= Constant matrix perturbation of a Weyl Hamiltonian
{title2=$E_\pm(\mathbf k)=d\pm|\hbar v\mathbf k+\mathbf b|$}
Adding a spatially constant <Hermitian matrix> $V=dI+\mathbf b\cdot\boldsymbol\sigma$ to an isotropic <Weyl Hamiltonian> preserves each momentum sector. By the <Hermitian two-by-two matrix in the Pauli basis> decomposition, the exact energies are $d\pm|\hbar v\mathbf k+\mathbf b|$. The touching point moves to $\mathbf k_*=-\mathbf b/(\hbar v)$ at energy $d$, without opening a gap in three-dimensional momentum space.
For a fixed $k_z=0$ slice, however,
$$
E_\pm=d\pm\sqrt{(\hbar vk_x+b_x)^2+(\hbar vk_y+b_y)^2+b_z^2}.
$$
This slice has minimum branch separation $2|b_z|$ if $b_z\ne0$, because it misses the shifted node. A one-dimensional cut can likewise look gapped while a crossing exists elsewhere. No nonzero two-by-two matrix anticommutes with all three <Pauli matrices>, which is another way to see why an independent constant mass matrix is unavailable for this isolated three-dimensional two-component model.
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