# Is the superconducting gap the same as the many-body gap in superconductor?

In conventional superconductors, it is well known that there is a gap $$\Delta_{\text{sc}}$$ to electronic or quasiparticle excitations. This is the gap that shows up when one measures the temperature dependence of the specific heat or in optical measurements of the frequency-dependent conductivity. In BCS theory, this gap is interpreted as the energy cost of breaking a Cooper pair into its two constituent electrons/quasiparticles.

More generally, a gapped quantum system is system with a many-body hamiltonian that, in the thermodynamic limit, has one or more degenerate ground states and a finite energy gap $$\Delta_{\text{mb}}$$ to any excited states. Since superconductors have $$\mathbf{Z}_2$$ topological order, they are gapped systems in this sense.

Question: Are the superconducting gap $$\Delta_{\text{sc}}$$ and the many-body gap $$\Delta_{\text{mb}}$$ the same, qualitatively or quantitatively?

If they are the same, how do we know? In other words, how do we know that the lowest-lying excitations correspond to separating a Cooper pair into quasiparticles?

If they are not the same, why not? What excitations are there with a lower energy than the quasiparticle excitations? How does $$\Delta_{\text{mb}}$$ compare quantitatively with $$\Delta_{\text{sc}}$$?