Past exam of the superconductivity and Quantum Coherence course of the University of Cambridge / 2013 / Paper 1 / 1 / b

In BCS theory, pairing mixes an electron state with a hole in its opposite-momentum partner. For a uniform spin-singlet gap, the uniform Bogoliubov--de Gennes Hamiltonian gives the momentum-pair eigenproblemHere ${\xi }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}-\mu$. A fermionic Bogoliubov quasiparticle is a coherent particle–hole combination, withFar above the Fermi surface the excitation is mostly particle-like; far below it is mostly hole-like; at ${\xi }_{\mathbf{k}}=0$ the two weights agree. The transformation preserves the fermionic canonical anticommutation relations. Its mean-field particle-number mixing does not mean that the full physical theory violates charge conservation.

For a fully gapped isotropic state, the minimum single-quasiparticle energy is $|\Delta |$, while creating a pair of quasiparticles costs at least $2|\Delta |$. This produces suppressed low-temperature thermal populations, an activated electronic heat capacity and a gap feature in superconducting tunneling. Gap nodes in an unconventional state instead allow low-energy excitations, changing the temperature dependence. These fermionic excitations are distinct from collective phase and amplitude oscillations of the paired condensate.