Bernd J. Schroers

We study the spectrum of Laplace operators on a one-parameter family of gravitational instantons of bi-axial Bianchi IX type coupled to an abelian connection with self-dual curvature. The family of geometries includes the Taub-NUT (TN), Taub-bolt and Euclidean Schwarzschild geometries and interpolates between them. The interpolating geometries have conical singularities along a submanifold of co-dimension two, but we prove that the associated Laplace operators have natural selfadjoint extensions and study their spectra. In particular, we determine the essential spectrum and prove that its complement, the discrete spectrum, is infinite. We compute some of these eigenvalues numerically and compare the numerical results with an analytical approximation derived from the asymptotic TN form of each of the geometries in our family.

The uniform motion of chiral magnetic skyrmions induced by a spin-transfer torque displays an intricate dependence on the skyrmions' topological charge and shape. We reveal surprising patterns in this dependence through simulations of the Landau-Lifshitz-Gilbert equation with Zhang-Li torque and explain them through a geometric analysis of Thiele's equation. Our results provide a universal geometrical understanding of the dependence of the skyrmion's velocity and Hall angle on the skyrmion's topological charge, shape, and orientation. The generality of our approach suggests the validity of our results for exchange-frustrated magnets, bubble materials, and other materials.

We obtain D_k ALF gravitational instantons by a gluing construction which captures, in a precise and explicit fashion, their interpretation as nonlinear superpositions of the moduli space of centred SU(2) monopoles, equipped with the Atiyah–Hitchin metric, and k copies of the Taub–NUT manifold. The construction proceeds from a finite set of points in euclidean space, reflection symmetric about the origin, and depends on an adiabatic parameter which is incorporated into the geometry as a fifth dimension. Using a formulation in terms of hyperKähler triples on manifolds with boundaries, we show that the constituent Atiyah–Hitchin and Taub–NUT geometries arise as boundary components of the five-dimensional geometry as the adiabatic parameter is taken to zero.

We present a novel approach to understanding the extraordinary diversity of magnetic skyrmion solutions. Our approach combines an original classification scheme with efficient analytical and numerical methods. We introduce the concept of chiral kinks to account for regions of disfavored chirality in spin textures, and classify two-dimensional magnetic skyrmions in terms of closed domain walls carrying such chiral kinks. In particular, we show that the topological charge of magnetic skyrmions can be expressed in terms of the constituent closed domain walls and chiral kinks. Guided by our classification scheme, we propose a method for creating hitherto unknown magnetic skyrmions which involves initial spin configurations formulated in terms of holomorphic functions and subsequent numerical energy minimization. We numerically study the stability of the resulting magnetic skyrmions for a range of external fields and anisotropy parameters, and provide quantitative estimates of the stability range for the whole variety of skyrmions with kinks. We show that the parameters limiting this range can be well described in terms of the relative energies of particular skyrmion solutions and isolated stripes with and without chiral kinks.

We give a succinct summary of the recently discovered solvable models of magnetic skyrmions in two dimensions, and of their general solutions. The models contain the standard Heisenberg term, the most general translation invariant Dzyaloshinskii-Moriya (DM) interaction term and, for each DM term, a particular combination of anisotropy and Zeeman potentials. We argue that simple mathematical features of the explicit solutions help understand general qualitative properties of magnetic skyrmion configurations in more generic models.

We introduce a family of models for magnetic skyrmions in the plane for which infinitely many solutions can be given explicitly. The energy defining the models is bounded below by a linear combination of degree and total vortex strength, and the configurations attaining the bound satisfy a first order Bogomol'nyi equation. We give explicit solutions which depend on an arbitrary holomorphic function. The simplest solutions are the basic Bloch and N\'eel skyrmions, but we also exhibit distorted and rotated single skyrmions as well as line defects, and configurations consisting of skyrmions and anti-skyrmions.

We introduce a family of models for magnetic skyrmions in the plane for which infinitely many solutions can be given explicitly. The energy defining the models is bounded below by a linear combination of degree and total vortex strength, and the configurations attaining the bound satisfy a first order Bogomol’nyi equation. We give explicit solutions which depend on an arbitrary holomorphic function. The simplest solutions are the basic Bloch and Néel skyrmions, but we also exhibit distorted skyrmions and anti-skyrmions as well as line defects and configurations consisting of skyrmions and anti-skyrmions.