Compactness results for static and dynamic chiral skyrmions near the conformal limit

Abstract

We examine lower order perturbations of the harmonic map problem from R 2 to S 2 including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizers, and their asymptotic limit. Finally we examine dynamic stability and compactness of almost minimizers in the context of the Landau-Lifshitz-Gilbert equation including spin-transfer torques arising from the interaction with an external current.

Gauged sigma models and magnetic Skyrmions

“…where h = 1/f is, like f , an arbitrary holomorphic map C → CP 1 . The simplest choice h = 0 leads to the Bloch hedgehog skyrmion with Q = −1 and the Belavin-Polyakov profile function θ(r) = 2 arctan r 2κ , as already noticed in [7]. Many other solutions are discussed in [9], and our Fig.…”

“…As far we are aware it is not known for which class of magnetisation fields n the general energy functional (1) is well-defined and finite. For the standard DM term (n, ∇ × n) and a certain class of potentials V , this question is answered in [6] and [7], where it was also pointed that, for analytical reasons, it is preferable to modify the energy functional by adding the boundary term…”

“…to obtain a well-defined variational problem. Clearly, (13) reduces to (12) for the simple gauge field (7). Adding the term (13) to the energy (3) has a number of advantages, at least from an analytical point of view.…”

“…The DM term (5) and the integrand of the boundary term (13) combine neatly into 3 a=1 2 i=1 D ai (∂ i n × (n −Â)) a in this case. For the simple case (7) with DM term κ(n, ∇ × n) and potential V (n) = κ 2 2 (1 − n 3 ) 2 , one checks that the matrix (18) is…”

Magnetic Skyrmions at Critical Coupling

“…Acknowledgements B B-S and CR acknowledge EPSRC-funded PhD studentships. BJS thanks Christof Melcher for correspondence and for pointing out reference [18], and Paul Sutcliffe for very helpful comments regarding the total energy of line defects in our model.…”