## Vortex patterns beyond hypergeometric.(English)Zbl 1267.82142

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 the linearized Ginzburg-Landau equations are determined.

### MSC:

 82D55 Statistical mechanics of superconductors 35Q56 Ginzburg-Landau equations