###### Abstract

The paper reports theoretical evidence that the multi-spin-axis magnetic
structure proposed in 1964 by van Laar actually does exist in CoO. This
tetragonal spin arrangement produces both the strong tetragonal and the weaker
monoclinic distortion experimentally observed in this material. The tetragonal
distortion is a product of the magnetostriction and dislocations of the Co
atoms necessary for the magnetic state to be an eigenstate of the system and
causes in turn a monoclinic-like distortion of the array of the oxygen atoms
comparable with the rhombohedral-like distortion of the oxygen atoms recently
proposed to be present in NiO and MnO. The monoclinic-like distortion has no
influence on the tetragonal magnetic structure which is generated by a special
nonadiabatic atomic-like motion of the electrons near the Fermi level
qualifying, in addition, CoO to be a Mott insulator.

###### keywords:

CoO; antiferromagnetic eigenstate; magnetic groups; Mott insulator; atomic-
like motion; nonadiabatic Heisenberg model; magnetic band; magnetic super band

xx 5 Received: date; Accepted: date; Published: date Magnetic Structure of CoO
Ekkehard Krüger Firstname Lastname, Firstname Lastname and Firstname Lastname
Correspondence<EMAIL_ADDRESS>
##  1 Introduction

Cobalt monoxide is antiferromagnetic with the Néel temperature $T_{N}=289$ K.
Just as the other isomorphic transition-metal monoxides MnO, FeO, and NiO, it
is a Mott insulator in both, the paramagnetic and the antiferromagnetic phase.
While above $T_{N}$, all the transition-metal monoxides possesses the fcc
structure $Fm3m=\Gamma^{f}_{c}O^{5}_{h}$ (225) (in parentheses the
international number), CoO occupies a special position in the magnetic phase:
the magnetic structures of MnO, FeO and NiO are known to be monoclinic base-
centered Rooksby (1948); Shull et al. (1951); Roth (1958); Cracknell and
Joshua (1969), but the magnetic structure of CoO is not fully understood. Two
different models of antiferromagnetic CoO are discussed in the literature:
first, the non-collinear multi-spin-axis magnetic structure with tetragonal
symmetry proposed in 1964 by van Laar van Laar (1965) and, second, a collinear
monoclinic structure similar to that of the other monoxides. The models can
hardly be distinguished by neutron diffraction data or by reverse Monte Carlo
refinements of these data Timm et al. (2016). They are considered as
alternative structures or as structures coexisting in antiferromagnetic CoO
Tomiyasu et al. (2004). The non-collinear structure is suggested by the marked
tetragonal distortion of CoO accompanying the antiferromagnetic state, and a
collinear structure could be associated with the additional small monoclinic
deformation unambiguously detected in the antiferromagnetic phase of CoO Jauch
et al. (2001).

The present paper reports theoretical evidence that the tetragonal multi-spin-
axis structure does exist in CoO and creates the tetragonal distortion of the
crystal. The additional small monoclinic deformation is not connected with the
magnetic order but is evidently a “monoclinic-like” deformation of the array
of the oxygen atoms, comparable to the rhombohedral-like distortion proposed
to exist in antiferromagnetic NiO Krüger (2019) and MnO Krüger (2020).

In the following Section 2 the magnetic group of the multi-spin-axis structure
is determined. Just as in NiO and MnO, a problem emerges in this context: a
system invariant under the already reported van Laar (1965); Timm et al.
(2016) type IV Shubnikov magnetic group $I_{c}4_{1}\\!/\\!acd$ given in
Equation (1) cannot possess antiferromagnetic eigenstates. As shown in Section
3, for this reason the crystal undergoes a marked tetragonal distortion going
beyond the magnetostriction. As demonstrated in Section 4, this tetragonal
distortion evidently produces a monoclinic-like deformation of the array of
the oxygen atoms in the deformed crystal.

In Section 5 the nonadiabatic Heisenberg model (NHM) shall be applied to the
electronic ground state of CoO. In Section 5.1, we will show that the
electrons of paramagnetic CoO may occupy an atomic-like state allowing the
system to be a Mott insulator. Having in Section 2 determined the active
magnetic group of antiferromagnetic CoO, we may apply the NHM to the
antiferromagnetic state of CoO, too. In Section 5.2, we will determine the
magnetic band related to the magnetic group of the antiferromagnetic state.
The electrons may perform an atomic-like motion in the nonadiabatic system
stabilizing the tetragonal non-collinear magnetic structure. In addition, this
atomic-like motion allows the electron system to be Mott insulating.

##  2 Magnetic group of the antiferromagnetic state

(a) (b)

Figure 1: Antiferromagnetic CoO with the magnetic group $M_{110}$ (Equation
(14)) based on the tetragonal body-centered Bravais lattice $\Gamma^{v}_{q}$.
The eight Co atoms in the unit cell are (arbitrarily) numbered from 1 to 8.
The three vectors $\bm{T}_{i}$ are the basic translations of $\Gamma^{v}_{q}$.
(a) Magnetic structure of CoO invariant under $M_{110}$. The O atoms are not
shown. The red arrows denote two components of the magnetic moments at the Co
atoms. The components parallel to the plain (001) lie in
$\pm(\bm{T}_{1}+\bm{T}_{3})$ or $\pm(\bm{T}_{2}+\bm{T}_{3})$ direction and are
all of the same size, the other components in $\pm z$ direction are also of
the same size. The figure is consistent with Figure 3 of van Laar (1965). (b)
Distortion of CoO invariant under $M_{110}$. The arrows denote the
displacements of the Co atoms stabilizing the antiferromagnetic structure. The
drawn displacements in $\pm(\bm{T}_{1}+\bm{T}_{3})$ or
$\pm(\bm{T}_{2}+\bm{T}_{3})$ direction form the only distortion of the
antiferromagnetic lattice invariant under $M_{110}$. Any displacement of the O
atoms, on the other hand, cannot happen because it would destroy the symmetry
of $M_{110}$.

The multi-spin-axis structure of CoO van Laar (1965) is indicated in Figure 1
(a) by red arrows. It is invariant under the space group $I4_{1}\\!/\\!acd$
(142) van Laar (1965) and under the type IV Shubnikov magnetic group Timm et
al. (2016):

$I_{c}4_{1}\\!/\\!acd=I4_{1}\\!/\\!acd+\\{K|{\textstyle\frac{1}{2}\frac{1}{2}}0\\}I4_{1}\\!/\\!acd,$
(1)

where $K$ denotes the anti-unitary operator of time-inversion. The unitary
subgroup $I4_{1}\\!/\\!acd$ has the tetragonal body-centered Bravais lattice
$\Gamma^{v}_{q}$ and contains (besides the pure translations) sixteen elements
which are expressible as products of the three generating elements

$\textstyle\\{C^{+}_{4z}|\frac{1}{2}00\\},\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\},\text{
and }\\{I|000\\}.$ (2)

We write the symmetry operations $\\{R|\alpha\beta\gamma\\}$ in the Seitz
notation: $R$ is a point group operation written with respect to the $x$, $y$,
and $z$ axes in Figure 1 and
$\alpha\bm{T}_{1}+\beta\bm{T}_{2}+\gamma\bm{T}_{3}$ the subsequent translation
Bradley and A.P.Cracknell (1972), where the basis translations $\bm{T}_{i}$
are also defined in Figure 1. We write the point group operations $R$ as
defined in Section 1.4 of Bradley and A.P.Cracknell (1972), here $R=E$ stands
for the identity operation, $R=C^{+}_{4z}$ for the anti-clockwise rotation
through $90^{\circ}$ about the $z$ axis, $R=I$ for the inversion, $R=C_{2a}$
for the rotation through $180^{\circ}$ about the $a$ axis as indicated in
Figure 1, and $R=\sigma_{da}$ for the reflection $IC_{2a}$. Since the magnetic
structure in Figure 1 (a) is invariant under the three generating elements (2)
and under the anti-unitary element
$\\{K|{\textstyle\frac{1}{2}\frac{1}{2}}0\\}$, it is invariant under the
complete magnetic group (1).

In what follows, the magnetic group $I_{c}4_{1}\\!/\\!acd$ is referred to as
$M_{142}$ because the unitary subgroup $I4_{1}\\!/\\!acd$ bears the
international number 142,

$M_{142}=I4_{1}\\!/\\!acd+\\{K|{\textstyle\frac{1}{2}\frac{1}{2}}0\\}I4_{1}\\!/\\!acd.$
(3)

Table 1 lists the character tables of the single valued irreducible
representations of $I4_{1}\\!/\\!acd$. In particular, the entry (a) below
$\\{K|000\\}$ means that all the one-dimensional representations of
$I4_{1}\\!/\\!acd$ follow Case (a) with respect to the gray magnetic group
$I4_{1}\\!/\\!acd+\\{K|000\\}I4_{1}\\!/\\!acd$. The Case (a) is defined by
Equation (7.3.45) of Bradley and A.P.Cracknell (1972), in the present case the
one-dimensional representations follow Case (a) because they are real.
According to Condition 1 in Krüger (2019), the antiferromagnetic state cannot
be an eigenstate of a system invariant under $M_{142}$ because
$I4_{1}\\!/\\!acd$ does not possess at least one one-dimensional
representation following Case (c) (as defined by Equation (7.3.47) of Bradley
and A.P.Cracknell (1972)).

These group-theoretical findings change drastically when we assume that the
subgroup $I4_{1}\\!/\\!cd$ (110) of $I4_{1}\\!/\\!acd$ (142) is the unitary
part of the magnetic group of antiferromagnetic CoO. $I4_{1}\\!/\\!cd$
contains one-half of the sixteen elements of $I4_{1}\\!/\\!acd$ (142) and may
be defined by the generating elements

$\textstyle\\{C^{+}_{4z}|\frac{1}{2}00\\}\text{ and
}\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\},$ (4)

see Table 3.7 of Bradley and A.P.Cracknell (1972). However, the origin used in
the Tables of Bradley and A.P.Cracknell (1972) for the space group
$I4_{1}\\!/\\!cd$ (110) is different from the origin used in the present paper
(and marked in Figure 1 by $O$). Thus, all the symmetry operations $S_{BC}$
used in Bradley and A.P.Cracknell (1972) for the group $I4_{1}\\!/\\!cd$ (110)
must be transformed by $\\{E|0\frac{1}{4}\frac{3}{4}\\}$ into the symmetry
operations $S_{paper}$ used in this paper,

$\textstyle
S_{paper}=\\{E|0\frac{1}{4}\frac{3}{4}\\}S_{BC}\\{E|0\frac{1}{4}\frac{3}{4}\\}^{-1}.$
(5)

Just as the group $I4_{1}\\!/\\!acd$, the group $I4_{1}\\!/\\!cd$ has the
tetragonal body-centered Bravais lattice $\Gamma^{v}_{q}$. From
$I4_{1}\\!/\\!cd$ we may derive two magnetic groups, the type IV Shubnikov
group

$\textstyle
M_{1}=I4_{1}\\!/\\!cd+\\{K|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd$ (6)

(with a black and white Bravais lattice), and the type III Shubnikov group

$\textstyle
M_{2}=I4_{1}\\!/\\!cd+\\{KI|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd$ (7)

(with an ordinary Bravais lattice), both leaving invariant the magnetic
structure. The one-dimensional representations at point $Z$ in Table 2 now
follow both requirements (i) and (ii) of Condition 1 of Krüger (2019) for the
magnetic group $M_{2}$, but no one-dimensional representation of
$I4_{1}\\!/\\!cd$ meets both requirements (i) and (ii) for $M_{1}$.

Thus, the type III Shubnikov group $M_{2}$ is the magnetic group of
antiferromagnetic CoO because it allows the system to possess
antiferromagnetic eigenstates. In what follows, we refer $M_{2}$ to as

$\textstyle
M_{110}=I4_{1}\\!/\\!cd+\\{KI|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd.$ (8)

##  3 Tetragonal distortion of antiferromagnetic CoO

In the tetragonally distorted crystal the eight Co and eight O atoms in the
unit cell are located at the positions

Co($000$) | Co($\frac{1}{2}1\frac{1}{2}$) | Co($1\frac{1}{2}1$) | Co($11\frac{1}{2}$)
---|---|---|---
Co($0\frac{1}{2}\frac{1}{2}$) | Co($\frac{1}{2}\frac{1}{2}0$) | Co($\frac{3}{2}11$) | Co($\frac{1}{2}\frac{1}{2}\frac{1}{2}$)
(9)

and

O($\frac{1}{4}\frac{1}{4}0$) | O($\frac{3}{4}\frac{5}{4}\frac{1}{2}$) | O($\frac{5}{4}\frac{3}{4}1$) | O($\frac{1}{4}\frac{1}{4}\frac{1}{2}$)
---|---|---|---
O($\frac{3}{4}\frac{3}{4}0$) | O($\frac{1}{4}\frac{3}{4}\frac{1}{2}$) | O($\frac{3}{4}\frac{5}{4}1$) | O($\frac{3}{4}\frac{3}{4}\frac{3}{2}$),
(10)

respectively, in the coordinate system defined by the basic translations
$\bm{T}_{1}$, $\bm{T}_{2}$, and $\bm{T}_{3}$ of $\Gamma^{v}_{q}$ given in
Figure 1. The positions (9) and (10) are the Wyckoff positions 16c and 16e
(with x=0) of the space group $I4_{1}\\!/\\!acd$ (142) van Laar (1965). It
should be noted that the unit cell of $\Gamma^{v}_{q}$ does not contain 16 Co
atoms and 16 O atoms as is often assumed (and is suggested by the notations
16c and 16e of the Wyckoff positions). In the present paper, the point in the
center of the tetrahedron is not (and must not be) an additional point within
the unit cell, but is a lattice point connected by translation symmetry with
the other lattice points. Thus, there are 8 Co and 8 O atoms in the unit cell.

Magnetostriction produces initially a distortion invariant under the magnetic
group $M_{142}$ in antiferromagnetic CoO. Since a system invariant under
$M_{142}$ does not possess antiferromagnetic eigenstates, the crystal must be
additionally distorted in such a way that the electronic Hamiltonian still
commutes with the symmetry operations of $M_{110}$, but does not commute with
the symmetry operations of ${M_{142}-M_{110}}$, cf. Section 3 of Krüger
(2019). The only distortion bringing the desired effect is a shift of the Co
atoms in $\pm(\bm{T}_{1}+\bm{T}_{3})$ and $\pm(\bm{T}_{2}+\bm{T}_{3})$
direction from their positions (9), as indicated by the arrows in Figure 1
(b). In fact, the magnetic structure together with the dislocations in Figure
1 (b) is invariant under the two generating elements (4) and under the anti-
unitary element $\\{KI|\frac{1}{2}\frac{1}{2}0\\}$ defining $M_{110}$, but the
dislocations are not invariant under the inversion $\\{I|000\\}$ being a
generating element of $M_{142}$. The oxygen atoms are not shifted from their
positions (10) because any dislocation of the oxygen atoms would be
incompatible with the magnetic group $M_{110}$.

This result may be understood by inspection of Figure 1, but also in terms of
Wyckoff positions: the type III Shubnikov group $M_{110}$ may be written in
the form Bradley and A.P.Cracknell (1972)

$\textstyle
M_{110}=I4_{1}\\!/\\!cd+\\{K|000\\}\Big{(}G-I4_{1}\\!/\\!cd\Big{)}$ (11)

where

$\textstyle G=I4_{1}\\!/\\!cd+\\{I|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd$
(12)

is an ordinary (unitary) space group. Equation (11) shows that the Wyckoff
positions of $M_{110}$ are equal to the Wyckoff positions of $G$, since the
operator $\\{K|000\\}$ of time inversion does not influence the atomic
positions. $G$ is equivalent to the group $I4_{1}\\!/\\!acd$ (142) because we
obtain the elements of $G$ when we transform the elements of
$I4_{1}\\!/\\!acd$ by the translation $\\{E|\frac{1}{4}\frac{1}{4}0\\}$. This
statement may be demonstrated by means of the generating elements

$\displaystyle\textstyle\\{C^{+}_{4z}|\frac{1}{2}00\\}$ $\displaystyle=$
$\displaystyle\textstyle\\{E|\frac{1}{4}\frac{1}{4}0\\}\\{C^{+}_{4z}|\frac{1}{2}00\\}\\{E|\frac{1}{4}\frac{1}{4}0\\}^{-1},$
$\displaystyle\textstyle\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}$
$\displaystyle=$
$\displaystyle\textstyle\\{E|\frac{1}{4}\frac{1}{4}0\\}\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}\\{E|\frac{1}{4}\frac{1}{4}0\\}^{-1},$
(13) $\displaystyle\textstyle\\{I|\frac{1}{2}\frac{1}{2}0\\}$ $\displaystyle=$
$\displaystyle\textstyle\\{E|\frac{1}{4}\frac{1}{4}0\\}\\{I|000\\}\\{E|\frac{1}{4}\frac{1}{4}0\\}^{-1},$

see Equation 1.5.12 and Table 3.2 of Bradley and A.P.Cracknell (1972).
Consequently, we receive the origin of $G$ by shifting the origin of
$I4_{1}\\!/\\!acd$ (142) about

$\textstyle\bm{t}_{0}=\frac{1}{4}\bm{T}_{1}+\frac{1}{4}\bm{T}_{2},$ (14)

that means, by shifting the origin from a Co atom to an O atom, see Figure 1.
Thus, the Wyckoff positions are interchanged: In the group $G$ and, hence, in
the magnetic group $M_{110}$, the Co atoms ly on the position $16e$ of the
group $I4_{1}\\!/\\!acd$ (142), and the O atoms on the position $16c$. The
position $16e$ contains a parameter (usually $x$) describing the dislocations
of the Co atoms indicated in Figure 1 (b), the O atoms on position $16c$ are
fixed. This fact plays an essential role in the following Section 4.

##  4 Monoclinic-like distortion

As is the case with antiferromagnetic NiO and MnO, also antiferromagnetic CoO
is clearly deformed by the magnetic structure but, additionally, by a slight
distortion seemingly incompatible with the magnetic structure. In NiO Krüger
(2019) and MnO Krüger (2020), this additional distortion is evidently produced
by the oxygen atoms which form an rhombohedral-like array within the
monoclinic magnetic group $M_{9}$ of antiferromagnetic NiO and MnO. This
distortion was called “inner distortion” of the magnetic group $M_{9}$ because
the symmetry of $M_{9}$ is not disturbed (and must not be disturbed) by the
rhombohedral-like distortion. In CoO, on the other hand, the magnetic group
$M_{110}$ is tetragonal and the additional distortion proved experimentally to
be monoclinic Jauch et al. (2001). This can be understood as follows:

Figure 1 shows the distorted antiferromagnetic structure of CoO with the
magnetic group $M_{110}$. The arrows in Figure 1 (b) specify the dislocations
of the cobalt atoms from their positions in Equation (9). These dislocations
stabilize, on the one hand, the antiferromagnetic structure (Section 2) and
produce, on the other hand, a strong tetragonal distortion of the crystal.
Within this distortion, adjacent Co atoms are dislocated either in the same or
in different directions and, consequently, their distance and, hence, their
mutual attraction varies around its value in the paramagnetic state.
Consequently, the crystal is tetragonally distorted in such a way that the
basic translations $\bm{T}_{1}$, $\bm{T}_{2}$, and $\bm{T}_{3}$ of
$\Gamma^{v}_{q}$ are no longer embedded in the cubic lattice since this
lattice no longer exists.

Figure 2: Oxygen atoms in the body-centered tetragonal Bravais lattice
$\Gamma^{v}_{q}$. The Co atoms are not shown.

The oxygen atoms, on the other hand, are not shifted from their positions in
the tetragonal body-centered lattice, that means, from their position given in
the list (10). Thus, there mutual distances are essentially the same as in the
paramagnetic phase. Just as in NiO and MnO, they form a periodic array within
the lattice of the Co atoms. We assume again that this array forms a Bravais
lattice which does not contain eight O atoms but only one O atom in the unit
cell. The array of the oxygen atoms is spanned by the vectors $\bm{\rho}_{1}$,
$\bm{\rho}_{2}$, and $\bm{\rho}_{3}$ in Figure 2. Though these vectors are
symmetry operations in the paramagnetic lattice, they are no translation
operators in the tetragonal distorted crystal. Nevertheless, they give an
approximate picture of the array of the oxygen atoms. In antiferromagnetic NiO
and MnO, the vectors $\bm{\rho}_{i}$ define an rhombohedral-like array of the
oxygen atoms because they form a trigonal basis within the monoclinic crystals
of NiO and MnO, see Section 4 of Krüger (2019). In CoO, on the other hand,
this basis is not trigonal because the vectors $\bm{\rho}_{i}$ have different
lengths: the length of $\bm{\rho}_{1}$ is different from the lengths of
$\bm{\rho}_{2}$ and $\bm{\rho}_{3}$ in the tetragonally deformed crystal,

$|\bm{\rho}_{1}|\neq|\bm{\rho}_{2}|=|\bm{\rho}_{3}|,$ (15)

because $\bm{\rho}_{1}$ lies in the bases area and $\bm{\rho}_{2}$ and
$\bm{\rho}_{3}$ ly in a side face of the tetrahedron, see Figure 2. Thus, the
vectors $\bm{\rho}_{i}$ define a monoclinic base-centered array of the oxygen
atoms, see Table 3.1 of Bradley and A.P.Cracknell (1972). As stated above,
this array is not exactly monoclinic because the vectors $\bm{\rho}_{i}$ are
no translational symmetry operations in the distorted crystal. Therefore I
call it a “monoclinic-like” distortion of antiferromagnetic CoO being an
“inner distortion” of the magnetic group $M_{110}$. This will say that the
monoclinic-like distortion of the oxygen atoms is not connected with any
change or modification of the tetragonal symmetry of the antiferromagnetic
state which remains invariant under the symmetry operations of the magnetic
group $M_{110}$.

##  5 Application of the Nonadiabatic Heisenberg Model

The NHM defines in narrow, partly filled electronic energy bands a strongly
correlated nonadiabatic atomic-like motion. The nonadiabatic localized states
defining this atomic-like motion are represented by symmetry-adapted and
optimally localized Wannier functions Krüger (2001). In the following
Subsection 5.1 we will show that the band structure of paramagnetic CoO
contains an “insulating band” Krüger (2020) whose Bloch functions can be
unitarily transformed into symmetry-adapted and optimally localized Wannier
functions including all the electrons at the Fermi level. In the second
Subsection 5.2 we will show that the band structure of antiferromagnetic CoO
encloses two “magnetic bands” Krüger (2019) related to the magnetic group
$M_{110}$ of the antiferromagnetic phase. The Bloch functions of these two
bands can be unitarily transformed into optimally localized Wannier functions
symmetry-adapted to $M_{110}$. They are even “magnetic super bands” Krüger
(2019) since all the electrons at the Fermi level belong to these two magnetic
band.

###  5.1 Atomic-like Electrons in Paramagnetic CoO

Figure 3: Conventional band structure (Section 5 of Krüger (2020)) of
paramagnetic fcc CoO as calculated by the FHI-aims program Blum et al. (2009);
Havu et al. (2009) using the length $a=4.260$ Å of the paramagnetic unit cell
as given in Bredow and Gerson (2000). The symmetry labels as defined in Table
A1 of Krüger (2019) are determined by the author (as described in Section 2 of
Krüger (2019)), the symmetry on the lines $\Sigma$ and $\Delta$ are defined by
Table 6 (a). The notations of the points and lines of symmetry in the
Brillouin zone for $\Gamma^{f}_{c}$ follow Figure 3.14 of Ref. Bradley and
A.P.Cracknell (1972). The insulating band is highlighted by the bold line.

Figure 3 shows the band structure of paramagnetic CoO. The band indicated by
the bold lines is characterized by Bloch functions with the symmetry:

$\Gamma^{+}_{3},\ L^{+}_{3},\ X^{+}_{1}+X^{+}_{3},\ W_{1}+W_{4}$ (16)

of band 5 listed Table 3 (a). Thus, the Bloch functions of this band can be
unitarily transformed into optimally localized Wannier functions adapted to
the cubic fcc symmetry and centered at the Co atoms. On the other hand, the
Bloch functions cannot be unitarily transformed in such a way that the Wannier
functions are centered at the O atoms since the symmetry of the Bloch
functions at point $L$ does not coincide with the symmetry of band 5 in Table
3 (b). The Bloch functions (16) define a band with two branches yielding two
degenerate Wannier functions with $\Gamma^{+}_{3}$ symmetry at each Co atom.
There symmetry may be called “$d$-like” since the energy band (16) originates
entirely from a $d$ orbital of Co, see Table 2.7 of Bradley and A.P.Cracknell
(1972). If this $d$-like band is half-filled, it defines an atomic-like motion
qualifying paramagnetic CoO to be a Mott insulator because it consists of all
the branches crossing the Fermi level. In the paramagnetic system the atomic-
like motion occurs solely between the Co atoms.

###  5.2 Atomic-like Electrons in Antiferromagnetic CoO

(a)

(b)

Figure 4:  The band structure of CoO given in Figure 3 folded into the
Brillouin zone for the tetragonal body-centered lattice $\Gamma^{v}_{q}$ of
the magnetic group $M_{110}$. As in Figure 3, the bands are calculated by the
FHI-aims program Blum et al. (2009); Havu et al. (2009). At points $\Gamma$
and $Z$, first the symmetry of the Bloch functions at the equivalent point in
the fcc Brillouin zone is given. Then, after the colon, the symmetry of these
Bloch functions in the tetragonal body-centered Brillouin zone is specified as
given in Table 5. On the line $\Sigma$,$F$, the symmetry labels $\Sigma_{1}$
and $\Sigma_{2}$ are determined by Table 6 (b). The Bloch functions
highlighted in red form two magnetic bands (i.e., twice the band in Table 4)
consisting of sixteen branches assigned to the eight cobalt and eight oxygen
atoms. The notations of the points and lines of symmetry follow Figure 3.10
(b) of Ref. Bradley and A.P.Cracknell (1972).

Table 4 lists the only magnetic band related to the magnetic group $M_{110}$
of antiferromagnetic CoO. Figure 4 shows the band structure of paramagnetic
CoO in Figure 3 as folded into the Brillouin zone $\Gamma^{v}_{q}$ of the
tetragonal body-centered magnetic structure. The band highlighted in Figure 3
by the bold line (the insulating band) becomes in Figure 4 twice the magnetic
band in Table 4. The Bloch functions defining the sixteen branches of these
two magnetic bands are highlighted in red. They can be unitarily transformed
into optionally three types of optimally localized Wannier functions symmetry-
adapted to $M_{110}$: Either (i) the Wannier functions are centered twice at
the eight Co atoms (that means two Wannier functions are at each Co atom), or
(ii) they are centered twice at the eight O atoms; or (iii) they are centered
at both the eight Co atoms and at the eight O atoms.

The atomic-like motion defined by the first and the second possibility (i) and
(ii) clearly has a higher Coulomb energy than the atomic-like motion belonging
to the third possibility (iii) because the Coulomb repulsion between two
electrons occupying localized states at the same atom is greater than the
repulsion of two electrons occupying localized states at different (adjacent)
atoms. Thus, the last case (iii) defines the energetically most favorable
nonadiabatic atomic-like motion which stabilizes the antiferromagnetic
structure with the magnetic group $M_{110}$ Krüger (2019, 1989). Moreover, the
two magnetic bands are magnetic super bands since they comprise all the
branches crossing the Fermi level and, thus, qualify antiferromagnetic CoO to
be a Mott insulator Krüger (2019). While in the paramagnetic system the
atomic-like motion occurs solely between the Co atoms (Section 5.1), it
comprises in the antiferromagnetic phase both, the Co and O atoms.

It should be noted that we can define in the band structure of
antiferromagnetic CoO other narrow bands differing (slightly) from the bands
highlighted in red, but coinciding also with the symmetry of the band in Table
4. They form likewise magnetic bands if they are half-filled. It is not easy
to say which bands represent best the nonadiabatic atomic-like motion in
antiferromagnetic CoO. In any case, the two highlighted bands are a first
approach to describing the atomic-like motion because, in this case, the
localized states in antiferromagnetic CoO most resemble the localized states
in the paramagnetic phase. At this stage, we realize that two magnetic bands
exist, but we do not know the exact combination of their branches.

##  6 Results

Under the assumption that the non-collinear multi-spin-axis magnetic structure
as proposed by van Laar is realized in antiferromagnetic CoO, essential
features of CoO can be understood:

  * •

The non-collinear magnetic structure is invariant under the magnetic group
$M_{142}=I_{c}4_{1}\\!/\\!acd$ (Equation 3). However, an electronic
Hamiltonian invariant under time inversion cannot possess magnetic eigenstates
(Section 2) if it commutes with all the symmetry operations of $M_{142}$. For
this reason, the crystal is markedly tetragonally deformed (see Section 3 and
Figure 1) in order that the group $M_{110}$ in Equation 8 becomes the magnetic
group of antiferromagnetic CoO.

  * •

This tetragonal deformation produces a monoclinic-like deformation of the
array of the oxygen atoms comparable with the rhombohedral-like deformation
proposed in NiO and MnO (Section 4). This monoclinic-like deformation does not
influence the magnetic structure, and thus, does not modify the magnetic group
$M_{110}$. Consequently, no inconsistency between the tetragonal and
monoclinic-like distortion remains.

  * •

In the band structure of CoO exist two magnetic bands related to $M_{110}$
(Section 5.2). One of the two bands yields Wannier functions centered at the
eight Co atoms, the other yields Wannier functions centered at the eight O
atoms. In each case, the Wannier functions are optimally localized and adapted
to the symmetry of the magnetic state. Thus, the electrons of this band may
perform a nonadiabatic atomic-like motion lowering their nonadiabatic
condensation energy $\Delta E$ (as defined in Equation (2.20) in Krüger
(2001)) by activating an exchange mechanism producing a magnetic structure
with the magnetic group $M_{110}$ (Section 5).

  * •

The two magnetic bands are magnetic super bands because they comprise all the
branches crossing the Fermi level. Thus, they qualify antiferromagnetic CoO to
be a Mott insulator (Section 5.2). The nonadiabatic atomic-like motion
involves both, the Co and the O atoms.

Independently of the special magnetic symmetry it was shown in Section 5.1
that

  * •

there exists an insulating band Krüger (2019) in the paramagnetic band
structure of CoO. The electrons of this band may perform a nonadiabatic
atomic-like motion that qualifies CoO to be a Mott insulator also in the
paramagnetic phase. The related optimally localized Wannier functions are
adapted to the cubic fcc symmetry of paramagnetic CoO, are centered on the Co
atoms, are twofold degenerate and have d-like $\Gamma^{+}_{3}$ symmetry. The
atomic-like motion occurs exclusively between the Co atoms.

##  7 Discussion

The results of the present paper as listed in the preceding Section 6 provide
strong evidence that the tetragonal non-collinear multi-spin-axis structure
exists in CoO. The foundations of the presented theory are

  1. (i)

the nonadiabatic Heisenberg model (NHM) published 20 years ago Krüger (2001),
and

  2. (ii)

the group-theoretical theorem, stating that in a system invariant under time
inversion, any magnetic eigenstate and its time-inverted state belong to a
two-dimensional irreducible co-representation of the magnetic group Krüger
(1989, 1999, 2016); Krüger and Strunk (2015).

Unfortunately, the three axioms of the NHM still are relatively unfamiliar
though they are easy to recognize. Nevertheless, since being introduced, the
NHM worked effectively in understanding superconductivity Krüger (2001, 2018);
Krüger and Strunk (2015), magnetism and Mott insulation Krüger (2019, 2020,
2020, 2016). The group-theoretical theorem (ii) proved already to be useful in
understanding distortions accompanying a magnetic state Krüger (1989); Krüger
and Strunk (2011); Krüger (2019, 2020).

This publication was supported by the Open Access Publishing Fund of the
University of Stuttgart.

###### Acknowledgements.

I am very indebted to Guido Schmitz for his continuing support of my work. The
author declares no conflict of interest. The following abbreviations are used
in this manuscript:
NHM | Nonadiabatic Heisenberg model
---|---
$E$ | Identity operation
$I$ | Inversion
$C^{+}_{4z}$ | anti-clockwise rotation through $90^{\circ}$ about the $z$ axis
$C_{2a}$ | Rotation through $180^{\circ}$ as indicated in Figure 1
$\sigma_{da}$ | Reflection $IC_{2a}$
$K$ | anti-unitary operator of time inversion

##  Appendix A Group-theoretical tables

This appendix provides Tables 1 – 6 along with notes to the tables.

Table 1:  Character tables of the single valued irreducible representations of
the tetragonal body-centered space group
$I4_{1}\\!/\\!acd=\Gamma^{v}_{q}D^{20}_{4h}$ (142).

$\Gamma(000)$
---
|  |  |  | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$ | $\\{C_{2b}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{C_{2x}|00\frac{1}{2}\\}$
| $\\{K|000\\}$ | $\\{E|000\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}00\\}$ | $\\{C_{2a}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{C_{2y}|0\frac{1}{2}0\\}$
$\Gamma^{+}_{1}$ | (a) | 1 | 1 | 1 | 1 | 1
$\Gamma^{+}_{2}$ | (a) | 1 | 1 | 1 | -1 | -1
$\Gamma^{+}_{3}$ | (a) | 1 | 1 | -1 | 1 | -1
$\Gamma^{+}_{4}$ | (a) | 1 | 1 | -1 | -1 | 1
$\Gamma^{+}_{5}$ | (a) | 2 | -2 | 0 | 0 | 0
$\Gamma^{-}_{1}$ | (a) | 1 | 1 | 1 | 1 | 1
$\Gamma^{-}_{2}$ | (a) | 1 | 1 | 1 | -1 | -1
$\Gamma^{-}_{3}$ | (a) | 1 | 1 | -1 | 1 | -1
$\Gamma^{-}_{4}$ | (a) | 1 | 1 | -1 | -1 | 1
$\Gamma^{-}_{5}$ | (a) | 2 | -2 | 0 | 0 | 0

$\Gamma(000)$ $(continued)$
---
|  |  | $\\{S^{+}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$
| $\\{I|000\\}$ | $\\{\sigma_{z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{S^{-}_{4z}|\frac{1}{2}00\\}$ | $\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{\sigma_{y}|0\frac{1}{2}0\\}$
$\Gamma^{+}_{1}$ | 1 | 1 | 1 | 1 | 1
$\Gamma^{+}_{2}$ | 1 | 1 | 1 | -1 | -1
$\Gamma^{+}_{3}$ | 1 | 1 | -1 | 1 | -1
$\Gamma^{+}_{4}$ | 1 | 1 | -1 | -1 | 1
$\Gamma^{+}_{5}$ | 2 | -2 | 0 | 0 | 0
$\Gamma^{-}_{1}$ | -1 | -1 | -1 | -1 | -1
$\Gamma^{-}_{2}$ | -1 | -1 | -1 | 1 | 1
$\Gamma^{-}_{3}$ | -1 | -1 | 1 | -1 | 1
$\Gamma^{-}_{4}$ | -1 | -1 | 1 | 1 | -1
$\Gamma^{-}_{5}$ | -2 | 2 | 0 | 0 | 0

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$
---
|  |  |  |  | $\\{C_{2a}|\frac{1}{2}\frac{1}{2}1\\}$ | $\\{C_{2a}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{I|000\\}$
| $\\{E|000\\}$ | $\\{E|001\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{3}{2}\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{C_{2b}|0\frac{1}{2}\frac{3}{2}\\}$ | $\\{C_{2b}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{I|001\\}$
$Z_{1}$ | 2 | -2 | 2 | -2 | 2 | -2 | 0
$Z_{2}$ | 2 | -2 | -2 | 2 | 0 | 0 | 0
$Z_{3}$ | 2 | -2 | 2 | -2 | -2 | 2 | 0
$Z_{4}$ | 2 | -2 | -2 | 2 | 0 | 0 | 0

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$ $(continued)$
---
|  |  |  | $\\{C_{2y}|0\frac{3}{2}0\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}01\\}$ | $\\{\sigma_{y}|0\frac{1}{2}0\\}$ | $\\{S^{-}_{4z}|\frac{1}{2}00\\}$
|  |  |  | $\\{C_{2y}|0\frac{1}{2}0\\}$ | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{y}|0\frac{1}{2}1\\}$ | $\\{S^{+}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$
| $\\{\sigma_{z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{3}{2}\\}$ | $\\{C_{2x}|00\frac{3}{2}\\}$ | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{3}{2}\\}$ | $\\{\sigma_{x}|00\frac{3}{2}\\}$ | $\\{S^{+}_{4z}|\frac{1}{2}\frac{1}{2}\frac{3}{2}\\}$
| $\\{\sigma_{z}|\frac{1}{2}0\frac{3}{2}\\}$ | $\\{\sigma_{da}|\frac{1}{2}\frac{3}{2}0\\}$ | $\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{C_{2x}|00\frac{1}{2}\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}00\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$ | $\\{S^{-}_{4z}|\frac{1}{2}10\\}$
$Z_{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$Z_{2}$ | 0 | 2 | -2 | 0 | 0 | 0 | 0
$Z_{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0
$Z_{4}$ | 0 | -2 | 2 | 0 | 0 | 0 | 0

Notes to Table 1:

  1. (i)

The notations of the points of symmetry follow Figure 3.10 (b) of Bradley and
A.P.Cracknell (1972).

  2. (ii)

Only the two points of symmetry $\Gamma$ and $Z$ invariant under the complete
space group are listed.

  3. (iii)

The character tables are determined from Table 5.7 in Bradley and
A.P.Cracknell (1972).

  4. (iv)

$K$ denotes the operator of time inversion. The entry (a) is determined by
Equation (7.3.51) of Bradley and A.P.Cracknell (1972) and indicates that the
related co-representations of the magnetic group
$I4_{1}\\!/\\!acd+\\{K|000\\}I4_{1}\\!/\\!acd$ follow Case (a) as defined in
Equation (7.3.45) of Bradley and A.P.Cracknell (1972).

Table 2:  Character tables of the single valued irreducible representations of
the tetragonal body-centered space group
$I4_{1}\\!/\\!cd=\Gamma^{v}_{q}C^{12}_{4v}$ (110).

$\Gamma(000)$
---
|  |  |  |  |  | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{\sigma_{y}|0\frac{1}{2}0\\}$
| $\\{K|000\\}$ | $\\{KI|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{K|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{E|000\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}00\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$
$\Gamma_{1}$ | (a) | (a) | (a) | 1 | 1 | 1 | 1 | 1
$\Gamma_{2}$ | (a) | (a) | (a) | 1 | 1 | 1 | -1 | -1
$\Gamma_{3}$ | (a) | (a) | (a) | 1 | 1 | -1 | 1 | -1
$\Gamma_{4}$ | (a) | (a) | (a) | 1 | 1 | -1 | -1 | 1
$\Gamma_{5}$ | (a) | (a) | (a) | 2 | -2 | 0 | 0 | 0

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$
---
|  |  |  |  |  |  |
| $\\{K|000\\}$ | $\\{KI|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{K|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{E|000\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{3}{2}\\}$ | $\\{E|100\\}$
$Z_{1}$ | (c) | (a) | (c) | 1 | 1 | -1 | -1
$Z_{2}$ | (c) | (a) | (c) | 1 | 1 | -1 | -1
$Z_{3}$ | (c) | (a) | (c) | 1 | 1 | -1 | -1
$Z_{4}$ | (c) | (a) | (c) | 1 | 1 | -1 | -1
$Z_{5}$ | (a) | (a) | (a) | 2 | -2 | 2 | -2

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$ $(continued)$
---
| $\\{\sigma_{db}|0\frac{1}{2}\frac{3}{2}\\}$ | $\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}1\\}$ | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}01\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$ | $\\{\sigma_{y}|0\frac{1}{2}0\\}$
| $\\{\sigma_{da}|\frac{1}{2}\frac{1}{2}0\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{C^{+}_{4z}|\frac{1}{2}00\\}$ | $\\{C^{-}_{4z}|\frac{1}{2}\frac{1}{2}\frac{3}{2}\\}$ | $\\{\sigma_{y}|0\frac{1}{2}1\\}$ | $\\{\sigma_{x}|00\frac{3}{2}\\}$
$Z_{1}$ | 1 | -1 | i | -i | i | -i
$Z_{2}$ | 1 | -1 | -i | i | -i | i
$Z_{3}$ | -1 | 1 | -i | i | i | -i
$Z_{4}$ | -1 | 1 | i | -i | -i | i
$Z_{5}$ | 0 | 0 | 0 | 0 | 0 | 0

$P(\frac{1}{4}\frac{1}{4}\frac{1}{4})$
---
|  |  |  |
| $\\{E|000\\}$ | $\\{E|010\\}$ | $\\{E|020\\}$ | $\\{E|030\\}$
$P_{1}$ | 2 | -2i | -2 | 2i

$P(\frac{1}{4}\frac{1}{4}\frac{1}{4})$ $(continued)$
---
| $\\{\sigma_{y}|0\frac{3}{2}1\\}$ | $\\{\sigma_{y}|1\frac{3}{2}1\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{3}{2}\\}$ | $\\{\sigma_{x}|00\frac{7}{2}\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$
| $\\{\sigma_{y}|0\frac{1}{2}0\\}$ | $\\{\sigma_{y}|0\frac{1}{2}1\\}$ | $\\{C_{2z}|\frac{1}{2}1\frac{3}{2}\\}$ | $\\{C_{2z}|\frac{3}{2}1\frac{3}{2}\\}$ | $\\{\sigma_{x}|00\frac{3}{2}\\}$ | $\\{\sigma_{x}|01\frac{3}{2}\\}$
$P_{1}$ | 0 | 0 | 0 | 0 | 0 | 0

$N(0\frac{1}{2}0)$
---
| $\\{E|000\\}$ | $\\{\sigma_{db}|0\frac{1}{2}\frac{1}{2}\\}$ | $\\{E|010\\}$ | $\\{\sigma_{db}|0\frac{3}{2}\frac{1}{2}\\}$
$N_{1}$ | 1 | i | -1 | -i
$N_{2}$ | 1 | -i | -1 | i

$X(00\frac{1}{2})$
---
|  |  | $\\{\sigma_{x}|00\frac{3}{2}\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{1}{2}\\}$ | $\\{\sigma_{y}|0\frac{1}{2}1\\}$
| $\\{E|000\\}$ | $\\{E|001\\}$ | $\\{\sigma_{x}|00\frac{1}{2}\\}$ | $\\{C_{2z}|\frac{1}{2}0\frac{3}{2}\\}$ | $\\{\sigma_{y}|0\frac{1}{2}0\\}$
$X_{1}$ | 2 | -2 | 0 | 0 | 0

Notes to Table 2:

  1. (i)

The notations of the points of symmetry follow Figure 3.10 (b) of Bradley and
A.P.Cracknell (1972).

  2. (ii)

The character tables are determined from Table 5.7 in Bradley and
A.P.Cracknell (1972).

  3. (iii)

$K$ denotes the operator of time inversion. The entries (a) and (c) are
determined by Equation (7.3.51) of Bradley and A.P.Cracknell (1972). They
indicate whether the related co-representations of the magnetic groups
$I4_{1}\\!/\\!cd+\\{K|000\\}I4_{1}\\!/\\!cd$,
$I4_{1}\\!/\\!cd+\\{KI|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd$, and
$I4_{1}\\!/\\!cd+\\{K|\frac{1}{2}\frac{1}{2}0\\}I4_{1}\\!/\\!cd$ follow Case
(a) or Case (c) as defined in Equations (7.3.45) and (7.3.47), respectively,
of Bradley and A.P.Cracknell (1972).

Table 3:  Excerpt of Table 1 of Krüger (2020) showing only Band 5 with optimally localized Wannier functions of $\Gamma^{+}_{3}$ symmetry centered at the Co atoms (Table (a)) and the O atoms (Table (b)).  (a) | Co($000$) | $\Gamma$ | $X$ | $L$ | $W$
---|---|---|---|---|---
Band 5 | $\Gamma^{+}_{3}$ | $\Gamma^{+}_{3}$ |  $X^{+}_{1}$ \+ $X^{+}_{3}$ | $L^{+}_{3}$ |  $W_{1}$ \+ $W_{4}$
(b) | O($\overline{\frac{1}{2}}\frac{1}{2}\frac{1}{2}$) | $\Gamma$ | $X$ | $L$ | $W$
Band 5 | $\Gamma^{+}_{3}$ | $\Gamma^{+}_{3}$ |  $X^{+}_{1}$ \+ $X^{+}_{3}$ | $L^{-}_{3}$ |  $W_{1}$ \+ $W_{4}$
Table 4:  Symmetry labels of the Bloch functions at the points of symmetry in
the Brillouin zone for $I4_{1}\\!/\\!cd$ (110) of the only energy band with
symmetry-adapted and optimally localized Wannier functions centered at the Co
$\big{(}$Table (a)$\big{)}$ as well as O $\big{(}$Table (b)$\big{)}$ atoms,
respectively.

(a) | Co | Co($000$) | Co($\frac{1}{2}1\frac{1}{2}$) | Co($1\frac{1}{2}1$) | Co($11\frac{1}{2}$) | Co($0\frac{1}{2}\frac{1}{2}$) | Co($\frac{1}{2}\frac{1}{2}0$) | Co($\frac{3}{2}11$) | Co($\frac{1}{2}\frac{1}{2}\frac{1}{2}$)
---|---|---|---|---|---|---|---|---|---
(b) | O | O($\frac{1}{4}\frac{1}{4}0$) | O($\frac{3}{4}\frac{5}{4}\frac{1}{2}$) | O($\frac{5}{4}\frac{3}{4}1$) | O($\frac{1}{4}\frac{1}{4}\frac{1}{2}$) | O($\frac{3}{4}\frac{3}{4}0$) | O($\frac{1}{4}\frac{3}{4}\frac{1}{2}$) | O($\frac{3}{4}\frac{5}{4}1$) | O($\frac{3}{4}\frac{3}{4}\frac{3}{2}$)
|  | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$ | $\bm{d}_{1}$

$(continued)$ |
---|---
(a) | (b) | $\\{KI|\frac{1}{2}\frac{1}{2}0\\}$ | $\Gamma$ | $Z$ | $N$ | $X$ | $P$
|  | OK |  $\Gamma_{1}$ \+ $\Gamma_{2}$ \+ $\Gamma_{3}$ \+ $\Gamma_{4}$ \+ 2$\Gamma_{5}$ |  $Z_{1}$ \+ $Z_{2}$ \+ $Z_{3}$ \+ $Z_{4}$ \+ 2$Z_{5}$ | 4$N_{1}$ \+ 4$N_{2}$ | 4$X_{1}$ | 4$P_{1}$

Notes to Table 4

  1. (i)

The symmetry of the band related to the Co atoms in Table (a) coincides fully
with the symmetry of the band related to the O atoms in Table (b).

  2. (ii)

The notations of the points of symmetry in the Brillouin zone for
$\Gamma^{v}_{q}$ follow Figure 3.10 (b) of Bradley and A.P.Cracknell (1972)
and the symmetry notations of the Bloch functions are defined in Table 2.

  3. (iii)

The bands are determined following Theorem 5 of Krüger and Strunk (2015).

  4. (iv)

The point groups $G_{0Co}$ and $G_{0O}$ of the positions Krüger and Strunk
(2015) of the Co and the O atoms contain only the identity operation:

$G_{0Co}=G_{0O}=\Big{\\{}\\{E|\bm{0}\\}\Big{\\}}.$ (17)

Thus, the Wannier functions at the atoms belong to the simple representation

| $\\{E|\bm{0}\\}$
---|---
$\bm{d}_{1}$ | 1

of $G_{0Co}$ and $G_{0O}$.

  5. (v)

The table defines a band consisting of eight branches with Bloch functions
that can be unitarily transformed into Wannier functions being:

     * •

as well localized as possible;

     * •

centered at the eight Co atoms (a) or at the eight O atoms (b); and

     * •

symmetry adapted to the unitary subgroup $I4_{1}\\!/\\!cd$ (110) of the
magnetic group $M_{110}$. In the present case this means that the symmetry
operations of $I4_{1}\\!/\\!cd$ effect an interchange of the eight Wannier
functions in the unit cell, cf. note (xi) of Table A2 in Krüger (2019).

     * •

The entry “OK” indicates that the Wannier functions follow not only Theorem 5,
but also Theorem 7 of Krüger and Strunk (2015). Consequently, they may be
chosen symmetry-adapted to the complete magnetic group $M_{110}$, cf. note
(xii) of Table A2 in Krüger (2019).

Table 5:  Compatibility relations between the Brillouin zone for the fcc space
group $Fm3m$ (225) of paramagnetic CoO and the Brillouin zone for the space
group $I4_{1}\\!/\\!cd$ (110) of the antiferromagnetic structure in distorted
CoO.

$\Gamma(000)$
---
$\Gamma^{+}_{1}$ | $\Gamma^{+}_{2}$ | $\Gamma^{-}_{2}$ | $\Gamma^{-}_{1}$ | $\Gamma^{+}_{3}$ | $\Gamma^{-}_{3}$ | $\Gamma^{+}_{4}$ | $\Gamma^{+}_{5}$ | $\Gamma^{-}_{4}$ | $\Gamma^{-}_{5}$
$\Gamma_{1}$ | $\Gamma_{4}$ | $\Gamma_{3}$ | $\Gamma_{2}$ |  $\Gamma_{1}$ \+ $\Gamma_{4}$ |  $\Gamma_{2}$ \+ $\Gamma_{3}$ |  $\Gamma_{2}$ \+ $\Gamma_{5}$ |  $\Gamma_{3}$ \+ $\Gamma_{5}$ |  $\Gamma_{1}$ \+ $\Gamma_{5}$ |  $\Gamma_{4}$ \+ $\Gamma_{5}$

$X(\frac{1}{2}0\frac{1}{2})$
---
$X^{+}_{1}$ | $X^{+}_{2}$ | $X^{+}_{3}$ | $X^{+}_{4}$ | $X^{+}_{5}$ | $X^{-}_{1}$ | $X^{-}_{2}$ | $X^{-}_{3}$ | $X^{-}_{4}$ | $X^{-}_{5}$
$\Gamma_{3}$ \+ $\Gamma_{5}$ |  $\Gamma_{1}$ \+ 2$\Gamma_{4}$ |  $\Gamma_{2}$ \+ $\Gamma_{5}$ | 2$\Gamma_{1}$ \+ $\Gamma_{4}$ |  $\Gamma_{2}$ \+ $\Gamma_{3}$ \+ 2$\Gamma_{5}$ |  $\Gamma_{4}$ \+ $\Gamma_{5}$ |  $\Gamma_{2}$ \+ 2$\Gamma_{3}$ |  $\Gamma_{1}$ \+ $\Gamma_{5}$ | 2$\Gamma_{2}$ \+ $\Gamma_{3}$ |  $\Gamma_{1}$ \+ $\Gamma_{4}$ \+ 2$\Gamma_{5}$

$L(\frac{1}{2}\frac{1}{2}\frac{1}{2})$
---
$L^{+}_{1}$ | $L^{+}_{2}$ | $L^{-}_{1}$ | $L^{-}_{2}$ | $L^{+}_{3}$ | $L^{-}_{3}$
$\Gamma_{2}$ \+ $\Gamma_{4}$ \+ $\Gamma_{5}$ |  $\Gamma_{1}$ \+ $\Gamma_{3}$ \+ $\Gamma_{5}$ |  $\Gamma_{1}$ \+ $\Gamma_{3}$ \+ $\Gamma_{5}$ |  $\Gamma_{2}$ \+ $\Gamma_{4}$ \+ $\Gamma_{5}$ |  $\Gamma_{1}$ \+ $\Gamma_{2}$ \+ $\Gamma_{3}$ \+ $\Gamma_{4}$ \+ 2$\Gamma_{5}$ |  $\Gamma_{1}$ \+ $\Gamma_{2}$ \+ $\Gamma_{3}$ \+ $\Gamma_{4}$ \+ 2$\Gamma_{5}$

$\Sigma_{M}(\frac{1}{4}\frac{1}{4}\frac{1}{2})$
---
$\Sigma_{1}$ | $\Sigma_{2}$ | $\Sigma_{3}$ | $\Sigma_{4}$
$Z_{1}$ \+ $Z_{2}$ \+ $Z_{5}$ |  $Z_{3}$ \+ $Z_{4}$ \+ $Z_{5}$ |  $Z_{3}$ \+ $Z_{4}$ \+ $Z_{5}$ |  $Z_{1}$ \+ $Z_{2}$ \+ $Z_{5}$

$W^{\prime\prime}(\frac{1}{4}\overline{\frac{1}{4}}\overline{\frac{1}{2}})$
---
$W_{1}$ | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$
$Z_{5}$ | $Z_{5}$ | $Z_{5}$ | $Z_{5}$ |  $Z_{1}$ \+ $Z_{2}$ \+ $Z_{3}$ \+ $Z_{4}$

$\Delta_{M}^{\prime}(\overline{\frac{1}{4}}\overline{\frac{1}{4}}0)$
---
$\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$ | $\Delta_{4}$ | $\Delta_{5}$
$Z_{3}$ \+ $Z_{4}$ |  $Z_{1}$ \+ $Z_{2}$ |  $Z_{1}$ \+ $Z_{2}$ |  $Z_{3}$ \+ $Z_{4}$ | 2$Z_{5}$

$\Delta_{M}(\frac{1}{4}0\frac{1}{4})$
---
$\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$ | $\Delta_{4}$ | $\Delta_{5}$
$X_{1}$ | $X_{1}$ | $X_{1}$ | $X_{1}$ | 2$X_{1}$

$\Lambda_{M}(\frac{1}{4}\frac{1}{4}\frac{1}{4})$
---
$\Lambda_{1}$ | $\Lambda_{2}$ | $\Lambda_{3}$
$N_{1}$ \+ $N_{2}$ |  $N_{1}$ \+ $N_{2}$ | 2$N_{1}$ \+ 2$N_{2}$

$W^{\prime}(\overline{\frac{3}{4}}\overline{\frac{1}{2}}\overline{\frac{1}{4}})$
---
$W_{1}$ | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$
$X_{1}$ | $X_{1}$ | $X_{1}$ | $X_{1}$ | 2$X_{1}$

Notes to Table 5:

  1. (i)

The upper rows list the representations of the little groups of the points of
symmetry in the Brillouin zone for $Fm3m$, and the lower rows list
representations of the little groups of the related points of symmetry in the
Brillouin zone for $I4_{1}\\!/\\!cd$.

The representations in the same column are compatible in the following sense:
Bloch functions that are basis functions of a representation $\bm{D}_{i}$ in
the upper row can be unitarily transformed into the basis functions of the
representation given below $\bm{D}_{i}$.

  2. (ii)

The notations of the points of symmetry follow Figures 3.14 and 3.10 (b),
respectively, of Bradley and A.P.Cracknell (1972).

  3. (iii)

The notations of the representations are defined in Table A4 of Krüger (2019)
and Table 2, respectively.

  4. (iv)

The compatibility relations are determined as described in great detail in
Krüger (1985).

Table 6:  Compatibility relations between points and lines in the Brillouin
zone for the fcc space group $Fm3m$ (225) of paramagnetic CoO (Table (a)) and
in the Brillouin zone for the space group $I4_{1}\\!/\\!cd$ (110) of the
antiferromagnetic structure of distorted CoO (Table (b)), as far as the are
useful for an understanding of Figures 3 and 4.

$\Gamma(000)$
---
$\Gamma^{+}_{1}$ | $\Gamma^{+}_{2}$ | $\Gamma^{-}_{2}$ | $\Gamma^{-}_{1}$ | $\Gamma^{+}_{3}$ | $\Gamma^{-}_{3}$ | $\Gamma^{+}_{4}$ | $\Gamma^{+}_{5}$ | $\Gamma^{-}_{4}$ | $\Gamma^{-}_{5}$
$\Delta_{1}$ | $\Delta_{3}$ | $\Delta_{4}$ | $\Delta_{2}$ |  $\Delta_{1}$ \+ $\Delta_{3}$ |  $\Delta_{2}$ \+ $\Delta_{4}$ |  $\Delta_{2}$ \+ $\Delta_{5}$ |  $\Delta_{4}$ \+ $\Delta_{5}$ |  $\Delta_{1}$ \+ $\Delta_{5}$ |  $\Delta_{3}$ \+ $\Delta_{5}$

$\Gamma(000)$
---
$\Gamma^{+}_{1}$ | $\Gamma^{+}_{2}$ | $\Gamma^{-}_{2}$ | $\Gamma^{-}_{1}$ | $\Gamma^{+}_{3}$ | $\Gamma^{-}_{3}$ | $\Gamma^{+}_{4}$ | $\Gamma^{+}_{5}$ | $\Gamma^{-}_{4}$ | $\Gamma^{-}_{5}$
$\Sigma_{1}$ | $\Sigma_{3}$ | $\Sigma_{4}$ | $\Sigma_{2}$ |  $\Sigma_{1}$ \+ $\Sigma_{3}$ |  $\Sigma_{2}$ \+ $\Sigma_{4}$ |  $\Sigma_{2}$ \+ $\Sigma_{3}$ \+ $\Sigma_{4}$ |  $\Sigma_{1}$ \+ $\Sigma_{2}$ \+ $\Sigma_{4}$ |  $\Sigma_{1}$ \+ $\Sigma_{3}$ \+ $\Sigma_{4}$ |  $\Sigma_{1}$ \+ $\Sigma_{2}$ \+ $\Sigma_{3}$

$X(\frac{1}{2}0\frac{1}{2})$
---
$X^{+}_{1}$ | $X^{+}_{2}$ | $X^{+}_{3}$ | $X^{+}_{4}$ | $X^{+}_{5}$ | $X^{-}_{1}$ | $X^{-}_{2}$ | $X^{-}_{3}$ | $X^{-}_{4}$ | $X^{-}_{5}$
$\Delta_{1}$ | $\Delta_{2}$ | $\Delta_{3}$ | $\Delta_{4}$ | $\Delta_{5}$ | $\Delta_{2}$ | $\Delta_{1}$ | $\Delta_{4}$ | $\Delta_{3}$ | $\Delta_{5}$

(a)

$\Gamma(010)$, $\Gamma(000)$
---
$\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$
$F_{1}$, $\Sigma_{1}$ |  $F_{2}$, $\Sigma_{2}$ |  $F_{1}$, $\Sigma_{1}$ |  $F_{2}$, $\Sigma_{2}$ |  $F_{1}$ \+ $F_{2}$, $\Sigma_{1}$ \+ $\Sigma_{2}$

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$
---
$Z_{1}$ | $Z_{2}$ | $Z_{3}$ | $Z_{4}$ | $Z_{5}$
$F_{1}$ | $F_{1}$ | $F_{2}$ | $F_{2}$ |  $F_{1}$ \+ $F_{2}$

$\Gamma(000)$
---
$\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | $\Gamma_{5}$
$\Lambda_{1}$ | $\Lambda_{2}$ | $\Lambda_{3}$ | $\Lambda_{4}$ | $\Lambda_{5}$

$Z(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}})$
---
$Z_{1}$ | $Z_{2}$ | $Z_{3}$ | $Z_{4}$ | $Z_{5}$
$\Lambda_{4}$ | $\Lambda_{2}$ | $\Lambda_{1}$ | $\Lambda_{3}$ | $\Lambda_{5}$

(b)

no

References

## References

  * Rooksby (1948) Rooksby, H.  A note on the structure of nickel oxide at subnormal AND elevated temperatures.  Acta Crystallographica 1948, 1, 226.  doi:black10.1107/S0365110X48000612.
  * Shull et al. (1951) Shull, C.G.; Strauser, W.A.; Wollan, E.O.  Neutron Diffraction by Paramagnetic and Antiferromagnetic Substances.  Phys. Rev. 1951, 83, 333–345.  doi:black10.1103/PhysRev.83.333.
  * Roth (1958) Roth, W.L.  Magnetic Structures of MnO, FeO, CoO, and NiO.  Phys. Rev. 1958, 110, 1333–1341.  doi:black10.1103/PhysRev.110.1333.
  * Cracknell and Joshua (1969) Cracknell, A.P.; Joshua, S.J.  The space group corepresentations of antiferromagnetic NiO.  Mathematical Proceedings of the Cambridge Philosophical Society 1969, 66, 493–504.  doi:black10.1017/S0305004100045229.
  * van Laar (1965) van Laar, B.  Multi-Spin-Axis Structure for CoO.  Phys. Rev. 1965, 138, A584–A587.  doi:black10.1103/PhysRev.138.A584.
  * Timm et al. (2016) Timm, L.; Tucker, M.G.; Keen, D.A.; Thygesen, P.M.M.; Saines, P.J.; Goodwin, A.L.  Exploration of antiferromagnetic CoO and NiOusing reverse Monte Carlo total neutron scattering refinements.  Phys. Scr. 2016, 91, 114004.  doi:black10.1088/0031-8949/91/11/114004.
  * Tomiyasu et al. (2004) Tomiyasu, K.; Inami, T.; Ikeda, N.  Magnetic structure of CoO studied by neutron and synchrotron x-ray diffraction.  Phys. Rev. B 2004, 70, 184411.  doi:black10.1103/PhysRevB.70.184411.
  * Jauch et al. (2001) Jauch, W.; Reehuis, M.; Bleif, H.J.; Kubanek, F.; Pattison, P.  Crystallographic symmetry and magnetic structure of CoO.  Phys. Rev. B 2001, 64, 052102.  doi:black10.1103/PhysRevB.64.052102.
  * Krüger (2019) Krüger, E.  Structural Distortion Stabilizing the Antiferromagnetic and Insulating Ground State of NiO.  Symmetry 2019, 12.  doi:black10.3390/sym12010056.
  * Krüger (2020) Krüger, E.  Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO.  Symmetry 2020, 12.  doi:black10.3390/sym12111913.
  * Bradley and A.P.Cracknell (1972) Bradley, C.; A.P.Cracknell.  The Mathematical Theory of Symmetry in Solids; Claredon, Oxford, 1972.
  * Krüger (2001) Krüger, E.  Nonadiabatic extension of the Heisenberg model.  Phys. Rev. B 2001, 63, 144403–1–13.  doi:black10.1103/PhysRevB.63.144403.
  * Krüger (2020) Krüger, E.  Wannier States of FCC Symmetry Qualifying Paramagnetic NiO to Be a Mott Insulator.  Symmetry 2020, 12.  doi:black10.3390/sym12050687.
  * Blum et al. (2009) Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M.  Ab initio molecular simulations with numeric atom-centered orbitals.  Computer Physics Communications 2009, 180, 2175 – 2196\.
  * Havu et al. (2009) Havu, V.; Blum, V.; Havu, P.; Scheffler, M.  Efficient O(N)O(N) integration for all-electron electronic structure calculation using numeric basis functions.  Computer Physics Communications 2009, 228, 8367 – 8379\.
  * Bredow and Gerson (2000) Bredow, T.; Gerson, A.R.  Effect of exchange and correlation on bulk properties of MgO, NiO, and CoO.  Phys. Rev. B 2000, 61, 5194–5201.  doi:black10.1103/PhysRevB.61.5194.
  * Krüger (1989) Krüger, E.  Stability and symmetry of the spin-density-wave-state in chromium.  Phys. Rev. B 1989, 40, 11090–11103.  doi:black10.1103/PhysRevB.40.11090.
  * Krüger (1999) Krüger, E.  Energy band with Wannier functions of ferromagnetic symmetry as the cause of ferromagnetism in iron.  Phys. Rev. B 1999, 59, 13795–13805.  doi:black10.1103/PhysRevB.59.13795.
  * Krüger (2016) Krüger, E.  Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn2As2.  Symmetry 2016, 8(10), 99.  doi:black10.3390/sym8100099.
  * Krüger and Strunk (2015) Krüger, E.; Strunk, H.P.  Group Theory of Wannier Functions Providing the Basis for a Deeper Understanding of Magnetism and Superconductivity.  Symmetry 2015, 7, 561–598.  doi:black10.3390/sym7020561.
  * Krüger (2001) Krüger, E.  Modified BCS Mechanism of Cooper Pair Formation in Narrow Energy Bands of Special Symmetry I. Band Structure of Niobium.  J. Supercond. 2001, 14, 469–489.  doi:black10.1023/A:1012231428443.
  * Krüger (2018) Krüger, E.  Constraining Forces Stabilizing Superconductivity in Bismuth.  Symmetry 2018, 10.  doi:black10.3390/sym10020044.
  * Krüger and Strunk (2011) Krüger, E.; Strunk, H.P.  The Structural Distortion in Antiferromagnetic LaFeAsO Investigated by a Group-Theoretical Approach.  J. Supercond. 2011, 24, 2103–2117.  doi:black10.1007/s10948-011-1177-6.
  * Krüger (1985) Krüger, E.  Symmetry of Bloch functions in the space group $D^{6}_{4h}$ of perfect antiferromagnetic chromium.  Phys. Rev. B 1985, 32, 7493 – 7501.  doi:black10.1103/PhysRevB.32.7493.