Bose–Einstein condensation and cuprate high-temperature superconductor

The rigorous formulations of boson (the ideal Bose gas) and fermion-pair Bose–Einstein (BE) condensations reveal that the two condensed states are different; the boson condensation given by the boson coherent state is a sound condensed state based on a large number of states corresponding to the grand canonical ensemble of the classical ideal gas (average particle number N) where the norm of the coherent state is equivalent to the grand (canonical) partition function $$\Xi _{0}=e^{N}$$ Ξ 0 = e N of the latter. The fermion-pair condensation is a very limited condensed state formed between holes and fermion-pairs and its condensate is a fermion-pair and hole condensate. The singlet-bond (SB) superconductivity theory for cuprate high-temperature superconductors finds the following; (1) the superconducting transition is a first-order transition, (2) the experimentally observed exponential behavior of the specific heat coefficient $$\gamma (T)\equiv C(T)/T$$ γ ( T ) ≡ C ( T ) / T near $$T_{c}$$ T c is caused by the high energy excitations of superconducting SB-pairs to the normal-state insulating immobile SB-pairs beyond the characteristic energy scale $$\sim k_{B}T_{c}$$ ∼ k B T c of the condensation energy, which is the same origin as that of the exponential $$\gamma (T)$$ γ ( T ) behavior in the BCS superconductivity, and (3) Josephson tunneling in d-wave superconductor Josephson junction cannot give rise to the so-called $$\pi $$ π -shift Josephson phase in both the underdoped and overdoped cuprate superconductor Josephson junctions. Graphical Abstract