Scaling behavior of superconductors

In our brief review, we will consider the general universal scaling properties of superconductors. The physics of superconductors, represented by both conventional and unconventional superconductors, has been the main topic of high-$T_c$ superconductor physics for over thirty years, revealing some of the properties of high-$T_c$ (or unconventional) superconductors. Scaling relationships lead to the identification of fundamental laws of nature and reveal the essence of superconductor physics. Advances in experimental technology allow us to collect important data, which in turn allow us to make definitive statements about the physical processes underlying strongly correlated Fermi systems. Basing on this observation, we analyze experimental facts that reveal the general scaling properties of both high-$T_c$ and ordinary superconductors, and theoretically explain that the Homes' law $\rho_{s0}= (1/2\pi\lambda_D)^2= T_c\sigma(T_c)$ is applicable to the both types of superconductors. Here $\rho_{s0}$ is the superconducting electron density, $\lambda_D$ is the zero-$T$ penetration depth, $\sigma$ is the normal state conductivity, $T$ is temperature and $T_c$ is the temperature of superconducting phase transition. Overall, these scaling relationships lead to the identification of fundamental laws of nature and reveal the essence of superconductor physics. All these observations support the theory of fermion condensation. Our theoretical results agree well with a body of diverse and seemingly unrelated experimental facts. They show that the topological fermion condensation quantum phase transition, generating flat bands, is an intrinsic property of strongly correlated Fermi systems and can be considered as a universal agent explaining their basic physics.