= Solution
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 $m=2\ne0$ and cannot be continuously replaced by a uniform nonvanishing planar texture while the surrounding loop is held fixed:
$$
\boxed{\text{Topologically nontrivial in a strictly planar nematic.}}
$$
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 $F_{\rm el}=(K\lambda_0^2/2)\int|\nabla\chi|^2$, 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:
$$
\Delta F_{\rm split}-\Delta F_{q=1}=-\frac{\pi K\lambda_0^2}{2}\log(d/r_0)+\text{finite core-energy difference}.
$$
The logarithmic gain eventually dominates the finite core cost. Hence
$$
\boxed{\text{Energetically favorable to split into two }+\tfrac12\text{ defects.}}
$$
This 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.
Back to article page
