How to write BdG equation for generalize Hamiltonian?

Question

I am reading this parer Majorana Corner Modes in a High-Temperature Platform,The Bogoliubov–de Gennes Hamiltonian is :

$\hat{H}=\sum_{k}\Psi^\dagger_kH(\vec{k})\Psi_k$with

$\Psi_k=(c_{a,k\uparrow},c_{b,k\uparrow},c_{a,k\downarrow},c_{b,k\downarrow},c_{a,-k\uparrow}^\dagger,c_{b,-k\uparrow}^\dagger,c_{a,-k\downarrow}^\dagger,c_{b,-k\downarrow}^\dagger)$ and Hamiltonian is $H(\vec{k})=M(\vec{k})\sigma_z\tau_z+A_xsin(k_x)\sigma_xs_z+A_ysin(k_y)\sigma_y\tau_z+\Delta(\vec{k})s_y\tau_y-\mu\tau_z$.

All of the $\sigma,\tau,s$ are Pauli Matrix,$s$ is in spin space,$\tau$ is about particle-hole symmetry and $\sigma$ represent the orbital.

1. Question1:

I want to solve this model through BdG;I have known use Fourier Transformation make this Hamiltonian in real space.But I cannot confirm the correct process about this,I want to get the BdG form for this Hamiltonian,both k-space and real space are made me happy.

I download the Supplemental Material for this paper which have the Hamiltonian in real space,however I wish I can realize the detail about how to do this.

2. Quextion2:

In article has two image I am a tiro in this,would someone willing to tell me how to plot these image?

Thanks for everyone who answer this question.

Answer 1

There are no many-body annihilation and creation operators in your matrix hamiltonian, so there is no way to tell if you are dealing with a superconductor or a simple band-theory model. Consequently there is no BdG version.