# How to write BdG equation for generalize Hamiltonian?

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.

1. 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.

• Although you added the annihilation and creations it's still not possible to to work out what you want because you have not specified how the $\sigma$ $\tau$ and $s$ matrices act on the entries in your eight-component field. I can guess what the $\sigma$ and $\tau$'s do, but the $s_{z,y}$'s are a mystery. – mike stone Mar 20 '19 at 13:02
• In the paper,I see that $s_i,\sigma_i,\tau_i$ are Pauli matrices in the spin $(\uparrow,\downarrow)$,orbital (a,b),and particle-hole space,respectively. – YuXuanLi Mar 20 '19 at 13:13