The symmetries of graphene were discussed intensively in the video, so let’s review them.

As we already said in our first week, graphene is the prototype of a system with sublattice symmetry, which makes the Hamiltonian block off-diagonal with respect to the two sublattices. The sublattice symmetry reads

\[\sigma_z\,H_0(\mathbf{k})\,\sigma_z = -H_0(\mathbf{k})\,.\]

Sublattice symmetry is only approximate, and it is consequence of the nearest neighbor tight-binding model. Just like the inversion symmetry mentioned in the video, it protects the Dirac points and needs to be broken in order to open a gap.

In addition to sublattice and inversion symmetry, the honeycomb lattice also has a three-fold rotation symmetry around the center of the unit cell. This symmetry is important to make the Dirac cones appear in the first place, but it will not play a role in all that follows.

Finally, there is time-reversal symmetry, which at the moment is perfectly preserved in our tight-binding model. Since we are not considering the spin degree of freedom of the electrons, the time-reversal symmetry operator in real space is just complex conjugation. In momentum space representation, time-reversal symmetry reads

\[ H_0(\mathbf{k}) = H_0^*(-\mathbf{k})\,.\]

It’s important to note that time-reversal symmetry sends \(\mathbf{K}\) into \(\mathbf{K}'\) and therefore it exchanges the two Dirac cones.

The product of (approximate) sublattice and time-reversal symmetries yields a further discrete symmetry, a particle-hole symmetry \(\sigma_z H^*(-\mathbf{k})\,\sigma_z = -H_0(\mathbf{k})\).