= Nearest-neighbour model of graphene
{title2=$E_\pm(\mathbf k)=\pm t|f(\mathbf k)|$}
Take a planar <honeycomb lattice> with two sites $A,B$ per primitive cell and one localized <atomic orbital> per site. In this explicitly idealized <tight-binding model>, all three nearest-neighbour hopping matrix elements are $-t$, with $t>0$; there is no on-site offset or further-neighbour hopping. Ignore <Electron> interactions and treat physical <spin> as a separate degeneracy. Let the bond length be $b$, and choose the three vectors from an $A$ site to its neighbours as
$$
\boldsymbol\delta_1=b(0,1),\qquad
\boldsymbol\delta_2=b(\sqrt3/2,-1/2),\qquad
\boldsymbol\delta_3=b(-\sqrt3/2,-1/2).
$$
The <Bloch state> amplitudes on $A,B$ form a two-component <vector>. A consistent phase convention gives the <Hamiltonian>
$$
H(\mathbf k)=-t\begin{pmatrix}0&f(\mathbf k)\\f(\mathbf k)^*&0\end{pmatrix},\qquad
f(\mathbf k)=\sum_{j=1}^3e^{i\mathbf k\cdot\boldsymbol\delta_j}
=e^{ibk_y}+2e^{-ibk_y/2}\cos(\sqrt3bk_x/2).
$$
Its characteristic equation is $E^2=t^2|f|^2$, giving the two displayed <energy bands>. The two-component structure comes from the two sites, rather than two values of physical <spin>. This is a particularly small matrix model in which a crystal's geometry creates an unexpected low-energy law.
\Image[/graphene-honeycomb-two-band-dirac-cone.png]
{title=A honeycomb lattice produces two touching electronic bands}
{description=Original nearest-neighbour model with the defined bond vectors. The surface shows the calculated bands near one zero of $f$, with distance in units of inverse bond length and energy in units of the hopping magnitude. These are model energies, not measured band data.}
{source=graphene-honeycomb-two-band-dirac-cone.py}
Historically, <Philip Russell Wallace> used a <tight-binding model> to investigate the <electronic band structure> of <graphite> in 1947. The material in that paper was layered <graphite>; the single-sheet model here isolates one simpler part of the problem.<The Band Theory of Graphite by Philip Russell Wallace>[Publisher abstract: tight-binding bands and Brillouin zones of graphite; published 1 May 1947]{ref}
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