#
Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn_{2}As_{2}

## Abstract

**:**

## 1. Introduction

#### 1.1. Nonadiabatic Heisenberg Model

## 2. Magnetic Bands in the Band Structure of ${\mathbf{BaMn}}_{\mathbf{2}}{\mathbf{As}}_{\mathbf{2}}$

#### 2.1. The Space Group $I\overline{4}2m$ (121) of the Antiferromagnetic Structure in Undistorted ${\mathit{BaMn}}_{2}{\mathit{As}}_{2}$

#### 2.2. The Space Group $P\overline{4}{2}_{1}c$ (114) of the Antiferromagnetic Structure in Distorted ${\mathit{BaMn}}_{2}{\mathit{As}}_{2}$

#### 2.3. Time-Inversion Symmetry

## 3. Physical Interpretation

- the experimentally observed [1] antiferromagnetic order together with a structural distortion not yet experimentally found;
- the semiconducting ground state; and
- the different magnetic structures in BaMn${}_{2}$As${}_{2}$ and BaFe${}_{2}$As${}_{2}$.

#### 3.1. The Antiferromagnetic Order and the Structural Distortion in BaMn${}_{2}$As${}_{2}$

- (i)
- no longer possesses the space group $I\overline{4}2m$;
- (ii)
- but is invariant under the space group $P\overline{4}{2}_{1}c$ and, additionally,
- (iii)
- is invariant under the anti-unitary operation $\{KI|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$ defining the magnetic group ${M}_{114}$ (4).

#### 3.2. The Semiconducting Ground State of BaMn${}_{2}$As${}_{2}$

#### 3.3. Different Magnetic Structures in BaMn${}_{2}$As${}_{2}$ and BaFe${}_{2}$As${}_{2}$

#### 3.4. Discussion

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Group-Theoretical Tables

$\mathbf{\Gamma}\mathbf{\left(}\mathbf{000}\mathbf{\right)}\mathbf{,}\mathit{Z}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\overline{\frac{\mathbf{1}}{\mathbf{2}}}\right)$ | |||||||
---|---|---|---|---|---|---|---|

${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{y}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{a}}$ | |||||

K | $\mathit{K}\mathit{I}$ | E | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{x}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{b}}$ | |

${\Gamma}_{1},{Z}_{1}$ | (a) | (a) | 1 | 1 | 1 | 1 | 1 |

${\Gamma}_{2},{Z}_{2}$ | (a) | (a) | 1 | 1 | 1 | −1 | −1 |

${\Gamma}_{3},{Z}_{3}$ | (a) | (a) | 1 | 1 | −1 | 1 | −1 |

${\Gamma}_{4},{Z}_{4}$ | (a) | (a) | 1 | 1 | −1 | −1 | 1 |

${\Gamma}_{5},{Z}_{5}$ | (a) | (a) | 2 | −2 | 0 | 0 | 0 |

$\mathit{P}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{\right)}$ | |||||||
---|---|---|---|---|---|---|---|

${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{y}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{a}}$ | |||||

K | $\mathit{K}\mathit{I}$ | E | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{x}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{b}}$ | |

${P}_{1}$ | (x) | (a) | 1 | 1 | 1 | 1 | 1 |

${P}_{2}$ | (x) | (a) | 1 | 1 | 1 | −1 | −1 |

${P}_{3}$ | (x) | (a) | 1 | 1 | −1 | 1 | −1 |

${P}_{4}$ | (x) | (a) | 1 | 1 | −1 | −1 | 1 |

${P}_{5}$ | (x) | (a) | 2 | −2 | 0 | 0 | 0 |

$\mathit{X}\left(00\frac{1}{2}\right)$ | ||||
---|---|---|---|---|

E | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{b}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{a}}$ | |

${X}_{1}$ | 1 | 1 | 1 | 1 |

${X}_{3}$ | 1 | 1 | −1 | −1 |

${X}_{2}$ | 1 | −1 | 1 | −1 |

${X}_{4}$ | 1 | −1 | −1 | 1 |

$\mathit{N}\left(0\frac{1}{2}0\right)$ | ||
---|---|---|

E | ${\mathit{C}}_{\mathbf{2}\mathit{y}}$ | |

${N}_{1}$ | 1 | 1 |

${N}_{2}$ | 1 | −1 |

- (i)
- The notations of the points of symmetry follow Figure 3.10b of Ref. [11].
- (ii)
- The character tables are determined from Table 5.7 of Ref. [11].
- (iii)
- K denotes the operator of time inversion. The entry (a) indicates that the related co-representations of the magnetic groups $I\overline{4}2m+\{K|000\}I\overline{4}2m$ and $I\overline{4}2m+\{KI|000\}I\overline{4}2m$ follow case (a) as defined in Equation (7.3.45) of Ref. [11] (and determined by Equation (7.3.51) of Ref. [11]). This information is interesting only in symmetry points invariant under the complete space group. (x) indicates that K does not leave invariant the point P.
- (iv)
- The one-dimensional representations at point P would be possible representations of a stable antiferromagnetic state because they comply with the demands in Section III C of Ref. [15].

$\mathbf{\Gamma}\left(000\right)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\Gamma}_{1}^{+}$ | ${\Gamma}_{2}^{+}$ | ${\Gamma}_{3}^{+}$ | ${\Gamma}_{4}^{+}$ | ${\Gamma}_{5}^{+}$ | ${\Gamma}_{1}^{-}$ | ${\Gamma}_{2}^{-}$ | ${\Gamma}_{3}^{-}$ | ${\Gamma}_{4}^{-}$ | ${\Gamma}_{5}^{-}$ |

${\Gamma}_{1}$ | ${\Gamma}_{2}$ | ${\Gamma}_{3}$ | ${\Gamma}_{4}$ | ${\Gamma}_{5}$ | ${\Gamma}_{3}$ | ${\Gamma}_{4}$ | ${\Gamma}_{1}$ | ${\Gamma}_{2}$ | ${\Gamma}_{5}$ |

$\mathit{N}\left(0\frac{1}{2}0\right)$ | |||
---|---|---|---|

${N}_{1}^{+}$ | ${N}_{1}^{-}$ | ${N}_{2}^{+}$ | ${N}_{2}^{-}$ |

${N}_{1}$ | ${N}_{1}$ | ${N}_{2}$ | ${N}_{2}$ |

$\mathit{X}\left(00\frac{1}{2}\right)$ | |||||||
---|---|---|---|---|---|---|---|

${X}_{1}^{+}$ | ${X}_{2}^{+}$ | ${X}_{3}^{+}$ | ${X}_{4}^{+}$ | ${X}_{1}^{-}$ | ${X}_{2}^{-}$ | ${X}_{3}^{-}$ | ${X}_{4}^{-}$ |

${X}_{1}$ | ${X}_{4}$ | ${X}_{3}$ | ${X}_{2}$ | ${X}_{3}$ | ${X}_{2}$ | ${X}_{1}$ | ${X}_{4}$ |

$\mathit{Z}\left(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}}\right)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

${Z}_{1}^{+}$ | ${Z}_{2}^{+}$ | ${Z}_{3}^{+}$ | ${Z}_{4}^{+}$ | ${Z}_{5}^{+}$ | ${Z}_{1}^{-}$ | ${Z}_{2}^{-}$ | ${Z}_{3}^{-}$ | ${Z}_{4}^{-}$ | ${Z}_{5}^{-}$ |

${Z}_{1}$ | ${Z}_{2}$ | ${Z}_{3}$ | ${Z}_{4}$ | ${Z}_{5}$ | ${Z}_{3}$ | ${Z}_{4}$ | ${Z}_{1}$ | ${Z}_{2}$ | ${Z}_{5}$ |

$\mathit{P}\left(\frac{1}{4}\frac{1}{4}\frac{1}{4}\right)$ | ||||
---|---|---|---|---|

${P}_{1}$ | ${P}_{2}$ | ${P}_{3}$ | ${P}_{4}$ | ${P}_{5}$ |

${P}_{1}$ | ${P}_{2}$ | ${P}_{3}$ | ${P}_{4}$ | ${P}_{5}$ |

- (i)
- The Brillouin zone for $I\overline{4}2m$ is identical to the Brillouin zone for $I4/mmm$.
- (ii)
- The upper rows list the representations of the little groups of the points of symmetry in the Brillouin zone for $I4/mmm$. The lower rows list representations of these groups in $I\overline{4}2m$.The representations in the same column are compatible in the following sense: Bloch functions that are basis functions of a representation ${\mathit{D}}_{i}$ in the upper row can be unitarily transformed into the basis functions of the representation given below ${\mathit{D}}_{i}$.
- (iii)

Mn | Mn($\frac{1}{4}\frac{3}{4}\frac{\mathbf{1}}{\mathbf{2}}$) | Mn($\frac{3}{4}\frac{1}{4}\frac{\mathbf{1}}{\mathbf{2}}$) | $\mathit{K}\mathit{I}$ | Γ | P | Z | X | N |
---|---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ | ${P}_{5}$ | ${Z}_{3}$ + ${Z}_{4}$ | ${X}_{2}$ + ${X}_{4}$ | ${N}_{1}$ + ${N}_{2}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{4}$ | OK | ${\Gamma}_{5}$ | ${P}_{3}$ + ${P}_{4}$ | ${Z}_{5}$ | ${X}_{1}$ + ${X}_{3}$ | ${N}_{1}$ + ${N}_{2}$ |

Band 3 | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{3}$ | OK | ${\Gamma}_{3}$ + ${\Gamma}_{4}$ | ${P}_{5}$ | ${Z}_{1}$ + ${Z}_{2}$ | ${X}_{2}$ + ${X}_{4}$ | ${N}_{1}$ + ${N}_{2}$ |

Band 4 | ${\mathit{d}}_{4}$ | ${\mathit{d}}_{2}$ | OK | ${\Gamma}_{5}$ | ${P}_{1}$ + ${P}_{2}$ | ${Z}_{5}$ | ${X}_{1}$ + ${X}_{3}$ | ${N}_{1}$ + ${N}_{2}$ |

As | As($\mathit{z}\mathit{z}0$) | As($\overline{\mathit{z}\mathit{z}}0$) | $\mathit{K}\mathit{I}$ | Γ | P | Z | X | N |
---|---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{4}$ | ${P}_{1}$ + ${P}_{4}$ | ${Z}_{1}$ + ${Z}_{4}$ | 2${X}_{1}$ | ${N}_{1}$ + ${N}_{2}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | OK | ${\Gamma}_{2}$ + ${\Gamma}_{3}$ | ${P}_{2}$ + ${P}_{3}$ | ${Z}_{2}$ + ${Z}_{3}$ | 2${X}_{3}$ | ${N}_{1}$ + ${N}_{2}$ |

Band 3 | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{4}$ | − | ${\Gamma}_{5}$ | ${P}_{5}$ | ${Z}_{5}$ | ${X}_{2}$ + ${X}_{4}$ | ${N}_{1}$ + ${N}_{2}$ |

Ba | Ba(000) | $\mathit{K}\mathit{I}$ | Γ | P | Z | X | N |
---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ | ${P}_{1}$ | ${Z}_{1}$ | ${X}_{1}$ | ${N}_{1}$ |

Band 2 | ${\mathit{d}}_{2}$ | OK | ${\Gamma}_{2}$ | ${P}_{2}$ | ${Z}_{2}$ | ${X}_{3}$ | ${N}_{2}$ |

Band 3 | ${\mathit{d}}_{3}$ | OK | ${\Gamma}_{3}$ | ${P}_{3}$ | ${Z}_{3}$ | ${X}_{3}$ | ${N}_{1}$ |

Band 4 | ${\mathit{d}}_{4}$ | OK | ${\Gamma}_{4}$ | ${P}_{4}$ | ${Z}_{4}$ | ${X}_{1}$ | ${N}_{2}$ |

- (i)
- (ii)
- The antiferromagnetic structure of undistorted BaMn${}_{2}$As${}_{2}$ has the space group $I\overline{4}2m$ and the magnetic group $M=I\overline{4}2m+\{KI|000\}I\overline{4}2m$ with K denoting the operator of time-inversion.
- (iii)
- Each row defines a band with Bloch functions that can be unitarily transformed into Wannier functions being
- as well localized as possible;
- centered at the stated atoms;
- and symmetry-adapted to the space group $I\overline{4}2m$ of the antiferromagnetic structure in undistorted BaMn${}_{2}$As${}_{2}$.

- (iv)
- The notations of the representations are defined in Table A1.
- (v)
- The bands are determined following Theorem 5 of Ref. [5].
- (vi)
- The Wannier functions at the Mn, As or Ba atom listed in the upper row belong to the representation ${\mathit{d}}_{i}$ included below the atom.
- (vii)
- The ${\mathit{d}}_{i}$ denote the one-dimensional representations of the “point groups of the positions” of the Mn, As and Ba atom (Definition 12 of Ref. [5]), ${S}_{4}$, ${C}_{2v}$, and ${D}_{2d}$, respectively, defined by

Mn Atoms | ||||
---|---|---|---|---|

E | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | i | −1 | −i |

${\mathit{d}}_{3}$ | 1 | −1 | 1 | −1 |

${\mathit{d}}_{4}$ | 1 | −i | −1 | i |

As Atoms | ||||
---|---|---|---|---|

E | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{a}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{b}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | 1 | −1 | −1 |

${\mathit{d}}_{3}$ | 1 | −1 | 1 | −1 |

${\mathit{d}}_{4}$ | 1 | −1 | −1 | 1 |

Ba Atom | |||||
---|---|---|---|---|---|

${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{y}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{a}}$ | |||

E | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{x}}$ | ${\mathsf{\sigma}}_{\mathit{d}\mathit{b}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 | 1 | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | 1 | 1 | −1 | −1 |

${\mathit{d}}_{3}$ | 1 | 1 | −1 | 1 | −1 |

${\mathit{d}}_{4}$ | 1 | 1 | −1 | −1 | 1 |

- (viii)
- The entry “OK” indicates whether the Wannier functions may even be chosen symmetry-adapted to the magnetic group $M=I\overline{4}2m+\{KI|000\}I\overline{4}2m$ of undistorted BaMn${}_{2}$As${}_{2}$ (see Theorem 7 of Ref. [5]).
- (ix)
- Hence, all the listed bands except for band 3 of As form magnetic bands as defined by Definition 16 of Ref. [5].
- (x)
- Each band consists of one or two branches (Definition 2 of Ref. [5]) depending on the number of the related atoms in the unit cell.

$\mathbf{\Gamma}\left(000\right)$ | |||||||
---|---|---|---|---|---|---|---|

$\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |||||

K | $\mathbf{\{}\mathit{K}\mathit{I}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}z}^{\mathbf{+}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |

${\Gamma}_{1}$ | (a) | (a) | 1 | 1 | 1 | 1 | 1 |

${\Gamma}_{2}$ | (a) | (a) | 1 | 1 | 1 | −1 | −1 |

${\Gamma}_{3}$ | (a) | (a) | 1 | 1 | −1 | 1 | −1 |

${\Gamma}_{4}$ | (a) | (a) | 1 | 1 | −1 | −1 | 1 |

${\Gamma}_{5}$ | (a) | (a) | 2 | −2 | 0 | 0 | 0 |

$\mathit{M}\left(\frac{1}{2}\frac{1}{2}0\right)$ | ||||||
---|---|---|---|---|---|---|

K | $\mathbf{\{}\mathit{K}\mathit{I}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{010}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{010}\mathbf{\}}$ | |

${M}_{1}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${M}_{2}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${M}_{3}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${M}_{4}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${M}_{5}$ | (a) | (a) | 2 | −2 | 2 | −2 |

$\mathit{M}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{0}\mathbf{\right)}$ $\mathbf{\left(}\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{e}\mathit{d}\mathbf{\right)}$ | ||||||
---|---|---|---|---|---|---|

$\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{010}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}\mathbf{|}\mathbf{010}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |

$\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}z}^{\mathbf{+}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |

${M}_{1}$ | 1 | −1 | i | −i | i | −i |

${M}_{2}$ | 1 | −1 | −i | i | −i | i |

${M}_{3}$ | −1 | 1 | −i | i | i | −i |

${M}_{4}$ | −1 | 1 | i | −i | −i | i |

${M}_{5}$ | 0 | 0 | 0 | 0 | 0 | 0 |

$\mathit{Z}\left(00\frac{1}{2}\right)$ | ||||||
---|---|---|---|---|---|---|

K | $\mathbf{\{}\mathit{K}\mathit{I}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{001}\mathbf{\}}$ | |

${Z}_{1}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${Z}_{2}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${Z}_{3}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${Z}_{4}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${Z}_{5}$ | (a) | (a) | 2 | −2 | 2 | −2 |

$\mathit{Z}\left(00\frac{1}{2}\right)$ $\left(\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{e}\mathit{d}\right)$ | ||||||
---|---|---|---|---|---|---|

$\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |

$\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}z}^{\mathbf{+}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | |

${Z}_{1}$ | 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 |

$\mathit{A}\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ | ||||||
---|---|---|---|---|---|---|

K | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | ||

${A}_{1}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${A}_{2}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${A}_{3}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${A}_{4}$ | (c) | (a) | 1 | 1 | −1 | −1 |

${A}_{5}$ | (b) | (a) | 2 | −2 | −2 | 2 |

$\mathit{A}\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ $\left(\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{e}\mathit{d}\right)$ | ||||||
---|---|---|---|---|---|---|

$\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{a}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | ||

$\mathbf{\{}{\mathit{S}}_{\mathbf{4}z}^{\mathbf{+}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathsf{\sigma}}_{\mathit{d}\mathit{b}}|\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | |||

${A}_{1}$ | −1 | 1 | i | −i | −i | i |

${A}_{2}$ | −1 | 1 | −i | i | i | −i |

${A}_{3}$ | 1 | −1 | −i | −i | i | i |

${A}_{4}$ | 1 | −1 | i | i | −i | −i |

${A}_{5}$ | 0 | 0 | 0 | 0 | 0 | 0 |

$\mathit{R}\mathbf{\left(}\mathbf{0}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\right)}$ | |||||
---|---|---|---|---|---|

$\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\mathbf{\}}$ | |||

$\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{001}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | |||

${R}_{1}$ | 2 | −2 | 0 | 0 | 0 |

$\mathit{X}\left(0\frac{1}{2}0\right)$ | |||||
---|---|---|---|---|---|

$\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{y}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{010}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{x}}\mathbf{|}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{3}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\}}$ | |||

$\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{010}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | |||

${X}_{1}$ | 2 | −2 | 0 | 0 | 0 |

- (i)
- The notations of the points of symmetry follow Figure 3.9 of Ref. [11].
- (ii)
- The character tables are determined from Table 5.7 of Ref. [11].
- (iii)
- K denotes the operator of time inversion. The entries (a), (b) and (c) indicate whether the related co-representations of the magnetic groups $P\overline{4}{2}_{1}c+\{K|000\}P\overline{4}{2}_{1}c$ and $P\overline{4}{2}_{1}c+\{KI|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}P\overline{4}{2}_{1}c$ follow case (a), (b) or (c) as defined in Equations (7.3.45), (7.3.46) and (7.3.47), respectively, of Ref. [11] (and determined by Equation (7.3.51) of Ref. [11]). This information is interesting only in symmetry points invariant under the complete space group.
- (iv)

$\mathbf{\Gamma}\left(000\right)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\Gamma}_{1}^{+}$ | ${\Gamma}_{2}^{+}$ | ${\Gamma}_{3}^{+}$ | ${\Gamma}_{4}^{+}$ | ${\Gamma}_{5}^{+}$ | ${\Gamma}_{1}^{-}$ | ${\Gamma}_{2}^{-}$ | ${\Gamma}_{3}^{-}$ | ${\Gamma}_{4}^{-}$ | ${\Gamma}_{5}^{-}$ |

${\Gamma}_{1}$ | ${\Gamma}_{2}$ | ${\Gamma}_{3}$ | ${\Gamma}_{4}$ | ${\Gamma}_{5}$ | ${\Gamma}_{3}$ | ${\Gamma}_{4}$ | ${\Gamma}_{1}$ | ${\Gamma}_{2}$ | ${\Gamma}_{5}$ |

$\mathit{X}\left(00\frac{1}{2}\right)$ | |||||||
---|---|---|---|---|---|---|---|

${X}_{1}^{+}$ | ${X}_{2}^{+}$ | ${X}_{3}^{+}$ | ${X}_{4}^{+}$ | ${X}_{1}^{-}$ | ${X}_{2}^{-}$ | ${X}_{3}^{-}$ | ${X}_{4}^{-}$ |

${M}_{5}$ | ${M}_{1}$ + ${M}_{2}$ | ${M}_{5}$ | ${M}_{3}$ + ${M}_{4}$ | ${M}_{5}$ | ${M}_{3}$ + ${M}_{4}$ | ${M}_{5}$ | ${M}_{1}$ + ${M}_{2}$ |

$\mathit{Z}\left(\frac{1}{2}\frac{1}{2}\overline{\frac{1}{2}}\right)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

${Z}_{1}^{+}$ | ${Z}_{2}^{+}$ | ${Z}_{3}^{+}$ | ${Z}_{4}^{+}$ | ${Z}_{5}^{+}$ | ${Z}_{1}^{-}$ | ${Z}_{2}^{-}$ | ${Z}_{3}^{-}$ | ${Z}_{4}^{-}$ | ${Z}_{5}^{-}$ |

${\Gamma}_{2}$ | ${\Gamma}_{1}$ | ${\Gamma}_{4}$ | ${\Gamma}_{3}$ | ${\Gamma}_{5}$ | ${\Gamma}_{4}$ | ${\Gamma}_{3}$ | ${\Gamma}_{2}$ | ${\Gamma}_{1}$ | ${\Gamma}_{5}$ |

$\mathit{P}\left(\frac{1}{4}\frac{1}{4}\frac{1}{4}\right)$ | ||||
---|---|---|---|---|

${P}_{1}$ | ${P}_{2}$ | ${P}_{3}$ | ${P}_{4}$ | ${P}_{5}$ |

${A}_{3}$ + ${A}_{4}$ | ${A}_{3}$ + ${A}_{4}$ | ${A}_{1}$ + ${A}_{2}$ | ${A}_{1}$ + ${A}_{2}$ | 2${A}_{5}$ |

${\mathbf{\Lambda}}_{\mathit{M}}\left(\frac{1}{4}\frac{1}{4}\overline{\frac{1}{4}}\right)$ line Λ | ||||
---|---|---|---|---|

${\Lambda}_{1}$ | ${\Lambda}_{2}$ | ${\Lambda}_{3}$ | ${\Lambda}_{4}$ | ${\Lambda}_{5}$ |

${Z}_{5}$ | ${Z}_{5}$ | ${Z}_{5}$ | ${Z}_{5}$ | ${Z}_{1}$ + ${Z}_{2}$ + ${Z}_{3}$ + ${Z}_{4}$ |

- (i)
- The Brillouin zone for $P\overline{4}{2}_{1}c$ lies within the Brillouin zone for $I4/mmm$.
- (ii)
- The upper rows list the representations of the little groups of the points of symmetry in the Brillouin zone for $I4/mmm,$ and the lower rows list representations of the little groups of the related points of symmetry in the Brillouin zone for $P\overline{4}{2}_{1}c$.The representations in the same column are compatible in the following sense: Bloch functions that are basis functions of a representation ${\mathit{D}}_{i}$ in the upper row can be unitarily transformed into the basis functions of the representation given below ${\mathit{D}}_{i}$.
- (iii)
- (iv)
- ${\Lambda}_{M}\left(\frac{1}{4}\frac{1}{4}\overline{\frac{1}{4}}\right)$ denotes the midpoint between Γ and Z in the Brillouin zone for $I4/mmm$.
- (v)
- The representations on the line Λ in the Brillouin zone for $I4/mmm$ are simple: the branch connecting ${\Gamma}_{5}$ and ${Z}_{5}$ in Figure 2 is labeled by the two-dimensional representation ${\Lambda}_{5}$, all the other branches are labeled by one of the one-dimensional representations ${\Lambda}_{1}$, ${\Lambda}_{2}$, ${\Lambda}_{3}$, or ${\Lambda}_{4}$.
- (vi)
- The compatibility relations are determined in the way described in great detail in Ref. [7].

Mn | Mn($\frac{\mathbf{1}}{\mathbf{2}}0\frac{1}{4}$) | Mn($0\frac{\mathbf{1}}{\mathbf{2}}\frac{1}{4}$) | Mn($1\frac{\mathbf{1}}{\mathbf{2}}\frac{3}{4}$) | Mn($\frac{\mathbf{1}}{\mathbf{2}}1\frac{3}{4}$) | $\{\mathit{K}\mathit{I}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$ | Γ |
---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ + ${\Gamma}_{3}$ + ${\Gamma}_{4}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | * | 2${\Gamma}_{5}$ |

$\left(\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{e}\mathit{d}\right)$ | |||||
---|---|---|---|---|---|

Mn | M | Z | A | R | X |

Band 1 | ${M}_{1}$ + ${M}_{2}$ + ${M}_{3}$ + ${M}_{4}$ | 2${Z}_{5}$ | 2${A}_{5}$ | 2${R}_{1}$ | 2${X}_{1}$ |

Band 2 | 2${M}_{5}$ | ${Z}_{1}$ + ${Z}_{2}$ + ${Z}_{3}$ + ${Z}_{4}$ | ${A}_{1}$ + ${A}_{2}$ + ${A}_{3}$ + ${A}_{4}$ | 2${R}_{1}$ | 2${X}_{1}$ |

As | As($00\mathit{z}$) | As($00\overline{\mathit{z}}$) | As($\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}},\frac{\mathbf{1}}{\mathbf{2}}+\mathit{z}$) | As($\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}},\frac{\mathbf{1}}{\mathbf{2}}-\mathit{z}$) | $\{\mathit{K}\mathit{I}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$ | Γ |
---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ + ${\Gamma}_{3}$ + ${\Gamma}_{4}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | * | 2${\Gamma}_{5}$ |

$\left(\mathit{c}\mathit{o}\mathit{n}\mathit{t}\mathit{i}\mathit{n}\mathit{u}\mathit{e}\mathit{d}\right)$ | |||||
---|---|---|---|---|---|

As | M | Z | A | R | X |

Band 1 | 2${M}_{5}$ | 2${Z}_{5}$ | ${A}_{1}$ + ${A}_{2}$ + ${A}_{3}$ + ${A}_{4}$ | 2${R}_{1}$ | 2${X}_{1}$ |

Band 2 | ${M}_{1}$ + ${M}_{2}$ + ${M}_{3}$ + ${M}_{4}$ | ${Z}_{1}$ + ${Z}_{2}$ + ${Z}_{3}$ + ${Z}_{4}$ | 2${A}_{5}$ | 2${R}_{1}$ | 2${X}_{1}$ |

Ba | Ba(000) | Ba($\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}$) | $\{\mathit{K}\mathit{I}|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}$ | Γ | M | Z | A | R | X |
---|---|---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}$ + ${\Gamma}_{2}$ | ${M}_{5}$ | ${Z}_{5}$ | ${A}_{3}$ + ${A}_{4}$ | ${R}_{1}$ | ${X}_{1}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{4}$ | OK | ${\Gamma}_{5}$ | ${M}_{1}$ + ${M}_{4}$ | ${Z}_{1}$ + ${Z}_{4}$ | ${A}_{5}$ | ${R}_{1}$ | ${X}_{1}$ |

Band 3 | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{3}$ | OK | ${\Gamma}_{3}$ + ${\Gamma}_{4}$ | ${M}_{5}$ | ${Z}_{5}$ | ${A}_{1}$ + ${A}_{2}$ | ${R}_{1}$ | ${X}_{1}$ |

Band 4 | ${\mathit{d}}_{4}$ | ${\mathit{d}}_{2}$ | OK | ${\Gamma}_{5}$ | ${M}_{2}$ + ${M}_{3}$ | ${Z}_{2}$ + ${Z}_{3}$ | ${A}_{5}$ | ${R}_{1}$ | ${X}_{1}$ |

- (i)
- (ii)
- The space group $P\overline{4}{2}_{1}c$ leaves invariant the experimentally observed [1] antiferromagnetic structure and defines the distortion of BaMn${}_{2}$As${}_{2}$ that possesses the magnetic super band consisting of band 1 of Mn, band 2 of As, and band 3 of Ba.
- (iii)
- The appertaining magnetic group reads as $M=P\overline{4}{2}_{1}c+\{KI|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}P\overline{4}{2}_{1}c$, where K still denotes the operator of time-inversion.
- (iv)
- The notations of the representations are defined in Table A4.
- (v)
- The bands are determined following Theorem 5 of Ref. [5].
- (vi)
- Each row defines a band with Bloch functions that can be unitarily transformed into Wannier functions being
- as well localized as possible;
- centered at the stated atoms; and
- symmetry-adapted to $P\overline{4}{2}_{1}c$.

- (vii)
- The Wannier functions at the Mn, As or Ba atom listed in the upper row belong to the representation ${\mathit{d}}_{i}$ included below the atom.
- (viii)
- The ${\mathit{d}}_{i}$ denote the representations of the “point groups of the positions” of the Mn, As and Ba atoms (Definition 12 of Ref. [5]), ${C}_{2}$, ${C}_{2}$, and ${S}_{4}$, respectively, defined by

Mn Atoms | ||
---|---|---|

$\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | −1 |

As Atoms | ||
---|---|---|

$\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | −1 |

Ba Atoms | ||||
---|---|---|---|---|

$\mathbf{\{}\mathit{E}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}z}^{\mathbf{+}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{C}}_{\mathbf{2}\mathit{z}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | $\mathbf{\{}{\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}\mathbf{|}\mathbf{000}\mathbf{\}}$ | |

${\mathit{d}}_{1}$ | 1 | 1 | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | i | −1 | −i |

${\mathit{d}}_{3}$ | 1 | −1 | 1 | −1 |

${\mathit{d}}_{4}$ | 1 | −i | −1 | i |

- (ix)
- The entry “OK” indicates whether the Wannier functions may even be chosen symmetry-adapted to the magnetic group $M=P\overline{4}{2}_{1}c+\{KI|\frac{1}{2}\frac{1}{2}\frac{1}{2}\}P\overline{4}{2}_{1}c$ (see Theorem 7 of Ref. [5]).
- (x)
- The asterisk “*” indicates that the Wannier functions may be chosen symmetry-adapted to the magnetic group M, but they do not allow that the magnetic moments are situated at the appertaining atoms. This complication (which has not yet been considered in Ref. [5]) may (but does not necessarily) occur only if the representations of the space group at point Γ are not one-dimensional, as it is the case in band 2 of both Mn and As, and in bands 2 and 4 of Ba. Consider, for example, band 2 of Mn and the two Mn($\frac{1}{2}0\frac{1}{4}$) and Mn($\frac{1}{2}1\frac{3}{4}$) atoms. The magnetic moments at the two positions A and B of these atoms are anti-parallel. Thus, the two Wannier functions ${w}_{A}\left(\overrightarrow{r}\right)$ and ${w}_{B}\left(\overrightarrow{r}\right)$ at these positions are complex conjugate, ${w}_{A}\left(\overrightarrow{r}\right)={w}_{B}^{*}\left(\overrightarrow{r}\right),$ and, hence, belong to co-representations ${\mathit{d}}_{A}$ and ${\mathit{d}}_{B}$ of the groups of these positions being also complex conjugate,$${\mathit{d}}_{A}={\mathit{d}}_{B}^{*}.$$
**N**defined by Theorem 7 of Ref. [5] takes the form $\mathbf{N}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$ in band 2 of Mn, yielding the two co-representations ${\mathit{d}}_{A}$ and ${\mathit{d}}_{B}$,E ${\mathit{C}}_{2\mathit{z}}$ $\mathit{K}{\mathsf{\sigma}}_{\mathit{x}}$ $\mathit{K}{\mathsf{\sigma}}_{\mathit{y}}$ ${\mathit{d}}_{A}$ 1 −1 1 −1 ${\mathit{d}}_{B}$ 1 −1 −1 1 Because ${\mathit{d}}_{A}$ and ${\mathit{d}}_{B}$ do not comply with Equation (8), the Wannier functions defined by band 2 of Mn do not form a magnetic band in antiferromagnetic BaMn${}_{2}$As${}_{2}$ since it is experimentally proven that the ordered magnetic moments lie at the Mn atoms. The Wannier functions defined by band 2 of As, on the other hand, form a magnetic band in BaMn${}_{2}$As${}_{2}$ because the As atoms do not bear ordered magnetic moments. - (xi)
- Each band consists of two or four branches (Definition 2 of Ref. [5]) depending on the number of the related atoms in the unit cell.

**Table A7.**Character tables of the single-valued irreducible representations of the tetragonal space group $P\overline{4}={\Gamma}_{q}{S}_{4}^{1}$ (81).

$\mathbf{\Gamma}\left(000\right)$, $\mathit{M}\left(\frac{1}{2}\frac{1}{2}0\right)$, $\mathit{Z}\left(00\frac{1}{2}\right)$, $\mathit{A}\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ | ||||||
---|---|---|---|---|---|---|

K | $\mathit{K}{\mathit{C}}_{\mathbf{2}\mathit{a}}$ | E | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{+}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{z}}^{\mathbf{-}}$ | |

${R}_{1}$ | (a) | (a) | 1 | 1 | 1 | 1 |

${R}_{2}$ | (c) | (a) | 1 | i | −1 | −i |

${R}_{3}$ | (a) | (a) | 1 | −1 | 1 | −1 |

${R}_{4}$ | (c) | (a) | 1 | −i | −1 | i |

- (i)
- The notations of the points of symmetry follow Figure 3.9 of Ref. [11].
- (ii)
- Only the points of symmetry invariant under the complete space group are listed.
- (iii)
- The character tables are determined from Table 5.7 in Ref. [11].
- (iv)
- K still denotes the operator of time inversion. The entries (a) and (c) indicate whether the related co-representations of the magnetic groups $P\overline{4}+\{K|000\}P\overline{4}$ and $P\overline{4}+\{K{C}_{2a}|000\}P\overline{4}$ follow case (a) or case (c) as defined in Equations (7.3.45) and (7.3.47), respectively, of Ref. [11] (and determined by Equation (7.3.51) of Ref. [11]).
- (v)
- The entries (a) and (c) for K and $K{C}_{2a}$ show that the representations ${R}_{2}$ and ${R}_{4}$ at any of the points Γ, M, Z, or A are possible representations of a stable antiferromagnetic state (see Theorem 1 of Ref. [6] or Section III C of Ref. [15]). This is important since ${M}_{81}$ (6) is the exact group of the magnetic structure in BaMn${}_{2}$As${}_{2}$.

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**Figure 1.**Experimentally observed [1] antiferromagnetic structure in undistorted (

**a**); and distorted (

**b**) BaMn${}_{2}$As${}_{2}$. While sufficient Ba atoms are depicted to recognize the orientation of the crystal, the Mn atoms are shown only within the respective unit cell. The As atoms are not included. The indicated (small) displacements of the Mn atoms in exact $\pm {\mathit{T}}_{3}$ direction realize the tetragonal primitive space group $P\overline{4}{2}_{1}c$. As argued in the text, they are required to stabilize the antiferromagnetic semiconducting state in BaMn${}_{2}$As${}_{2}$.

**Figure 2.**Band structure of BaMn${}_{2}$As${}_{2}$ as calculated by the “Fritz Haber Institute ab initio molecular simulations” (FHI-aims) program [13,14], using the structure parameters given in Ref. [1]. The space group of BaMn${}_{2}$As${}_{2}$ is the tetragonal group $I4/mmm$ (139) [1], the given symmetry labels are determined by the author. The notations of the points and lines of symmetry in the Brillouin zone for ${\Gamma}_{q}^{v}$ follow Figure 3.10b of Ref. [11], and the symmetry labels are defined in Table 2 of Ref. [6]. ${E}_{F}$ denotes the Fermi level. The band structure of BaMn${}_{2}$As${}_{2}$ essentially coincides with the band structure of BaFe${}_{2}$As${}_{2}$ (depicted in Figure 2 of Ref. [6]) when the Fermi level is moved upwards to the dashed line.

**Figure 3.**Band structure of BaMn${}_{2}$As${}_{2}$ as given in Figure 2 with symmetry labels of the space group $I\overline{4}2m$ (121) of the antiferromagnetic structure in undistorted BaMn${}_{2}$As${}_{2}$. The symmetry labels are determined from Table A2. The labels highlighted in

**red**define band 2 of Mn in Table A3.

**Figure 4.**The band structure of BaMn${}_{2}$As${}_{2}$ as given in Figure 2 folded into the Brillouin zone for the tetragonal primitive Bravais lattice ${\Gamma}_{q}$ of the space group $P\overline{4}{2}_{1}c$ (114). The symmetry labels are defined in Table A4 and are determined from Figure 2 by means of Table A5. The notations of the points of symmetry follow Figure 3.9 of Ref. [11]. ${E}_{F}$ denotes the Fermi level. The lines and symmetry labels highlighted in

**red**form the magnetic “super” band of the experimentally observed [1] antiferromagnetic structure in BaMn${}_{2}$As${}_{2}$. Whenever a

**black**and a

**red**line overlap, the

**red**line lies on the top.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Krüger, E.
Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn_{2}As_{2}. *Symmetry* **2016**, *8*, 99.
https://doi.org/10.3390/sym8100099

**AMA Style**

Krüger E.
Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn_{2}As_{2}. *Symmetry*. 2016; 8(10):99.
https://doi.org/10.3390/sym8100099

**Chicago/Turabian Style**

Krüger, Ekkehard.
2016. "Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn_{2}As_{2}" *Symmetry* 8, no. 10: 99.
https://doi.org/10.3390/sym8100099