Introduction to superconductivity and self-duality as a cooperation mechanism to complexity emergence

Abstract

Initially we conduct a review of superconductivity and examine a variety of topics, includ- ing the Fermi-Landau theory, the generic Landau theory of phase transition with a focus on Ginzburg-Landau, the Fhrölich model, Bardeen-Cooper-Schrieffer, and Bogoliubov theories, as well as their relation to the coherent Glauber states. Next, we establish the connection between microscopic theories and GL, a result pioneered by Gor’kov, and recent developments in the Extended Ginzburg-Landau theory by A.Shanenko and A.Vagov et al. - a step beyond Gor’kov, providing a self-consistent expansion valid further away from the critical temperature. These results are reproduced by formulating an alternative time-saving method for computing higher-order Landau theories of superfluid phase transition (in the absence of the induction- field coupling). This is accomplished through the formulation of a diagrammatic dictionary and a concise collection of rules. The primary original contribution of this work, though, is the description of novel semi-analytic solutions to the self-dual superconducting solutions at the Bogomol’nyi point (κ = 1/√2) and their correspondence to the appearance of patterns similar to those in U.Krägeloh’s (1969) pioneering measurement in "Flux line lattices in the intermediate state of superconductors near κ = 1/√2". The semi-analytic solutions are coined stripe, bubble and donut. They exhibit stable thermodynamics beyond κ = 1/√2, in the ‘intertype’ domain, as we predict from the Extended Ginzburg Landau theory. We observe the results in the time-dependent Ginzburg-Landau model starting from configurations similar to the semi-analytic solutions as ab initio ansatz. The time-evolved solutions qualitatively co- incide with Krägeloh’s experimental results. The obtained results allow us to cast doubt on a widely accepted view of how complexity develops. We present a phenomenology in which ’cooperation’ rather than ’competition’ is the appropriate keyword for justifying the complexity emergence.