 Constant matrix perturbation of a Weyl Hamiltonian

Adding a spatially constant Hermitian matrix $V=dI+\mathbf{b}\cdot \mathbf{\sigma }$ 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 $d\pm |\hslash v\mathbf{k}+\mathbf{b}|$. The touching point moves to ${\mathbf{k}}_{\ast }=-\mathbf{b}/(\hslash v)$ at energy $d$, without opening a gap in three-dimensional momentum space.

For a fixed $k_z=0$ slice, however,This slice has minimum branch separation $2|b_z|$ if $b_z\ne 0$, 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.