Statics and dynamics of magnetic vortices and of Nielsen–Olesen (Nambu) strings

Abstract

We review recent works on statics and dynamics of magnetic vortices in the Ginzburg–Landau model of superconductivity and of Nielsen–Olesen (Nambu) strings in the Abelian–Higgs model of particle physics.

Non-topological Vortex Configurations in the ABJM Model

“…To this purpose, in [34] the authors derived self-dual equations of BPS-type [8,56] for the massdeformed ABJM model with gauge group U (N ) × U (N ). When N = 2 such BPS equations reduce to the Abelian Higgs equation [22,30,50] whose existence theory is completely understood [30]. While for N > 2, the BPS equations give rise to more general systems of the type describing non-Abelian vortex configurations, for which existence and uniqueness results were established recently in [27].…”

Multi-Vortex Non-radial Solutions to the Magnetic Ginzburg-Landau Equations

“…In addition, Gustafson, Sigal and Tzaneteas [14] suggest that for magnetic vortices, stationary multi-vortex configurations of degrees ±1 occur with discrete symmetry group. In this paper, we prove rigorously that this last conjecture is true.…”

“…where n is an integer, z ∈ R 2 and γ : R 2 → R. For reviews and books on the Ginzburg-Landau equations of superconductivity, one can refer to [3,5,7,14,16,22,23], for example.…”

“…Stationary states of superconductors occupying (for simplicity) the plane R 2 , are described by pairs (ψ, A), where ψ : R 2 → C is the order parameter and A : R 2 → R 2 is the magnetic potential. These states satisfy the system of equations −∆ A ψ + λ(|ψ| 2 The Ginzburg-Landau equations on the plane model superconductors in R 3 which are spatially homogeneous in one direction, when neglecting boundary effects [14]. They also describe equilibrium states of the U (1) Higgs model of particle physics [16].…”

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

“…1 The Ginzburg-Landau theory is reviewed in every book on superconductivity and most of the books on solid state or condensed matter physics. For reviews of rigorous results see the papers [11,12,20,29] and the books [28,17,21,27] 2 In the problem we consider here it is appropriate to deal with Helmholtz free energy at a fixed average magnetic field b := 1|Ω| Ω curl A, where |Ω| is the area or volume of Ω.…”