Violation of Self-Duality for Topological Solitons due to Soliton-Soliton Interaction on a Cylindrical Geometry

Abstract

We study classical Heisenberg spins on an infinite elastic cylinder. In the continuum limit the Hamiltonian of the system is given by the nonlinear o. model. We investigate the periodic, cylindrically symmetric solution of the sine-Gordon equation (the Euler-Lagrange equation for this Hamiltonian).

Self-organization, condensation, and annihilation of topological vortices and antivortices in a multiferroic

“…In addition, beautiful optical patterns in liquid crystals such as the Schlieren texture of a nematic and the cholesteric fingerprint texture are due to assemblies of topological defects, called disclinations, in ordered arrangements of mesoscopic molecules (6,7). Large-scale spatial configurations of these topological defects have been investigated only in a limited degree (8)(9)(10)(11). Exceptions include the cases of supercurrent vortices or liquid crystals, but they tend to exhibit either trivial or rather irregular configurations.…”

Self-dual bending theory for vesicles

“…Conversely, when the shape tensor violates the local property of self-duality/anti-selfduality then the bending Hamiltonian is frustrated, i.e. an extra energy contribution is spontaneously created [12,[26][27][28][29] that tends to vanish globally in the system. Before following this matter any further, we turn our attention to the spherical vesicles of revolution which are of pertinent interest.…”

“…As a matter of fact, topology gives rise to new physics [24,25] with some of the following relevant features: (i) metastable configurations (mostly solitons) fall into distinct classes, of which the trivial configuration (vacuum) belongs only to one class; (ii) there exists no superposition principle, i.e. resultant configurations form complicated ones; (iii) frustration phenomena may arise from topological obstruction leading to both global and localized effects [26][27][28][29].…”

Topological magnetic solitons on a paraboloidal shell

“…However, if there are several length scales in the problem, a geometrical frustration is induced in the system and the skyrmion assume a width depending on the geometrical parameters of the surface. This phenomenon was first discussed in the context of an elastic cylinder in which periodic solitons (skyrmions) would produce deformations on the surface in order to minimize the magnetic energy [15]. Therefore, the paraboloid curvature stabilizes the skyrmion and change its width due to the two characteristic lengths associated with the paraboloid, that is, height and radius.…”

“…Thus, 2π-skyrmions are obtained from Eq. (15). The skyrmion width can not be defined from this general surface parametrization.…”