Topological solitons and geometrical frustration

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Abstract
We study classical Heisenberg spins coupled by an isotropic or an anisotropic spin-spin interaction on an infinite elastic cylinder. In the continuum limit, the Hamiltonian of the system is given by a nonlinear σ model. We investigate the cylindrically symmetric solutions of the sine-Gordon equation (the Euler-Lagrange equation for this Hamiltonian). The periodic solution as well as the anisotropic one-soliton solution do not satisfy the self-dual equations of Bogomol’nyi [Sov. J. Nucl. Phys. 24, 449 (1976)] which are a necessary condition to reach the minimum energy configuration in each homotopy class. This generates geometrical frustration and produces a geometric effect: a shrinking of the cylinder coupled with nontrivial spin distributions.