Critical Fluctuations in Superconductors: A Functional-Integral Approach

Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form ∫Dψe^-βF[ψ], where F is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient b of the fourth-order term in F, however, shows that b is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar {c} {c}{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices γ=2, ν=1. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for $T>0$ and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at Tc, with critical index α=-1. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in F. As a result we conjecture that the Hartree approximation of Mar {c} {c}{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar {c} {c}{}elja.