Unified description of collective modes in superconductors and semiconductors with an exciton condensed phase

It is shown that the Bethe–Salpeter approach, the Bardeen, Cooper, and Schrieffer (BCS) based vertex method, and a generalized random‐phase approximation (GRPA) to the many‐electron problem in the presence of a condensed quantum phase yield the same theoretical excitation spectrum $\omega (Q)$ to collective modes. This spectrum reveals a secondary peak in optical absorption in semiconductors that can be understood as signaling the existence of an excitonic Bose–Einstein condensate (BEC). The analysis shows as well that there is an additional, non‐trivial linearly‐dispersive “ moving” Cooper‐pair solution for superconductors in both weak and strong coupling.