Ginzburg–Landau theory of superconductivity at fractal dimensions
Аннотация
The post-Gaussian effective potential in $D=2+2\ensuremath{\epsilon}$ dimensions is evaluated for the Ginzburg--Landau theory of superconductivity. Two- and three-loop integrals for the post-Gaussian correction terms in $D=2+2\ensuremath{\epsilon}$ dimensions are calculated and $\ensuremath{\epsilon}$ expansions for these integrals are constructed. In $D=2+2\ensuremath{\epsilon}$ fractal dimensions the Ginzburg--Landau parameter turned out to be sensitive to $\ensuremath{\epsilon}$ and the contribution of the post-Gaussian term is larger than that for $D=3$. Adjusting $\ensuremath{\epsilon}$ to the recent experimental data on $\ensuremath{\kappa}(T)$ for the high-${T}_{c}$ cuprate superconductor ${\mathrm{Tl}}_{2}{\mathrm{Ca}}_{2}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{10}$, we found that $\ensuremath{\epsilon}=0.21$ is the best choice for this material. The result clearly shows that, in order to understand high-${T}_{c}$ superconductivity, it is necessary to include the fluctuation contribution as well as the contribution from the dimensionality of the sample. The method gives a theoretical tool to estimate the effective dimensionality of samples.