Table 2 Spin-Laue classification for \({{\rm{AA}}}^{{\prime} }\)-stacked bilayer MnPS3 under parallel (↑↓/↑↓) and antiparallel (↑↓/↓↑) magnetic orderings
From: Engineering altermagnetism via layer shifts and spin order in bilayer MnPS3
Offset/structure | Phase (Zeng et al.) | MSG | Spin-Laue group | Generators/comment |
|---|---|---|---|---|
\({{\rm{AA}}}^{{\prime} }\uparrow \downarrow /\uparrow \downarrow\) (Parallel) ordering | ||||
(0,0) | AM (Type III/1\({\overline{3}}^{2}m\)) | \(P\overline{6}{\prime} {2}^{{\prime} }m\) (189.221) | \(\{[h| | E]:h\in H\}\cup [{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); \([{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) connector present. |
(0,1/3) | AM (Type III/22/2mx) | \({C}^{{\prime} }m\) (8.32) | \(\{[h| | E]:h\in H\}\cup [{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); mirror-type connector \([{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\). |
(1/3,1/3) | AM (Type III/22/2mx) | \({C}^{{\prime} }m\) (8.32) | \(\{[h| | E]:h\in H\}\cup [{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); mirror-type connector \([{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\). |
(1/3,2/3) | AFM (Type II) | P321 (150) | {[h∣∣E]: h ∈ H} ∪ [P∣∣T] | Effective \({\mathcal{P}}{\mathcal{T}}\) via inversion pairing; no splitting. |
(1/2,0) | AFM (Type II) | Aem2 (39) | {[h∣∣E]: h ∈ H} ∪ [P∣∣T] | \(2{\bf{t}}\in {\mathcal{L}}\) restores an effective \({\mathcal{P}}{\mathcal{T}}\); spin degeneracy observed. |
\({{\bf{AA}}}^{{\prime} }\uparrow \downarrow /\downarrow \uparrow\) (Antiparallel) ordering | ||||
|---|---|---|---|---|
(0,0) | AM (Type III/1\({\overline{3}}^{2}m\)) | \(P\overline{6}{\prime} 2{m}^{{\prime} }\) (189.223) | \(\{[h| | E]:h\in H\}\cup [{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); \([{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) connector survives. |
(0,1/3) | AM (Type III/22/2mx) | \({C}^{{\prime} }m\) (8.32) | \(\{[h| | E]:h\in H\}\cup [{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); mirror-type connector \([{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\). |
(1/3,1/3) | AM (Type III/22/2mx) | \({C}^{{\prime} }m\) (8.32) | \(\{[h| | E]:h\in H\}\cup [{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); mirror-type connector \([{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\). |
(1/3,2/3) | AM (Type III/1\({\overline{3}}^{2}m\)) | \(P321{\prime}\) (150.25) | \(\{[h| | E]:h\in H\}\cup [{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); \([{C}_{2}| | {C}_{{\rm{spin}}}^{2}]\) connector present. |
(1/2,0) | AM (Type III/2m2m1m) | \(A{\prime} e{\prime} m2\) (39.197) | \(\{[h| | E]:h\in H\}\cup [{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) | Broken \({\mathcal{P}}{\mathcal{T}}\); mirror-type connector with spin inversion yields Type III altermagnetism. |
