# Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statement

#### 1.2. Organization of the Paper

## 2. Magnetic Group of the Antiferromagnetic State

## 3. Rhombohedral-Like Distortion

## 4. Interpretation of the Experimental Findings of Goodwin et al.

- (i)
- The significant shifts of the Mn atoms in the $\pm ({\mathit{T}}_{2}-{\mathit{T}}_{3})$ direction realize the magnetic group ${M}_{9}$ and stabilize in this way the antiferromagnetic structure; see Section 2.
- (ii)
- The observed displacements of the Mn atoms in Equation (10) are greatest in the ${\mathit{T}}_{1}$ direction; they are even 12 times greater than in the $({\mathit{T}}_{2}+{\mathit{T}}_{3})$ direction. This corroborates my supposition [5] that the mutual attraction between Mn atoms with opposite shifts in the $\pm ({\mathit{T}}_{2}-{\mathit{T}}_{3})$ direction is mainly responsible for the rhombohedral-like deformation of the crystal. The displacements are maximal in the ${\mathit{T}}_{1}$ direction since in this direction, they are parallel to the plane ${\sigma}_{db}$ and, thus, do not destroy the magnetic group ${M}_{9}$, as illustrated by the red line in Figure 1.

## 5. Conventional Band Structure

## 6. Symmetry-Adapted and Optimally Localized Wannier Functions in MnO

#### 6.1. Optimally Localized Wannier Functions Symmetry-Adapted to the Paramagnetic fcc Structure

- (i)
- (ii)
- (iii)
- The point group ${G}_{0Mn}$ of the positions [19] of the Mn atoms is equal to the full cubic point group ${O}_{h}$. The Wannier functions belong to the representation of ${G}_{0Mn}$ included below the atom.

#### 6.2. Optimally Localized Wannier Functions Symmetry-Adapted to the Antiferromagnetic Structure

- (i)
- (ii)
- The bands are determined by means of Theorem 5 of [19].
- (iii)
- The point groups ${G}_{0Mn}$ and ${G}_{0O}$ of the positions [19] of the Mn respective O atoms contain, in each case, only the identity operation:$${G}_{0Mn}={G}_{0O}=\left\{\left\{E\right|\mathbf{0}\}\right\}.$$Thus, the Wannier functions at the Mn or O atoms belong to the simple representation:

of ${G}_{0Mn}$ and ${G}_{0O}$.$\left\{E\right|\mathbf{0}\}$ ${\mathit{d}}_{1}$ 1 - (iv)
- (v)
- The entry “OK” indicates that the Wannier functions follow not only Theorem 5, but also Theorem 7 of [19]. Consequently, they may not only be chosen symmetry-adapted to the space group $Cc$, but also to the complete magnetic group ${M}_{9}$.

## 7. Results

- (i)
- The insulating ground state of both paramagnetic and antiferromagnetic MnO,
- (ii)
- the stability of the antiferromagnetic state,
- (iii)
- the rhombohedral-like deformation in the antiferromagnetic phase,

- The antiferromagnetic state in MnO is evidently stabilized by strongly correlated atomic-like electrons in a magnetic band. The magnetic band in MnO is even a magnetic super band because it comprises all the electrons at the Fermi level. Thus, the special atomic-like motion in this band qualifies antiferromagnetic MnO to be a Mott insulator.
- The Bloch functions of a (roughly) half filled energy band in the paramagnetic band structure of MnO can be unitarily transformed into optimally localized Wannier functions symmetry-adapted to the fcc symmetry of the paramagnetic phase. These Wannier functions are situated at the Mn atoms, have d symmetry, and comprise all the electrons at the Fermi level. Thus, the atomic-like motion represented by these Wannier functions qualifies also paramagnetic MnO to be a Mott insulator.
- The magnetic structure is stabilized by a shift of the Mn atoms in the $\pm ({\mathit{T}}_{2}-{\mathit{T}}_{3})$ direction. These shifts evidently produce the rhombohedral-like deformation of the crystal because the attraction between the Mn atoms increases slightly when the Mn atoms are shifted in opposite directions. This concept presented in [5] was corroborated by the experimental observations of Goodwin et al. [15].
- The rhombohedral-like distortion does not possess a rhombohedral (trigonal) space group, but is an inner distortion of the base-centered monoclinic magnetic group ${M}_{9}$ in Equation (4). The group ${M}_{9}$, on the other hand, must not be broken because it stabilizes the antiferromagnetic structure.

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NHM | Nonadiabatic Heisenberg model |

E | Identity operation |

I | Inversion |

${C}_{2b}$ | Rotation through $\pi $, as indicated in Figure 1 |

${\sigma}_{db}$ | Reflection $I{C}_{2b}$ |

K | Anti-unitary operator of time inversion |

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**Figure 1.**Manganese atoms in distorted antiferromagnetic MnO. The O atoms are not shown. The Mn atoms bear magnetic moments parallel or antiparallel to $\left[11\overline{2}\right]$. The atoms marked by the same color (red or green) have parallel magnetic moments, and two atoms marked by different colors bear antiparallel moments. The vectors ${\mathit{T}}_{i}$ are the basic translations of ${\Gamma}_{m}^{b}$.

**Figure 2.**Conventional (Section 5) band structure of paramagnetic fcc MnO as calculated by the FHI-aims program [16,17], using the length $a=4.426$ A of the fcc paramagnetic unit cell given in [2]. The symmetry labels (as defined in Table A1 of [5]) are determined by the author. The notations of the points of symmetry follow Figure 3.14 of [18]. The band highlighted by the bold lines forms an insulating band of d symmetry consisting of five branches.

**Figure 3.**The band structure of MnO given in Figure 2 folded into the Brillouin zone for the monoclinic base-centered Bravais lattice ${\Gamma}_{m}^{b}$. The symmetry labels (as defined in Table A4 of [5]) are obtained from Table A5 of [5]. The notations of the points of symmetry are defined in Figure 3.4 of [18]. The bold lines highlight the magnetic super band consisting of six branches. The midpoint ${\Lambda}_{\mathrm{M}}$ of the line $\overline{\Gamma Z}$ is equivalent to the points ${W}^{\prime}\left(\overline{\frac{1}{4}}\frac{1}{4}\frac{1}{2}\right)$ and ${\Sigma}^{\prime}\left(\frac{1}{4}\overline{\frac{1}{4}}0\right)$ in the Brillouin zone for the paramagnetic fcc lattice. The number “1” on the line $\overline{A\Gamma V}$ indicates that here, only one branch belongs to the magnetic super band.

**Table 1.**Symmetry labels of two energy bands in the Brillouin zone for paramagnetic MnO with Bloch functions that can be unitarily transformed into optimally localized Wannier functions symmetry-adapted to the space group $Fm3m$ (225) and centered at the Mn atoms.

Mn (000) | $\mathbf{\Gamma}$ | X | L | W | |
---|---|---|---|---|---|

Band 5 | ${\Gamma}_{3}^{+}$ | ${\Gamma}_{3}^{+}$ | ${X}_{1}^{+}$ + ${X}_{3}^{+}$ | ${L}_{3}^{+}$ | ${W}_{1}$ + ${W}_{4}$ |

Band 8 | ${\Gamma}_{5}^{+}$ | ${\Gamma}_{5}^{+}$ | ${X}_{4}^{+}$ + ${X}_{5}^{+}$ | ${L}_{1}^{+}$ + ${L}_{3}^{+}$ | ${W}_{3}$ + ${W}_{5}$ |

**Table 2.**Symmetry labels of all energy bands in the Brillouin zone for antiferromagnetic MnO with Bloch functions that can be unitarily transformed into optimally localized Wannier functions symmetry-adapted to the magnetic group ${M}_{9}=Cc+K\left\{{C}_{2b}\right|\mathbf{0}\}Cc$ and centered at the Mn (Table (a)) or O (Table (b)) atoms, respectively.

(a) Mn | Mn${}_{1}$ (000) | Mn${}_{2}$ $\left(\overline{\frac{1}{2}}00\right)$ | $\mathit{K}\left\{{\mathit{C}}_{2\mathit{b}}\right|0\}$ | $\mathbf{\Gamma}$ | A | Z | M | L | V |
---|---|---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ | ${A}_{1}$ + ${A}_{2}$ | ${Z}_{1}$ + ${Z}_{2}$ | ${M}_{1}$ + ${M}_{2}$ | 2${L}_{1}$ | 2${V}_{1}$ |

(b) O | O${}_{1}$ $\left(\overline{\frac{1}{4}}\frac{1}{2}\overline{\frac{1}{2}}\right)$ | O${}_{2}$ $\left(\overline{\frac{3}{4}}\frac{1}{2}\overline{\frac{1}{2}}\right)$ | $K\left\{{C}_{2b}\right|\mathbf{0}\}$ | $\Gamma $ | A | Z | M | L | V |

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ | ${A}_{1}$ + ${A}_{2}$ | ${Z}_{1}$ + ${Z}_{2}$ | ${M}_{1}$ + ${M}_{2}$ | 2${L}_{1}$ | 2${V}_{1}$ |

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Krüger, E.
Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO. *Symmetry* **2020**, *12*, 1913.
https://doi.org/10.3390/sym12111913

**AMA Style**

Krüger E.
Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO. *Symmetry*. 2020; 12(11):1913.
https://doi.org/10.3390/sym12111913

**Chicago/Turabian Style**

Krüger, Ekkehard.
2020. "Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO" *Symmetry* 12, no. 11: 1913.
https://doi.org/10.3390/sym12111913