Coulomb repulsion-induced hyperbolic pairing as a mechanism of high-$T_c$ superconductivity

The hyperbolic metric of the dispersion law (the effective mass tensor components of carriers are opposite in sign) in the vicinity of the Fermi contour in high-$T_c$ superconducting cuprates in the case of repulsive interaction gives rise to a superconducting state characterized by the condensate of pairs with a large total momentum (hyperbolic pairing). The gain in the energy of the superconducting state over the normal state is due to the fact that a change in the kinetic energy of pairs (because of the negative light component of the effective mass) dominates over the change in the potential energy (corresponding to energy loss). The shift of the chemical potential upon the transition to the superconducting phase is substantial in this case. With increasing repulsive interaction, the superconducting gap $\Delta_K$ increases and the resulting gain in energy changes to an energy loss at a certain critical value of the repulsive potential. The low temperature $T_c$ of the superconducting transition and the large value of $\Delta_K$ in this region of potential values are the reasons for the high value of the $2\Delta/T_c$ ratio and for the developed quantum fluctuations that are observed in underdoped cuprate superconductors.