= Reduced BCS pairing Hamiltonian
The <reduced BCS pairing Hamiltonian> retains scattering of opposite-momentum, opposite-spin <Cooper pairs>:
$$
H=\sum_{k,\sigma}\epsilon_kc_{k\sigma}^\dagger c_{k\sigma}+\sum_{k,k'}V_{k,k'}b_k^\dagger b_{k'},\qquad b_k^\dagger=c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger.
$$
A <mean-field approximation> introduces the <BCS anomalous average> and pairing field $\Delta_k=-\sum_{k'}V_{k,k'}\langle b_{k'}\rangle$. The subtraction $-\sum V_{k,k'}\langle b_k^\dagger\rangle\langle b_{k'}\rangle$ corrects the double counting of the mean-field interaction energy.
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