Nematic director ($\mathbf{n}\sim -\mathbf{n}$)

A nematic director is a unit vector specifying the local unoriented axis of a nematic liquid crystal. Head-tail symmetry identifies $\mathbf{n}$ with $-\mathbf{n}$, so the order-parameter space is a Real projective space rather than a sphere.

A nematic disclination is a topological line defect of the unoriented director field. In a two-dimensional sample its point cross-section can have half-integer winding because $\mathbf{n}$ and $-\mathbf{n}$ represent the same state.

For a continuous director angle around a closed counterclockwise circuit, the enclosed topological charge is $q=\Delta \theta /(2\pi )$. Since $\theta$ is defined modulo $\pi$, elementary nematic disclinations can have $q=\pm 1/2$.

A two-dimensional director winding can sometimes unwind when the director may tilt out of the plane. For a three-dimensional nematic the fundamental group of the order-parameter space is ${\pi }_1({\mathbb{RP}}^2)=\mathbb{Z}/2\mathbb{Z}$, so two disclination lines are topologically equivalent to none.

In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to $F=(K/2)\int |\nabla \theta {|}^2$ up to the convention-dependent effective modulus.