A superconductor with 4-fermion attraction perturbed by magnetic impurities

Abstract

A superconductor with 4-fermion attraction, considered by Ma\'{c}kowiak and Tarasewicz is modified by adding to the Hamiltonian a long-range magnetic interaction $V$ between conduction fermions and localized distinguishable spin 1/2 magnetic impurities. $V$ has the form of a reduced s-d interaction. An upper and lower bound to the system's free energy density $f(H,\beta)$ is derived and the two bounds are shown to coalesce in the thermodynamic limit. The resulting mean-field equations for the gap $\Delta$ and a parameter $y$, characterizing the impurity subsystem are solved and the solution minimizing $f$ is found for various values of magnetic coupling constant $g$ and impurity concentration. The phase diagrams of the system are depicted with five distinct phases: the normal phase, unperturbed superconducting phase, perturbed superconducting phase with nonzero gap in the excitation spectrum, perturbed gapless superconducting phase and impurity phase with completely suppressed superconductivity.