Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape

Abstract

We study a nonlocal Allen-Cahn type problem for vector fields of unit length, arising from a model for domain walls (called Néel walls) in ferromagnetism. We show that the nonlocal term gives rise to new features in the energy landscape; in particular, we prove existence of energy minimisers with prescribed winding number that would be prohibited in a local model.

Separation of domain walls with nonlocal interaction and their renormalised energy by $Γ$-convergence in thin ferromagnetic films

“…Some of these observations are obviously consistent with our results [19,20] about the existence/nonexistence of minimisers of the functional E 1 for a prescribed winding number. In particular, for winding numbers giving rise to the situation of statement 4, we find that E 1 has no minimiser.…”

“…This is because otherwise, the attraction of next-to-neighbouring walls may dominate the (comparatively small) repulsion of neighbouring walls. This is a phenomenon that was exploited in other papers of the authors [19,20] (in a somewhat different model with fixed ) to construct energy minimisers comprising several Néel walls. For the problem studied here, a plausible consequence (not proved here) is that for small transition angles, several Néel walls may in the limit 0 collapse to a single composite Néel wall, corresponding to a jump of more than 2π in the lifting (i.e., if α ∈ (0, π) is small, under the condition (8), we may have a jump of size |ω n | ≥ 2π).…”

“…Topological Néel walls are expected when we minimise E subject to the above boundary data (which amounts to prescribing the topological degree). Indeed, it can be shown, with arguments similar to a related model [19,Lemma 3.1], that there are exactly as many Néel walls for minimisers as the topology requires and thus their signs necessarily alternate. When passing to the limit, we expect that the positions of the Néel walls converge to a minimiser of W ( • , d ± N ) if such a minimiser exists.…”

“…where Ḣ1/2 (R) is the homogeneous Sobolev space of fractional order 1/2 and we write x 1 for the independent variable for reasons that will become clear later. For details about the background of this model, and how it is derived from the full micromagnetic energy, we refer to previous work [8,9,30,31,7,21,16,5,29,32,18,19,20]. We note, however, that the first term on the right hand side in (1) is called exchange energy and the second term is called stray field energy or magnetostatic energy in the theory of micromagnetics.…”

“…Indeed, in many of the proofs in Section 3, we have to consider u * a,d and a limiting anisotropy energy jointly. This is somewhat surprising, as a variant of the Pohozaev identity [19,Proposition 1.1] implies that for critical points of E , the exchange energy and the anisotropy energy are of the same magnitude, while the stray field energy is larger by a factor of order | log |.…”