Gamma-convergence and the emergence of vortices for Ginzburg–Landau on thin shells and manifolds

Abstract

We analyze the Ginzburg-Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in R 3 . We establish -convergence to a reduced Ginzburg-Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold M is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field H c1 is simply (4π/(area of M)) ln κ for large values of the Ginzburg-Landau parameter κ. Merging this with the -convergence result, we also obtain the same asymptotic value for H c1 in 3d valid for large κ and sufficiently thin shells.

Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films

“…Thus the minimizing matrix Q 2 is arbitrary as long as it satisfies the trace constraint (8). If, in addition, min{2c 2 + c 4 , c 3 + c 4 } > 0 the energy is still minimized by x given by (13) with c 2 = 0, i.e.,…”

“…In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. To achieve this goal we employ the tool of Γ-convergence that has proved successful in tackling problems of dimension reduction in other settings, such as elasticity [12] and Ginzburg-Landau theory [13]. We work in the domain Ω × (0, h) where 0 < h ≪ 1 and Ω ⊂ R 2 is bounded and Lipschitz.…”

The generalized point-vortex problem and rotating solutions to the Gross–Pitaevskii equation on surfaces of revolution

“…From (3.7), we have To obtain the convergence result as ε → 0, we adapt the vortex-ball construction by Jerrard [9] and Sandier [15] to the setting on M. Details of adjusting this technology to the setting of geodesic balls on a manifold can be found in Sec. 5 of [7]. Recall that for Ω ⊂ M with smooth boundary ∂Ω, the degree of a smooth function u : Ω → C around ∂Ω is defined by…”

Persistence of superconductivity in thin shells beyond Hc1

“…The results in [4] and [5] cover only a moderate regime; in these works the intensity of the applied field is H c1 + O(ln ln κ) and thus the number of vortices remains bounded as κ goes to infinity.…”