A superconductor with 4-fermion attraction perturbed by magnetic impurities

TL;DR

This paper investigates a superconductor with 4-fermion attraction modified by magnetic impurities, deriving bounds on free energy, solving mean-field equations, and mapping phase diagrams with five distinct phases.

Contribution

It introduces a model combining 4-fermion attraction with magnetic impurity interactions, deriving bounds, and analyzing phase transitions in this complex system.

Findings

Identification of five distinct phases including gapless and impurity phases.

Derivation of bounds on free energy density that converge in the thermodynamic limit.

Solution of mean-field equations revealing phase diagram structure.

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.