Rigorous study of the gap equation for an inhomogeneous superconducting state nearTc

A rigorous analytic study of the self-consistent gap equation (symobolically $\Delta ={\mathcal{F}}_T\left[\Delta \right]$), for an inhomogeneous superconducting state, is presented in the Bogoliubov formulation. The gap function $\Delta \left(\overrightarrow{\mathrm{r}}\right)$ is taken to simulate a planar normal-superconducting phase boundary: $\Delta \left(\overrightarrow{\mathrm{r}}\right)={\Delta }_{\infty }tanh\left(\frac{\alpha {\Delta }_{\infty }z}{v_F}\right)\ominus \mathrm{}\left(z\right)\mathrm{}$, where ${\Delta }_{\infty }\mathrm{}\left(T\right)\mathrm{}$ is the equilibrium gap, $v_F$ is the Fermi velocity, and $\ominus \mathrm{}\left(z\right)\mathrm{}$ is a unit step function. First a special space integral of the gap equation $\propto \int {0+}^{\infty }({\mathcal{F}}_T-\Delta )\left(\frac{d\Delta }{\mathrm{dz}}\right)dz$ is evaluated essentially exactly, except for a nonperturbative WKBJ approximation used in solving the Bogoliubov-de Gennes equations. It is then expanded near the transition temperature $T_c$ in power of ${\Delta }_{\infty }\propto {(1-\frac{T}{T_c})}^{\frac{1}{2}}$, demonstrating an exact cancellation of a subseries of "anomalous-order" terms. The leading surviving term is found to agree in order, but not in magnitude, with the Ginzburg-Landau-Gor'kov (GLG) approximation. The discrepancy is found to be linked to the slope discontinuity in our chosen $\Delta$. A contour-integral technique in a complex-energy plane is then devised to evaluate the local value of ${\mathcal{F}}_T-\Delta$ exactly. Our result reveals that near $T_c$ this method can reproduce the GLG result essentially everywhere, except within a BCS coherence length [not $\xi \mathrm{}\left(T\right)\mathrm{}$!] from a singularity in $\Delta$, where ${\mathcal{F}}_T-\Delta$ can have a singular contribution with an "anomalous" local magnitude, not expected from the GLG approach. This anomalous term precisely accounts for the discrepancy found in the special integral of the gap equation as mentioned above, and likely explains the ultimate origin of the anomalous terms found in the free energy of an isolated vortex line by Cleary.