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Effects of critical fluctuations and dimensionality on the jump in specific heat at the superconducting transition temperature: Application to YBa2Cu_3 O_7 - δ, Bi_2 Sr_2CaCu_2 O_8+δand KOs_2 O_6 compounds

We report on a study of the superconducting order parameter thermodynamic fluctuations in $\mathrm{YBa}_2\mathrm{Cu}_3\mathrm{O}_{7 - δ}$, $\mathrm{Bi}_{2}\mathrm{Sr}_{2}\mathrm{CaCu}_{2}\mathrm{O}_{8+ δ}$ and $\mathrm{KOs}_2\mathrm{O}_6$ compounds. A non-perturbative technique within the framework of the renormalized Gaussian approach is proposed. The essential features are reported (analytically and numerically) through Ginzburg-Landau (GL) model-based calculations which take into account both the dimension and the microscopic parameters of the system. By presenting a self-consistent approach (SCA) improvement on the GL theory, a technique for obtaining corrections to the asymptotic critical behavior in terms of non universal parameters is developed. Therefore, corrections to the specific heat and the critical transition temperature for one-, two- and three-dimensional samples are found taking into account the fact that fluctuations occur at all length scales as the critical point of a system is approached. The GL model in the free-field approximation and the 3D-XY model are suitable for describing the weak and strong fluctuation regimes respectively. However, with a modified quadratic coefficient, the renormalized GL model is able to explain certain experimental observations including the specific heat of complicated systems, such as the cuprate superconductors and the $β$-pyrochlore oxides. It is clearly shown that the enhancement, suppression or rounding of the specific heat jump of high-$T_c$ cuprate superconductors at the transition are indicative of the order parameter thermodynamic fluctuations according to the dimension and the nature of interactions.