Fermi-condensate quantum phase transition in high-$T_c$ superconductors

The effect of a quantum phase transition associated with the appearance of fermionic condensation in an electron liquid on the properties of superconductors is considered. It is shown that the electron system in both superconducting and normal states exhibits characteristic features of a quantum protectorate after the point of this Fermi-condensate quantum phase transition. The single-particle spectrum of a superconductor can be represented by two straight lines corresponding to two effective masses $M^*_{FC}$ and $M^*_{L}$. The $M^*_{FC}$ mass characterizes the spectrum up to the binding energy $E_0$, which is of the order of the superconducting gap in magnitude, and determines the spectrum at higher binding energies. Both effective masses are retained in the normal state; however, $E_0\simeq4\,$ T. These results are used to explain some remarkable properties of high-$T_c$ superconductors and are in good agreement with recent experimental data.