Moduli spaces and breather dynamics of analytic solutions in chiral magnets without Heisenberg exchange interaction

TL;DR

This work analyzes the restricted chiral-magnet model with vanishing Heisenberg exchange, recasting the Landau-Lifshitz dynamics as inviscid fluid flow and solving via characteristics. It uncovers an infinite-dimensional moduli space of static Skyrmions generated by contour-preserving maps and shows a string-like effective description of the moduli, where circles of constant $n_3$ form a nonintersecting cloud whose centers can slide freely. The study also constructs breather-like axisymmetric solutions, extends the construction to 3D with helix-like pathlines, and proves Hopfions do not exist in the static restricted model. The results provide a geometric, moduli-space perspective on Skyrmions in the absence of Heisenberg exchange, with implications for near-BPS dynamics and potential applications to higher-dimensional solitons and related first-order systems.

Abstract

We investigate the special case of the chiral magnet with vanishing Heisenberg exchange energy, whose axisymmetric Skyrmion solution has previously been found. The dynamical equations of this model look like inviscid fluid flow, and by investigating path-lines of this flow we can construct explicit static and dynamic solutions. We find an infinite-dimensional family of static Skyrmions that are related to the axisymmetric Skyrmion by co-ordinate transformations thus discovering a new moduli space, and further infinite-dimensional families of axisymmetric and non-axisymmetric breather-like supercompactons. We call the maps generating such moduli space "contour-preserving" maps.

Figures (6)

• Figure 1: "Streamlines" of constant $n_3$ for positive and negative $R(n_3)$.
• Figure 2: Axisymmetric static Skyrmion-like solution by assembling a series of concentric circles for positive $R(n_3)$.
• Figure 3: Pictorial illustration of the string-like model.
• Figure 4: Three solutions of restricted magnetic Skyrmions, with different non-trivial "string motion" in the case of the Zeeman potential $h=1$, $u=0$, giving rise to a supercompacton. The four rows correspond to the collection of circles and the fields $n_3$, $n_1$ and $n_2$, respectively. In this figure $k=1$.
• Figure 5: Constant $n_3$ contours at different points in time. Note that while the magnetization is always tangent to its path-line, for $ht-\Phi\neq n\pi$ it is not tangent to the circle of constant $n_3$, in contrast to the axisymmetric static solution.