According to this source, it is proved that, in absence of spin-orbit coupling, spatial inversion symmetry (as a part of point-group symmetry which operates as $\hat{S}\psi(\vec{r})=\psi(-\vec{r})$) creates the degeneracy below,

$$ E_n(\hat{S}\vec{k})=E_n(-\vec{k})=E_n(\vec{k}) $$

The other one, which is time-reversal symmetry, cause the same degeneracy as above, regardless the presence of space-inversion symmetry, it said.

Now I move on to this paper (in part IIB), it states that, in absence of spin-orbit coupling, double degeneracy only arise if both time-reversal(T) and space-inversion(P) symmetry are simultaneously present, or their combined $PT$-symmetry. Additionally, it said that, if only $T$-symmetry present, double degeneracy only arice at TRIM points, and no degeneracy are there if there's only $P$-symmetry present.

Why are they contradictory? Or what I missed here?


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