Structural Distortion Stabilizing the Antiferromagnetic and Semiconducting Ground State of BaMn$_2$As$_2$

We report evidence that the experimentally found antiferromagnetic structure as well as the semiconducting ground state of BaMn$_2$As$_2$ are caused by optimally-localized Wannier states of special symmetry existing at the Fermi level of BaMn$_2$As$_2$. In addition, we find that a (small) tetragonal distortion of the crystal is required to stabilize the antiferromagnetic semiconducting state. To our knowledge, this distortion has not yet been established experimentally.


I. INTRODUCTION
The electronic ground state of BaMn 2 As 2 shows resemblances but also striking differences, as compared with the ground state of the isostructural compound BaFe 2 As 2 .Both materials become antiferromagnetic below the respective Néel temperature.However, while the magnetic moments in BaMn 2 As 2 are orientated along the tetragonal c axis [1], see Fig. 1, they are orientated perpendicular to this axis in BaFe 2 As 2 [2].
Also very interesting is the observation that, unlike BaFe 2 As 2 , BaMn 2 As 2 is a small band gap antiferromagnetic semiconductor [3,4].No structural transformation or distortion of BaMn 2 As 2 in this antiferromagnetic semiconducting state was experimentally detected [1].
The present paper reports evidence that the remarkable features of the electronic ground state of BaMn 2 As 2 are connected with optimally-localized Wannier functions existing at the Fermi level of BaMn 2 As 2 .These Wannier functions are adapted to the symmetry of the antiferromagnetic structure and constructed from Bloch functions of well-defined symmetry forming narrow "magnetic bands" as defined in Ref. [5] (see Definition 16 ibidem).
In Section II.1 we shall identify the tetragonal space group I42m (121) as the space group of the antiferromagnetic structure observed in BaMn 2 As 2 (the number in parenthesis is the international number) and determine the magnetic group M 121 of this structure.We will show that no magnetic band related to M 121 exists in the band structure of BaMn 2 As 2 .The situation changes drastically when in Section II.2 we shall consider a slightly distorted crystal.We will define and verify the existence of a magnetic "super" band in distorted antiferromagnetic BaMn 2 As 2 .This super band consists of three magnetic bands as defined in Ref. [5] with Wannier functions situated at the Ba, the Mn, and the As atoms, respectively.
Our group-theoretical results in Section II will be physically interpreted in Section III.We will argue in Section III.1 that a small tetragonal distortion of the crystal is required to stabilize the antiferromagnetic semiconducting ground state of BaMn 2 As 2 .This distortion alters the space group I42m = Γ v q D 11 2d of the undistorted antiferromagnetic crystal into the space group P 42 1 c = Γ q D 4 2d (which is still tetragonal) and may be realized by the displacements of the Mn atoms depicted in Fig. 1 (b).These displacements are evidently so small that they have not yet been experimentally verified [1].In Section III.2 we shall show that evidently the magnetic super band is responsible for the small band gap in the antiferromagnetic semiconducting ground state, and in Section III.3 why the space groups of the magnetic structures in BaFe 2 As 2 and BaMn 2 As 2 differ so strikingly.

I.1. Nonadiabatic Heisenberg model
The existence of magnetic bands in the band structure of BaMn 2 As 2 is physically interpreted within the nonadiabatic Heisenberg model (NHM) [6].The second postulate of the NHM (Equation (2.19) of [6]) states that in narrow bands (i.e. in band satisfying Equation (2.13) of [6]) the electrons may lower their total correlation energy by condensing into an atomic-like state as it was described by Mott [7] and Hubbard [8]: the electrons occupy the localized states as long as possible and perform their band motion by hopping from one atom to another.Within the NHM, however, the localized states are not represented by (hybrid) atomic orbitals but consequently by symmetry-adapted optimally-localized Wannier states.The electrons are strongly correlated in this atomic-like state, leading to the consequence that a consistent description of the localized Wannier states must involve the nonadiabatic motion of the atomic cores [6].
Hence, the nonadiabatic localized functions representing these nonadiabatic Wannier states depend on an additional coordinate characterizing the motion of the atomic cores.Fortunately, these mathematically complicated functions need not be explicitly known.They can be simply managed within the group-theoretical NHM because they have the same symmetry as the related adiabatic optimally-localized Wannier functions as defined in Ref. [5].In this context we speak of "adiabatic" Wan- Experimentally observed [1] antiferromagnetic structure in undistorted (a) and distorted (b) BaMn2As2.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 ±T3 direction realize the tetragonal primitive space group P 421c.
nier functions if they do not dependent on the nonadiabatic motion of the atomic cores.
The total correlation energy of the electron system decreases by the nonadiabatic condensation energy ∆E defined in Equation (2.20) of [6] at the condensation into the nonadiabatic atomic-like state.

II. MAGNETIC BANDS IN THE BAND STRUCTURE OF BaMn2As2
II.1.The space group I42m (121) of the antiferromagnetic structure in undistorted BaMn2As2 Removing from the space group I4/mmm of BaMn 2 As 2 all the symmetry operations not leaving invariant the magnetic moments of the Mn atoms, we obtain the group I42m (121) as the space group of the antiferromagnetic structure in undistorted BaMn 2 As 2 .Just as I4/mmm, the group I42m has the tetragonal bodycentered Bravais lattice Γ v q .The group I42m may be defined by the two "generating elements" see Table 3.7 of Ref. [9].Just as in all our papers, we write the symmetry operations {R|pqr} in the Seitz notation detailed in the textbook of Bradley and Cracknell [9]: R stands for a point group operation (as defined, e.g., in Table 1.4 ibidem) and pqr denotes the subsequent translation t = pT 1 +qT 2 +rT 3 , where the T 1 , T 2 , and T 3 denote the basic vectors of the respective Bravais lattice given in Fig. 1.A (magnetic) structure is invariant under a space group G if it is already invariant under the generating elements of G.The two generating symmetry operations (1) leave invariant the atoms of BaMn 2 As 2 since both operations are elements of I4/mmm.By means of Fig. 1 (a) we can realize that they additionally leave invariant the magnetic structure, cf.Sec.3.1 of Ref. [10].
The associated magnetic group reads as where K and I denote the operator of time inversion and the inversion, respectively.{KI|000} leaves invariant both the atoms and the magnetic structure since {I|000} ∈ I4/mmm.With consideration of the change of symmetry by the magnetostriction, but neglecting all other magnetic interactions, we receive from the band structure of BaMn 2 As 2 given in Fig. 2 the band structure of antiferromagnetic undistorted BaMn 2 As 2 depicted in Fig. 3.All the possible magnetic bands (Definition 16 of Ref. [5]) in the magnetic group M 121 are listed in Table 3.The "best" magnetic band would be band 2 of Mn as highlighted in Fig. 3 by the red labels.
Band 2 of Mn, however, is not a magnetic band in BaMn 2 As 2 because it misrepresents the Bloch functions at parts of the Fermi level.Between the N 1 and P 3 states it jumps over the Fermi level simulating in this way Bloch states at the Fermi level which do not exist.The same situation we have between the Z 5 state and the two X 3 , X 1 states.Along the lines F, Σ the Γ 5 state is connected with Energy (eV) 2. Band structure of BaMn2As2 as calculated by the FHI-aims program [11,12], using the structure parameters given in Ref. [1].The space group of BaMn2As2 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 Γ v q follow Fig. 3.10 (b) of Ref. [9], and the symmetry labels are defined in Table 2 of Ref. [13].EF denotes the Fermi level.The band structure of BaMn2As2 essentially concides with the band structure of BaFe2As2 (depicted in Fig. 2 of Ref. [13]) when the Fermi level is moved upwards to the dashed line.
two Bloch states at the Fermi level, which, however, are not connected to Z 5 .
We could try to render ineffective these unfavorable jumps by adding further bands to the Mn band as it will be successful in the space group P 42 1 c considered in the following Section II.2.By means of Table 3 we may satisfy ourselves that this procedure is not possible.For instance, we neither can add band 1 nor band 2 of As to band 2 of Mn because there is neither a Γ 4 state nor an additional P 3 state available in die band structure.

II.2. The space group P 421c (114) of the antiferromagnetic structure in distorted BaMn2As2
The situation described in the preceding Section II.1 changes drastically when we consider the space group P 42 1 c.This group has no longer the the tetragonal bodycentered Bravais lattice Γ v q , but the tetragonal primitive lattice Γ q and may be defined by the two generating elements see Table 3.7 of Ref. [9] (note that the basis vectors now are given in Fig. 1 (b)).As well as the generating elements of I42m (1) they leave invariant both the positions of the atoms and the magnetic structure since the vec- 2 ) is a lattice vector in Γ v q .In addition, the generating elements (3) leave invariant the displacements of the Mn atoms depicted in Fig. 1 (b).Thus these displacements "realize" the space group P 42 1 c in the sense that the electrons now move in a potential adapted to the symmetry of the distorted crystal.The group P 42 1 c represents only a small distortion of the crystal because it is still tetragonal and possesses the same point group as the space group I42m of antiferromagnetic undistorted BaMn 2 As 2 .It is only the translation t = ( 1 Energy (eV) 3. Band structure of BaMn2As2 as given in Fig. 2 with symmetry labels of the space group I42m (121) of the antiferromagnetic structure in undistorted BaMn2As2.The symmetry labels are determined from Tab. 2 and the red labels define band 2 of Mn in Tab. 3.
may be written as Folding the band structure of BaMn 2 As 2 as given in Fig. 2 into the Brillouin zone for P 42 1 c we receive the band structure depicted in Fig. 4. All the magnetic bands in the magnetic group M 114 are listed in Table 6.Now we have a very interesting situation not yet considered in our former papers: We are able to assign optimally-localized symmetry-adapted Wannier functions to all the atoms in the unit cell of Γ q , meaning that we have a band of ten branches with Wannier functions at the two Ba, the four Mn and the four As atoms.Such a magnetic band related to all the atoms in the unit cell we call magnetic "super" band.It is highlighted in red in Fig. 4 and consists of band 1 of Mn, band 2 of As and band 3 of Ba in Tab.6 and, hence, is defined by the symmetry labels While the magnetic super band in BaMn 2 As 2 satisfies the condition (38) of Theorem 5 of Ref. [5] in all the point of symmetry, there are little complications on the lines Σ and ∆ in two branches: The four circles in Fig. 4 mark regions with unavoidable transitions from ∆ 2 to ∆ 1 , ∆ 1 to ∆ 2 , Σ 1 to Σ 2 , and Σ 2 to Σ 1 symmetry.These transitions clearly destroy the P 42 1 c symmetry of the Wannier functions.However, since these transitions only occur in two lines of two branches fare away from the Fermi level, we assume that the magnetic structure of BaMn 2 As 2 may be described with high accuracy in the space group P 42 1 c.Nevertheless, these transitions produce an additional small distortion of the crystal going beyond the displacements of the Mn atoms depicted in Fig. 1 (b).This additional small distortion is not considered in this paper.However, we should keep in mind (see Note (v) of Table 7) that the Wannier functions are exactly adapted to the space group P 4 = Γ q S 1 4 (81) since in this space group the mentioned complications on the lines Σ and ∆ disappear.The related exact magnetic group is a subgroup of M 114 (4).First, there exist two subgroups of M 114 defined by the two anti-unitary operations {KC 2a |000} and {KI| 1 2 1 2 1 2 }, respectively.A detailed examination shows that only the group allows an additional distortion of the crystal. -1 The band structure of BaMn2As2 as given in Fig. 2 folded into the Brillouin zone for the tetragonal primitive Bravais lattice Γq of the space group P 421c (114).The symmetry labels are defined in Table 4 and are determined from Fig. 2 by means of Table 5.The notations of the points of symmetry follow Fig. 3.9 of Ref. [9].EF denotes the Fermi level.The red lines and red symmetry labels form the magnetic "super" band of the experimentally observed [1] antiferromagnetic structure in BaMn2As2.It is related to all the atoms of BaMn2As2, that means that it is related to two Ba, four Mn and four As atoms in the unit cell of Γq, and, hence, consists of ten branches.Whenever a black and a red line overlap, the red line lies on the top.The four circles mark the regions with transitions from ∆2 to ∆1, ∆1 to ∆2, Σ1 to Σ2, and Σ2 to Σ1, respectively, on the lines ∆ and Σ.These transitions slightly destroy the symmetry of related the Wannier functions in the space group P 421c.However, since these transitions only occur in two lines of two branches fare away from the Fermi level, we assume that we may describe the magnetic structure of BaMn2As2 with high accuracy in the space group P 421c.Nevertheless, these transitions produce a small additional distortion of the crystal going beyond the distortion depicted in Fig. 1 (b).

II.3. Time-inversion symmetry
The time-inversion symmetry is no essential object in antiferromagnetic BaMn 2 As 2 : all the three space groups I4/mmm (139), P 42 1 c (114), and P 4 (81) possess onedimensional representations allowing a stable magnetic state with the magnetic group M 121 (2), M 114 (4), and M 81 (6), respectively, see Tables 1, 4, and 7 and the notes to these tables.Consequently, time-inversion symmetry influences neither the antiferromagnetic structure nor the structural distortions in BaMn 2 As 2 (as it is the case, for instance, in BaFe 2 As 2 [13]).Time-inversion symmetry only forbids magnetic moments located at the As atoms, see Note (x) of Table 6.

III. PHYSICAL INTERPRETATION
The existence and the properties of the roughly halffilled magnetic super band in the band structure of BaMn 2 As 2 yield to an understanding of three phenomena that shall be considered in this section: 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 .
III.1.The antiferromagnetic order and the structural distortion in BaMn2As2 In a material possessing a narrow, roughly half-filled magnetic band or super band related to a magnetic group M , the NHM defines a nonadiabatic Hamiltonian H n representing atomic-like electrons (Section I.1) within this band [6].An important feature of H n is that it only commutes with the symmetry operations of M , but does not commute with the remaining symmetry operations of the paramagnetic group of the crystal (this follows from the fact that the optimally-localized Wannier functions in a magnetic band may be chosen symmetry-adapted to M , but cannot be chosen symmetry-adapted to the complete paramagnetic group, cf.Section 1 of Ref. [5]).
Thus, the electrons in such a narrow, roughly half-filled magnetic band or super band may gain the nonadiabatic condensation energy ∆E (Section I.1) by condensing into an atomic-like state only if the electrons really move in a potential with the magnetic group M , that is, only if a magnetic structure with the magnetic group M really exists.As a consequence, the electrons activate in the nonadiabatic system a spin dependent exchange mechanism producing a magnetic structure with the magnetic group M [14,15].
In the case of BaMn 2 As 2 , the group I42m (121) is the space group of the antiferromagnetic structure in undistorted BaMn 2 As 2 .However, within this group there does not exist a magnetic band, see Sec.II.1.Indeed, a magnetic band, even a magnetic super band, exists in the space group P 42 1 c (114) of distorted BaMn 2 As 2 , see Sec.II.2.
Hence, in BaMn 2 As 2 the electron system cannot condense into the atomic-like state by the production of the magnetic structure alone but must additionally produce a spatial distortion of the crystal realizing -together with the magnetic structure -the magnetic group M 114 (4).This is achieved by the displacements of the Mn atoms depicted in Fig. 1 (b).Consequently, the magnetically ordered ground state of BaMn 2 As 2 is accompanied by the displacements of the Mn atoms depicted in Fig. 1  (b).

III.2. The semiconducting ground state of BaMn2As2
The magnetic super band defines not only Wannier functions situated at all the atoms of BaMn 2 As 2 , but it also comprises all the Bloch states at the Fermi level, see Fig. 4. Thus, if the magnetic super band is exactly half filled, the nonadiabatic Hamiltonian H n produces very specific atomic-like electrons: at any atom of BaMn 2 As 2 there exists a localized Wannier state occupied by exactly one electron and besides these atomic-like electrons there do not exist band-like electrons which would be able to transport electrical current.Thus, H n possesses a semiconducting ground state since the atomic-like state is separated from any band-like state by the nonadiabatic condensation energy ∆E mentioned in Section I.1.The experimental observation of an insulating ground state in BaMn 2 As 2 suggests that, indeed, the magnetic super band is exactly half-filled.

III.3. Different magnetic structures in BaMn2As2
and BaFe2As2 While both compounds BaMn 2 As 2 and BaFe 2 As 2 exhibit an antiferromagnetic ordering below the respective Néel temperature, the space groups of the magnetic structures are quite different: in BaMn 2 As 2 the space group of the magnetic structure is the tetragonal group P 42 1 c (114) with magnetic moments oriented along the tetragonal c axis, and in BaFe 2 As 2 it is the orthorhombic group Cmca with magnetic moments orientated perpendicular to the c axis, see Fig. 1 of [1] and Fig. 3 of [2], respectively.
This surprising experimental observation can be understood comparing the band structures of both compounds as given in Fig. 2 and Fig. 2 of Ref. [13].The band structure of BaFe 2 As 2 is very similar to the band structure of BaMn 2 As 2 , the essential difference is the position of the Fermi level: we may approximate the band structure of BaFe 2 As 2 by the band structure of BaMn 2 As 2 by shifting the Fermi level upwards by about 0.3 eV as it is indicated in Fig. 2.
A magnetic (super) band may be physically active only if the band is nearly half-filled.The band width of the (red) magnetic super band in Fig. 4 may be approximated by 2σ, where denotes the standard deviation of the N = 6 • 10 energy values E k in the six points of symmetry of the magnetic super band.Thus, the Fermi level is shifted in BaFe 2 As 2 nearly to the top of the magnetic super band.Hence, in BaFe 2 As 2 this band is far from being half-filled and determines neither the magnetic structure nor produces an isolating ground state in BaFe 2 As 2 .Instead, the magnetic structure in BaFe 2 As 2 is determined by the nearly half-filled magnetic band presented in Fig. 3 of Ref. [13] which is related to the space group Cmca of the magnetic structure experimentally found in BaFe 2 As 2 [2].

IV. CONCLUSIONS
This paper emphasizes the importance of the nonadiabatic condensation energy ∆E defined in Equation (2.20) of Ref. [6] (and already mentioned in Section I.1) which is evidently responsible for the striking electronic features of BaMn 2 As 2 .∆E is released at the transition from an adiabatic band-like motion of the electrons to the nonadiabatic strongly-correlated atomic-like motion.
This finding is in accordance with former observations on a great number of superconducting and magnetic materials (see Section 1 of [5]) suggesting that superconductivity and magnetism are always connected with superconducting (Definition 22 of Ref. [5]) and magnetic bands, respectively.Thus, in superconducting and magnetic bands, the nonadiabatic condensation energy ∆E may evidently produce superconductivity and mag-netism, respectively, and in some cases even a small band gap semiconductor.

ACKNOWLEDGMENTS
I am very indebted to Guido Schmitz for his support of my work.

Appendix: Group-theoretical tables
This appendix provides Tables 1 -7 along with notes to the tables.TABLE 1. Character tables of the irreducible representations of the tetragonal space group I42m = Γ v q D 11 2d (121) of the experimentally observed [1] antiferromagnetic structure in BaMn2As2. 1 (i) The notations of the points of symmetry follow Fig. 3.10 (b) of Ref. [9].
(ii) The character tables are determined from Table 5.7 of Ref. [9].
(iii) K denotes the operator of time inversion.The entry (a) indicates that the related corepresentations of the magnetic groups I42m + {K|000}I42m and I42m + {KI|000}I42m follow case (a) as defined in equation (7.3.45) of Ref. [9] (and determined by equation (7.3.51) of Ref. [9]).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. [14].TABLE 3. Representations at the points of symmetry in the space group I42m = Γ v q D 11 2d (121) of all the energy bands of antiferromagnetic BaMn2As2 with symmetry-adapted and optimally localized Wannier functions centered at the Mn, As, and Ba atoms, respectively.
Mn Mn( 1 Notes to Table 3 (i) z = 0.36 . . .[1]; the exact value of z is meaningless in this table.
(ii) The antiferromagnetic structure of undistorted BaMn2As2 has the space group I42m and the magnetic group M = I42m + {KI|000}I42m 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 I42m of the antiferromagnetic structure in undistorted BaMn2As2.
(iv) The notations of the representations are defined in Table 1.
(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 di included below the atom.
(vii) The di denote the one-dimensional representations of the "point groups of the positions" of the Mn, As and Ba atom (Definition 12 of Ref. [5]), S4, C2v, and D 2d , respectively, as defined by the tables Mn atoms As atoms The entry "OK" indicates whether the Wannier functions may even be chosen symmetry-adapted to the magnetic group M = I42m + {KI|000}I42m of undistorted BaMn2As2, 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.TABLE 4. Character tables of the irreducible representations of the tetragonal space group P 421c = ΓqD 4 2d (114) of the experimentally observed [1] antiferromagnetic structure in distorted BaMn2As2.Γ(000) Notes to Table 4 (i) The notations of the points of symmetry follow Fig. 3.9 of Ref. [9].
(ii) The character tables are determined from Table 5.7 of Ref. [9].
(iv) The entries (a) and (c) for K and {KI| 1 2 1 2 1 2 } show that all the one-dimensional representations at M , Z, or A are possible representations of a stable antiferromagnetic state, see Appendix A of Ref. [16] or Section III C of Ref. [14].TABLE 6. Representations at the points of symmetry in the space group P 421c (114) of all the energy bands of distorted antiferromagnetic BaMn2As2 with symmetry-adapted and optimally localized Wannier functions centered at the Mn, As, or Ba atoms, respectively.
(ii) The space group P 421c leaves invariant the experimentally observed [1] antiferromagnetic structure and defines the distortion of BaMn2As2 that possesses the magnetic super band consisting of band 1 of Mn, band 2 of As, and band 3 of Ba.
(iv) The notations of the representations are defined in Table 4.
(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 421c.

Notes to Table 6 (continued)
(vii) The Wannier functions at the Mn, As or Ba atom listed in the upper row belong to the representation di included below the atom.
(viii) The di denote the representations of the "point groups of the positions" of the Mn, As and Ba atoms (Definition 12 of Ref. [5]), C2, C2, and S4, respectively, as defined by the tables Mn atoms {E|000} {C2z|000} As atoms {E|000} {C2z|000} The entry "OK" indicates whether the Wannier functions may even be chosen symmetry-adapted to the magnetic group M = P 421c + {KI| 1 2 1 2 1 2 }P 421c, 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( 1 2 0 1 4 ) and Mn( 1 2 1 3 4 ) atoms.The magnetic moments at the two positions A and B of these atoms are anti-parallel.Thus, the two Wannier functions wA( r) and wB( r) at these positions are complex conjugate, wA( r) = w * B ( r), and, hence, belong to co-representations dA and dB of the groups of these positions being also complex conjugate, dA = d * B . (A.1) The matrix N defined by Theorem 7 of Ref.Because dA and dB do not comply with Equation (A.1), the Wannier functions defined by band 2 of Mn do not form a magnetic band in antiferromagnetic BaMn2As2 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 BaMn2As2 because the As atoms do not bear ordered magnetic moments.
(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. [9].
(v) The entries (a) and (c) for K and KC2a show that the representations R2 and R4 at any of the points Γ, M , Z, or A are possible representations of a stable antiferromagnetic state, see Appendix A of Ref. [16] or Section III C of Ref. [14].This is important since M81 (6) is the exact group of the magnetic structure in BaMn2As2.
FIG. 1.Experimentally observed[1] antiferromagnetic structure in undistorted (a) and distorted (b) BaMn2As2.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 ±T3 direction realize the tetragonal primitive space group P 421c.

2 1 2 1 2 } 2 1 2 1 2 }
is no longer a symmetry operation in P 42 1 c.At first, the two anti-unitary operations {KI|000} and {KI| 1 may define the magnetic group of the magnetic structure since both operations leave invariant the magnetic structure.However, only {KI| 1 leaves additionally invariant the displacement of the Mn atoms depicted in Fig1 (b).These displacements, however, are required to realize the space group P 42 1 c.Hence, the magnetic group of antiferromagnetic distorted BaMn 2 As 2 2 of Mn, yielding the two co-representations dA and dB defined by the table