BCS theory

Definition

Consider spin-$1/2$ electrons with creation and annihilation operators $c^\dagger_{k\sigma}, c_{k\sigma}$ for momentum $k$ and spin $\sigma \in \set{\uparrow, \downarrow}$, single-particle energies $\varepsilon_k$, chemical potential $\mu$ and $\xi_k = \varepsilon_k - \mu$. The reduced BCS Hamiltonian is $$H = \sum_{k,\sigma} \xi_k \, c^\dagger_{k\sigma} c_{k\sigma} - \sum_{k, k'} V_{kk'} \, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow} c_{-k'\downarrow} c_{k'\uparrow},$$ where, in the BCS model, $V_{kk'} = V > 0$ if $\abs{\xi_k} \le \hbar\omega_D$ and $\abs{\xi_{k'}} \le \hbar\omega_D$, and $V_{kk'} = 0$ otherwise, for a cutoff energy $\hbar\omega_D$. The BCS state is $$\lvert \Psi \rangle = \prod_k \big(u_k + v_k \, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\big) \lvert 0 \rangle, \qquad \abs{u_k}^2 + \abs{v_k}^2 = 1.$$ Minimising the energy over such states at zero temperature gives the gap equation $$\Delta_k = \sum_{k'} V_{kk'} \frac{\Delta_{k'}}{2E_{k'}}, \qquad E_k = \sqrt{\xi_k^2 + \abs{\Delta_k}^2},$$ where $E_k$ is the energy of a quasiparticle excitation.