex cases, one can always define a MPG. Radaelli and Chapon (2007).
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