On the Bose-Einstein condensation of nonzero-momentum cooper pairs. A second-order phase transition

Abstract

Based on the BCS Hamiltonian, the normal-to-super phase transition is investigated, approaching the critical temperatureT c from the high-temperature side. Nonzero-momentum Cooper pairs, that is, pairs of electrons (holes) with antiparallel spins and nearly opposite momenta aboveT c in the bulk limit, are shown to move like independent bosons with the energye vs. momentump relatione=1/2vF ν, whereν F represents the Fermi velocity (1/2m*ν F 2 ≡e F≡Fermi energy). The system of free Cooper pairs undergoes a phase transition of the second order with the critical temperatureT c given byk B T c=1/2(π2ħ3 ν F 3 n/1.20257)1/3 wheren is the number density of Cooper pairs. The ratio of the jump of the heat capacity, ΔC, to the maximum heat capacity,C s, is a universal constant: ΔC/C s=0.60874; this number is close to the universal constant 0.588 obtained by the finite-temperature BCS theory. The physical significance of these results is discussed, referring to the well-known BCS theory, which treats the many-Cooper-pair ground state exactly and the thermodynamic state belowT c approximately. An explanation is proposed on the question why sodium should remain normal down to 0 K, based on the band structures with the hypothesis that the supercondensate composed of zero-momentum “electron” and “hole” Cooper pairs is electrically neutral.