Background

My question concerns Beenakker’s paper on "Specular Andreev Reflection in Graphene": cond-mat/0604594 (arxiv)/Phys.Rev.Lett. 97, 067007 (The same topic is also discussed in Rev. Mod. Phys. 80, 1337)

In this paper the conductance is calculated for a N-S interface using the Blonder-Tinkham-Klapwijk formalism. The starting point is the BdG equation
begin{equation}
begin{bmatrix}
H-E_F & Delta
Delta^* & E_F – THT^{-1}
end{bmatrix}
begin{bmatrix}
u
v
end{bmatrix} = epsilonbegin{bmatrix}
u
v
end{bmatrix}.
end{equation}
$E_F,Delta$, and $epsilon$ is the Fermi energy, superconducting gap, and excitation energy respectively. For simplicity neglect the proximity effect such that we can use a step-potential $Delta = Delta_0 theta(x)$. $u$ and $v$ are electron and hole components.
Here $H$ is the single-particle Hamiltonian in a hexagonal lattice (graphene)
begin{equation}
H = begin{bmatrix}
H_+ & 0
0 & H_-
end{bmatrix},
end{equation}
with
begin{equation}
H_{pm} = -i hbar v left(sigma_x partial_x pm sigma_y partial_yright).
end{equation}
In this notation $v$ and $sigma_{x,y}$are the Fermi velocity, and Pauli matrices respectively.

The problem

It is easy to derive this equation by starting from a nearest neighbour Tight-binding Hamiltonian of the form
begin{equation}
H = – t sum_{langle ij rangle} c_i^{dagger} c_j – E_F sum_i c_i^{dagger} c_i + sum_{i sigma}left(sigma Delta c_{i,sigma}^{dagger} c_{i,-sigma}^{dagger} + h.c.right)
end{equation}
as done here cond-mat/1308.0017 (arxiv). It is assumed that the lattice is hexagonal with two sublattices A and B. In order to do so one assumes translational invariance in both the $x$ and $y$ directions and performs the usual Fourier transformations
begin{equation}
A_{mathbf{k} sigma} = frac{1}{sqrt{N_A}}sum_i A_{i sigma} e^{-i mathbf{k}cdot mathbf{r}}.
end{equation}
Here $A (B)$ denotes operators living on the A (B) sublattice. One also has to perform the replacement $mathbf{p} = – i hbar nabla$, and double the degrees of freedom (but these steps are not important for my question).

However, I’m not sure if this method is rigorous (although it seems it is extensively used in the literature). My issue is that to perform the Fourier transformation I thought it was necessary to assume translational invariance in the $x$ direction. However, this symmetry is broken by the $Delta = Delta_0 theta(x)$ term. For simplicity we assume that $t$ and $E_F$ are constants.

• Are there any assumptions that validate ignoring the broken translational symmetry?
• If not is there perhaps a better way to re-derive the BdG equation starting from the Tight-binding model?