Table 1 Spin-Laue classification for AA-stacked bilayer MnPS₃ under parallel (↑↓/↑↓) and antiparallel (↑↓/↓↑) configurations

(0,0)

AFM (Type II)

\(P\overline{3}1{m}^{{\prime} }\) (162.75)

{[h∣∣E]: h ∈ H} ∪ [P∣∣T]

\(H=\overline{3}m\); \({\mathcal{P}}{\mathcal{T}}\) intact; centrosymmetric; no splitting.

(0,1/3)

AFM (Type II)

\(C{2}^{{\prime} }/{m}^{{\prime} }\) (12.60)

Same as above

H = 2/m; \({\mathcal{P}}{\mathcal{T}}\) preserved.

(1/3,1/3)

AFM (Type II)

\(C{2}^{{\prime} }/{m}^{{\prime} }\) (12.60)

Same as above

Same as (0,1/3).

(1/3,2/3)

AFM (Type II)

\(P{\overline{3}}^{{\prime} }\) (147.15)

Same as above

\({\mathcal{P}}{\mathcal{T}}\) exact; inversion preserved; no splitting.

(1/2,0)

AFM (Type II)

\(C{2}^{{\prime} }/{m}^{{\prime} }\) (12.60)

Same as above

Same as (0,1/3).

(0,0)

AM (Type III / ¹\({\overline{3}}^{2}m\))

P31m (162.73)

\(\{[h| | E]:h\in H\}\cup \{[{C}_{2z}| | {C}_{{\rm{spin}}}^{2}],[{C}_{2,[110]}| | {C}_{{\rm{spin}}}^{2}]\}\)

\({\mathcal{P}}{\mathcal{T}}\) broken; \([{C}_{2z}| | {C}_{{\rm{spin}}}^{2}]\) and \([{C}_{2,[110]}| | {C}_{{\rm{spin}}}^{2}]\) act as connectors, enforcing alternating splitting.

(0,1/3)

AM (Type III / ²m²m¹m)

\(C2/{m}^{{\prime} }\) (12.58)

\(\{[h| | E]:h\in H\}\cup \{[{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\}\)

\({\mathcal{P}}{\mathcal{T}}\) broken; mirror connector \([{m}_{x}| | {C}_{{\rm{spin}}}^{2}]\) present.

(1/3,1/3)

AM (Type III / ²m²m¹m)

\(C2/{m}^{{\prime} }\) (12.58)

Same as above

Same as (0,1/3).

(1/3,2/3)

Quasi-AM (SST-4)

P3 (144)

{[h∣∣E]: h ∈ H}

\({\mathcal{P}}{\mathcal{T}}\) broken; no valid \([a| | {C}_{{\rm{spin}}}^{2}]\) survives; unprotected splitting at Γ.

(1/2,0)

AM (Type III / ²m²m¹m)

\(C2/{m}^{{\prime} }\) (12.58)

Same as (0,1/3)

\({\mathcal{P}}{\mathcal{T}}\) broken; mirror-type connector survives.