I would like to numerically find the edge modes of a $p_x$ + $i p_y$ BdG Hamiltonian. The lattice version is given by
H = $\sum\left[-t \left(c_{m+1,n}^{\dagger} c_{m,n} + \text{h.c} \right) – t\left(c_{m,n+1}^{\dagger} c_{m,n} + \text{h.c} \right) – \mu\,c_{m,n}^{\dagger}c_{m,n} + \left(\Delta c_{m+1,n}^{\dagger} c_{m,n}^{\dagger} + \Delta^* c_{m,n} c_{m+1,n}\right) + \left(i\Delta c_{m,n+1}^{\dagger} c_{m,n}^{\dagger} -i \Delta^* c_{m,n} c_{m,n+1}\right)\right]$
where $c_{m,n}$ is the annihilation operator for a spin polarised fermion on site (m,n).
While I understand how to take this system and put it on a finite lattice if there are only hopping terms ($c^{\dagger}c$ terms), how would I do that for terms such as $c^{\dagger}c^{\dagger}$?
Specifically, I want to find the spectrum of this system if it has periodic boundary conditions in one direction and open boundary conditions in the other.
Best Answer
First you need to bring it into the following form:
$H=\Psi^\dagger h \Psi$
Here $\Psi$ is a big column vector:
$\Psi=(\dots, c_{m,n}, \dots, c_{m,n}^\dagger, \dots)^T$
Basically, the first half of $\Psi$ are all annihilation operators, and the second half are all creation ones. If the number of sites is $N$, the size of $\Psi$ is $2N$. So $h$ is a $2N\times 2N$ matrix. To bring it into this form, one has to do a little bit of work, to rewrite all $c_i^\dagger c_j$ term as $-c_j c_i^\dagger$, etc. But this is not too difficult.
If you do everything correctly, $h$ is a Hermitian matrix and you can now go and diagonalize it. The results are of course the energies of the Bogoliubov quasiparticles and their forms are given by the unitary transformation that diagonalize $h$.