# BCS quadratic Hamiltonian: confused with operators

I'm working through J. Annet book: Superconductivity, Superfluids and Condensates.

In chapter 6 (BCS theory) the effective Hamiltonian in quadratic form is introduced:

$$\sum_{k\sigma} (\epsilon_k - \mu)c^{+}_{k\sigma}c_{k\sigma} - \sum_{k} (\Delta^{*} c_{-k\downarrow} c_{k\uparrow} + \Delta c^{+}_{k\uparrow} c^{+}_{-k\downarrow} ) \\$$

and then rewritten in a matrix form:

$$\sum_k{ \begin{pmatrix} c^{+}_{k\uparrow} & c_{-k\downarrow} \end{pmatrix} \begin{pmatrix} \epsilon_k - \mu & -\Delta \\ -\Delta^{*} & -(\epsilon_k - \mu) \end{pmatrix} \begin{pmatrix} c_{k\uparrow} \\ c^{+}_{-k\downarrow} \end{pmatrix}}$$

If I now start with the matrix form, multiply it out, collect terms and compare with the first one I see that I require:

$$c^{+}_{-k\downarrow}c_{-k\downarrow} = c^{+}_{k\downarrow}c_{k\downarrow}$$

I can't see why this is the case. Have I done something wrong or am I missing something?

• This is not the case. You must use $\sum_k \epsilon_k c^\dagger_{-k}c_{-k}=\sum_k \epsilon_k c^\dagger_{k}c_{k}$ by changing the dummy variable $k\to -k$, and using $\epsilon_{-k}= \epsilon_k$.