Functional Integral Methods in the Study of Critical Fluctuations in Superconductors .

Klaus D. Usadel

Lorenz Bartosch

一 高山

Alexei M. Tsvelik, Fabian H. L. Eßler

We consider the single particle spectral function for a two-dimensional clean superconductor in a regime of strong critical thermal phase fluctuations. In the limit where the maximum of the superconducting gap is much smaller than the Fermi energy we obtain an exact expression for the spectral function integrated over the momentum component perpendicular to the Fermi surface.

Kyozi Kawasaki, T. Tsuzuki

R. F. Hassing, John W. Wilkins

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 $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, 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\ifmmode \check{c}\else \v{c}\fi{}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 $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. 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 ${T}_{c}$, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. 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\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.