 Dirac point

A Dirac point is a momentum-space point where two energy bands touch and their leading splitting is linear in displacement from that point. A local two-band Hamiltonian often takes the form $H=E_{D}I+\hslash v_x{q}_x{\sigma }_x+\hslash v_y{q}_y{\sigma }_y$, where $\mathbf{q}$ is measured from the touching point and the Pauli matrices act on a two-component internal basis.

Its energy eigenvalues are $E_D\pm \hslash \sqrt{v_x^2{q}_x^2+v_y^2{q}_y^2}$, producing a Dirac cone. The two components can label sublattices rather than real spin. Adding a symmetry-allowed term $m{\sigma }_z$ opens a gap $2|m|$, illustrating why the symmetries protecting the touching matter. Distinct Dirac valleys in graphene are regions around different Dirac points.