# What is continuum limit (low energy limit) in condense matter physics?

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?

• 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. – Sebastian Riese Aug 17 '15 at 21:32
• Done, but I cannot state continuum limit without graphene because I just don't understand it.@SebastianRiese – Chong Wang Aug 18 '15 at 3:02
• my second question here: physics.stackexchange.com/q/201024 – Chong Wang Aug 18 '15 at 3:18

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.