# Topological order in Weyl Semimetal

Is the topological phase in a Weyl semimetal is intrinsic or symmetry protected? How can we realize that?

If symmetry protected, which symmetry protects the topological phase of non-centrosymmetric Weyl semimetals and magnetic Weyl semimetals?

$$\hat{H}(\mathbf{k})=d_0(\mathbf{k})+d_1(\mathbf{k})\sigma_1+d_2(\mathbf{k})\sigma_2+d_3(\mathbf{k})\sigma_3$$
where $$\sigma_i$$ are the Pauli matrices. The eigenvalues are given by $$E_{\pm}=d_0\pm\sqrt{d_1^2+d_2^2+d_3^2}$$, so that a band crossing occurs when $$d_1^2+d_2^2+d_3^2=0$$. This equation involves three parameters, and in a three dimensional system, $$\mathbf{k}=(k_1,k_2,k_3)$$ has three components, so you can generically tune the $$\mathbf{k}$$ vector to make the two bands degenerate without the need of any symmetry.
Having established this, if you consider a system with both time reversal and inversion symmetry, then every band is doubly degenerate (spin up and spin down electrons have the same energy at every $$\mathbf{k}$$-point). This means that potential band crossings would involve four (rather than two) bands, and be Dirac rather than Weyl points. So a pre-requisite before you can have a Weyl point is that you need to break the spin degeneracy caused by time reversal and inversion symmetries. This can be accomplished by breaking one of these two symmetries.