Recently,I am learning BdG method in superconductor system,I have some question about particle-hole symmetry during construct Hamiltonian matrix for this system.

In Hamiltonian if spin orbital term is $$ H_{S O}= \sum_{\mathbf{i}}\left(i \lambda \Phi_{\mathbf{i} \mathbf{j}} c_{\mathbf{i} \uparrow}^{\dagger} c_{\mathbf{i}+\hat{x} \downarrow}+i \lambda \Phi_{\mathbf{i} \mathbf{j}} c_{\mathbf{i} \downarrow}^{\dagger} c_{\mathbf{i}+\hat{x} \uparrow}\right. +\lambda \Phi_{\mathbf{i} \mathbf{j}} c_{\mathbf{i} \uparrow}^{\dagger} c_{\mathbf{i}+\hat{y} \downarrow}-\lambda \Phi_{\mathbf{i} \mathbf{j}} c_{\mathbf{i} \downarrow}^{\dagger} c_{\mathbf{i}+\hat{y} \uparrow}+\mathrm{H.c.} )$$

I just can write a matrix about "Particle" part,I cannot write the "Hole" part

So,how to write the "Hole" part about this Hamiltonian,or what the meaning for Particle-Hole symmetry in superconductor system.

Please give me some tips or something helpful.


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