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 $\int D\psi e^{-\beta F\left[\psi \right]}$, 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 Marčelja 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 $\gamma =2\mathrm{}$, $\nu =1\mathrm{}$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for $T>\mathrm{}0\mathrm{}$ 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 $T_c$, with critical index $\alpha =-1\mathrm{}$. 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 Marčelja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Marčelja.