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BCS Theory

Developed by Bardeen, Cooper, and Schrieffer, this theory explains superconductivity through the formation of Cooper pairs mediated by lattice vibrations.
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The statement of the theorem

Consider the BCS mean-field Hamiltonian for the pairing interaction: \nH=∑k,σξkck,σ†ck,σ+12∑k(Δkck,↑c−k,↓+Δ∗kc−k,↓†ck,↑†)+12∑k(Δk∗c−k,↓ck,↑+Δ∗kck,↑†c−k,↓†)\mathcal{H} = \sum_{\mathbf{k}, \sigma} \xi_{\mathbf{k}} c_{\mathbf{k}, \sigma}^{\dagger} c_{\mathbf{k}, \sigma} + \frac{1}{2} \sum_{\mathbf{k}} \left( \Delta \mathbf{k} c_{\mathbf{k}, \uparrow} c_{-\mathbf{k}, \downarrow} + \Delta^* \mathbf{k} c_{-\mathbf{k}, \downarrow}^{\dagger} c_{\mathbf{k}, \uparrow}^{\dagger} \right) + \frac{1}{2} \sum_{\mathbf{k}} \left( \Delta \mathbf{k}^* c_{-\mathbf{k}, \downarrow} c_{\mathbf{k}, \uparrow} + \Delta^* \mathbf{k} c_{\mathbf{k}, \uparrow}^{\dagger} c_{-\mathbf{k}, \downarrow}^{\dagger} \right) \nThe superconducting gap Δ\Delta is determined self-consistently by the gap equation:\n\Delta = \lambda \sum_{\mathbf{k}} \frac{1}{2 \sqrt{\xi_{\mathbf{k}}^2 + |\Delta \mathbf{k}|^2}} \tanh\left(\frac{\sqrt{\xi_{\mathbf{k}}^2 + |\Delta \mathbf{k}|^2}}}{2 k_B T}\right) \nwhere λ\lambda is the coupling constant, ξk\xi_{\mathbf{k}} is the electron dispersion, and TT is the temperature.