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

TL;DR

This paper investigates how the geometry of curved surfaces with cylindrical symmetry influences the stability and properties of topological spin excitations, such as vortices and solitons, in the Heisenberg and XY models.

Contribution

It provides explicit calculations of vortex energy and anisotropy parameters on catenoid and hyperboloid surfaces, revealing geometry-dependent effects on spin excitations.

Findings

Vortex energy depends on surface geometry.

Anisotropy stabilizing vortices varies with surface shape.

Sine-Gordon equation predicts $$-solitons in the isotropic regime.

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 $\pi$-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.