Sarma-Bogomol'nyi equations in superconductivity

TL;DR

This work analyzes how the second-order Ginzburg-Landau equations for superconductivity can be reduced to first-order Sarma-Bogomol'nyi (SB) equations at the Bogomol'nyi point $\kappa=1/\sqrt{2}$. It compares Sarma's operator method, which derives SB relations directly from the GL equations without assuming topological defects, with Bogomol'nyi's energy-minimization approach, which relies on flux quantization. The authors show that Sarma's method is more general and that a combined Bogomol'nyi-Sarma procedure yields SB equations without cylindrical symmetry, broadening the scope of Bogomol'nyi-type reductions. These results imply that Bogomol'nyi equations may provide a wider class of solutions beyond topological defects and could be relevant to other field theories exhibiting similar structures.

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.