Weyl Hamiltonian ($H=v\mathbf{\sigma }\cdot \mathbf{p}$)

The isotropic Weyl Hamiltonian acts on a two-component spinor as $H=v\mathbf{\sigma }\cdot \mathbf{p}$. For a plane wave of momentum $\hslash \mathbf{k}$, its two energy eigenvalues are $E_{\pm }=\pm \hslash |v||\mathbf{k}|$. For $v>0$ the corresponding Bloch vectors point along and opposite $\mathbf{k}$, respectively. If the Pauli matrices describe physical spin one-half, the mean spin is $\pm (\hslash /2)\hat{\mathbf{k}}$; in an effective two-band model the same algebra can describe an internal pseudospin instead. The bands meet linearly at the origin. ^[ref]

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.