 BCS quasiparticle spectrum

The BCS quasiparticle spectrum is the positive excitation energy obtained by diagonalizing the paired particle-hole block. Here the weak coupling shell model has $\Delta \ll {\Omega }_D$: the minimum lies inside the pairing shell, and states outside it cost more. To derive it for a uniform isotropic real $\Delta >0$, measure the normal-state energy from the chemical potential, ${\xi }_k={ϵ}_k-\mu$, and use the same sign convention as the BCS coherence factors:Its eigenvalues are ${\epsilon }_{\pm }=\pm E_k$, where $E_k=\sqrt{{\xi }_k^2+{\Delta }^2}$. The positive $E_k$ is the cost of creating a quasiparticle above the BCS ground state. The lower branch $-E_k$ is its particle-hole partner in this doubled matrix description, not an excitation that lowers the ground state energy without bound.

The location and value of the minimum follow directly:The unique minimum as a function of $\xi$ is $E(0)=\Delta$, at the normal-state Fermi surface. On an isotropic three-dimensional Fermi surface it occurs for every direction of $k$, rather than at a unique point of momentum space. Near that minimum,Farther away, $E(\xi )\simeq |\xi |$ when $\Delta \ll |\xi |\ll {\Omega }_D$. For the illustrative $\lambda =0.25$ example, a state at $\xi =\Delta$ costs $\sqrt{2}\Delta \simeq 1.03644\mathrm{meV}$ rather than the minimum $0.732871\mathrm{meV}$.