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In Weyl semimetals time reversal symmetry (TRS) or centrosymmetry has to be broken (CS). It is stated, there are at least four Weyl points in TRS systems. I tried understanding this looking at the semiclassical equation of motion

$\dot{\vec{r}} = \vec{v_k} + \dot{\vec{k}} \times \vec{\Omega_k}$

Under time reversion $t \rightarrow -t$ , $ \vec{v_k} \rightarrow -\vec{v}_{-k}$ and $\dot{\vec{k}} \rightarrow \dot{\vec{k}}$. For TRS, it follows that the Berry curvature should transform such that $\vec{\Omega_k} \rightarrow -\vec{\Omega}_{-k}$. In that case, the equation above is satisfied.

However, I don't see the argument, why there have to be at least four Weyl points in TRS systems, since the Berry curvature transforms by switching sign and thus describes a Weyl point with opposite chirality at $-\vec{k}$.

Is there TRS if the equation above is satisfied after applying the transformation? Or does the TRS actually mean, that the Berry curvature should not transform like mentioned above. I guess I amm mixing up the transformation itself and the corresponding symmetry.

In some papers I read that for TRS systems the Berry curvature holds

$ \vec{\Omega_k} = - \vec{\Omega_{-k}}$

That is, there should be a Weyl point with opposite chirality and no-go theorem is not violated. Why there has to be at least four Weyl points? On the other hand I found in one paper for TRS systems

$\vec{\Omega_k} = \vec{\Omega_{-k}}$

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