# Impact of Dzyaloshinskii-Moriya and anisotropic exchange interactions on the
cubic kagome antiferromagnets Mn3X and Mn3AB

J. S. R. McCoombs  Department of Physics and Atmospheric Science, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5  A. Zelenskiy  Department of
Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia,
Canada B3H 3J5  M. L. Plumer  Department of Physics and Atmospheric Science,
Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5  Department of
Physics and Physical Oceanography, Memorial University of Newfoundland, St.
John’s, Newfoundland, Canada A1B 3X7  B. W. Southern  Department of Physics
and Astronomy, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2  T.
L. Monchesky  Department of Physics and Atmospheric Science, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5

###### Abstract

We perform a symmetry analysis of the ABC-stacked kagome planes of Mn atoms
common to the $L1_{2}$ Mn3X and antiperovskite Mn3AB alloys. In addition to
the single-ion-anisotropy and Kitaev-type anisotropic exchange known to
stabilize $None$ spin structures in these materials, our analysis results in a
staggered Dzyaloshinskii-Moriya interaction and further gamma-type anisotropic
exchange between nearest-neighbor spins. The presence of these new terms is
shown not to affect the energetics of the $None$ ground-states which explains
their absence in prior minimal magnetic models. We go on to show, however,
that their influence becomes apparent when spin-wave excitations are
considered. We highlight these effects by calculating inelastic neutron
scattering cross-sections to illustrate experimental means of probing the
existence and relative strengths of these cloaked interactions.

##  I Introduction

Both the Cu3Au-type Mn3X alloys with X $\in$ {Rh, Pt, Ir} and the
antiperovskite Mn3AB alloys with A $\in$ {Ga, Zn, Ni, Sn, Ag, Rh, Pt, etc.}
and B $\in$ {C, N} can exist in a cubic configuration belonging to the space
group $Pm\bar{3}m$ with Mn ions occupying the face centers of the cubic unit
cell, X/A atoms occupying the unit cell corners, and either a $B$ atom or a
vacancy at the cell center (Fig. 1). In both structures, the octahedrally-
arranged Mn ions form an ABC stacked array of 2D kagome layers which are
contained to the {111}-family of planes. As is the case for the heavily
studied 2D kagome system, the magnetic interactions in these 3D systems can be
highly frustrated leading to a highly degenerate manifold of magnetic ground-
states Wills (2001); Hemmati  _et al._ (2012) when only nearest neighbor,
isotropic exchange interactions are considered. By contrast, many of these
Mn3X and Mn3AB compounds are found with an ordered $\mathbf{q}=0$, $None$ non-
collinear Néel ordering Krén  _et al._ (1968) which is stabilized either by
further-range exchange, anisotropic exchange, single-ion anisotropy (SIA), or
a combination thereof.

In the Mn3AB systems, the onset of $None$ ordering is usually associated with
a dramatic, first-order increase in lattice dimension, though, with certain
chemical doping procedures this transition can be smoothed out into a more
continuous expansion Takenaka  _et al._ (2008); Takenaka and Takagi (2005);
Matsuno  _et al._ (2009); Iikubo  _et al._ (2008). The prospect of a
controlled negative thermal expansion (NTE) has propelled interest in this
class of materials. On the other hand, the Mn3X alloys have been heavily
studied in part as effective exchange-biasing layers in giant
magnetoresistance (GMR) sensors Tsunoda  _et al._ (2010) with the ordered
$L1_{2}$ Cu_3Au-type structures enhancing the already large unidirectional
magnetic anisotropy characteristic of disordered $\gamma$-type MnIr, MnRh, and
MnPt alloys allowing for thinner pinning layers in GMR heterostructures.

In both the Mn3X and Mn3AB systems, non-collinear, antiferromagnetic spin
arrangements have also garnered much interest recently due to the impact this
type of spin-arrangement can have on various transport phenomenon. Chen et al.
Chen  _et al._ (2014) first predicted the possibility of a non-zero anomalous
Hall effect (AHE) in the antiferromagnetic metal Mn3Ir and the effect has
since been experimentally realized in the related AB-stacked Mn3X compounds
Mn3Ge and Mn3Sn, the Cu3Au-type compounds Mn3Ir and Mn3Pt as well as the
antiperovskites Mn3GaN and Mn3Ni_1-xCu_xN Nayak  _et al._ (2016); Nakatsuji
_et al._ (2015); Iwaki  _et al._ (2020); Liu  _et al._ (2018); Hajiri  _et
al._ (2019); Zhao  _et al._ (2019) solidifying the crucial symmetry breaking
role these non-collinear spin arrangements play in enabling AHE even in the
absence of a ferromagnetic moment.

Non-collinear and particularly non-coplanar spin arrangements have also
recently been studied for the impact that their magnetic excitations can have
on thermal transport properties. The thermal Hall effect has been demonstrated
in magnetic insulators with either non-coplanar spin arrangements or certain
forms of the Dyzaloshinskii-Moriya (DM) interaction leading to
dissipationless, transverse thermal current carried by spin-waves in an
analogous fashion to the electronic Hall effect Katsura  _et al._ (2010);
Owerre (2017); Lu  _et al._ (2019). Though the materials we consider here are
metals, our analysis remains valid for insulating magnetic structures with the
same symmetries.

In this paper we derive a classical magnetic Hamiltonian for the ABC stacked
kagome planes of Mn ions in both the Cu3Au-type Mn3X compounds and the
antiperovskite Mn3AB compounds based on symmetry considerations. Using this
model we explore the space of possible ground-state configurations building
upon the known $\mathbf{q}=0$ non-collinear configurations. We show that
though the presence of the DM interaction in these systems does not alter the
ground-state configurations in the $\mathbf{q}=0$ regime; its influence is
activated when considering the magnonic excitations of these ground-states. We
provide calculations which illustrate the influence the DM interaction has on
the spin wave dispersion in these materials and demonstrate that the magnitude
of the DM interaction should be measurable using inelastic neutron scattering.

##  II The symmetry-allowed magnetic Hamiltonian

For the purpose of this paper we consider only classical, bilinear,
Heisenberg-type spin-spin interactions. We use the symmetries of the
crystallographic space group to identify all allowable single-site and two-
site interactions that have exchange paths contained within the first crystal
unit cell. There are three Mn atoms in the unit cell shown in Fig. 1, which
sit on three different sublattice positions. Each Mn atom has eight nearest
neighbors, four from each of the other two sub-lattices. The spin-Hamiltonian
can therefore be compactly expressed as

$\mathcal{H}=\frac{1}{2}\sum_{\diamond}\sum_{ij}\mathbf{S}_{i}^{T}\mathcal{J}_{ij}\mathbf{S}_{j}$
(1)

where the first sum runs over the octahedra (effectively the crystal unit
cells), $\diamond$ and the second sum runs over the sub-lattice indices,
$i,j\in(1,2,3,1^{\prime},2^{\prime},3^{\prime})$. $\mathbf{S}$ is an
18-dimensional vector containing the six 3-dimensional spins that reside on
the faces of the unit-cell,

$\mathbf{S}^{T}=\left(\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{S}_{3},\mathbf{S}_{1}^{\prime},\mathbf{S}_{2}^{\prime},\mathbf{S}_{3}^{\prime}\right)$
(2)

and $\mathcal{J}$ is the $\left(18\times 18\right)$ coupling matrix

$\mathcal{J}=\left[\begin{matrix}\mathcal{J}_{11}&\mathcal{J}_{12}&\mathcal{J}_{13}&\mathcal{J}_{11^{\prime}}&\mathcal{J}_{12^{\prime}}&\mathcal{J}_{13^{\prime}}\\\\[5.0pt]
\mathcal{J}_{21}&\mathcal{J}_{22}&\mathcal{J}_{23}&\mathcal{J}_{21^{\prime}}&\mathcal{J}_{22^{\prime}}&\mathcal{J}_{23^{\prime}}\\\\[5.0pt]
\mathcal{J}_{31}&\mathcal{J}_{32}&\mathcal{J}_{33}&\mathcal{J}_{31^{\prime}}&\mathcal{J}_{32^{\prime}}&\mathcal{J}_{33^{\prime}}\\\\[5.0pt]
\mathcal{J}_{1^{\prime}1}&\mathcal{J}_{1^{\prime}2}&\mathcal{J}_{1^{\prime}3}&\mathcal{J}_{1^{\prime}4}&\mathcal{J}_{1^{\prime}2^{\prime}}&\mathcal{J}_{1^{\prime}3^{\prime}}\\\\[5.0pt]
\mathcal{J}_{2^{\prime}1}&\mathcal{J}_{2^{\prime}2}&\mathcal{J}_{2^{\prime}3}&\mathcal{J}_{2^{\prime}1^{\prime}}&\mathcal{J}_{2^{\prime}2^{\prime}}&\mathcal{J}_{2^{\prime}3^{\prime}}\\\\[5.0pt]
\mathcal{J}_{3^{\prime}1}&\mathcal{J}_{3^{\prime}2}&\mathcal{J}_{3^{\prime}3}&\mathcal{J}_{3^{\prime}1^{\prime}}&\mathcal{J}_{3^{\prime}2^{\prime}}&J_{3^{\prime}3^{\prime}}\end{matrix}\right]$
(3)

with

$\mathcal{J}_{ij}=\left[\begin{matrix}\mathcal{J}_{ij}^{xx}&\mathcal{J}_{ij}^{xy}&\mathcal{J}_{ij}^{xz}\\\\[5.0pt]
\mathcal{J}_{ij}^{yx}&\mathcal{J}_{ij}^{yy}&\mathcal{J}_{ij}^{yz}\\\\[5.0pt]
\mathcal{J}_{ij}^{zx}&\mathcal{J}_{ij}^{zy}&\mathcal{J}_{ij}^{zz}\end{matrix}\right].$
(4)

Figure 1: The crystal structure of (a) the Cu3Au-type Mn3X compounds with the
red Mn ions occupying the cubic faces and the black X atoms occupying the unit
cell corners (b) the antiperovskite Mn3AB compounds with the red Mn ions
occupying the cubic faces, the black A atoms occupying the unit cell corners
and the yellow B atom occupying the body-center of the unit cell.

The coupling matrix, $\mathcal{J}$, contains all twelve nearest neighbor (NN)
bonds shown in Fig. 2(a), all three single-site interactions, and all three
next-nearest-neighbor (NNN) bonds. The factor of $\frac{1}{2}$ in front of
$\mathcal{H}$ is due to the usual double-counting.

By definition, the coupling matrix, $\mathcal{J}$, is equal to its transpose,
thus, in general, it could contain 171 unique parameters. However, the spin-
Hamiltonian must be invariant under all symmetry operations of the crystal,
and so must be invariant under the generating operations of point group
$O_{h}$. This restriction imposes five equations that $\mathcal{J}$ must obey,
along with the condition that $\mathcal{J}=\mathcal{J}^{T}$. The remaining
requirements are as follows:

$\displaystyle C_{2}\mathcal{J}C_{2}^{T}$ $\displaystyle=\mathcal{J},$ (5)
$\displaystyle C_{2}^{\prime}\mathcal{J}C_{2}^{\prime T}$
$\displaystyle=\mathcal{J},$ $\displaystyle C_{3}\mathcal{J}C_{3}^{T}$
$\displaystyle=\mathcal{J},$ $\displaystyle
C_{2}^{\prime\prime}\mathcal{J}C_{2}^{\prime\prime T}$
$\displaystyle=\mathcal{J},$
$\displaystyle\mathcal{I}\mathcal{J}\mathcal{I}^{T}$
$\displaystyle=\mathcal{J}.$

The operation $C_{2}$ refers to a $\pi$ rotation about the $[001]$
crystallographic direction, $C_{2}^{\prime}$ is a $\pi$ rotation about
$[100]$, $C_{3}$ is a $2\pi/3$ rotation about $[111]$, $C_{2}^{\prime\prime}$
is a $\pi$ rotation about $[110]$ and $\mathcal{I}$ is the inversion
operation. Upon application of these symmetries, the 171 free parameters
contained in $\mathcal{J}$ are reduced to just eight. These are most
conveniently displayed in the representative single-site $\mathcal{J}_{11}$,
NN $\mathcal{J}_{23}$, and NNN $\mathcal{J}_{11^{\prime}}$ sub-matrices:

$\displaystyle\mathcal{J}_{11}$
$\displaystyle=\left[\begin{matrix}\mathcal{J}_{11}^{xx}&0&0\\\\[5.0pt]
0&\mathcal{J}_{11}^{xx}&0\\\\[5.0pt] 0&0&\mathcal{J}_{11}^{zz}\\\\[5.0pt]
\end{matrix}\right],$ (6) $\displaystyle\mathcal{J}_{23}$
$\displaystyle=\left[\begin{matrix}\mathcal{J}_{12}^{xx}&\mathcal{J}_{12}^{xy}&0\\\\[5.0pt]
\mathcal{J}_{12}^{yx}&\mathcal{J}_{12}^{xx}&0\\\\[5.0pt]
0&0&\mathcal{J}_{12}^{zz}\\\\[5.0pt] \end{matrix}\right],$
$\displaystyle\mathcal{J}_{11^{\prime}}$
$\displaystyle=\left[\begin{matrix}\mathcal{J}_{11^{\prime}}^{xx}&0&0\\\\[5.0pt]
0&\mathcal{J}_{11^{\prime}}^{xx}&0\\\\[5.0pt]
0&0&\mathcal{J}_{11^{\prime}}^{zz}\\\\[5.0pt] \end{matrix}\right],$

where all other $\mathcal{J}_{ij}$ can be obtained by an appropriate rotation.
Following the decomposition procedure outlined in Bertaut (1963), we can re-
write the spin Hamiltonian in terms of more familiar bilinear coupling terms
as follows:

$\displaystyle\mathcal{H}=\mathcal{H}_{J}+\mathcal{H}_{D}+\mathcal{H}_{A}+\mathcal{H}_{B}+\mathcal{H}_{K}+\mathcal{H}_{J^{\prime}}.$
(7)

The various terms contributing to this general form of the Hamiltonian are
summarized as:

$\displaystyle\mathcal{H}_{J}$
$\displaystyle=\frac{J}{2}\sum_{\diamond}\sum_{ij}\mathbf{S}_{i}\dotproduct\mathbf{S}_{j},$
(8) $\displaystyle\mathcal{H}_{D}$
$\displaystyle=\frac{D}{2}\sum_{\diamond}\sum_{ij}\hat{\mathbf{D}}_{ij}\dotproduct\mathbf{S}_{i}\times\mathbf{S}_{j},$
$\displaystyle\mathcal{H}_{A}$
$\displaystyle=\frac{A}{2}\sum_{\diamond}\sum_{ij}\left(\hat{\mathbf{n}}_{ij}\dotproduct\mathbf{S}_{i}\right)\left(\hat{\mathbf{n}}_{ij}\dotproduct\mathbf{S}_{j}\right),$
$\displaystyle\mathcal{H}_{B}$
$\displaystyle=\frac{B}{2}\sum_{\diamond}\sum_{ij}\left(\hat{\mathbf{l}}_{ij}\dotproduct\mathbf{S}_{i}\right)\left(\hat{\mathbf{m}}_{ij}\dotproduct\mathbf{S}_{j}\right)+\left(\hat{\mathbf{m}}_{ij}\dotproduct\mathbf{S}_{i}\right)\left(\hat{\mathbf{l}}_{ij}\dotproduct\mathbf{S}_{j}\right),$
$\displaystyle\mathcal{H}_{K}$
$\displaystyle=\frac{K}{2}\sum_{\diamond}\sum_{i}\left(\hat{\mathbf{n}}_{i}\dotproduct\mathbf{S}_{i}\right)^{2},$
$\displaystyle\mathcal{H}_{J^{\prime}}$
$\displaystyle=\frac{J^{\prime}}{2}\sum_{\diamond}\sum_{ii^{\prime}}\mathbf{S}_{i}\dotproduct\mathbf{S}_{i}^{\prime},$

where the various unit-vectors,
$\hat{\mathbf{D}}_{ij},\hat{\mathbf{n}}_{ij},\hat{\mathbf{m}}_{ij},\hat{\mathbf{l}}_{ij},$
and $\hat{\mathbf{n}}_{i}$ are defined in Eq. 32.

This decomposition of the Hamiltonian via decomposition of the coupling
matrix, $\mathcal{J}$, leads to terms representing NN isotropic exchange
$(\mathcal{H}_{J})$, DM interaction $(\mathcal{H}_{D})$, Kitaev-type
anisotropic exchange, $(\mathcal{H}_{A})$, symmetric, off-diagonal,
anisotropic exchange (sometimes referred to as Gamma-type exchange)
$(\mathcal{H}_{B})$, single-ion anisotropy $(\mathcal{H}_{K})$ and NNN
isotropic exchange $\mathcal{H}_{J^{\prime}}$ where we have six free
parameters, $J,J^{\prime},D,A,B$ and $K$ (reduced from eight since one
parameter amounts to an uninteresting constant energy shift and we have
assumed for the scope of this paper that $A^{\prime}=0$, neglecting the
second-order Kitaev-type term). These six free parameters of our model can be
defined in terms of the eight in Eq. 6 as

$\displaystyle J$ $\displaystyle=\mathcal{J}_{12}^{xx}$ (9) $\displaystyle
J^{\prime}$
$\displaystyle=\mathcal{J}_{11^{\prime}}^{xx}=\mathcal{J}_{11^{\prime}}^{zz}$
$\displaystyle D$
$\displaystyle=\frac{\mathcal{J}_{12}^{yx}-\mathcal{J}_{12}^{xy}}{2}$
$\displaystyle A$ $\displaystyle=\mathcal{J}_{12}^{zz}-\mathcal{J}_{12}^{xx}$
$\displaystyle B$
$\displaystyle=\frac{\mathcal{J}_{12}^{yx}+\mathcal{J}_{12}^{xy}}{2}$
$\displaystyle K$ $\displaystyle=\mathcal{J}_{11}^{zz}-\mathcal{J}_{11}^{xx}.$

Of particular interest is the presence of the DM interaction between
neighboring Mn ions. The revealed directions of $\mathbf{D}_{ij}$ through our
symmetry analysis of $\mathcal{J}$ (shown in Fig. 2(a)) are consistent with
Moriya’s rules Moriya (1960). They are also reminiscent of the form of the DM
coupling in the isolated kagome plane geometry described by Elhajel et al.
Elhajal  _et al._ (2002) which included both an in-plane and out-of-plane
component for the DM vectors. The presence of this in-plane component can be
most easily justified by noting that the kagome planes are not themselves
mirror planes, as is the case in the AB-stacked systems which consequently do
not exhibit an in-plane component. Discussion of the DM interaction in both
Mn3X and Mn3AB systems is almost completely lacking likely due to energetic
cancellations in the most common T1 and $\Gamma_{5g}$ configurations as is
pointed out by Chen et al. Chen  _et al._ (2020). However, as we will argue,
the impact of this chiral term is not necessarily vanishing in these
materials.

Figure 2: (a) The magnetic Mn ions within the unit cell along with double-
sided orange arrows indicating the easy/hard-axis directions on each of the
three sites along with blue arrows indicating the direction of the DM vector
at the bisector of each nearest-neighbor bond. (b) A top down view of the unit
cell highlighting the directions of $\hat{\mathbf{D}}_{ij}$ in the top of the
octahedron. (c) The T1 or $\Gamma_{4g}$ spin configuration. (d) The T2 or
$\Gamma_{5g}$ spin configuration.

It is worth noting that the spin-structures of two representative examples
from each of the classes, Mn_3X and Mn_3AB, can be explained rather simply
using Eq. 8. The Cu3Au-type structure Mn3Ir was described by Szunyogh et al.
Szunyogh  _et al._ (2009) using a combination of $\mathcal{H}_{J}$,
$\mathcal{H}_{A}$ and $\mathcal{H}_{K}$. The authors found that the Kitaev-
type term and the SIA could effectively be rolled into a single, effective,
easy-axis anisotropy term with a negative-valued $K_{\text{eff}}$ where the
easy axis for each Mn atom corresponded to the normal direction of the cubic
face it occupies as is shown in Fig. 2(a). The combination of easy-axis
anisotropy and isotopic NN exchange of this form was investigated further by
LeBlanc et al. LeBlanc  _et al._ (2014) who showed that the ground-state spin
structure of such a model should be the canted $None$ T1 (or $\Gamma_{4g}$)
structure exhibiting a small net moment along one of its [111] directions.
Figure 2(c) shows the T1 structure with negligible canting. Likewise, the
structure for the antiperovskite Mn3GaN was described by Bertaut et al.
Bertaut  _et al._ (1968) using the subset of Eq. 7 including
$\mathcal{H}_{J}$, $\mathcal{H}_{J^{\prime}}$ and $\mathcal{H}_{K}$. In this
case, the SIA axes shown in Fig. 2 are hard axes since the value for $K$ was
determined to be positive. This energy penalty attributed to $S_{1z}$,
$S_{2x}$ and $S_{3y}$ along with the ferromagnetic NNN coupling
$(J^{\prime}<0)$ leads to another $None$ structure in which each spin is
confined to both the (111) plane and its respective cubic face which the
authors dubbed the $\Gamma_{5g}$ configuration (also sometimes referred to as
T2), as shown in Fig. 2(d). These two configurations will serve as
representative examples of the two most commonly reported ground-state spin
configurations in Mn3X and Mn3AB compounds with the goal of this paper being
an expansion upon these simple examples of non-collinear order in Mn_3X and
Mn_3AB systems with a particular emphasis on understanding the cloaked
influence of the DM interactions in these materials.

It is important to point out that the coupling matrix $\mathcal{J}$ that was
used to determine terms in the Hamiltonian contains $\left(3\times 3\right)$
coupling matrices for each bond in the unit-cell. However, to simply describe
spin-configurations in these systems, it is convenient to work in a three spin
basis since each unit cell contains only three Mn atoms. This is most
conveniently achieved by working with the Fourier transformed
${\mathcal{J}(\mathbf{q})}$, where each element is defined as

$\displaystyle\mathcal{J}_{ij}(\mathbf{q})=\sum_{\mathbf{d}}\mathcal{J}_{ij}(\mathbf{d})e^{-i\mathbf{q}\cdot\mathbf{d}},$
(10)

with $\mathbf{d}=\mathbf{r}-\mathbf{r^{\prime}}$ being the vector which points
from the Bravais lattice point of the unit cell containing first atom $i$ to
the Bravais lattice point of the unit cell containing the second $j$. By doing
this, we retain all the information contained in Eq. 3 while using a basis
which conforms with the crystal structure. For example, in the case
${\mathbf{q}=0}$, we expect every Mn atom on sublattice 1 to point in a common
direction and likewise for sublattices 2 and 3.

The Fourier transformed $\mathcal{J}(\mathbf{q})$ is as follows:

$\displaystyle\mathcal{J}(\mathbf{q})$
$\displaystyle=\left[\begin{matrix}\mathcal{J}_{11}+\mathcal{J}_{11^{\prime}}\gamma_{11^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{12}\gamma_{12}\left(\mathbf{q}\right)+\mathcal{J}_{12^{\prime}}\gamma_{12^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{13}\gamma_{13}\left(\mathbf{q}\right)+\mathcal{J}_{13^{\prime}}\gamma_{13^{\prime}}\left(\mathbf{q}\right)\\\\[6.0pt]
~{}\mathcal{J}_{21}\gamma_{21}\left(\mathbf{q}\right)+\mathcal{J}_{21^{\prime}}\gamma_{21^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{22}+\mathcal{J}_{22^{\prime}}\gamma_{22^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{23}\gamma_{23}\left(\mathbf{q}\right)+\mathcal{J}_{23^{\prime}}\gamma_{23^{\prime}}\left(\mathbf{q}\right)~{}\\\\[6.0pt]
\mathcal{J}_{31}\gamma_{31}\left(\mathbf{q}\right)+\mathcal{J}_{31^{\prime}}\gamma_{31^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{32}\gamma_{32}\left(\mathbf{q}\right)+\mathcal{J}_{32^{\prime}}\gamma_{32^{\prime}}\left(\mathbf{q}\right)&\mathcal{J}_{33}+\mathcal{J}_{33^{\prime}}\gamma_{33^{\prime}}\left(\mathbf{q}\right)\\\
\end{matrix}\right],$ (11)

where, for example,

$\displaystyle\gamma_{12}\left(\mathbf{q}\right)=1+e^{i2\pi\left(h-l\right)},$
(12)

with $\mathbf{q}=\left(2\pi h/a,2\pi k/a,2\pi l/a\right)$ and the
$\mathcal{J}_{ij}$ are the same as in Eq. 6.

We can now use the Fourier transformed $\mathcal{J}\left(\mathbf{q}\right)$ to
gain insight into the periodicity of the ground-state spin structure for
various combinations of the six free material parameters, $J,J^{\prime},D,A,B$
and $K$. We employ the Luttinger-Tisza method Luttinger and Tisza (1946);
Lyons and Kaplan (1960) to minimize the Fourier-transformed Hamiltonian,

$\displaystyle\mathcal{H}=\frac{1}{2}\sum_{\mathbf{q}}\sum_{i,j}\mathbf{S}_{i}^{T}(\mathbf{q})\mathcal{J}_{ij}(\mathbf{q})\mathbf{S}_{j}(\mathbf{-q})$
(13)

under the “weak” constraint which states that the sum of the magnitudes of all
spins must equal the number of spins. Under this weak constraint, minimization
of (13) amounts to an eigenvalue problem in which the minimum energy
eigenvalue of $\mathcal{J}(\mathbf{q})$, denoted $\lambda_{0}(\mathbf{q})$,
corresponds to the ground-state energy for a given $\mathbf{q}$. We note that
it is possible to obtain unphysical spin-configurations with non-unit-length
spins while operating with the weak constraint. To validate a given solution,
one must first check that each spin is unit-length.

##  III Minimum energy ordering vectors

In practice, we expect the minimum energy state to have a wavevector that is
along one of the high–symmetry directions of the lattice. Consequently, we
evaluate the minimum energy eigenvalue $\lambda_{0}(\mathbf{q})$ along the
path $\Gamma-X-M-\Gamma-R~{}||~{}X-R-M$ through the first Brillouin zone (Path
1 in Fig. 3).

Figure 3: The first Brillouin zone of the cubic system. The direction of the
vector spin chirality is indicated by the arrow and is assumed to always be
along the [111] direction without loss of generality.

As an example, consider the model described in section II for Mn3GaN with
$J=1$, $K=0.2$ and $J^{\prime}=-0.5$. The minimum energy
$\lambda_{0}(\mathbf{q})$ of $\mathcal{J}(\mathbf{q})$ is plotted as the
solid, black line in Fig. 4 which demonstrates a clear energy minimum at the
$\Gamma$-point. This $\mathbf{q}=0$ minimum energy state is consistent with
the $\Gamma_{5g}$ ordering of this model described by Bertaut et al. (though
it is also consistent with the $\mathbf{q}=0$ T1 state and analysis of the
eigenvectors is necessary to distinguish between the two (see section IV)). By
contrast, if $K$ and $J^{\prime}$ are set to zero, $\lambda_{0}(\mathbf{q})$
becomes degenerate along the entire branch $\Gamma-X$ as can be seen in Fig. 4
following the red, dashed line, indicating a highly degenerate ground-state
manifold similar to previous studies Harris  _et al._ (1992); Chernyshev and
Zhitomirsky (2015) in which so long as the angle between each spin in the
system is $None$ the state qualifies as a viable magnetic ground-state.

Figure 4: (Solid black) The minimum energy eigenvalues
$\lambda_{0}(\mathbf{q})$ for the coupling matrix $\mathcal{J}(\mathbf{q})$
with $J=1,K=0.2,$ and $J^{\prime}=-0.5$ and all other parameters set to zero.
The clear minimum exists exclusively at the $\Gamma$-point, consistent with
the $\mathbf{q}=0$ order known for this set of parameters. (Red dashed) The
minimum energy eigenvalues $\lambda_{0}(\mathbf{q})$ for the coupling matrix
$\mathcal{J}(\mathbf{q})$ with $J=1$ and all other parameters set to zero.
$\lambda_{0}(\mathbf{q})$ is degenerate along the entire branch $\Gamma-X$
indicating a high degree of degeneracy.

We wish to study the extended model described by our general
$\mathcal{J}(\mathbf{q})$ in the case of antiferromagnetic nearest-neighbor
interactions, $J=1$. The minimum-energy $\mathbf{q}$ are summarized in the
two-dimensional slices through configuration space shown in Fig. 5 for this
situation. There are a total of six different scenarios that can manifest. The
different scenarios are labeled with the Roman numerals (I-VI). In cases
I-III, the energy is minimized by a single $\mathbf{q}$ with
$\mathbf{q}_{\text{I}}\in\Gamma$, $\mathbf{q}_{\text{II}}\in M$ and
$\mathbf{q}_{\text{III}}\in R$. In cases IV-VI we find varying degrees of
degeneracy throughout the BZ. Configuration IV has a flat “dispersion” in the
slice $\Gamma-X$, V has a similar behavior in the slice $M-R$ and VI has a
degenerate $\lambda_{0}(\mathbf{q})$ for the whole BZ-path shown in Fig. 3.

The phase boundaries between regions exhibit mixed degeneracy of the energy
minimizing $\mathbf{q}$ on either side of the boundary indicative of the
continuous transitions between regions. For example, at the phase boundary
between regions IV and V, scenario VI is realized. Also worth noting is that a
small, negative $J^{\prime}$ breaks all degeneracies, with regions labeled IV
switching to I and regions labeled V switching to II.

Figure 5: Two dimensional slices through the four-dimensional sub-phase-space
containing the parameters $A,B,D$ and $K$. The various regions identified
correspond to one of the six possible energy-minimizing $\mathbf{q}$-vectors.
Case I corresponds to a minimum at the $\Gamma$-point, II to minima at the
$M$-points, III to minima at the $R$-points, IV to a macroscopic degeneracy
between $\Gamma$ and $X$, V to a macroscopic degeneracy between $M$ and $R$
and VI to a macroscopic degeneracy over the entire reciprocal space path.

##  IV Diagonalization of the $\mathbf{q}=0$ Hamiltonian

We now explore the possible phases of our spin-system in the specific case
$\mathbf{q}=0$ given the symmetry-allowed Hamiltonian found in section II
which is shown in Eq. (14). Importantly, all dependence on $D$ and $B$ has
dropped out in the $\mathbf{q}=0$ limit. This is consistent with the two
$\mathbf{q}=0$ configurations described in section II which had no DM
interaction or Gamma-type exchange term present in their descriptions. In
order to do this in a convenient and expedient manner we follow the method
detailed by Essafi et al. Essafi  _et al._ (2017). Since the crystal
structures of Cu3Au-type Mn3X and antiperovskite Mn3AB belong to space group
$Pm\bar{3}m$ with corresponding point-group symmetry $O_{h}$, the spin-
Hamiltonian must also be invariant under all operations of $O_{h}$. These
operations are detailed in Table 1 and Eq. 31. We have used this property once
already to find the form of the coupling matrix, $\mathcal{J}$. Now we use it
to aid in the diagonalization of $\mathcal{J}$.

$\mathcal{J}(\mathbf{q}=0)=\displaystyle\left[\begin{matrix}6J^{\prime}&0&0&4J&0&0&4A+4J&0&0\\\
0&6J^{\prime}&0&0&4A+4J&0&0&4J&0\\\ 0&0&6J^{\prime}+2K&0&0&4J&0&0&4J\\\
4J&0&0&6J^{\prime}+2K&0&0&4J&0&0\\\ 0&4A+4J&0&0&6J^{\prime}&0&0&4J&0\\\
0&0&4J&0&0&6J^{\prime}&0&0&4A+4J\\\ 4A+4J&0&0&4J&0&0&6J^{\prime}&0&0\\\
0&4J&0&0&4J&0&0&6J^{\prime}+2K&0\\\
0&0&4J&0&0&4A+4J&0&0&6J^{\prime}\end{matrix}\right].$ (14)

property once already to find the form of the coupling matrix, $\mathcal{J}$.
Now we use it to aid in the diagonalization of $\mathcal{J}$.

The $\left(9\times 9\right)$ symmetry operations, $\Gamma(g)$, are
representative of the ten classes of $O_{h}$, where
${g\in\left(E,C_{3},C_{2},C_{4},C_{2}^{\prime},i,S_{4},S_{6},\sigma_{h},\sigma_{d}\right)}$.
They are, however, a reducible representation. By decomposition of the
$\Gamma$ representation via a unitary transformation
$\mathcal{U}^{\dagger}\Gamma(g)\mathcal{U}$, the $\Gamma$ representation can
be decomposed into an irreducible representation. The transformed operations,
$\mathcal{U}^{\dagger}\Gamma(g)\mathcal{U}$, will be block-diagonal, all with
the same block structure. In fact, any $\left(9\times 9\right)$ matrix which
is invariant under $O_{h}$ will also be block-diagonalized by $\mathcal{U}$,
including the coupling matrix of interest, $\mathcal{J}$.

Our first task is to find the relevant characters of the $\Gamma$
representation operators, $\chi(g)$. These characters are simply the trace of
one representative operation from each class, $g$. To simplify matters, we
note that the full operation $\Gamma(g)$ is the tensor product of the
corresponding spin permutation matrix $\mathcal{P}(g)$ and the spin rotation
matrix $\mathcal{R}(g)$:

$\displaystyle\Gamma(g)=\mathcal{P}(g)\otimes\mathcal{R}(g),$ (15)

thus the corresponding character is just the product of the permutation
operation’s character and the rotation operation’s character:

$\displaystyle\chi(g)=\chi_{\mathcal{P}}(g)\chi_{\mathcal{R}}(g).$ (16)

Operation | $\chi_{R}$ | $\chi_{P}$ | $\chi$
---|---|---|---
$C_{3}:\\{3^{+}_{111}\\}$ | 0 | 0 | 0
$C_{2}:\\{2_{110}\\}$ | -1 | 0 | 0
$C_{4}:\\{4^{+}_{001}\\}$ | 1 | 2 | 2
$C_{2}^{\prime}:\\{2_{001}\\}$ | -1 | 2 | -2
$\mathcal{I}:\\{-1\\}$ | 3 | 0 | 0
$S_{4}:\\{-4^{+}_{001}\\}$ | 1 | 0 | 0
$S_{6}:\\{-3^{+}_{111}\\}$ | 0 | 0 | 0
$\sigma_{h}:\\{m_{001}\\}$ | -1 | 4 | -4
$\sigma_{d}:\\{m_{110}\\}$ | -1 | 2 | -2
Table 1: Representative symmetry operations from each class of $O_{h}$ with
their corresponding permutation characters $\chi_{P}$, rotation characters
$\chi_{R}$ and total characters $\chi$.

We can now use these characters to find a decomposition,

$\displaystyle\Gamma=\oplus_{\text{I}}\gamma_{\text{I}}\Gamma_{\text{I}}$ (17)

of the spin configuration into its irreducible representations where the
coefficients, $\gamma_{I}$ are

$\displaystyle\gamma_{\text{I}}=\frac{1}{n}\sum_{g\in
O_{h}}\chi_{\text{I}}(g)\chi(g)$ (18)

where $n$ is is the total number of symmetry elements, in this case $n=48$,
and $\chi_{\text{I}}(g)$ are the characters summarized in Table 2. This
results in the following decomposition for the 9-D representation:

$\displaystyle\Gamma=2T_{1g}\oplus T_{2g}.$ (19)

As a result, we expect that our coupling matrix should be block-diagonalizable
into three three-dimensional blocks, two which transform as $T_{1g}$ and one
which transforms as $T_{2g}$. It is possible to describe 9 basis vector
configurations that transform as the representations $T_{1g}$ and $T_{2g}$.
These nine symmetry-adapted-order-parameters (SAOP’s) will be convenient for
interpreting the ground-state spin configuration of the $\mathbf{q}=0$ spin-
Hamiltonian. They are defined as follows in the
$\mathbf{S}=\left[\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{S}_{3}\right]$ basis:

$\displaystyle\mathbf{T}_{1g}^{Fx}$ $\displaystyle=\left[100,100,100\right]$
(20) $\displaystyle\mathbf{T}_{1g}^{Fy}$
$\displaystyle=\left[010,010,010\right]$ $\displaystyle\mathbf{T}_{1g}^{Fz}$
$\displaystyle=\left[001,001,001\right]$ $\displaystyle\mathbf{T}_{1g}^{Ax}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[100,-200,100\right]$
$\displaystyle\mathbf{T}_{1g}^{Ay}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[010,010,0-20\right]$
$\displaystyle\mathbf{T}_{1g}^{Az}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[00-2,001,001\right]$
$\displaystyle\mathbf{T}_{2g}^{x}$
$\displaystyle=\sqrt{\frac{3}{2}}\left[100,000,-100\right]$
$\displaystyle\mathbf{T}_{2g}^{y}$
$\displaystyle=\sqrt{\frac{3}{2}}\left[010,0-10,000\right]$
$\displaystyle\mathbf{T}_{2g}^{z}$
$\displaystyle=\sqrt{\frac{3}{2}}\left[000,001,00-1\right].$

Whereas configurations from the $T_{1g}^{F\alpha}$ block conform to the strong
constraint with unit-length spins, configurations from the $T_{1g}^{A\alpha}$
and $T_{2g}$ blocks do not. To realize physical spin-configurations in these
blocks, linear combinations must be formed. Two example physical
configurations are as follows,

$\displaystyle\mathbf{T}_{1g}^{A}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(\mathbf{T}_{1g}^{Ax}+\mathbf{T}_{1g}^{Ay}+\mathbf{T}_{1g}^{Az}\right)$
(21) $\displaystyle\mathbf{T}_{2g}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(-\mathbf{T}_{2g}^{x}+\mathbf{T}_{2g}^{y}-\mathbf{T}_{2g}^{z}\right)$

and are shown in Fig. 2(c,d) where it becomes obvious that the $T_{1g}^{A}$
configuration corresponds to T1($\Gamma_{4g}$) ordering while $T_{2g}$
corresponds to T2($\Gamma_{5g}$). Rewriting the coupling matrix,
$\mathcal{J}(\mathbf{q}=0)$ in the SAOP basis reveals the utility in working
in such a basis. The transformed $\mathcal{J}$ is

$\mathcal{J^{\prime}}(\mathbf{q}=0)=\left[\begin{matrix}P&0&0&S&0&0&0&0&0\\\
0&P&0&0&S&0&0&0&0\\\ 0&0&P&0&0&S&0&0&0\\\ S&0&0&Q&0&0&0&0&0\\\
0&S&0&0&Q&0&0&0&0\\\ 0&0&S&0&0&Q&0&0&0\\\ 0&0&0&0&0&0&R&0&0\\\
0&0&0&0&0&0&0&R&0\\\ 0&0&0&0&0&0&0&0&R\\\ \end{matrix}\right]$ (22)

with

$\displaystyle P$ $\displaystyle=\frac{8A}{3}+8J+6J^{\prime}+\frac{2K}{3}$
(23) $\displaystyle Q$
$\displaystyle=\frac{4A}{3}-4J+6J^{\prime}+\frac{4K}{3}$ $\displaystyle R$
$\displaystyle=-4A-4J+6J^{\prime}$ $\displaystyle S$
$\displaystyle=\frac{2\sqrt{2}}{3}\left(2A-K\right).$

This form of the coupling matrix is nearly diagonal, and one can gain insight
as to which eigenvectors, or SAOP’s, will need to mix in order to create the
minimal energy state given a prescribed set of $J,A,J^{\prime},$ and $K$.
Namely, in the case where $A=K=0$, we can see immediately that for $J<0$ the
ground state belongs to the $T_{1g}^{F}$ family with some $SO(3)$ global
rotational symmetry as is the case in the classical 3D Heisenberg ferromagnet.
Meanwhile, if $J>0$, we find that the $T_{1g}^{A}$ and $T_{2g}$
antiferromagnetic states minimize the energy and are degenerate with no mixing
between $T_{1g}^{A}$ and $T_{1g}^{F}$ since the off diagonal terms are zero.
If $A$ and/or $K$ are not zero, we expect these two modes to intermix. To move
forward, we must take one more step to remove the mixing between our
mathematically intuitive but physically arbitrary SAOP’s in the $T_{1g}^{F}$
and $T_{1g}^{A}$ families. To do this we create six new basis vectors which
are simply linear combinations of $\mathbf{T}_{1g}^{F\alpha}$ and
$\mathbf{T}_{1g}^{A\alpha}$:

$\displaystyle\mathbf{T}_{1g}^{\text{I}\alpha}$
$\displaystyle=\frac{\xi\mathbf{T}_{1g}^{F\alpha}+\mathbf{T}_{1g}^{A\alpha}}{\sqrt{1+\xi^{2}}}$
(24) $\displaystyle\mathbf{T}_{1g}^{\text{II}\alpha}$
$\displaystyle=\frac{\mathbf{T}_{1g}^{F\alpha}+\zeta\mathbf{T}_{1g}^{A\alpha}}{\sqrt{1+\zeta^{2}}}.$

where the constants $\xi$ and $\zeta$ are defined in Eq. (26). With these
slightly more cumbersome order parameters defined, the $\mathbf{q}=0$
Hamiltonian now becomes fully diagonal with the coupling matrix re-defined as

$\mathcal{J^{\prime\prime}}(\mathbf{q}=0)=\left[\begin{matrix}P^{\prime}&0&0&0&0&0&0&0&0\\\
0&P^{\prime}&0&0&0&0&0&0&0\\\ 0&0&P^{\prime}&0&0&0&0&0&0\\\
0&0&0&Q^{\prime}&0&0&0&0&0\\\ 0&0&0&0&Q^{\prime}&0&0&0&0\\\
0&0&0&0&0&Q^{\prime}&0&0&0\\\ 0&0&0&0&0&0&R&0&0\\\ 0&0&0&0&0&0&0&R&0\\\
0&0&0&0&0&0&0&0&R\\\ \end{matrix}\right],$ (25)

with $P^{\prime}$ and $Q^{\prime}$ defined in Eq. 26 and the three ${(3\times
3)}$ blocks now made very apparent. Thus, whichever of $P^{\prime}$,
$Q^{\prime}$ or $R$ is minimized for a given set of $J,A,J^{\prime},K$ will
saturate the spin-configuration in its corresponding normalized linear
combination of SAOP’s.

Returning to the two examples in Section II, we can immediately explain the
canted T1 ordering for Mn3Ir since for ${J=1,K<0,A=0,J^{\prime}=0}$, the
minimum energy eigenvalue is $P^{\prime}$ which means the spin configuration
should be $\mathbf{T}_{1g}^{\text{I}}$ ($T1$ plus canting toward the $[111]$
direction). Likewise, for Mn3GaN with ${J=1,K>0,A=0,J^{\prime}<0}$, we
immediately find the minimum energy eigenvalue to be $R$ which corresponds to
the $\mathbf{T}_{2g}$ ($\Gamma_{5g}$) configuration.

$\displaystyle P^{\prime}$
$\displaystyle=2A+2J+6J^{\prime}+K-\sqrt{4A^{2}+8AJ-4AK+36J^{2}-4JK+K^{2}}$
(26) $\displaystyle Q^{\prime}$
$\displaystyle=2A+2J+6J^{\prime}+K+\sqrt{4A^{2}+8AJ-4AK+36J^{2}-4JK+K^{2}}$
$\displaystyle\xi$
$\displaystyle=\frac{2\sqrt{2}\left(-2A+K\right)}{2A+18J-K+3\sqrt{4A^{2}+8AJ-4AK+36J^{2}-4JK+K^{2}}}$
$\displaystyle\zeta$
$\displaystyle=\frac{2A+18J-K-3\sqrt{4A^{2}+8AJ-4AK+36J^{2}-4JK+K^{2}}}{2\sqrt{2}\left(-2A+K\right)}.$

Using this methodology, we can take our analysis one step further to examine
the amounts of $T_{1g}^{F}$ and $T_{1g}^{A}$ that make up our newly defined
$T_{1g}^{I}$ state. Figure 6 shows a similar picture to Fig. 5(e) with the
added insight of just how much canting along the [111] direction is induced
for various values of $A$ and $K$. Importantly, this is also a measure of the
scalar spin chirality,
$\chi_{ijk}=\mathbf{S}_{i}\cdot\left(\mathbf{S}_{j}\times\mathbf{S}_{k}\right)$
of the configuration. Notably, for nearly the entire portion of the phase
diagram corresponding to the $T_{1g}$ configuration, $\chi_{ijk}$ is non-zero
due to coupling with the $\mathbf{T}_{1g}^{F}$ mode inducing canting toward
the [111] direction while for the $T_{2g}$ region, $\chi_{ijk}$ is identically
zero since no coupling with the ferromagnetic mode is permitted, resulting in
a strictly coplanar arrangement. In both cases the vector spin chirality,
${\bf\it\kappa}_{ijk}\propto\mathbf{S}_{i}\times\mathbf{S}_{j}+\mathbf{S}_{j}\times\mathbf{S}_{k}+\mathbf{S}_{k}\times\mathbf{S}_{i}$
is directed along the [111] crystallographic direction, reflecting the shared
handedness of the two configurations.

Figure 6: Phase diagram for the parameters $A$ and $K$ with $J=1$ and
$J^{\prime}=-0.1$. The intensity in the region labeled $T_{1g}$ corresponds to
the weight of the $T_{1g}^{F}$ component in the linear combination
$T_{1g}^{I}$ which is associated with the amount of ferromagnetic canting
toward the [111] direction. The region labeled $T_{2g}$ has no ferromagnetic
contribution, it couples to neither of the $T_{1g}$ modes.

##  V Influence on spin-wave dispersion

As was made clear in section IV, the presence of the DM interaction and gamma-
type anisotopic exchange have no impact on the ground-state spin
configurations in the $\mathbf{q}=0$ regime. The energetic contributions due
to these terms cancel out identically in these ferromagnetic sub-lattice
configurations. However, as we will show, one should still be able to glean
information on their presence and and even qualitative measurements of their
magnitudes by considering the elementary magnetic excitations of the
${\mathbf{q}=0}$ ground-states.

Following the methods outlined in Ref. Toth and Lake (2015) and Ref. Petit
(2011), we calculate first the spin wave dispersion along the paths outlined
in Fig. 3. It is important to note that we consider here a system which has
spontaneously aligned in such a way that the vector-spin chirality (or the
magnetic moment for $T_{1g}$ with $K<0$) is aligned solely with the [111]
crystallographic direction. This alignment breaks some of the crystallographic
symmetries and as a result, the point-group symmetry of the system is reduced
to $D_{3d}$ for coplanar arrangements and $S_{6}$ for non-coplanar
arrangements. Consequently, it is necessary to probe both paths through the
first cubic Brillouin zone indicated in Fig. 3 since the points labeled
$M^{\prime}$, for example, indicate wavevectors that lie within the chosen
kagome planes, perpendicular to $\vec{\kappa}$, while points labeled $M$
indicate a wavevector with components both perpendicular and parallel to
$\vec{\kappa}$.

Returning to our two representative examples (Fig. 7(a,c)), we see that the
$T_{2g}$ ground state with $K>0$ retains the characteristic dispersionless
mode between $\Gamma$ and $X$ while the $T_{1g}$ state with $K<0$ breaks the
continuous degeneracy due to the induced ferromagnetic component resulting in
a nearly dispersionless mode. Importantly, in both situations, each of the
three spin-wave modes are degenerate along the two paths through the Brillouin
zone. In addition, all three modes are degenerate at both the $M/M^{\prime}$
and $R/R^{\prime}$ points. These degeneracies are lifted as soon as a small DM
term of the form detailed in Eq. 6 is introduced, leading to a path-dependent
spectrum, a gapped spin-wave spectrum at $M/M^{\prime}$ and $R/R^{\prime}$ as
well as mode crossings that are not confined to the BZ boundaries.

Figure 7: Spin-wave dispersion for (a) the $T_{1g}$ ground-state with
$K=-0.1$, $D=0$, (b) the $T_{1g}$ ground-state with $K=-0.1$ and $D=-0.2$, (c)
the $T_{2g}$ ground-state with $K=0.1$, $D=0$, and (d) the $T_{2g}$ ground-
state with $K=0.1$, $D=-0.2$. In all cases $J=1$ and the remaining parameters
are set to zero.

In the interest of exploring possible ways to measure this DM induced effect,
we provide calculations of inelastic neutron scattering intensities. Again
following Refs. Toth and Lake (2015); Petit (2011), we calculate the
perpendicular component of the $\left(3\times 3\right)$ dynamical spin-spin
correlation matrix,

$\small\mathcal{S}(\mathbf{k},\omega)=\sum_{\alpha\beta}\left(\delta_{\alpha\beta}-\frac{q_{\alpha}q_{\beta}}{q^{2}}\right)\mathcal{S}_{\alpha\beta}\left(\mathbf{k},\omega\right)$
(27)

where

$\small\mathcal{S}_{\alpha\beta}(\mathbf{k},\omega)=\frac{1}{2\pi
N}\sum_{mi,nj}e^{i\mathbf{k}(\mathbf{r}_{mi}-\mathbf{r}_{nj})}\int_{-\infty}^{\infty}d\tau
e^{-i\omega\tau}\left<S_{mi}^{\alpha}S_{nj}^{\beta}\right>,$ (28)

$\alpha,\beta\in\left(x,y,z\right)$, $m,n$ are unit cell indices, $N=3$, and
$i,j$ again label sublattices.

We do this calculation for a path containing $M^{\prime}$ and $R$ since these
are effectively the local $x$ and $z$ directions within the coplanar/near-
coplanar kagome planes. The path we follow highlights markedly higher
intensity in the second Brillouin zones consistent with the elastic neutron
scattering intensity appearing for $h,k,\ell$ all even and (nearly) vanishing
for $h,k,\ell$ all odd as was reported in LeBlanc  _et al._ (2014). As can be
seen in Fig. 8, the effect of the DM interaction in the $T_{1g}$ configuration
is qualitatively obvious since both modes with appreciable intensity at the
$R$ point become non-degenerate. Alternatively, for the $T_{2g}$ state, the
two modes with appreciable intensity remain degenerate along the path
$\Gamma-R$ making the effect of the DM interaction less obvious. Focusing on
the $M^{\prime}$ point, the influence of the DM interaction can again be
observed with the effect again far more obvious for the $T_{1g}$
configuration. Nonetheless, the influence of the DM interaction should be
measurable at $M^{\prime}$ in both configurations since all three spin wave
modes have appreciable intensity in both cases.

Figure 8: Relative magnitudes of the dynamical correlation function
$\mathcal{S}(\mathbf{k},\omega)$ for (a) the $T_{1g}$ configuration with
$K=-0.1,D=0$, (b) the $T_{1g}$ configuration with $K=-0.1,D=-0.2$, (c) the
$T_{2g}$ configuration with $K=0.1,D=0$ and (d) the $T_{2g}$ configuration
with $K=0.1,D=-0.2$.

With the effect of the DM interaction being so prevalent in the slice
$\mathcal{S}\left(\mathbf{k}=R,\omega\right)$, we think it useful to
investigate the dependence on the splitting of the two visible modes as a
function of the DM strength. We define $\Delta E$ as the difference in energy
of the top and intermediate modes of the $T_{1g}$ structure at the $R$-point.
Figure 9 shows the relationship between $\Delta E$ and $D$ for
$D\in\left(-0.5,0\right)$ which is linear, outlining a potential means of
comparing the relative DM strength in different T1-ordered materials. We also
note the relevance of this effect as it pertains to certain optical
measurements such as the work carried out in Ref. Rodríguez-Suárez and Rezende
(2021).

Figure 9: The difference in $\omega$ of the two visible spin wave modes for
the T1 structure at the $R$ point as a function of $D$.

Though single crystal INS data for either the Mn3X or the Mn3AB systems does
not exist to our knowledge, experiments have been conducted on polycrystalline
samples of the Cu3Au-type Mn3Ir LeBlanc  _et al._ (2021). In the
interpretation of this data, the authors made use of not just nearest-neighbor
isotropic exchange $J$ and effective SIA $K_{\text{eff}}$, but also included
further neighbor isotropic exchange interactions reaching out to the fourth
nearest-neighbor Mn ions. The basis of including these interactions and the
initial guesses of their various strengths were owing to a density functional
theory (DFT) study by Szunyogh et al. Szunyogh  _et al._ (2009) which
predicted alternating ferro/antiferromagnetic couplings, $J_{1},J_{2},J_{3}$
and $J_{4}$ along with a value for $K_{\text{eff}}$. The authors fit this
five-parameter model to the powder INS data and found discrepancies with the
values predicted by density functional theory (DFT). We wish to point out here
that the influence of the DM interaction could substantively alter the powder
INS calculations even when its magnitude is relatively weak and should be
considered when interpreting the spin-wave spectra in this material.To
highlight this, in Fig. 10 we show calculations of the powder INS spectrum for
two situations. The first is the case with all values set to the DFT
predictions (normalized so that $J_{1}=1$) while the second includes the same
values for $J_{1},J_{2},J_{3},J_{4}$ and $K_{\text{eff}}$ but also includes DM
coupling with value $D=-0.2$. Importantly, the presence of the DM interaction
affects not only the total magnon bandwidth, causing higher energy
excitations, but also affects the width of the high energy parallel “tails”,
increasing this width in $E$ from $\approx 1J$ to $\approx 1.4J$ as can be
seen in Fig. 10.

Figure 10: Comparison of calculated powder INS intensity profiles for (a) the
$J_{1},J_{2},J_{3},J_{4}$ model with $D=0$ and (b) the
$J_{1},J_{2},J_{3},J_{4}$ model with $D=-0.2$.

##  VI Conclusions

To summarize, we have conducted a symmetry analysis of $L1_{2}$ Mn3X and Mn3AB
compounds to construct a classical magnetic Hamiltonian which includes terms
due to NN and NNN isotropic exchange, NN Kitaev-type anisotropic exchange, NN
Gamma-type anisotropic exchange, NN DM interaction and single-ion anisotropy.
The magnetic ground states we predict using our model are consistent with
previous results obtained considering only NN/NNN isotropic exchange and
single-ion anisotropy with the off-diagonal $\mathcal{H}_{D}$ and
$\mathcal{H}_{B}$ terms making no energetic contribution in large swaths of
the ground-state phase diagrams. We show, however, that these terms can
strongly influence the elementary excitations of these spin-structures.
Reduction in the Brillouin zone symmetry leads to path-dependent spin-wave
dispersion in a similar manner to the AB-stacked system as described in
Zelenskiy  _et al._ (2022). Additionally, gaps are formed at the Brillouin
zone boundaries which have linear dependence on the presupposed DM
interaction’s magnitude. Our inelastic neutron scattering calculations show
that both effects should be measurable and highlight a route for quantifying
and comparing the relative influence of these cloaked interactions in
different Mn3X and Mn3AB compounds.

##  VII Acknowledgments

The authors would like to thank D. Kalliecharan, B. D. MacNeil, and C. D.
Rudderham for their valuable discussions and insights. This work was supported
by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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##  Appendix A Symmetry Operations

In section II we made use of the generating operations of point group $O_{h}$.
These operations can be found in Aroyo  _et al._ (2006), but we detail their
application to our system here. To apply the generating operations to our
coupling matrix $\mathcal{J}_{ij}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)$
we must first understand the spin permutations that result from each
operation. The full transformation $\mathcal{T}$ associated with each
generating operation $g$ is the tensor product between the permutation matrix
$\mathcal{P}$ and the associated axial-vector rotation matrix $\mathcal{R}$:

$\mathcal{T}(g)=\mathcal{P}(g)\otimes\mathcal{R}(g).$ (29)

These permutation and rotation matrices for the six-spin basis used in section
II are as follows:

$\displaystyle\mathcal{P}(C_{2})$
$\displaystyle=\left[\begin{matrix}1&0&0&0&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ 0&0&0&1&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{matrix}\right],\mathcal{R}(C_{2})=\left[\begin{matrix}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{matrix}\right]$ (30)
$\displaystyle\mathcal{P}(C_{2}^{\prime})$
$\displaystyle=\left[\begin{matrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&1&0&0&0\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&0&0&0&1\end{matrix}\right],\mathcal{R}(C_{2}^{\prime})=\left[\begin{matrix}-1&0&0\\\
0&1&0\\\ 0&0&-1\end{matrix}\right]$ $\displaystyle\mathcal{P}(C_{3})$
$\displaystyle=\left[\begin{matrix}0&0&1&0&0&0\\\ 1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&0\end{matrix}\right],\mathcal{R}(C_{3})=\left[\begin{matrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{matrix}\right]$
$\displaystyle\mathcal{P}(C_{2}^{\prime\prime})$
$\displaystyle=\left[\begin{matrix}0&0&0&1&0&0\\\ 0&0&1&0&0&0\\\
0&1&0&0&0&0\\\ 1&0&0&0&0&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&1&0\end{matrix}\right],\mathcal{R}(C_{2}^{\prime\prime})=\left[\begin{matrix}0&1&0\\\
1&0&0\\\ 0&0&-1\end{matrix}\right]$ $\displaystyle\mathcal{P}(\mathcal{I})$
$\displaystyle=\left[\begin{matrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{matrix}\right],\mathcal{R}(\mathcal{I})=\left[\begin{matrix}1&0&0\\\
0&1&0\\\ 0&0&1\end{matrix}\right]$ .

In section IV we made use of representative symmetry operations from each
class of point group $O_{h}$. The character table for $O_{h}$ is as follows:

$O_{h}$ | $E$ | $8C_{3}$ | $6C_{2}$ | $6C_{4}$ | $3C_{2}^{\prime}$ | $\mathcal{I}$ | $6S_{4}$ | $8S_{6}$ | $3\sigma_{h}$ | $6\sigma_{d}$
---|---|---|---|---|---|---|---|---|---|---
$A_{1g}$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$
$A_{2g}$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $+1$ | $-1$ | $+1$ | $+1$ | $-1$
$E_{g}$ | $+2$ | $-1$ | $0$ | $0$ | $+2$ | $+2$ | $0$ | $-1$ | $+2$ | $0$
$T_{1g}$ | $+3$ | $0$ | $-1$ | $+1$ | $-1$ | $+3$ | $+1$ | $0$ | $-1$ | $-1$
$T_{2g}$ | $+3$ | $0$ | $+1$ | $-1$ | $-1$ | $+3$ | $-1$ | $0$ | $-1$ | $+1$
$A_{1u}$ | $+1$ | $+1$ | $+1$ | $+1$ | $+1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$A_{2u}$ | $+1$ | $+1$ | $-1$ | $-1$ | $+1$ | $-1$ | $+1$ | $-1$ | $-1$ | $+1$
$E_{u}$ | $+2$ | $-1$ | $0$ | $0$ | $+2$ | $-2$ | $0$ | $+1$ | $-2$ | $0$
$T_{1u}$ | $+3$ | $0$ | $-1$ | $+1$ | $-1$ | $-3$ | $-1$ | $0$ | $+1$ | $+1$
$T_{2u}$ | $+3$ | $0$ | $+1$ | $-1$ | $-1$ | $-3$ | $+1$ | $0$ | $+1$ | $-1$
Table 2: Character table for point group $O_{h}$.

The permutation and rotation matrices associated with the representative
operations $g$ from each class are as follows:

$\displaystyle\mathcal{P}(C_{3})$ $\displaystyle=\left[\begin{matrix}0&0&1\\\
1&0&0\\\
0&1&0\end{matrix}\right],\mathcal{R}(C_{3})=\left[\begin{matrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{matrix}\right]$ (31) $\displaystyle\mathcal{P}(C_{2})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right],\mathcal{R}(C_{2})=\left[\begin{matrix}0&1&0\\\
1&0&0\\\ 0&0&-1\end{matrix}\right]$ $\displaystyle\mathcal{P}(C_{4})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right],\mathcal{R}(C_{4})=\left[\begin{matrix}0&-1&0\\\
1&0&0\\\ 0&0&1\end{matrix}\right]$ $\displaystyle\mathcal{P}(C_{2}^{\prime})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right],\mathcal{R}(C_{2}^{\prime})=\left[\begin{matrix}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{matrix}\right]$ $\displaystyle\mathcal{P}(\mathcal{I})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right],\mathcal{R}(\mathcal{I})=\left[\begin{matrix}1&0&0\\\
0&1&0\\\ 0&0&1\end{matrix}\right]$ $\displaystyle\mathcal{P}(S_{4})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right],\mathcal{R}(S_{4})=\left[\begin{matrix}0&-1&0\\\
1&0&0\\\ 0&0&1\end{matrix}\right]$ $\displaystyle\mathcal{P}(S_{6})$
$\displaystyle=\left[\begin{matrix}0&0&1\\\ 1&0&0\\\
0&1&0\end{matrix}\right],\mathcal{R}(S_{6})=\left[\begin{matrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{matrix}\right]$ $\displaystyle\mathcal{P}(\sigma_{h})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right],\mathcal{R}(\sigma_{h})=\left[\begin{matrix}-1&0&0\\\
0&-1&0\\\ 0&0&1\end{matrix}\right]$ $\displaystyle\mathcal{P}(\sigma_{d})$
$\displaystyle=\left[\begin{matrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right],\mathcal{R}(\sigma_{d})=\left[\begin{matrix}0&1&0\\\
1&0&0\\\ 0&0&-1\end{matrix}\right]$ .

with $\mathcal{P}(g)$ in the $[\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{S}_{3}]$
basis and $\mathcal{R}(g)$ in the $[S_{i}^{x},S_{i}^{y},S_{i}^{z}]$ basis.

##  Appendix B Unit vectors

In section II, the symmetry-allowed magnetic Hamiltonian was defined as 8. The
various unit vectors,
$\hat{\mathbf{D}}_{ij},\hat{\mathbf{n}}_{i},\hat{\mathbf{n}}_{ij},\hat{\mathbf{l}}_{ij},\hat{\mathbf{m}}_{ij}$,
used to define the couplings are as follows:

$\displaystyle\mathbf{D}_{12}=\mathbf{D}_{1^{\prime}2^{\prime}}=D\hat{\mathbf{y}}$
(32)
$\displaystyle\mathbf{D}_{1^{\prime}2}=\mathbf{D}_{12^{\prime}}=-D\hat{\mathbf{y}}$
$\displaystyle\mathbf{D}_{31}=\mathbf{D}_{3^{\prime}1^{\prime}}=D\hat{\mathbf{x}}$
$\displaystyle\mathbf{D}_{31^{\prime}}=\mathbf{D}_{3^{\prime}1}=-D\hat{\mathbf{x}}$
$\displaystyle\mathbf{D}_{23}=\mathbf{D}_{2^{\prime}3^{\prime}}=D\hat{\mathbf{z}}$
$\displaystyle\mathbf{D}_{23^{\prime}}=\mathbf{D}_{2^{\prime}3}=-D\hat{\mathbf{z}}$
$\displaystyle\hat{\mathbf{n}}_{1}=\hat{\mathbf{n}}_{1^{\prime}}=\hat{\mathbf{z}}$
$\displaystyle\hat{\mathbf{n}}_{2}=\hat{\mathbf{n}}_{2^{\prime}}=\hat{\mathbf{x}}$
$\displaystyle\hat{\mathbf{n}}_{3}=\hat{\mathbf{n}}_{3^{\prime}}=\hat{\mathbf{y}}$
$\displaystyle\hat{\mathbf{n}}_{12}=\hat{\mathbf{n}}_{12^{\prime}}=\hat{\mathbf{n}}_{21^{\prime}}=\hat{\mathbf{n}}_{1^{\prime}2^{\prime}}=\hat{\mathbf{y}}$
$\displaystyle\hat{\mathbf{n}}_{13}=\hat{\mathbf{n}}_{13^{\prime}}=\hat{\mathbf{n}}_{31^{\prime}}=\hat{\mathbf{n}}_{1^{\prime}3^{\prime}}=\hat{\mathbf{x}}$
$\displaystyle\hat{\mathbf{n}}_{23}=\hat{\mathbf{n}}_{23^{\prime}}=\hat{\mathbf{n}}_{32^{\prime}}=\hat{\mathbf{n}}_{2^{\prime}3^{\prime}}=\hat{\mathbf{z}}$
$\displaystyle\hat{\mathbf{l}}_{12}=\hat{\mathbf{l}}_{12^{\prime}}=\hat{\mathbf{l}}_{21^{\prime}}=\hat{\mathbf{l}}_{1^{\prime}2^{\prime}}=\hat{\mathbf{x}}$
$\displaystyle\hat{\mathbf{l}}_{13}=\hat{\mathbf{l}}_{13^{\prime}}=\hat{\mathbf{l}}_{31^{\prime}}=\hat{\mathbf{l}}_{1^{\prime}3^{\prime}}=\hat{\mathbf{y}}$
$\displaystyle\hat{\mathbf{l}}_{23}=\hat{\mathbf{l}}_{23^{\prime}}=\hat{\mathbf{l}}_{32^{\prime}}=\hat{\mathbf{l}}_{2^{\prime}3^{\prime}}=\hat{\mathbf{x}}$
$\displaystyle\hat{\mathbf{m}}_{12}=\hat{\mathbf{m}}_{12^{\prime}}=\hat{\mathbf{m}}_{21^{\prime}}=\hat{\mathbf{m}}_{1^{\prime}2^{\prime}}=\hat{\mathbf{z}}$
$\displaystyle\hat{\mathbf{m}}_{13}=\hat{\mathbf{m}}_{13^{\prime}}=\hat{\mathbf{m}}_{31^{\prime}}=\hat{\mathbf{m}}_{1^{\prime}3^{\prime}}=\hat{\mathbf{z}}$
$\displaystyle\hat{\mathbf{m}}_{23}=\hat{\mathbf{m}}_{23^{\prime}}=\hat{\mathbf{m}}_{32^{\prime}}=\hat{\mathbf{m}}_{2^{\prime}3^{\prime}}=\hat{\mathbf{y}}.$


those of vanilla VQE plotted in red. On average, even pre-training a single
set of fragmented circuits reduces the final error by orders of magnitude. The
performance gain due to fragmented pre-training grows even larger as $T$ is
increased, and notably, the amount of error reduction grows with system size.
These results indicate that increasing the batch size $T$ will continue to
reduce the error on average. In general, larger batch sizes are necessary to
produce the same average error as the system size is increased. This can be
understood by the fact that the number of possible ways to partition the
system can increase with $N$, although exact scaling depends heavily on how
the fragmentation is constrained. In Fig. 14(b), we consider the fraction of
cases where the final percent error of the fragment-initialized optimization
is larger than that of vanilla VQE. This corresponds to the number of times
that pre-training produced a solution less optimal than that produced by
random initialization. Although this fraction is significant when only one set
of fragmented circuits is trained (particularly for large system sizes), by
increasing $T$ to 5, the fraction is reduced by nearly an order of magnitude.