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In condensed matter theory, I can sometimes encounter such a term as continuum limit, also known as low energy limit. I have a question about this term, let me illustrate my question through an example.

Graphene, the famous two dimensional material, has two dirac cones at the corner of its Brillouin Zone. In continuum limit(lattice constant $\to$ 0), only electron states near the two dirac cones participate in the dynamics.

Here's my question: Why only electron states near the two dirac cones participate in the dynamics and what does it have to do with continuum limit?

Can anyone help me with this?

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    $\begingroup$ Would you consider to split the question in two (one on the continuum limit and its significance and the other on the continuum limit of graphene), the way the question is stated I feel it is two broad and therefore difficult to answer coherently. $\endgroup$ – Sebastian Riese Aug 17 '15 at 21:32
  • $\begingroup$ Done, but I cannot state continuum limit without graphene because I just don't understand it.@SebastianRiese $\endgroup$ – Chong Wang Aug 18 '15 at 3:02
  • $\begingroup$ my second question here: physics.stackexchange.com/q/201024 $\endgroup$ – Chong Wang Aug 18 '15 at 3:18
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In the continuum limit the lattice spacing $a$ goes to zero, therefore the Brillouin zone grows to infinity. If the Fermi velocity shall remain constant, the hopping parameter has to be rescaled as $t \propto 1/a$ (remember that the bandwidth is on the scale of $t$ and $v_F = \nabla_k E(\vec k)$), therefore only the features close to the Dirac points remain at finite energy.

In this fashion the continuum limit linearizes the spectrum (by only retaining the portion infinitesimally close to the Dirac points), and the continuum Hamiltonian can written in terms of two types of fermions with linear dispersion that respectively live around one or the other Dirac cone.

A continuum limit is equivalent to a low energy limit as the results obtained from it are the same, as the results obtained by considering only long wavelength excitations. Another way to understand this is, that all elementary low energy excitations are holes or electrons near the Fermi level, so it is obvious that an approximation reducing the spectrum to the linear parts near the Fermi level gives a correct low energy description.

So to answer the last part of your question: States that are not close to one of the Dirac cones can participate in the dynamics, but not for the low energy properties, that are usually considered (as one is interested in the low temperature behaviour, or transport phenomena at low temperature and small fields).

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