Schwarzian Derivative of Third Chaotic Transition in Mercury Based Superconductors

Abstract

The superconductivity displays some nonlinear quantum properties such as quantum chaotic transitions, effective mass of quasi-particles. The calculation of the effective mass equation of the mercury based superconductor has a crucial step in order to determine chaotic behavior of the sample. The sample has three quantum chaotic points; the critical transition (Tc), paramagnetic Meissner transition (TPME) and the quantum gravitational transition temperatures (TQG). Tc and TPME were already investigated by means of Schwarzian derivative and it was found that the Schwarzian derivatives are both negative at two chaotic points. In the study, the third quantum chaotic point of the system, TQG is investigated by the Schwarzian derivative of the effective mass of the sample. As is known, the object’s mass varies in a gravitational field. Hence, it is determined that the Schwarzian derivative of the effective mass, which has the negative value, shows the quantum chaotic transition in the system. At the vicinity of TQG where the first derivative of the net effective mass has a maximum value, the plasma frequency shifts from microwave to infrared. As a result, it is proposed that the Schwarzian derivative is a convenient mathematical method for precise prediction of chaotic points and transitions in superconducting systems.