Singular lines and singular points of ferromagnetic spin systems and of nematic liquid crystals

The spins in a ferromagnet define a vector field. Their simplest line singularities are therefore disclinations of unit strength, and their point singularities are of the types analysed by Poincaré. A nematic liquid crystal is normally unpolarised. As a result, a vector placed along the nematic axis at a point may be carried continuously through « good » liquid crystal and return inverted. The simplest line singularities are therefore disclinations of half unit strength. A double circuit restores the vector to its original orientation. The director field thus has the character of the wave function of a spinor. The extension of this idea to three dimensions allows a preliminary description of the point singularities of nematic structures.