On geometry-dependent vortex stability and topological spin excitations on curved surfaces with cylindrical symmetry

Abstract

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.

Topologically stable magnetization states on a spherical shell: Curvature-stabilized skyrmions

“…This is a fundamental difference between our approach and a number of previous studies, where soliton solutions were found on curvilinear shells, yet anisotropy was either neglected, 49,[65][66][67] or it was spatially uniform lacking any correlation with the geometry. 50,68,69 Our approach is based on the fundamental behavior of magnetically ordered media, where spin-orbit couplings provide the vital link between nontrivial curved geometry and the spin-system. Therefore, any realistic assessment of possible magnetization states in curved geometries must include the geometrically allowed anisotropic couplings.…”

Topological magnetic solitons on a paraboloidal shell

“…If we insert a hole at the origin of the paraboloid, the winding number defined as Q S = (4π) −1 sin ΘdΘdΦ is given by Q S = 1/(e 2ζ + 1), obtaining fractional skyrmions, as well as other truncated non-simply-connected surfaces [33,37,38] these solutions do not have topological stability, once the spin sphere mapping is not completely done. Similar arguments are applied when we study the above model on a paraboloid limited by its maximum radius, r ∈ [0, R].…”

“…However, the energy of a vortex depends on Q 2 , such that solutions with Q > 1 are energetically unstable [38]. In this way, from now on, we consider only Q = 1, which can be viewed in Fig.…”

Berry phases and zero-modes in toroidal topological insulator

“…Examples include the integer quantum Hall effect, topological insulators [1], topological superconductors and topological superfluids [2]. On the other hand, geometrical features of the real and momentum spaces are very important to describe the physical properties of several condensed matter systems, like the nematic order in liquid crystals [3], graphene [4,5] and magnetic systems [6][7][8][9][10].…”