We study Heisenberg model of classical spins lying on the toroidal support, whose internal and external radii are r and R, respectively. The isotropic regime is characterized by a fractional soliton solution. Whenever the torus size is very large, R → ∞, its charge equals unity and the soliton effectively lies on an infinite cylinder. However, for R = 0 the spherical geometry is recovered and we obtain that configuration and energy of a soliton lying on a sphere. Vortex-like configurations are also supported: in a ring torus (R > r) such excitations present no core where energy could blow up. At the limit R → ∞ we are effectively describing it on an infinite cylinder, where the spins appear to be practically parallel to each other, yielding no net energy. On the other hand, in a horn torus (R = r) a singular core takes place, while for R < r (spindle torus) two such singularities appear. If R is further diminished until vanish we recover vortex configuration on a sphere.

The energetics associated to the ferromagnetic, vortex, and onionlike magnetization configurations are explicitly computed in the toroidal geometry. The analysis reveals that the vortex appears to be the most prominent of such states, minimizing total energy in every torus with internal radius r≳10 nm, or even in smaller ones provided that R/ℓex≳1.5 (R is the torus external radius and ℓex is the exchange length). This possibility of having very small nanomagnets comprising a vortex-type state, might have importance in higher density binary logic and/or storage and in novel mechanisms for cancer therapy applications.

Understanding the domain wall dynamics is an important issue in modern magnetism. Here we present results of domain wall displacement in curved cylindrical nanowires at a constant magnetic field. We show that the average velocity of a transverse domain wall increases with curvature. Contrary to what it is observed in stripes, in a curved wire the transverse domain wall oscillates along and rotates around the nanowire with the same frequency. These results open the possibility of new oscillation-based applications.

We study the nonlinear σ-model in an external magnetic field applied on curved surfaces with rotational symmetry. The Euler-Lagrange equations derived from the Hamiltonian yield the double sine-Gordon equation (DSG) provided the magnetic field is tuned with the curvature of the surface. A 2π skyrmion appears like a solution for this model and surface deformations are predicted at the sector where the spins point in the opposite direction to the magnetic field. We also study some specific examples by applying the model on three rotationally symmetric surfaces: the cylinder, the catenoid and the hyperboloid.

We study the Heisenberg Model on cylindrically symmetric curved surfaces. Two kinds of excitations are considered. The first is given by the isotropic regime, yielding the sine-Gordon equation and π-solitons are predicted. The second one is given by the XY model, leading to a vortex turning around the surface. Helical states are also considered, however, topological arguments can not be used to ensure its stability. The energy and the anisotropy parameter which stabilizes the vortex state are explicitly calculated for two surfaces: catenoid and hyperboloid. The results show that the anisotropy and the vortex energy depends on the underlying geometry.

We show that the curvature of nanomagnets can be used to induce chiral textures in the magnetization field. Among the phenomena related to the interplay between geometry and magnetic behavior at nanomagnets, an effective curvature-induced chiral interaction has been recently predicted. In this work, it is shown that a magnetization configuration consisting of two structures with opposite winding numbers (vortex and antivortex) appear as remanent states in hollow toroidal nanomagnets. It is shown that these topological configurations are a result of a chiral interaction induced by curvature. In this way, the obtained results present a new form to produce stable vortices and antivortices by using nanomagnets with variable curvature.

A discussion on the interaction between skyrmions in a bi-layer system connected by a non-magnetic metal is presented. From considering a free charge carrier model, we have shown that the Ruderman-Kittel-Kasuya-Yosida (RKKY ) interaction can induce attractive or repulsive interaction between the skyrmions depending on the spacer thickness. We have also shown that due to an increasing in RKKY energy when the skyrmions are far from each other, their widths are diminished. Finally, we have obtained analytical solutions to the skyrmion position when the in-plane distance between the skyrmions is small and it is shown that an attractive RKKY interaction yields a skyrmion precessory motion. This RKKY-induced coupling could be used as a skyrmion drag mechanism to displace skyrmions in multilayers.

During the last years, topologically protected collective modes of the magnetization have called much attention. Among these, skyrmions and merons have been the object of intense study. In particular, topological skyrmions are objects with an integer skyrmion number Q while merons have a half-integer skyrmion charge q.In this work, we consider a Q = 1 skyrmion, composed by a meron and an antimeron (bimeron), displacing in a ferromagnetic racetrack, disputing a long-distance competition with its more famous counterpart, the typical Q = 1 cylindrically symmetrical skyrmion. Both types of topological structures induce a Magnus force and then are subject to the Hall effect. The influence of the Dzyaloshinskii-Moriya interaction (DMI) present in certain materials and able to induces DMI-skyrmions is also analyzed. Our main aim is to compare the motions (induced by a spin-polarized current) of these objects along with their own specific racetracks. We also investigate some favorable factors which are able to give breath to the competitors, impelling them to remain in the race for longer distances before their annihilation at the racetrack lateral border. An interesting result is that the DMI-skyrmion loses this hypothetical race due to its larger rigidity.

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