Foundations of Physics

, Volume 46, Issue 8, pp 915–942 | Cite as

Bose–Einstein Condensation of Nonideal Cooper Pairs in the Hartree–Fock–Popov Theory



The Hartree–Fock–Popov theory of interacting Bose particles is generalized to the Cooper-pair system with a screened Coulomb repulsive interaction in high-temperature superconductors. At zero temperature, it is found that the condensate density \(n_c(0)\) of Cooper pairs is of the order \(n_c(0)\simeq 10^{18}\) cm\(^{-3}\), consistently with the fact that a small fraction of the total p holes participate in pairing. We find that the phonon velocity c(0) at zero temperature is of the order \(c(0)\simeq 10\) km s\(^{-1}\). The computation shows that the transition temperature \(T_c\) is a dome-shaped function of the p hole concentration \(\delta \), which is consistent with experiments. At finite temperature, we find that the condensate fraction \(n_c(T)/n\) decreases continuously from \(n_c(0)/n\) to zero as the temperature increases from zero to the transition temperature \(T_c\). For higher temperatures, we find that the repulsive interaction between Cooper pairs drives more Cooper pairs into the condensate. The computation reveals that the phonon velocity c(T) decreases continuously from c(0) to zero as the temperature increases from zero to the transition temperature \(T_c\). The Cooper-pair system undergoes a first-order phase transition from the normal state to the BEC state.


Boson condensation Cooper pairing Hartree–Fock–Popov theory 



This work was supported by the National Natural Science Foundation of China under Grants Nos. 10174024 and 10474025.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of PhysicsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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