Symmetry properties of wave functions in magnetic crystals
| space operator | time reversal | Full space group | ||
| Paramagnetic | H | θ | G=H+Hθ | |
| magnetic (below TN) | ℋ | G'=ℋ+ℋa0 (a0=v0θ) | ||
Extra timer-reversal degeneracy
u belongs to ℋ
u={σ|τ}
σk=k+Kq
v0={ρ0|τ0}
ρ0k=-k+Kq'
Relation between representation of ℋ and G'
| case 1 | case 2 | case 3 | ||
| ℋk | Δ(i)(u) | Δ(i)(u) | Δ(i)(u), Δ(i)(v0-1uv0)*=Δ(j)(u) | |
| G'k | D(i)(u) | D(i)(u) | D(i)(u) | |
| Relation | D(i)(u) =Δ(i)(u) | D(i)(u) =2Δ(i)(u) | D(i)(u) =Δ(i)(u)+Δ(j)(u) | |
| degeneracy | no new degeneracy | degeneracy doubled | Δ(i)(u) and Δ(j)(u) are now degenerate | |
| Criteria: ∑χΔ(i){u'u} where u=u={σ|τ} u'=v0-1uv0 The sum is over σ, and there are M of them. The ni is the dimension of Δ(i)(u) ω=1 for single group, =-1 for double group | ωχ{Δ(i)(v02)}M/ni | -ωχ{Δ(i)(v02)}M/ni | 0 | |
| If v02={E|Rn} or {E'|Rn} where E=ωE | ωM | -ωM | 0 |
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