Bogoliubov- de Gennes Hamiltonian and Anti-Commutation Rule

Question

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.

Answer 1

You are correct. There is a $c$-number energy shift: $$ \hat H_{\rm Bogoliubov}= a^\dagger_i H_{ij}a_j +\frac 12 \Delta_{ij} a^\dagger_i a^\dagger_j +\frac 12 \Delta^{\dagger}_{ij} a_i a_j\\ = \frac12 \left(\matrix{ a^\dagger _i &a_i}\right)\left(\matrix{ H_{ij}& \phantom {-}\Delta_{ij}\cr \Delta^{\dagger}_{ij}& -H^T_{ij}}\right) \left(\matrix{ a_j\cr a^\dagger_j}\right) +\frac 12 {\rm tr}(H). $$ Keeping track of the ${\rm tr} (H)$ is not important for quasiparticle energies, and so it commonly omitted, but it is essential when you are interested in the dynamics of the condensate.