XY-model on planar and space curves

We study the XY-model on a planar curve with a segment with constant curvature κ0 and a space curve with a segment with both constant curvature κ0 and torsion τ0. 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 θ1 and θ2. 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 τ0 the static soliton solution can be more stable than the trivial (θ=nπ/2) solution. The curvature introduces nonlinearity in the problem thereby localizing the energy in the region with nonzero curvature.