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Ginzburg-Landau minimizers with high topological degrees in an annulus

Amandine Aftalion, Rémy Rodiac

TL;DR

The article establishes a sharp dichotomy for Ginzburg-Landau minimizers in an annulus with rapidly increasing outer boundary degree: below the critical scale $d_{oldsymbol{ ext{ε}}} hicksim rac{| ext{ln}oldsymbol{ ext{ε}}|}{2 ext{ln}(R_2/R_1)}$ the minimizer forms a giant vortex, while above this scale vortices proliferate and concentrate near the outer boundary as ${oldsymbol{ ext{ε}}} o 0$. The authors develop two complementary upper bounds (giant vortex and boundary-boundary circle of vortices) and a matching lower-bound analysis by extending the domain and reducing to a 1D mean-field problem $F^a(j)$. They use the vortex-balls construction, Jacobian estimates, and symmetry reduction to rigorously compute the limiting current and vorticity measures, proving that in the high-degree regime the boundary circle hosts most vortices while the interior remains giant-vortex-dominated. The results illuminate how geometry and large boundary degree drive vortex organization, with potential implications for fermionic rings and related quantum fluids. The work combines refined variational techniques with careful boundary extensions to capture boundary-vortex concentration and provides precise energy scalings and limiting profiles.

Abstract

Motivated by recent experiments on fermionic rings, we study the asymptotic behaviour of minimizers of the Ginzburg-Landau (GL) energy in an annulus with a Dirichlet data which depends on the GL parameter on the outer boundary. We show that there is a critical degree of order $|\ln \varepsilon|$ under which the ground state displays a giant vortex and above which minimizers exhibit a combination of a giant vortex and vortices which tend to the outer boundary as the GL parameter tends to zero. Our analysis relies on the construction of suitable upper and lower bounds, on the extension to a slightly bigger annulus and on the minimization of the mean-field energy appearing in the lower bound. In order to be able to derive the minimum of this energy we use the symmetry of the domain and criticality with respect to inner variations.

Ginzburg-Landau minimizers with high topological degrees in an annulus

TL;DR

The article establishes a sharp dichotomy for Ginzburg-Landau minimizers in an annulus with rapidly increasing outer boundary degree: below the critical scale the minimizer forms a giant vortex, while above this scale vortices proliferate and concentrate near the outer boundary as . The authors develop two complementary upper bounds (giant vortex and boundary-boundary circle of vortices) and a matching lower-bound analysis by extending the domain and reducing to a 1D mean-field problem . They use the vortex-balls construction, Jacobian estimates, and symmetry reduction to rigorously compute the limiting current and vorticity measures, proving that in the high-degree regime the boundary circle hosts most vortices while the interior remains giant-vortex-dominated. The results illuminate how geometry and large boundary degree drive vortex organization, with potential implications for fermionic rings and related quantum fluids. The work combines refined variational techniques with careful boundary extensions to capture boundary-vortex concentration and provides precise energy scalings and limiting profiles.

Abstract

Motivated by recent experiments on fermionic rings, we study the asymptotic behaviour of minimizers of the Ginzburg-Landau (GL) energy in an annulus with a Dirichlet data which depends on the GL parameter on the outer boundary. We show that there is a critical degree of order under which the ground state displays a giant vortex and above which minimizers exhibit a combination of a giant vortex and vortices which tend to the outer boundary as the GL parameter tends to zero. Our analysis relies on the construction of suitable upper and lower bounds, on the extension to a slightly bigger annulus and on the minimization of the mean-field energy appearing in the lower bound. In order to be able to derive the minimum of this energy we use the symmetry of the domain and criticality with respect to inner variations.

Paper Structure

This paper contains 14 sections, 25 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Main ideas
  5. Notations
  6. Acknowledgements:
  7. Upper bounds
  8. Extending to apply the vortex-balls construction
  9. The case $d_{\varepsilon}$ of order $|\ln {\varepsilon}|$: proof of Theorem \ref{['th:main1']}
  10. Lower bound of the energy
  11. Minimization of the function $F$
  12. Proof of the main results
  13. The case $d_{\varepsilon} < \alpha_c|\ln {\varepsilon}|$: proof of Theorem \ref{['th:uniqueness_d_small']}
  14. The case $d_{\varepsilon} \gg |\ln {\varepsilon}|$: proof of Theorem \ref{['th:main_0_case_d_big']}

Key Result

Theorem 1.1

Let $(u_{\varepsilon})_{\varepsilon}$ be a family of minimizers of $E_{\varepsilon}$ in ${{\mathcal{I}}}_{\varepsilon}$. Let We recall that $\text{\boldmath $e_\theta$}=(-\sin \theta,\cos \theta)$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • proof
  • Proposition 2.4
  • ...and 33 more