Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 3 / d

For a strictly planar nematic director, the order-parameter manifold at fixed nonzero amplitude is ${\mathbb{RP}}^1$, homeomorphic to a circle. Its fundamental group is $\mathbb{Z}$. The topological charge of a two-dimensional nematic disclination is $q=\Delta \chi /(2\pi )$ for director angle $\chi$, defined modulo $\pi$; the integer winding of the physical doubled angle $2\chi$ is $m=2q$. Thus a $q=+1$ texture has $m=2\ne 0$ and cannot be continuously replaced by a uniform nonvanishing planar texture while the surrounding loop is held fixed:It can nevertheless split into two $q=+1/2$ defects, because their charges add to one. Such splitting preserves the total winding and is not topological unwinding. Allowing escape into the third dimension changes the manifold to ${\mathbb{RP}}^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$; the doubled winding of this integer defect is then trivial. That escape is excluded by the stated two-dimensional director model.

Energetically, the defect is not the preferred isolated charge-one configuration at large separation of scales. At constant amplitude, the planar nematic divergence-elasticity identity gives $F_{\mathrm{el}}=(K{\lambda }_0^2/2)\int |\nabla \chi {|}^2$, so a charge $q$ has self-energy $\pi K{\lambda }_0^2{q}^2\log (L/r_0)$. Separating two $+1/2$ defects by $d$, with $r_0\ll d\ll L$, makes the far field still charge one but reduces the near-field logarithm:The logarithmic gain eventually dominates the finite core cost. HenceThis is energetic fission of an integer nematic disclination. It proves failure of global energetic stability, not the absence of every microscopic kinetic barrier or a full linear stability spectrum of the radial core. Core structure, boundaries or pinning can make a charge-one texture metastable over a finite time or length scale.