# Magnetic octupoles as the order parameter for unconventional
antiferromagnetism

Sayantika Bhowal  Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27,
8093 Zurich, Switzerland  Nicola A. Spaldin  Materials Theory, ETH Zurich,
Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland

###### Abstract

We show that time-reversal symmetry broken, centrosymmetric antiferromagnets
with non-relativistic spin-splitting are conveniently described in terms of
the ferroic ordering of magnetic octupoles. The magnetic octupoles are the
lowest-order ferroically ordered magnetic quantity in this case, and so are
the natural order parameter for the transition into the magnetically ordered
state. They provide a unified description of the broken time-reversal symmetry
and the non-relativistic spin splitting as well as a platform for manipulating
the latter, and account for other phenomena, such as piezomagnetism,
characteristic of this class of antiferromagnets. Unusually for
antiferromagnets, we show that the magnetic octupoles cause a non-zero
magnetic Compton scattering, providing a route for their direct experimental
detection. We illustrate these concepts using density-functional and model
calculations for the prototypical non-relativistic spin-split antiferromagnet,
rutile-structure manganese difluoride, MnF2.

##  I Introduction

The behavior that we now know as antiferromagnetism was first noticed around
100 years ago, when peaks in both specific heat and magnetic susceptibility
were observed in materials such as MnO that have zero net magnetic moment [1,
2]. Soon after, Néel proposed a model in which local magnetic dipole moments
of equal magnitude on two sublattices order in an antiparallel fashion [3].
While the predictions of the Néel model were consistent with the observations
[4], another ten years elapsed before neutron diffraction provided the first
direct evidence of antiferromagnetic ordering of magnetic dipoles [5].

Usually the order parameter, $\vec{L}$, of an antiferromagnet (AFM) is defined
in terms of the difference between the local magnetic dipole moments,
$\vec{M}_{1}$ and $\vec{M}_{2}$, on the two sublattices,
$\vec{L}=\vec{M}_{1}-\vec{M}_{2}$. Such a definition is conceptually
intuitive, but lacks the convenience provided by ferroic order parameters such
as the magnetization, $\vec{M}$ in ferromagnets or the electric polarization,
$\vec{P}$ in ferroelectrics. For example, the antiferromagnetic vector does
not provide information about the conjugate field required to select for a
particular antiferromagnetic domain, and fails to distinguish between
antiferromagnets that do or do not break time-reversal symmetry. The magnetic
dipoles, however, are just one of the terms in a multipole expansion of the
energy of a general magnetization density in a magnetic field. They are
generally the lowest-order local multipole on an atomic site, which makes them
appealing for classifying magnetic order, but there is no fundamental reason
why they should necessarily be the best choice. In particular, when the
magnetic dipoles order antiferromagnetically, higher order multipoles that
order ferroically might be more suitable.

Indeed, such a higher-order-multipole description is now established in the
case of AFMs that break both time-reversal ($\mathcal{T}$) and space-inversion
($\mathcal{I}$) symmetries, and which are classified by the ferroic ordering
of their local magnetoelectric multipoles [6, 7, 8]. The magnetoelectric
multipoles make up the next-order term, beyond the magnetic dipoles, in the
multipole expansion of the magnetic interaction energy (see Eqn. 1), and so
depend linearly on both position $r$ and magnetic moment $\mu$. All such
$\mathcal{T}$ and $\mathcal{I}$-broken AFMs therefore exhibit a linear
magnetoelectric response, in which an applied electric field induces a
magnetization linear in the field strength and vice versa [9]. Their conjugate
field is the product of electric and magnetic fields, which is exploited in
so-called magnetoelectric annealing to select for a particular
antiferromagnetic domain in magnetoelectric devices [10, 11]. The ferroic
ordering of magnetoelectric multipoles also plays a crucial role in
antiferromagnetic spintronics [12, 13] and skyrmionics [14, 15], and can give
rise to unconventional transport properties [16].

Recently there has been renewed interest in a class of AFMs that break time-
reversal symmetry and exhibit a spin-splitting of their energy bands that is
not of relativistic origin (in conventional antiferromagnets the bands are
doubly spin degenerate). First invoked in 1964 [17], such non-relativistic
spin-splitting (NRSS) is typically much larger than relativistic Rashba-like
spin splitting, and can be substantial in materials containing only light
elements. An important recent development was the articulation of guiding
principles for realizing such unconventional magnetism in materials [18, 19,
20, 21, 22] so that proposed unconventional properties of both fundamental and
technological importance [23, 20], including efficient spin-current generation
[24, 25, 26], spin-splitting torque [27, 28], giant magnetoresistance [29],
spontaneous Hall effect [30, 31, 32] and superconductivity [33], chiral
magnons [34] could now be within reach. A new name was even introduced –
altermagnet – to describe this class of AFMs [23] [see Fig. 1 (a)-(c)]. Note
that many NRSS AFMs are centrosymmetric and so are not described by ferroic
ordering of magnetoelectric multipoles.

Here we show that such time-reversal symmetry broken, centrosymmetric AFMs are
conveniently described in terms of the ferroic ordering of magnetic octupoles.
The magnetic octupoles form the next term in the magnetic multipole expansion
after the magnetoelectric multipoles, and are the lowest-order ferroically
ordered magnetic quantity in this case. They are the natural order parameter
for the transition into the magnetically ordered state, and provide a
convenient and unified description of the broken $\mathcal{T}$ symmetry and
the non-relativistic spin splitting. Importantly for potential device
applications, they provide a platform for manipulating the spin-splitting.
They also account for other phenomena displayed by this class of AFMs such as
the piezomagnetic effect [35, 36] and strong magnetic anisotropy [37], and
allow us to predict new behaviors such as an anti-piezomagnetism. Finally, we
show that, unusually for an antiferromagnet, they will have a non-zero
magnetic Compton scattering, providing a route for their direct experimental
detection.

Figure 1: Non-relativistic spin-split (NRSS) antiferromagnets and ferro-type
magnetic octupolar order. (a)-(c) shows the spin splitting of the bands for
conventional ferromagnets (Zeeman splitting), antiferromagnets (degenerate
spin-polarized bands), and the recently discovered AFMs with NRSS (symmetric
in $\vec{k}$) of their bands. (d) and (e) show the antiferro-type magnetic
dipolar (arrows, (d)) order and ferro-type magnetic ${\cal O}_{32}^{-}$
octupolar (colored anisotropic octupolar magnetic distribution, (e)) order in
MnF2 respectively.

We illustrate our ideas using rutile-structure manganese difluoride, MnF2.
MnF2 has been widely explored as a classic example of a two sub-lattice AFM
over the past century [38, 39, 40], and was recently identified as a prototype
centrosymmetric AFM with NRSS [18]. Importantly, Mn ions of opposite spin
orientation have inequivalent fluorine environments (Fig. 1 (d)). This results
in identical, ferroically ordered ${\cal O}_{32^{-}}$ magnetic octupoles at
each Mn site (Fig. 1 (e)), which cause the broken $\mathcal{T}$ symmetry in
spite of the AFM spin compensation.

The remaidner of the manuscript is organized as follows. We begin by briefly
describing the crystal and magnetic structures of MnF2 in Section II. This is
followed by our discussion of the ordered magnetic octupoles and their role in
NRSS in Section III. In Section IV we predict new behaviors resulting from the
ferromagneto-octupolar order that await experimental verification, and propose
magnetic Compton scattering as a route to the direct detection of magnetic
octupoles. Finally, we summarize our results and discuss promising future
directions in Section V.

##  II Crystal and Magnetic Structure

MnF2 crystallizes in the centrosymmetric tetragonal rutile structure with the
space group symmetry $P4_{2}/mnm$ ($D_{4h}$ point group) [41, 18]. As depicted
in Fig. 1 (d), the unit cell contains two formula units, with two Mn atoms at
the corner and the center of the unit cell, octahedrally coordinated by the F
atoms. Importantly, the F environment around the Mn atom at the center is
rotated by 90∘ around the $z$ axis with respect to that at the corner Mn atom.
As a result of this non-equivalent F environment, the Mn sites, although
equivalent, are not related by a lattice translation. This has a crucial
impact on the symmetry of the AFM structure of MnF2 (magnetic space group
$P4_{2}~{}^{\prime}/mnm^{\prime}$) below the Néel temperature (T${}_{\rm
N}=67$ K [42]), where the collinear Mn spins align antiparallely along [001]
[43] [see Fig. 1 (d)]. Such a magnetic configuration breaks the $\cal T$
symmetry despite the vanishing magnetization, since time-reversal plus
translation is not a symmetry of the antiferromagnetic configuration.

##  III Magnetic Octupole and Non-relativistic Spin Splitting

The broken $\cal T$ symmetry in the absence of any net magnetization is
indicative of the existence of magnetic multipole of higher order than the
magnetic dipole. Such multipoles appear in the expansion of the interaction
energy ${\cal E}_{\rm int}$ between a spatially varying magnetic field
$\vec{H}(\vec{r})$ and a magnetization density $\vec{\mu}(\vec{r})$ [44, 8],

$\displaystyle{\cal E}_{\rm int}$ $\displaystyle=$
$\displaystyle-\int\vec{\mu}(\vec{r})\cdot\vec{H}(\vec{r})d^{3}r$ (1)
$\displaystyle=$
$\displaystyle\underbrace{-\int\vec{\mu}(\vec{r})\cdot\vec{H}(0)d^{3}r}_{\text{Dipolar~{}term}}\underbrace{-\int
r_{i}\mu_{j}(\vec{r})\partial_{i}H_{j}(0)d^{3}r}_{\rm
Magnetoelectric~{}multipolar~{}term}\underbrace{-\int
r_{i}r_{j}\mu_{k}(\vec{r})\partial_{i}\partial_{j}H_{k}(0)d^{3}r}_{\rm
Octupolar~{}term}~{}...~{}~{}~{}.$

In a compensated AFM, the net magnetization
$\vec{M}=\int\vec{\mu}(\vec{r})d^{3}r$ is absent and so the conventional
dipolar Zeeman term which is the first term in the above expansion, does not
contribute. Furthermore, the presence of inversion symmetry in centrosymmetric
antiferromagnets with symmetric NRSS with respect to $\vec{k}$ forbids the
existence of any ferro-type magnetoelectric multipole ${\cal M}_{ij}=\int
r_{i}\mu_{j}(\vec{r})d^{3}r$ since these break inversion symmetry, forming the
second term in Eq. (1). This makes the first symmetry-allowed ferro-type
magnetic multipole the inversion-symmetric rank-3 magnetic octupole, ${\cal
O}_{ijk}=\int r_{i}r_{j}\mu_{k}(\vec{r})d^{3}r$, which forms the third term in
the above expansion. While this simple symmetry argument suggests that the
magnetic octupole is the first allowed net nonzero magnetic multipole in a
centrosymmetric AFM with NRSS, its existence can only be confirmed through an
explicit computational analysis of the multipoles.

In the following we take MnF2 as a representative system for such
unconventional AFMs and explicitly analyze the multipoles in the system. In
particular, we focus on the magnetic octupole, show its possible manipulation
via structural and magnetic modifications, and correlate it to the
characteristic non-relativistic spin splitting of the energy bands.

###  III.1 Multipole Analysis

In order to compute the atomic-site multipoles, we decompose the density
matrix $\rho_{lm,l^{\prime}m^{\prime}}$, computed using density-functional
theory as implemented in the Elk code (see the computational details in
Appendix A), into tensor moments [8]. Since the desired parity even multipoles
(as the structure has inversion symmetry) have contributions from even
$l+l^{\prime}$ terms, we evaluate both $d-d$ and $p-p$ matrix element
contributions. We consider both $\cal T$ even (charge) and odd (magnetic)
multipoles.

The computed magnetic octupoles, ${\cal O}_{32^{-}}$ and ${\cal O}_{30}$, and
electric quadrupoles, ${\cal Q}_{22^{-}}$ and ${\cal Q}_{20}$, are shown in
Fig. 2 (a) and (c) as the relative strength of the spin-orbit coupling
constant, $\lambda_{r}$, is varied. As we can see from this variation, the
magnetic octupoles are non-zero even without the presence of the spin-orbit
coupling. It is also clear from Fig. 2 that the magnitudes of the magnetic
octupoles depend on $\lambda_{r}$, whereas the values of the quadrupoles
remain invariant, suggesting that the quadrupoles have only structural origin
while the magnetic octupoles may have both structural and magnetic
dependencies. We note that a pure structural origin of quadrupoles is not a
general case for any systems with quadrupolar distortion, e.g., the
quadrupoles in the iso-space-group compound Ba2MgReO6 (with a canted
antiferromagnetic spin configuration) are reported to have a strong spin
dependence [45].

The computed magnetic multipoles show the presence of a ferro-type magnetic
octupole ${\cal O}_{32^{-}}$ (Fig. 1e) with real-space representation
$xym_{z}$ (where $m_{z}$ is the $z$ component of the magnetic moment) at the
Mn sites, belonging to the totally symmetric irreducible representation
$A_{1g}$. The existence of the magnetic octupole is consistent with the
$B_{1g}^{-}$ active representation of MnF2 as well as the symmetry analysis
described earlier. The magnetic octupole ${\cal O}_{30}$ with real-space
representation $(3z^{2}-r^{2})m_{z}$, also has a non-zero value at the Mn
sites, however, they have anti-parallel alignment between the Mn sites,
resulting in an absence of net ${\cal O}_{30}$ octupole moment. We also found
non-zero magnetic octupole components ${\cal Q}^{(\tau)}_{x^{2}-y^{2}}$ and
$t^{(\tau)}_{z}$ with ferro and antiferro-type alignments respectively, which
we will discuss later in Section IV.1 and in Appendix D. In addition to these
magnetic octupoles, the crystal structure of MnF2 also hosts electric
quadrupoles, ${\cal Q}_{20}$ with $(3z^{2}-r^{2})$ distortion and ${\cal
Q}_{22^{-}}$, representing the $xy$-structural distortion, which have ferro-
and antiferro-type alignments respectively.

Figure 2:  Variation of magnetic octupoles at the Mn atoms as the relative
strength of the spin-orbit coupling $\lambda_{r}$ is varied for the (a)
crystal structure of MnF2 and (b) for the modified structure. The same
variation of the charge quadrupoles for the (c) crystal structure of MnF2 and
(d) for the modified structure. Here $\lambda_{r}=\lambda_{f}/\lambda$, with
$\lambda_{f}$ and $\lambda$ being the enforced value of the spin-orbit
coupling constant in the calculation and its actual value in the material
respectively. The band-decomposed charge densities for the top valence band in
Fig. 3 (a) of MnF2 in the $a$-$b$ plane for the (e) crystal structure of MnF2
and (f) for the modified structure. (g) and (h) are the same, showing the
band-decomposed magnetization densities. The opposite MnF6 octahedral
rotations are indicated in black arrows.

For a physical understanding and better visualization, we further compute the
band-decomposed charge and spin densities for the top valence band (which also
undergoes NRSS) in the electronic structure of MnF2, shown in Fig. 3 (a) and
the results are shown in Fig. 2 (e) and (g). As apparent from the figure, the
charge density around the Mn atoms is highly anisotropic in the $x-y$ plane, a
signature of the existing ${\cal Q}_{22^{-}}$ quadrupole. Interestingly, the
spin density around the Mn atoms, shown in Fig. 2 (d), follows the anisotropic
charge density, indicating a correlation between the spin anisotropy
(quantified by the magnetic octupoles) and the charge anisotropy (quantified
by the electric quadrupoles). This further justifies the dependence of the
octupoles on $\lambda_{r}$.

###  III.2 Magnetic Octupolar Domains: Correlation to Structure and Spin

We now show how the coupling between magnetic octupoles and charge quadrupoles
determine the magnetic octupolar domains. Since the magnetic octupoles are
linked to the NRSS, as we show in the next section, the understanding of the
magnetic octupolar domain is also useful in manipulating the NRSS.

We begin by changing the fluorine environment around the Mn atoms, without
affecting the spin arrangements at the Mn sites. Specifically, we change the
coordinates of the F ions from $4f:(x,x,0)\rightarrow 4g:(x,-x,0)$, which
alters the F-Wyckoff site symmetry from $4f$ to $4g$, while keeping the space-
group symmetry unchanged. We note that structurally (without considering the
magnetism), the new structure is equivalent to the original crystal structure
of MnF2 (shown in Fig. 1 (c)), with a shifted origin at (0.5,0.5,0.5), so that
the central and corner Mn atoms are exchanged in the new structure. This
results in a 90∘ alternation of MnF6 octahedral rotation and, hence, their
distortion in the $x-y$ plane (see Figs. 2 (e)-(h)). Correspondingly, this
leads to a reversal of sign in the computed antiferro-type ${\cal Q}_{22^{-}}$
quadrupole for the modified structure, as depicted in Fig. 2 (d). This is also
evident from the changes in the anisotropic charge density distribution around
the Mn atoms in the modified structure as shown in Fig. 2 (f). Note that the
modified fluorine environment, however, does not affect the distortion of the
MnF6 octahedra along $z$ direction, and the sign of the ferro type ${\cal
Q}_{20}$ quadrupole, therefore, remains unaltered (see Fig. 2 (d)).

In order to see the impact of the charge quadrupoles on the magnetic
octupoles, we further analyze the magnetic multipoles of this modified
structure. Interestingly, our computed octupoles show that the sign of the
${\cal O}_{32^{-}}$ octupole reverses whereas that of ${\cal O}_{30}$ remains
as it is (see Fig. 2 (b)), showing the reversal of the magnetic octupolar
domain. Corresponding changes in the anisotropic magnetization density around
the Mn atoms are shown in Fig. 2 (h). This also emphasizes the correlation
between ${\cal O}_{32^{-}}$ octupole and ${\cal Q}_{22^{-}}$ quadrupole; and
${\cal O}_{30}$ octupole and ${\cal Q}_{20}$ quadrupole. Note that the ${\cal
O}_{32^{-}}$ and ${\cal O}_{30}$ octupoles remain ferro- and antiferro-type
respectively. This suggests selecting the magnetic octupolar domain by only
changing the position of the non-magnetic F atoms, without affecting the
magnetic Mn atom’s position or its spin arrangements, emphasizing a strong
interplay between lattice and the magnetic configuration, quantified by the
magnetic octupole.

We close this section by describing the manipulation of the magnetic octupole
by flipping the direction of all the Mn-spins while keeping the same
antiferromagnetic arrangement between the Mn atoms. Physically, it corresponds
to a different antiferromagnetic domain. The reversal of the Mn-spins results
in a reversal of the sign of both octupoles, in contrast to the previous case
where only the ferro-type ${\cal O}_{32^{-}}$ octupole reverses its sign. Note
that, in this case electric quadrupoles remain the same, as they do not depend
on the spin arrangement. It is interesting to point out that this manipulation
of the magnetic octupole has in fact important physical implications. For
example, the two antiferromagnetic domains with ferro-type octupoles of
opposite sign can also be visualized as two separate ferro-octupolar domains.
Such re-visualization is particularly useful in describing important physical
properties that are characteristics of the magnetic octupoles as well as
understanding the conjugate fields for creating such ferro-octupolar domains,
as we discuss later in Section. IV.

###  III.3 Relevance to Non-relativistic Spin-splitting

Next we link the unconventional spin-splitting of the energy bands in the
Brillouin zone (BZ) of the antiferromagnetic MnF2 to the ferro-octupolar order
using the reciprocal-space representation of the ferro-type octupole. Since
the magnetic octupole can be manipulated by modifying the F-environment or the
spin arrangements, as discussed above, we show that these changes can also be
used to manipulate the spin-splitting of the bands.

Our calculated antiferromagnetic band structure of MnF2 both in the presence
and absence of spin-orbit interaction is depicted in Fig. 3(a). As we see from
the band structure, there is a significant energy splitting between the up-
and down-spin bands along $\Gamma\rightarrow$ M direction in the BZ of MnF2.
Interestingly, the splitting is present even without the spin-orbit
interaction, and inclusion of spin-orbit interaction does not affect the
energy splitting along that direction, in agreement with the reported band
structure in Ref. 18. The splitting is large compared to the typical
relativistic Rashba-type spin-splitting and does not require any broken
inversion symmetry of the structure [46].

To understand this unconventional spin-splitting in MnF2, we analyze the
reciprocal-space representation of the ferro-type ${\cal O}_{32^{-}}$
octupole. The reciprocal space representations of the multipoles have been
used successfully to describe the band asymmetries in the BZ of
noncentrosymmetric materials [12, 47, 48]. In contrast to the odd-parity
multipoles, for which the real and reciprocal-space representations are rather
counter-intuitive, for the even-parity multipoles, such as the ${\cal
O}_{32^{-}}$ magnetic octupole, that are relevant here the analysis is much
more straightforward. The reciprocal space representation in this case can
simply be obtained by replacing $\vec{r}\rightarrow\vec{k}$ so that the
reciprocal-space representation of ${\cal O}_{32^{-}}$ octupole ($xym_{z}$ in
real space) is $k_{x}k_{y}m_{z}$. This immediately explains the splitting
between up and down spin-polarized bands, with the spin polarization along
$\hat{z}$, along $[110]$ direction in the momentum space, e.g.,
$\Gamma\rightarrow$ M and A$\rightarrow$ Z directions in momentum space. Note
that such splitting will occur along any momentum direction with non-zero
$k_{x}$ and $k_{y}$.

Figure 3: Spin splitting in MnF2. (a) Band structure of MnF2 in both absence
and presence of spin-orbit interaction, depicting the spin splitting along
$M\rightarrow\Gamma$ direction. The up and down spin-polarized bands in
absence of spin-orbit coupling are shown in solid blue and red lines
respectively and the bands in presence of spin-orbit interaction are indicated
in green dots. (b) The corresponding band structure along the same high-
symmetry $k$ path for the hypothetical modified structure (see text for
details), showing the reversal of spin splitting along $M\rightarrow\Gamma$.
(c) Band structure of MnF2 showing the reversal of spin splitting as the
momentum direction changes from [1$\bar{1}$0] to [110]. (d) A comparison of
the DFT result and the tight-binding analytical expression, Eq. (6), for the
energy splitting between the two top-most spin-polarized bands in (a).

Interestingly, since the reciprocal space representation is an even function
of $\vec{k}$, the resulting spin-splitting should also be symmetric, in
contrast to the anti-symmetric spin splitting in the Rashba interaction. This
indeed is the case e.g., in MnF2 with identical spin splitting along $[110]$
and $[\bar{1}\bar{1}0]$ directions in the momentum space. In addition from the
representation $k_{x}k_{y}m_{z}$, we also expect the spin-splitting to reverse
as the direction in the momentum space changes from [110] to [1$\bar{1}$0]
($d$-wave spin splitting). Indeed, the computed DFT band structure depicts
such reversal of spin-splitting under ${\cal C}_{4}$ rotation of the momentum
direction, as shown in Fig. 3 (b). The representation analysis, therefore,
confirms that the ferro-type ordering of the ${\cal O}_{32^{-}}$ octupoles is
responsible for the spin splitting of the energy bands, analogous to the spin
splitting of bands in a conventional ferromagnet with ferro-type magnetic
dipole. Note that the atomic-site magnetic octupole, discussed here, is
distinct from the cluster and bond multipoles predicted by Hayami et. al. [49,
50] for spin splitting in collinear antiferromagnets. The magnetic octupole
description has the advantage that it naturally occurs in a magnetic multipole
expansion and also describes the order parameter for such unconventional
antiferromagnetism, as discussed above.

To further verify the role of ${\cal O}_{32^{-}}$ octupole in generating the
spin-splitting, we analyze the spin-splitting of the bands for the case of
structural modification, discussed in the previous section, for which the
${\cal O}_{32^{-}}$ octupole switches sign. As expected, in this case, the
spin splitting also reverses (see Fig. 3 (c)). Similar reversal of the spin-
splitting also occurs for the opposite magnetic dipolar domain (not shown
here), in which the magnetic ${\cal O}_{32^{-}}$ octupoles also switch sign.

###  III.4 Role of microscopic parameters in the spin splitting

Having shown that the ferro-type ordering of the magnetic octupoles generates
the spin-splitting in MnF2, to determine the role of different microscopic
parameters, such as electronic hopping, exchange splitting, etc., on the
strength of the spin splitting, we next carry out a low-energy tight-binding
(TB) analysis. For this purpose, we construct a minimal four-band TB model in
the Bloch function basis with the order of the basis set in the sequence
$\\{{\rm Mn1}-d_{xz},{\rm Mn1}-d_{yz},{\rm Mn2}-d_{xz},{\rm Mn2}-d_{yz}\\}$.
The Hamiltonian reads as follows

${\cal
H}_{t}=\alpha(\vec{k})\mathbb{I}+\beta(\vec{k})\Sigma_{z}\otimes\sigma_{x}+\gamma(\vec{k})\Sigma_{x}\otimes\sigma_{0}+\delta(\vec{k})\Sigma_{x}\otimes\sigma_{x}.$
(2)

Here, $\mathbb{I}$ is a $4\times 4$ identity matrix, $\vec{\Sigma}$ and
$\vec{\sigma}$ are the Pauli matrices in the sublattice bases of Mn1 and Mn2
and in the orbital bases of $d_{xz}$ and $d_{yz}$ respectively, and
$\sigma_{0}$ is an identity matrix in the orbital bases. The choice of the
orbitals is governed by the predominant orbital characters of the top pair of
valence bands along $\Gamma\rightarrow$ M in the BZ of MnF2 (see Fig. 6 in
Appendix C). The functions $\alpha(\vec{k}),\beta(\vec{k}),\gamma(\vec{k})$,
and $\delta(\vec{k})$ are determined by the effective $d$-$d$ hopping
parameters ($t_{i}$, $i=1,4$) and the on-site energies ($\varepsilon_{i}$,
$i=1,2$) of the orbitals and their explicit functional forms are given below,

$\displaystyle\alpha(\vec{k})$ $\displaystyle=$
$\displaystyle\varepsilon_{1}+2t_{1}\cos(k_{z}c)$
$\displaystyle\beta(\vec{k})$ $\displaystyle=$
$\displaystyle\varepsilon_{2}+2t_{2}\cos(k_{z}c)$
$\displaystyle\gamma(\vec{k})$ $\displaystyle=$ $\displaystyle
8t_{3}\cos\Big{(}\frac{k_{x}a}{2}\Big{)}\cos\Big{(}\frac{k_{y}a}{2}\Big{)}\cos\Big{(}\frac{k_{z}c}{2}\Big{)}$
$\displaystyle\delta(\vec{k})$ $\displaystyle=$
$\displaystyle-8t_{4}\sin\Big{(}\frac{k_{x}a}{2}\Big{)}\sin\Big{(}\frac{k_{y}a}{2}\Big{)}\cos\Big{(}\frac{k_{z}c}{2}\Big{)}.$
(3)

Here, $a$ and $c$ are the lattice constants of the tetragonal structure. For
simplicity, we consider electronic hoppings only up to second nearest neighbor
and the realistic TB parameters are extracted from the DFT band structure of
MnF2 using the NMTO downfolding technique [51].

The diagonalization of the Hamiltonian in Eq. 2, gives us the four energy
eigenvalues,

$\displaystyle{\cal E}^{-}_{\pm}(\vec{k})$ $\displaystyle=$
$\displaystyle\alpha(\vec{k})-\\{\beta(\vec{k})^{2}+(\delta(\vec{k})\pm\gamma(\vec{k}))^{2}\\}^{1/2}$
$\displaystyle{\cal E}^{+}_{\pm}(\vec{k})$ $\displaystyle=$
$\displaystyle\alpha(\vec{k})+\\{\beta(\vec{k})^{2}+(\delta(\vec{k})\pm\gamma(\vec{k}))^{2}\\}^{1/2}.$
(4)

Analysis of the corresponding eigenvectors shows that there is an energy
splitting $\Delta{\cal E}={\cal E}^{-}_{+}(\vec{k})-{\cal E}^{-}_{-}(\vec{k})$
between bands of dominant Mn1 and Mn2 sublattice contributions. Note that such
energy splitting between bands of different sublattice characters is present
prior to including the effect of antiferromagnet exchange splitting $J$. We
now show that the inclusion of $J$ translates the sublattice splitting of the
bands into the spin splitting of the bands.

To include the effect of the antiferromagnetic exchange, we rewrite the
Hamiltonian (2) in the basis of $\\{{\rm Mn1}-d_{xz}\uparrow,{\rm
Mn1}-d_{yz}\uparrow,{\rm Mn2}-d_{xz}\uparrow,{\rm Mn2}-d_{yz}\uparrow,{\rm
Mn1}-d_{xz}\downarrow,{\rm Mn1}-d_{yz}\downarrow,{\rm
Mn2}-d_{xz}\downarrow,{\rm Mn2}-d_{yz}\downarrow\\}$ and add the exchange term
${\cal H}^{\rm AFM}_{\rm ex}=J{\cal
S}_{z}\otimes(\Sigma_{z}\otimes\sigma_{0})$ to it. The full Hamiltonian is
given by,

${\cal H}={\cal S}_{0}\otimes{\cal H}_{t}+J{\cal
S}_{z}\otimes(\Sigma_{z}\otimes\sigma_{0})~{}~{}~{}~{}.$ (5)

Here, $\vec{\cal S}$ and ${\cal S}_{0}$ are the Pauli matrices and the
identity matrix in the spin basis. The exchange splitting energy between up
and down spin polarized bands, $2J\approx 5$ eV, extracted from the computed
spin-polarized densities of states of MnF2.

By diagonalizing the Hamiltonian (5), we obtain the energy eigenvalues and
focus on the spin-polarized top most valence bands, with energies ${\cal
E}_{\uparrow}$ and ${\cal E}_{\downarrow}$. We note that their eigenvalues are
identical to those of ${\cal E}^{-}_{\pm}$ in Eq. (III.4), except that
$\beta(\vec{k})\rightarrow J+\beta(\vec{k})$. Physically, this means that the
two Mn sublattices, that primarily contribute to those bands, have opposite
spin polarization in the presence of antiferromagnetism and, therefore, they
lead to the spin splitting of the bands. The explicit analytical form of the
energy splitting is given by,

$\displaystyle\Delta{\cal E}_{s}$ $\displaystyle=$ $\displaystyle{\cal
E}_{\uparrow}-{\cal E}_{\downarrow}$ (6) $\displaystyle=$
$\displaystyle\\{(J+\beta)^{2}+(\delta-\gamma)^{2}\\}^{1/2}-\\{(\beta+J)+(\delta+\gamma)^{2}\\}^{1/2}$
$\displaystyle\approx$
$\displaystyle\frac{32}{{\cal\epsilon}}t_{3}t_{4}\sin(k_{x}a)\sin(k_{y}a).$

Here, in obtaining the last equality we have used the fact that
$\epsilon\gg(\delta+\gamma)$, where $\epsilon=J+\beta\approx
J+\varepsilon_{2}+2t_{2}$, using Eq. III.4 and ignoring terms that are second
order in $k_{z}$ or higher. Note that this approximation in $\epsilon$ becomes
exact in the $k_{z}=0$ plane, which contains the desired $\Gamma\rightarrow$ M
momentum direction of spin-splitting. For a realistic set of parameters
(listed in Appendix B), we compare the analytical result in Eq. (6) to the DFT
computed energy splitting of the spin-polarized bands. As depicted in Fig. 3
(d), they agree reasonably with each other, suggesting that our minimal model
captures the essential physics of the spin-splitting in MnF2.

We pause here and analyze the obtained analytical relation in Eq. (6) for the
spin-splitting energy. First of all, it is clear from Eq. (6), that
$\Delta{\cal E}_{s}(\vec{k})=\Delta{\cal E}_{s}(-\vec{k})$, i.e., it is
symmetric in $\vec{k}$, but changes sign under
$(k_{x},k_{y})\rightarrow(k_{x},-k_{y})$, consistent with the computed DFT
bands. Secondly, we see that the splitting energy $\Delta{\cal E}_{s}$ depends
directly on the inter-sublattice hopping parameters, $t_{3}$ (intra-orbital)
and $t_{4}$ (inter-orbital) in the absence of which $\Delta{\cal E}_{s}$
vanishes. This emphasizes the crucial role of interaction between the two
sublattices, which in combination with the antiferromagnetic exchange,
generates the spin splitting. Physically, this indicates that a structure-spin
correlation, a reminiscent of the magnetic octupole as discussed before, is
responsible for the spin splitting. It is interesting to point out that the
inter-sublattice hopping $t_{4}$ (as well as the product $t_{3}t_{4}$) in MnF2
is a symmetric hopping and it changes sign as the direction of hopping changes
from [11z] to [1$\bar{1}$z] with $z\neq 0$ leading to symmetric spin
splitting. This is analogous to the antisymmetric hopping in a nonmagnetic
broken-inversion symmetric system that gives rise to Rashba-like antisymmetric
spin splitting of the energy bands [48]. Finally, the TB analysis also
provides a microscopic understanding of the reversal of spin splitting
described before. For the modified structure, the onsite energy
$\varepsilon_{2}$ and the hopping $t_{2}$ change sign, leading to a sign
change in $\beta$. While, the sign change does not affect the energy
eigenvalues ${\cal E}^{-}_{\pm}(\vec{k})$ in Eq. III.4 (since the dependence
on $\beta$ comes in even power), it reverses the dominant sublattice
contributions in the corresponding eigenvectors (as also evident from the full
DFT band structure, depicted in the appendix C), resulting in a reversal of
sublattice splitting of bands. Since, the sublattice splitting later
transforms into the spin splitting, this consequently leads to the reversal of
the spin splitting. The reversal of spin-splitting for the other
antiferromagnetic domain follows directly from the resulting sign change in
the antiferromagnetic exchange $J$, which, in turn, alters the spin
polarization of the bands.

Overall, the TB analysis provides a crucial insight into the roles of
different microscopic parameters in generating the unconventional spin
splitting of the energy bands in MnF2. The TB analysis further serves as a
link between the proposed “modern” ferro-octupolar order and the conventional
antiferromagnetic dipolar order.

##  IV Implications of Magnetic Octupole

The next step is to identify the implications of the existing magnetic
octupole in determining the physical properties of a centrosymmetric AFM with
NRSS as well as its possible direct measurements. Here, we point out (A) the
resulting physical properties, piezo and anti-piezomagnetic effects and (B)
the possible detection of magnetic octupoles using the magnetic Compton
scattering effect. Once again, we take MnF2 as our example material for
illustration. We show that the existing ferro-type magnetic octupole ${\cal
O}_{32^{-}}$ describes the well-known piezomagnetic effect in MnF2. More
interestingly, however, the knowledge of the antiferro-type magnetic octupole
${\cal O}_{30}$ helps us to predict a previously unknown anti-piezomagnetic
effect. In addition to the underlying fundamental physics and technological
applications of these effects, we also propose magnetic Compton scattering as
an experimental technique for the detection of the apparently hidden magnetic
octupoles. The corresponding measurement set-up as a guidance for future
experiments are also discussed.

###  IV.1 Piezo and Anti-piezomagnetic Effects

The piezomagnetic effect, describes changes in magnetization due to an applied
stress or changes in shape due to an applied magnetic field. It is
particularly promising for applications because it provides a means for
manipulating magnetism via strain engineering in antiferromagnets. In
addition, since it is a linear coupling in contrast to the quadratic coupling
in the commonly used magnetostriction, it also allows for magnetization
switching. The recent demonstration that the dynamically excited optical
phonons can induce the symmetry-breaking lattice distortions required in the
piezomagnetic effect[52, 53, 54], has revived interest. Such optically induced
strain would overcome the limitation of a large mechanical strain and lead to
practical applications in memory and spintronic devices.

The piezomagnetic effect has been predicted and experimentally shown for some
AFMs with NRSS [35, 52, 36]. In this section, we show that the piezomagnetic
effect is the result of ferroic ordering of magnetic octupoles, and illustrate
our ideas for the specific example of MnF2. In addition, we predict an
antipiezomagnetic effect in MnF2 resulting from the antiferro-type ${\cal
O}_{30}$ magnetic octupole.

General symmetry description- We begin by correlating the symmetries of the
magnetic octupole and the piezomagnetic response. We note that both are rank-3
tensors and have the same symmetry, breaking time-reversal symmetry while
keeping inversion symmetry intact. To correlate the elements of the
piezomagnetic response to the magnetic octupole, we analyze the non-zero
elements in the magnetic octupole tensor ${\cal
O}_{ijk}=\int\mu_{i}r_{j}r_{k}d^{3}r$ following the tensor decomposition
reported in Ref. [55]. Note that in general $i,j,k$ are the dummy indices and
to be consistent with the indices of the piezomagnetic response tensor
$\Lambda_{ijk}$, here we associate the index $i$ to the magnetization and $j$
and $k$ to spatial coordinates so that the octupole ${\cal O}_{ijk}$ is
symmetric under the exchange of $j$ and $k$ indices by construction.

The octupole ${\cal O}_{ijk}$ can be decomposed into a totally symmetric
tensor ${\cal S}_{ijk}$ of dimension 10 and an 8 dimensional non-symmetric
residue tensor ${\cal R}_{ijk}$, that account for the 18 independent elements
of ${\cal O}_{ijk}$ [55]. The totally symmetric tensor ${\cal S}_{ijk}$ can be
further decomposed into a traceless totally symmetric part $\tilde{\cal
S}_{ijk}$ and a trace part ${\cal T}_{ijk}$ of dimensions 7 and 3 respectively
and the residue tensor ${\cal R}_{ijk}$ into two irreducible components
$\tilde{\cal R}^{(5)}_{ijk}$ and $\tilde{\cal R}^{(3)}_{ijk}$ of dimensions 5
and 3 respectively, so that ${\cal O}_{ijk}=\tilde{\cal S}_{ijk}+{\cal
T}_{ijk}+\tilde{\cal R}^{(5)}_{ijk}+\tilde{\cal R}^{(3)}_{ijk}$. The explicit
forms of each of these irreducible components are given in Ref. [55]. Note
that the 7 independent components of the totally symmetric traceless part
$\tilde{\cal S}_{ijk}$ can be built from the spherical harmonics with $l=3$
and, hence, these components are often exclusively referred to as the magnetic
octupole, in contrast to the entire ${\cal O}_{ijk}$ tensor.

We now explicitly consider the case of MnF2, which is known to exhibit a
piezomagnetic effect [56, 57, 58, 35] with the non-zero elements of the
piezomagnetic response tensor $\Lambda_{ijk}$,
$\Lambda_{xyz}=\Lambda_{yxz}\neq\Lambda_{zxy}$ so that,

$\displaystyle{\cal M}_{x}=\Lambda_{xyz}\sigma_{yz},{\cal
M}_{y}=\Lambda_{yxz}\sigma_{xz},{\cal M}_{z}=\Lambda_{zxy}\sigma_{xy},$ (7)

where $\vec{\cal M}$ is the magnetization that results from the application of
shear stress $\sigma_{ij}$. We show next that the non-zero components of the
piezomagnetic response of MnF2 correlate to the ferro-type ${\cal O}_{32^{-}}$
octupole.

Analyzing the different elements of $\tilde{\cal S}_{ijk}$, we see that the
${\cal O}_{32^{-}}$ octupole, which has a ferro-type ordering in MnF2, appears
only when $i=x,j=y,k=z$ and for the permutation of the indices. $\tilde{\cal
S}_{ijk}$ being symmetric, all these elements are equal in magnitude.
Similarly, analyzing the elements of other irreducible components, we find
that the only other multipole that has a ferro-type ordering in MnF2 is ${\cal
Q}^{(\tau)}_{x^{2}-y^{2}}$, identified as the $x^{2}-y^{2}$ quadrupole
component of the toroidal moment density
$\tau(\vec{r})=\vec{r}\times\vec{\mu}(\vec{r})$. This leads to non-zero
elements in the residue tensor $\tilde{\cal R}^{(5)}_{ijk}$, with $\tilde{\cal
R}^{(5)}_{xyz}=\tilde{\cal R}^{(5)}_{yzx}\neq\tilde{\cal R}^{(5)}_{zxy}$.
Combining the ferro-type magnetic octupole components in MnF2, and the tensor
decomposition of the the magnetic octupole ${\cal O}_{ijk}$, we obtain ${\cal
O}_{xyz}={\cal O}_{yxz}\neq{\cal O}_{zxy}$. This nicely correlates with the
symmetry allowed as well as experimentally observed components of
piezomagnetic response for MnF2 in Eq. 7, confirming the one-to-one
correlation between the piezomagnetic response and the magnetic octupole
tensor.

Finally we also predict an anti-piezomagnetic response in MnF2 due to the
antiferro-type ${\cal O}_{30}$ octupole. Upon application of stress we expect
an additional change in the Mn spin moments which is, however, opposite for
the two Mn atoms so that their contributions cancel each other, leading to a
zero net magnetization. Here, the tensor decomposition guides us in predicting
which spin components will change due to an applied strain with a certain
orientation. Therefore, we follow the same procedure as before and analyze
first the elements of the symmetric traceless $\tilde{\cal S}_{ijk}$ to
identify the elements of $\tilde{\cal S}_{ijk}$ in which the ${\cal O}_{30}$
octupole appears. These are $\tilde{\cal S}_{xxz}=\tilde{\cal
S}_{yyz}\neq\tilde{\cal S}_{zzz}$. The elements with symmetric permutation of
these indices are also allowed. For these same elements of the residue tensors
$\tilde{\cal R}^{(5)}_{ijk}$ and $\tilde{\cal R}^{(3)}_{ijk}$, we find that
only $\tilde{\cal R}^{(3)}_{xxz}=\tilde{\cal R}^{(3)}_{yyz}\neq\tilde{\cal
R}^{(3)}_{zxx}=\tilde{\cal R}^{(3)}_{zyy}$ are non-zero due to the existence
of the antiferro-type multipole $t_{z}^{(\tau)}$, defined as the $z$ component
of the moment of the toroidal moment density, in MnF2. This means that if both
${\cal O}_{30}$ and $t_{z}^{(\tau)}$ were ferro-type, we would have the
following non-zero components in the piezomagnetic response

$\displaystyle\Lambda_{xxz}=\Lambda_{yyz},\Lambda_{zxx}=\Lambda_{zyy},\text{and}~{}~{}\Lambda_{zzz}~{}~{}.$
(8)

However, since in fact ${\cal O}_{30}$ and $t_{z}^{(\tau)}$ have antiferro-
type arrangement in MnF2, the first equality in the above equation instead
indicates that a spin component along $\hat{x}$ ($\hat{y}$) will develop at
individual Mn sites if we apply a shearing stress $\sigma_{xz}$
($\sigma_{yz}$) to the structure, with the developed spin components having an
anti-parallel alignment between the Mn sites, so that there is no net
magnetization along $\hat{x}$ ($\hat{y}$). We refer to this effect as an anti-
piezomagnetic effect due to the generation of anti-parallel spin components
upon application of stress in analogy to the piezomagnetic effect where
parallel spin moments are generated to give rise to a net change in
magnetization.

Figure 4:  Piezo- and antipiezo-magnetic effects in MnF2. The variation of (a)
the net magnetic moment along the $y$ direction and (b) the individual Mn
magnetic moments along $x$ direction as the shear strain $\sigma_{xz}$ is
varied. The variations of (c) the net magnetic moment along $x$ direction and
(d) the individual Mn magnetic moments along $y$ direction as a function of
the shear strain $\sigma_{yz}$, depicting the piezomagnetic and
antipiezomagnetic effects driven by ferro-type and antiferro-type magnetic
octupoles in MnF2. For the piezomagnetic effects in (a) and (c), the
variations are shown for two different magnetic domains (in green and blue
data points) while for anti-piezomagnetic effect the variations of local spin
magnetic moments (in green and blue data points) at two Mn atoms are shown. In
both cases, variations are also shown for two different strengths of the spin-
orbit coupling constant, viz., $\lambda_{r}=1$ (solid line) and
$\lambda_{r}=2$ (dashed line). The parameter $\lambda_{r}$ is defined at the
caption of Fig. 2.

DFT results for MnF2- Next, to computationally verify our symmetry-guided
prediction of an anti-piezomagnetic effect and to better understand the
microscopic details of both piezo- and antipiezo-magnetic effects, we
explicitly study the effect of a shear stress $\sigma_{xz}$ ($\sigma_{yz}$) on
the magnetism of MnF2 within the DFT framework. For each value of strain, we
relax the internal atomic coordinates while fixing the lattice constants to
the strained values. The results of our calculations are depicted in Fig. 4.
As is clear from Fig. 4 (a) and (c), application of shear stress $\sigma_{xz}$
($\sigma_{yz}$) generates a net moment along $\hat{y}$ ($\hat{x}$), as
expected due to the piezomagnetic effect. In addition, as shown in Fig. 4 (b)
and (d), a tiny spin component appears along $\hat{x}$ ($\hat{y}$) at the
individual Mn sites with an antiparallel orientation at the neighboring Mn
site, corresponding to the predicted anti-piezomagnetic effect.

We see from Fig. 4 that both piezo- and antipiezo-magnetic responses are
linear in nature. Also, in both cases the generated moments reverse their
directions in the opposite antiferromagnetic domain. Such a reversal of moment
direction is consistent with experimental reports and can be understood from
the fact that both ${\cal O}_{32^{-}}$ and ${\cal O}_{30}$ octupoles have
opposite signs in the opposite antiferromagnetic domains.

Further, to understand the importance of spin-orbit interaction on these
effects, we artificially doubled the strength of the spin-orbit coupling in
our calculation and as depicted in Fig. 4, this results in an enhancement in
the generated moment for both cases. This suggests that, unlike the magnetic
Compton scattering described in the next section, both piezo- and antipiezo-
magnetic effects are relativistic effects. Physically, this can be understood
from the fact that the stress applied to the structure needs to be coupled to
the magnetization density of the system, which is mediated via spin-orbit
interaction. We note that the dependence on the spin-orbit coupling strength
also helps to predict the hierarchy of the piezo and antipiezo-magnetic
effects in different materials. For example, the relativistic piezo- and
antipiezo-magnetic effects are expected to be much stronger in CoF2 compared
to MnF2 due to the strong spin-orbit interaction of the Co atoms in the
former.

The predicted anti-piezomagnetic effect should be experimentally observable by
detecting the resulting spin canting in the presence of a uniform stress.
While the early experiments [57], indeed, indicated rotation of spins upon
application of a shear stress so that an antiparallel spin-component is
generated in addition to a net magnetization in a piezomagnetic effect,
confirmation of the antipiezomagnetic effect would require further
measurements to verify the linear generation and the switching of canted
moments. Another possibility of experimental verification would be to apply a
dynamical stress, causing opposite stresses on the two Mn sublattices, so that
the anti-piezomagnetic effect would lead to a net magnetization. Our work,
correlating the piezo- and antipiezo-magnetic effects to the magnetic
octupoles serves as a guideline for future observation and manipulation of
spin arrangements using strain [52, 53].

###  IV.2 Direct Detection of Magnetic Octupoles: Magnetic Compton Profile in
an Antiferromagnet

The Compton scattering [59] of x-ray photons, which was an early confirmation
of quantum mechanical behavior, is a widely used technique today in fields as
diverse as radio-biology, astrophysics, and condensed matter physics. In
condensed matter systems, it is used to measure the electron density in
momentum space or in an extension known as magnetic Compton scattering, the
spin-dependent electron momentum density [60],

$J_{\rm
mag}(p_{z})=\int\int[\rho_{\uparrow}(\vec{p})-\rho_{\downarrow}(\vec{p})]dp_{x}dp_{y}~{}~{}~{}.$
(9)

Here, $J_{\rm mag}$ is the magnetic Compton profile (MCP), the key quantity
measured in the magnetic Compton scattering measurements, and
$\rho_{\uparrow}(\vec{p})$ and $\rho_{\downarrow}(\vec{p})$ are respectively
the up- and down- spin-polarized electron density in momentum space.

Magnetic Compton scattering has been extensively applied to ferri- and ferro-
magnetic systems (with non-zero magnetization) [61, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73] to extract spin polarizations at Fermi surfaces [70,
71, 72]. In one of our recent works, we proposed that a spin-polarized
electron density can also exist in the momentum space of non-magnetic systems,
provided that the inversion symmetry is broken, leading to a MCP [48]. To
date, however, MCP has not been proposed or measured in conventional
antiferromagnets. Because the up and down spin-polarized bands are degenerate,
leading to vanishing spin-polarized electron density in momentum space. Here
we show that the spin-splitting of the energy bands in antiferromagnets with
ferro-type magnetic octupoles results in a non-zero MCP, despite the zero net
magnetization. This, in turn, facilitates the MCP as a direct probe for
existence of ferro-type magnetic octupoles.

To verify the non-zero MCP for our example material MnF2, we explicitly
compute the MCP using the methods implemented in the ELK code (see the
computational details in Appendix A). The computed MCP of MnF2 along the [110]
direction in momentum space is shown in Fig. 5(a). This is to our knowledge
the first identification of a MCP in an AFM. We note that the MCP is present
even without including spin-orbit effects, as expected due to the non-
relativistic spin splitting in MnF2. Note also that the integral of the MCP is
zero, consistent with the net vanishing moment in the system.

The characteristics of the computed MCP are quite different from those of
nonmagnetic ferroelectrics. First, the computed MCP is symmetric in $\vec{p}$
in contrast to the antisymmetric MCP in ferroelectrics [48]. This follows from
the symmetric and antisymmetric spin splitting in MnF2 and ferroelectrics
respectively. More importantly, however, the magnitude of the MCP in MnF2 is
larger by about an order of magnitude compared to the computed values for the
ferroelectrics, PbTiO3 and GeTe [48]. This again is associated with the large
NRSS of the bands in contrast to the weak relativistic spin-splitting of the
bands in ferroelectrics. Finally, as shown in Fig. 5 (a), the MCP in MnF2
changes sign as the momentum direction is changed from the [110] to the
[1$\bar{1}$0] direction, unlike the case of ferroelectrics for which the
profile, being antisymmetric, switches sign as $\vec{p}\rightarrow-\vec{p}$.
Such sign reversal of the MCP in MnF2 is understandable from the reversal of
the spin splitting as the momentum direction changes from [110] to
[1$\bar{1}$0] (see Fig. 3 (b)).

Since the magnetic octupole leads to the spin-splitting of the bands, which,
in turn, gives rise to the MCP, the MCP provides a direct measurement of the
existence of ferroically ordered magnetic octupoles in MnF2. For further
confirmation we compute the MCPs for the cases of the reversed structure and
the other AFM domain (described in section III.2), for both of which the
ferro-type ${\cal O}_{32^{-}}$ magnetic octupole reverses sign. Indeed, the
computed MCPs, shown in Fig. 5 (a), reverse the sign of their profile, in
agreement with our expectation.

Figure 5: Magnetic Compton profiles (MCPs) of MnF2 along (a) [110] and
[1$\bar{1}$0] and (b) [111] and [1$\bar{1}$1] directions in the momentum
space. The reversal of the profiles is apparent from (a) and (b) as the
momentum direction changes by ${\cal C}_{4}$ rotation. Panel (a) also depicts
the MCPs along [110] direction, $J_{\rm mag}^{\rm RS}(p_{110})$ and $J_{\rm
mag}^{\rm RD}(p_{110})$, for the hypothetical modified structure and for the
reversed magnetic domain respectively. In both cases, the MCP switches sign
compared to the profile of MnF2 along the same momentum direction.

Proposed Experimental Setup. The measurement setup needed to detect magnetic
octupoles using MCP will be similar to that of a conventional magnetic Compton
scattering experiment with circularly polarized light. Generally, the
measurements are performed in back-scattering geometry with either parallel
spin and momentum directions or along a momentum direction that has at least
one component along the direction of the spin polarization. Since the spin
polarization direction in MnF2 is along $\hat{z}$, we further compute the MCPs
along the [111] direction in reciprocal space. As depicted in Fig. 5 (b), the
computed MCP, although smaller compared to that along [110], still has a much
larger magnitude compared to the case of a ferroelectric, suggesting that it
is likely discernible in experiments.

We note that since the two antiferromagnetic domains lead to opposite spin
splitting, it is crucial to carry out the measurements on a single
antiferromagnetic domain of MnF2. Such a single antiferromagnetic domain can
be obtained by the simultaneous application of a uniaxial stress and a
magnetic field while cooling the sample through the Néel temperature T${}_{\rm
N}\approx$ 67 K [58, 35]. It is interesting to point out here that the
combination of stress and magnetic field is in fact the conjugate field of a
magnetic octupole, and as described before each of the antiferromagnetic
domains can indeed be identified as a ferro-octupolar domain. Such a single
magnetic domain is also referred to as a piezomagnetic domain due to its close
connection to the piezomagnetic effect in MnF2 [58], driven by the ferroic
magnetic octupoles as discussed in the previous section.

##  V Discussions and Outlook

To summarize, using MnF2 as an example material, we have shown that the order
parameter of centrosymmetric antiferromagnets with NRSS is the magnetic
octupole, since it is the lowest-order ferroically ordered magnetic quantity
in this case. The ferromagneto-octupolar ordering provides a convenient
description of the NRSS and reveals the conjugate field – a combined shear
stress and magnetic field – which can in turn manipulate the NRSS through
selection of the magnetic domain. The magnetic octupole description explains
the reported piezomagnetic response of such systems, and allows us to predict
an as-yet-unobserved nonlinear magnetoelectric effect as well as an
antipiezomagnetic effect resulting from an additional antiferroic arrangement
of different magnetic octupoles. Finally, we propose magnetic Compton
scattering for the direct detection of magnetic octupoles in such
unconventional antiferromagnets.

We note that centrosymmetric antiferromagnets with NRSS may also have higher-
order ferroically ordered even-parity magnetic multipoles in addition to their
ferromagneto-octupolar order. These higher-order multipoles are relevant for
describing NRSS with $g$-wave or $i$-wave symmetry. For example, Fe2O3 in its
low-temperature state with magnetic moments oriented along the symmetry axis,
which is reported to have a $g$-wave spin splitting [23], allows for a
magnetic triakontadipole in addition to the magnetic octupole. The connection
between this rank-5 even-parity magnetic multipole and the corresponding
$g$-wave NRSS is an interesting direction for future study.

In addition to providing important insight into the newly discovered
unconventional antiferromagnets with NRSS, the results presented here are
relevant for the prolonged effort to reveal and detect the magnetic octupolar
phase [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 55, 84], as well as for
potential applications through strain engineering of antiferromagnetism via
the piezo- or antipiezo-magnetic effect in spintronic devices. We note that
magnetic octupoles are also likely to be relevant for the reported spin-phonon
interaction [85, 86] and surface magnetization [87] in MnF2, and could shed
light on the reported strong magnetic anisotropy in doped FeSb2 [37].

Merging the seemingly disconnected fields of hidden order, antiferromagnetic
spintronics, and inelastic scattering techniques, our work opens up new
directions for exploration which we hope will motivate both theoretical and
experimental investigation in the near future.

## Acknowledgements

We thank Steve Collins, Jon Duffy, Urs Staub, and Andrea Urru for stimulating
discussions. NAS and SB were supported by the ERC under the EU’s Horizon 2020
Research and Innovation Programme grant No 810451 and by the ETH Zurich.
Computational resources were provided by ETH Zurich’s Euler cluster, and the
Swiss National Supercomputing Centre, project ID eth3.

## Appendix

###  V.1 Computational Details

The electronic structure of MnF2 has been computed using the linearized
augmented plane wave (LAPW) method as implemented in the ELK code [88, 89]. We
use the LDA+SOC+$U$ formalism, with $U_{\rm eff}=5$ eV at the Mn site [18]. A
basis set of $l_{max(apw)}=8$, a $5\times 5\times 7$ k-point sampling of the
Brillouin zone are used to achieve self-consistency. The product of the
muffin-tin radius (2.4, and 2 a.u. for Mn and F respectively) and the maximum
reciprocal lattice vector is taken to be 7. The magnetic Compton profile and
the atomic-site multipoles are computed using the extended versions [90, 8] of
the Elk code. The spin-polarized electron momentum densities are calculated
and projected onto the selected momentum directions ($\vec{p}$) to obtain the
magnetic Compton profile following the implementations, reported in Ref. [90].
The computed MCP is scaled to the factor that normalizes the valence
contribution of the total Compton profile to the total number of valence
electrons per formula unit of MnF2 in the calculation, which is 29 in this
case. For the computation of atomic-site multipoles, the density matrix
$\rho_{lm,l^{\prime}m^{\prime}}$ is decomposed into the tensor moments, of
which the parity even tensor moments have contributions from $l=l^{\prime}$
terms. We, therefore, evaluate both $d-d$ and $p-p$ matrix element
contributions for the multipoles at the Mn site. The electronic structure of
MnF2 is also computed within the plane-wave based projector augmented wave
(PAW) [91, 92] method as implemented in the Vienna ab initio simulation
package (VASP) [93, 94] and the results agree well with that computed using
the ELK code. The atomic relaxations in presence of shear strain in the
piezomagnetic effect are carried out until the Hellman-Feynman forces on each
atom becomes less than 0.01 eV/Å.

Figure 6: Nonmagnetic band structure of MnF2, showing the $d_{xz}$ and
$d_{yz}$ orbital contributions for the (a) Mn1 and (b) Mn2 sublattices. (c)
and (d) depict the same for the hypothetical modified structure, indicating a
reversal of Mn1 and Mn2 sublattice contributions for the modified structure.

###  V.2 Tight-binding Parameters

The realistic tight-binding parameters of the Hamiltonian (2), i.e., the
effective $d-d$ hoppings $t_{i}$ ($i=1,4$) and the onsite energies
$\varepsilon_{i}$ ($i=1,2$) in Eq. III.4 are extracted from the DFT
calculations by downfolding the effect of the F-$p$ orbitals using the Nth
order muffin-tin orbital (NMTO) method [51]. The computed parameters are
listed in Table 1.

Table 1: Tight-binding parameters of the Hamiltonian (2), extracted using NMTO method. $d$-$d$ hopping parameters (Ry) | Onsite energies (Ry)
---|---
$t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $\varepsilon_{1}$ | $\varepsilon_{2}$
0.0036 | -0.0038 | 0.0040 | 0.0034 | -0.1385 | -0.0339

###  V.3 DFT band structures in absence of magnetism

The computed band structure in absence of magnetism is shown in Fig. 6,
depicting the splitting between bands of two different sublattice
contributions along $\Gamma\rightarrow$ M. For example, for the pair of bands
around 1 eV along $\Gamma\rightarrow$ M, the top band has predominant
contributions from the Mn1 sublattice while the bottom band is predominantly
of Mn2 sublattice character. For the modified structure, described in Section
III.2, the computed atom and orbital projected band structure shows that the
band structure remains identical except that the sublattice characters of the
same pair of bands are reversed.

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