= BCS quasiparticle spectrum
{c}
{title2=$E_k=\sqrt{\xi_k^2+|\Delta|^2}$}
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>, $\xi_k=\epsilon_k-\mu$, and use the same sign convention as the <BCS coherence factors>:
$$
\mathcal H_k=\begin{pmatrix}\xi_k&-\Delta\\-\Delta&-\xi_k\end{pmatrix},\qquad
\det(\mathcal H_k-\varepsilon I)=\varepsilon^2-\xi_k^2-\Delta^2=0.
$$
Its <eigenvalues> are $\varepsilon_\pm=\pm E_k$, 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:
$$
\frac{dE}{d\xi}=\frac{\xi}{\sqrt{\xi^2+\Delta^2}},\qquad
\frac{d^2E}{d\xi^2}=\frac{\Delta^2}{(\xi^2+\Delta^2)^{3/2}}>0.
$$
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,
$$
E(\xi)=\Delta+\frac{\xi^2}{2\Delta}+O(\xi^4/\Delta^3).
$$
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 $\sqrt2\Delta\simeq1.03644\ \mathrm{meV}$ rather than the minimum $0.732871\ \mathrm{meV}$.
\Image[/bcs-pairing-shell-and-quasiparticle-spectrum.png]
{title=Opposite-momentum Cooper pairing opens a quasiparticle gap}
{description=Original schematic section through a spherical <Fermi surface>, alongside the local isotropic <BCS quasiparticle spectrum>. The shell thickness is exaggerated and its geometry is schematic. Both members of a shown <Cooper pair> have nonzero opposite <momenta>. The spectrum uses $x=\xi/\Delta$ only within the pairing shell; the lower branch is the redundant particle-hole partner, while positive excitations cost at least $\Delta$.}
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