BCS anomalous average
= BCS anomalous average
{c}
{title2=$\langle c_{-k\downarrow}c_{k\uparrow}\rangle$}
The BCS anomalous average is the pair expectation $\langle b_k\rangle=\langle c_{-k\downarrow}c_{k\uparrow}\rangle$. In the <BCS ground state> it is $u_kv_k=\Delta_k/(2E_k)$ for real gap at zero temperature. At temperature $T$ it becomes $\Delta_k\tanh(E_k/(2k_BT))/(2E_k)$. Substitution into the pairing-field definition gives the <BCS gap equation>.