Coupled Bogoliubov equations for electrons and phonons

The interaction between fermions and bosons is central to modern physics. To describe systems with such interactions, the authors apply Wick's theorem to the square of the Fr\"ohlich Hamiltonian and keep all terms up to quadratic order. What emerges are two coupled Bogoliubov equations: one for electrons and one for phonons. Remarkably, the self-consistent solution to these equations accurately describes both the renormalized gap in insulators and the superconducting gap in metals.