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In much of the literature floating around, it is commonly implied that an important part of obtaining a Weyl semimetal phase is to break either time reversal symmetry or inversion symmetry.

However, none of the literature I found seems to give a clear answer on why these symmetries seem mutually exclusive in Weyl semimetals, it is just taken for granted that you can't break both symmetries.

Why is this assumed?

I had thought that all these broken symmetries did was to lift the spin or sublattice degeneracy, which in principle should still permit a Weyl node pair.

Is there something that forces a gap to appear when inversion and time reversal symmetries are broken?

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