A unified picture of the Bose–Einstein condensation in Fermi and Bose systems

We show that all the Bose–Einstein (BE) condensations of Bose and Fermi systems can be formulated using the common ‘boson’ operator (valid below the BE condensation temperature $$T_{c}$$ T c ) with the coherent translational momentum $${{\varvec{q}}}=0$$ q = 0 (center-of-mass momentum in fermion system). The BE condensation is identified as the condensate of hopping bosons in Bose system and hopping fermion-pairs in Fermi system and the condensation energy is given by those average hopping energy times their condensate. We demonstrate that this scheme is actually working in all the BE condensations of strong-coupling fermion-pair system, weak-coupling Cooper-pair BCS system and the ideal Bose gas system. We find that the change of cuprate high- $$T_{c}$$ T c superconductivity from the underdoped to the overdoped regime is the doping-induced crossover from strong- to weak-coupling BE condensation arising from the same origin of the attractive superexchange interaction. We also study the BE condensation in the large attractive $$-U$$ - U Hubbard model and obtain a new BE-condensed ground state which is different from the one considered hitherto. We briefly comment on the more recently discovered BE condensations and their crossovers in the ultra-cold alkaline Fermi atom gases. Graphical abstract