XY-model on planar and space curves

Abstract

We study the XY-model on a planar curve with a segment with constant curvature κ 0 and a space curve with a segment with both constant curvature κ 0 and torsion τ 0 . In the first case the bent segment breaks the rotational invariance of the XY-model and thus we get a fractional static sine-Gordon soliton interpolating between the two states θ 1 and θ 2 . In the second case the helical segment breaks the helicity of the model and thus creating a ground state and a metastable state spin configuration with a fractional static soliton. For sufficiently large τ 0 the static soliton solution can be more stable than the trivial ( θ = n π / 2 ) solution. The curvature introduces nonlinearity in the problem thereby localizing the energy in the region with nonzero curvature.