Continuum limit of graphene

Question

I am studying the continuum limit as described in section II B of this paper. The tight binding Hamiltonian for graphene is given by

$$ H = -t \sum_{\langle i,j \rangle, \sigma}\left(a^\dagger_{\sigma,i}b_{\sigma , j} + \text{h.c.}\right)$$

as show in Eq. (4). It states that in order to derive a theory that is valid close to the Dirac points, we approximate the fields as

$$ a_n \approx e^{-i K \cdot R_n} a_{1,n} +e^{-iK' \cdot R_n} a_{2,n}$$

and similarly for $b_n$, as shown in Eq. (17). Upon substituting this into the Hamiltonian $H$, we arrive at

$$ H \approx -iv_F\int dx dy \left( \psi_1^\dagger \sigma \cdot \nabla \psi_1 + \psi^\dagger_2 \sigma^* \cdot \nabla \psi_2\right)$$

where $ \sigma = (\sigma^x , \sigma^y)$ and $\psi_i^\dagger = (a_i^\dagger,b_i^\dagger)$, as shown in Eq. (18).

My questions

1. The answer to this question states that as the lattice spacing $\delta \rightarrow 0$, only the features close to the Dirac points remain at finite energy. However, this was not given any more detail. How do I show this mathematically? What is our definition of "close to the Dirac points"?

2. Upon substituting the approximations for $a_n,b_n$ into the Hamiltonian $H$, I find cross terms such as $a^\dagger_1 b_2 $, however the continuum limit quoted above does not have terms like this, i.e., no coupling between $1$ and $2$. I suppose a physical reason for this is that a low-energy process should not evolve the system near one Dirac point to the other, however is there a precise mathematical reason why this is the case?

Answer 1

In the energy space, the Fermi level in graphene lies at or near the Dirac point (it can be somewhat shifted from this point by doping or a backgate potential). In the situations where we are interested only in the low energy excitations, it is reasonable to expan the energy in terms of the electron momentum deviations from this Fermi point.

Remark 1: Note that this is by no means unique to graphene: the linearization of the dispersion relation near the Fermi surface is a routine procedure when describing processes in metals, whereas in semiconductors one similarly employs the effective mass approximation. However, in semiconductors the expansion is done near the band minimum/maximum, since the Fermi level lies in the gap, whereas in metals the Fermi surface is not reduced to a single point, as in graphene.

Remark 2: Note that we are talking about the expansion in $k$-space (note also that we speak of the Bloch momentum, rather than the real momentum). This fact might be somewhat obscured by the Hamiltonian being transformed back into the position space. However, the wave function in this Hamiltonian should be understood as the envelope function, which describes only long wavelength dynamics.

Getting back to your questions:

1. Since we limit ourselves to small $k$, we are talking about long wave lengths, i.e., we are interested in scales much bigger than the lattice spacing. Indeed, the Bloch wave length becomes comparable to the lattice spacing only at the edges of the Brillouin zone ($k=\pi/a$ in the Krönig-Penny model). Thus, taking lattice spacing to zero is just another way of taking the same limit - it makes the wave vectors corresponding to the edges of the Brillouin zone to be very large.
2. By now it is probably clear why there are no processes connecting two Dirac points: although these processes are energetically allowed, they correspond to very large wave vectors. If, e.g., we expect the energy excitations to be induced by an applied electric field, this field would have to change in space on the scale of the lattice constant, which is not the case. (E.g., electromagnetic radiation has wavelengths ranging in hundreds of nanometers or even micrometers.)