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In the context of the BCS theory of superconductivity, which energy gap is one referring to when using the BCS gap equation

$\Delta_{\textbf{k}}=-\sum_{\textbf{k'}}V_{\textbf{k},\textbf{k'}}\text{g}_{\textbf{k'}}$

with the Gorkov amplitude $\text{g}_{\textbf{k}}=\left\langle \hat{c}_{-\textbf{k}\uparrow}\hat{c}_{\textbf{k}\downarrow}\right\rangle$? In which energy spectrum does this gap occur? Is it the one of Cooper pairs or Bogoliubov quasiparticles? I am somewhat confused at the moment, so any handwaving explanation would be really helpful.

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It's the energy gap of the single-particle spectrum. If one tries to probe the system with a STM, for example, you will get no tunneling below this gap. It can be read from the retarded Green function which in matrix form is $$ g^r(\omega) = -i\frac{\pi\rho_0}{\sqrt{(\omega^2-\Delta^2)}}\left(\begin{array}{cc}\omega & \Delta \\ \Delta & \omega \end{array} \right) $$ where the basis is $[\psi(x)_\uparrow, -\psi^{\dagger}(x)_\downarrow]$. So the single-particle density of states, when read off the green function $\rho(\epsilon) = -{\rm Im}\{g^r(\epsilon)\}/\pi$ has a gap for $|\omega|<\Delta$.

The spectrum of the Cooper pairs doesn't have a gap. Adding or removing a Cooper pair from the ground state costs zero energy.

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  • $\begingroup$ Then how is this gap leading to the well known properties of superconductivity? $\endgroup$
    – Milarepa
    Jan 14 '20 at 19:59

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