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As we know, there are two distinct Dirac points for the free electrons in graphene. Which means that the energy spectrum of the 2$\times$2 Hermitian matrix $H(k_x,k_y)$ has two degenerate points $K$ and $K^{'}$ in BZ.

According to the von Neumann-Wigner theorem (no-crossing theorem): To make two eigenvalues of a Hermitian matrix (depending on some independent real parameters) cross, generally speaking, we need to change at least 3 parameters. But in the 2D graphene case, the variation of only 2 parameters $k_x,k_y$ can cause the energy levels cross.

So I want to know whether there are some physical or mathematical reasons for the existence of Dirac points in graphene.

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3 Answers 3

up vote 1 down vote accepted

Your use of the no-crossing idea is correct - we do not expect level crossings in two dimension to appear unless protected by symmetry. The symmetries in this case are the symmetries of the honeycomb lattice and time reversal. The protection of level crossings by symmetry is ubiquitous in solid-state.

I should add that the existence of these Dirac point is actually slightly more robust than would be implied by simple symmetry arguments. The band structure will still have Dirac cones if one applies any perturbation that does not violate parity, time reversal and is not extremely strong[1]. This is because of the interplay of the Berry's curvature and the Dirac point, which I could find a reference for if you would like.

[1] Extremely strong means that if I imagined increasing the strength of this perturbation up from zero it would drag the Dirac cones from the $K$, $K'$ points into each other. This would mean a perturbation of energy about the bandwidth, which is several electron-volts.

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Thanks for your brilliant answer. Could you show me some relevant references? –  Kai Li Jan 15 at 9:10

Wikipedia says:

The eigenvalues of a Hermitian matrix depending on $N$ continuous real parameters cannot cross except at a manifold of $N-2$ dimensions.

Since the Hamiltonian has $N=2$ parameters ($k_x$, $k_y$), the crossing manifold has a dimension $N-2=0$, which is a point. So it's, in principle, allowed for graphene to have degenerate states (there are also a lot of other degenerate states if you look at the whole bandstructure). This is definitely only a necessary, not a sufficient condition (e.g. one could look at bilayer graphene which does not have this degeneracy).

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Thanks a lot for your answer. Yeah, the criterion you present just gives a possibility of degeneracy. But what I most concern here is that the underlying mechanism to cause degeneracy in 2D graphene. –  Kai Li Apr 5 '13 at 15:41
And the von Neumann-Wigner theorem also says that:For the real symmetric matrix case, the minimal number of real parameters we need to tune to make level crossing reduces to 2. So I want to know the 2D graphene case whether has something to do with the real symmetric matrix case in von Neumann-Wigner theorem ? –  Kai Li Apr 5 '13 at 15:55
1) I would say symmetry - I never came up with a better explanation. 2) No, because the matrix is not real. The matrix you need to solve to obtain the bandstructure is complex and Hermitian. –  Rafael Reiter Apr 5 '13 at 15:58
Maybe, but can you specify what kinds of symmetries make degeneracy possible? –  Kai Li Apr 5 '13 at 16:08
How about this explanation:As we know, there are at most $n^2$ independent real parameters for a $n\times n$ Hermitian matrix , and if we want use von Neumann-Wigner theorem, the $n\times n$ Hermitian matrix which we deal with should has $n^2$ independent real parameters, then the theorem works. –  Kai Li Apr 5 '13 at 16:50

the Dirac Point on Graphene is protected by hidden symmetry. And it is explained very well in the paper arXiv:1406.3800. It is not that easy to understand the hidden symmetry. Personally speaking, I thought it is combination of inversion, time reversal and reflection symmetry, though the hidden symmetry in that paper has a totally different form with my understanding.

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