# Continuum limit of graphene

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?

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.
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.