I encounter a problem which is related to Bogoliubov-de Gennes(BdG) Hamiltonian. Suppose I use BdG theory to transform my many body Hamiltonian into the following form:
\begin{equation} \hat{H} = \sum_{k \sigma} \xi_{k} c^{\dagger}_{k \sigma} c_{k \sigma} - \sum_{k} ( \Delta^{*} c_{-k \downarrow} c_{k \uparrow} + \Delta c^{\dagger}_{k \uparrow} c^{\dagger}_{k \downarrow}) \end{equation}
Then, we can rewrite our Hamiltonian into matrix form by introducing a vector of creation and annihilation operator $\phi_k = ( c_{k \uparrow} ~~ c^{\dagger}_{-k \downarrow})^{T}$:
\begin{equation} \hat{H} = \sum_k ( c^{\dagger}_{k \uparrow} ~~ c_{-k \downarrow}) \begin{pmatrix} \xi_{k} & - \Delta \\ -\Delta^{*} & -\xi_{k} \end{pmatrix} \begin{pmatrix} c_{k \uparrow} \\ c^{\dagger}_{-k \downarrow} \end{pmatrix} \end{equation}
However, when I expand matrix form Hamiltonian to see whether I can recover the original one, I encounter a problem in the $-\xi_{k}$ term. Firstly, the expansion of such matrix form Hamiltonian is as follow:
\begin{equation} \hat{H} = \sum_{k} \xi_{k} c^{\dagger}_{k \uparrow} c_{k \uparrow} -(\Delta^{*} c_{-k \downarrow}c_{k \uparrow} + \Delta c^{\dagger}_{k \uparrow}c^{\dagger}_{-k \downarrow}) - \xi_{k} c_{-k \downarrow} c^{\dagger}_{-k \downarrow} \end{equation}
Since $k$ and $-k$ are dummy variables, I can re-label the term $c_{-k \downarrow}c^{\dagger}_{-k \downarrow} \rightarrow c_{k \downarrow}c^{\dagger}_{k \downarrow}$. But when I use the anti-commutation rules to normal order the $ c_{k \downarrow}c^{\dagger}_{k \downarrow}$, I got a trouble since $\{c_{k \downarrow}, c^{\dagger}_{k \downarrow} \} = c_{k \downarrow} c^{\dagger}_{k \downarrow} + c^{\dagger}_{k \downarrow} c_{k \downarrow} = \delta_{k,k} \delta_{\downarrow, \downarrow} = 1$, implying that $ c_{k \downarrow} c^{\dagger}_{k \downarrow} = 1 - c^{\dagger}_{k \downarrow} c_{k \downarrow}$. Therefore, my $- \xi_{k}$ term would become like the following:
\begin{equation} \begin{split} -\xi_{k} c_{-k \downarrow} c^{\dagger}_{-k \downarrow} &\rightarrow -\xi_{k}(1 - c^{\dagger}_{k \downarrow} c_{k \downarrow}) \\ &= \xi_{k} c^{\dagger}_{k \downarrow} c_{k \downarrow} - \xi_{k} \end{split} \end{equation}
This means that the expansion of the matrix form Hamiltonian will acquire a constant energy shift which is inconsistent with the original one. Could anyone help me to find out the mistake that I had made in the whole derivation? Thank you.