Sarma-Bogomol'nyi equations in superconductivity

TL;DR

This paper compares two methods for deriving first-order equations in superconductivity models, highlighting Sarma's approach as more general and potentially revealing new solutions beyond topological defects.

Contribution

It clarifies the relationship between Sarma's and Bogomol'nyi's methods, showing Sarma's approach does not require topological defect assumptions, broadening the scope of solutions.

Findings

Both methods rely on the same operator identity.

Sarma's method does not assume the presence of topological defects.

Bogomol'nyi equations may lead to a wider class of solutions.

Abstract

Topological defects occurring in nonlinear classical field theories are described by a system of second-order differential equations. A breakthrough was made in 1976 by E. B. Bogomoln'yi who demonstrated that in several field theories these equations can be reduced to first-order provided the coupling constants take on particular values. One of the examples involved a string in the Abelian Higgs model which is equivalent to the Abrikosov flux line of the Ginzburg-Landau theory of superconductivity. In a similar vein, in the 1966 textbook Superconductivity of Metals and Alloys P. G. de Gennes explained how to reduce the second-order Ginzburg-Landau equations to first-order at a particular value of the Ginzburg-Landau parameter by a method due to G. Sarma. We analyze the two ways of arriving at the first-order Sarma-Bogomol'nyi equations and conclude that while they both rely on the same operator identity, Sarma's method is free of the assumption that there is a topological defect. The implication is that Bogomol'nyi equations found in other field theories may be a source of a wider range of solutions beyond topological defects.