Vortex patterns beyond hypergeometric
type: Item
instance of: scholarly article
title: Vortex patterns beyond hypergeometric
zbMATH Open document ID: 1267.82142
published in: Journal of Geometry and Symmetry in Physics
publication date: 2013-01-28T00:00:00Z
review text: The paper studies the existence of confined vortex loops in a superconducting infinite space, where the magnetic field is generated by a point-like magnetic dipole placed at the origin. First, a theoretical formalism based on the functional of free-energy in the Ginzburg-Landau (GL) theory is introduced. In the approximation of the weak magnetic field, the Euler-Lagrange PDEs arising from the functional reduce to a magnetic Schrödinger equation. The exact solutions are obtained in the dipole coordinates \((a, b)\) using dimensionless variables. In order to obtain a separation of the dipole variables, the solutions are studied for \(a >> b\), which describes points close to the \(z\)-axis. The gradient of the order parameter function is considered to be directed mainly radially and orthogonally to the magnetic dipole field lines. In this case, the coordinate surfaces \(a = \mathrm{const}\) describe the vortex surfaces. The exact solution of the dipole equation is obtained in the form of Heun functions and is sufficient to prove the occurrence of spontaneous vortex phases with a mutual interconnection of vortices at the origin. The analytic solutions of the linearized dipole equation are investigated by mapping it to a double confluent Heun equation and then taking linear combinations of two solutions of the dipole equation. In order to generate physical solutions for the full nonlinear GL problem, the nonlinear dipole equation is introduced in the Gibbs free energy equation and this integral is minimized in the space of unknown parameters. The author proves that it is enough to investigate order parameters constructed using only two linear solutions. It is proved that multi-vortex states are possible even without the presence of an external field and that they are characterized by confined vortex loops. For the minimization of the Gibbs free energy functional, the Hessian matrix method is used. Finally, the symmetries of the confined phase starting from solutions of …
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35Q56
zbMATH DE Number: 6131403
zbMATH Keywords: Ginzburg-Landau theory
zbMATH Keywords: Gibbs free energy functional
zbMATH Keywords: dipole equation
zbMATH Keywords: Heun functions
zbMATH Keywords: vortex loops
author: Andrei Ludu
reviewed by: Ivan A. Parinov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6131403