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A real symmetric traceless $2$ by $2$ matrix has eigenvalues $\lambda$ and $-\lambda$ and can be writtenfor a unit nematic director $\mathbf{n}\sim -\mathbf{n}$. Since $Q_{ij}{Q}_{ji}=2{\lambda }^2$, the uniform Landau free energy isFor $a<0<b$, its nonzero minima satisfyand henceThe signs $\pm {\lambda }_0$ amount to exchanging the two orthogonal eigenvectors, so the chosen convention takes ${\lambda }_0\ge 0$.

The bulk potential gives fluctuations in $\lambda$ a nonzero restoring force, whereas slowly rotating the nematic director costs only gradients. At lengths much larger than the amplitude correlation length, it is therefore the leading gradient expansion approximation to set $\lambda ={\lambda }_0$ while retaining $\mathbf{n}(\mathbf{r})$.

DifferentiatinggivesSubstitution into the elastic term yields

The anchoring conditions are $\theta (0)=0$ modulo $\pi$ and $\theta (L)=\pi /2$ modulo $\pi$, since the nematic director identifies angles differing by $\pi$. Their difference can therefore be $m\pi /2$ for any odd integer $m$. The Euler-Lagrange equation ofis ${\theta }^{\prime \prime }=0$, so every stationary solution has the formIts free energy per unit length in the $y$ direction isThe smallest possible $m^2$ is one, giving exactly the two degenerate global minima $m=1$ and $m=-1$.