Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / a / Solution

A polar liquid crystal distinguishes the two ends of each constituent. Its orientational order is described by a vector $\mathbf{p}$, and $\mathbf{p}$ and $-\mathbf{p}$ represent different states. A nematic liquid crystal has head-tail symmetry, so its director obeys $\mathbf{n}\sim -\mathbf{n}$. Its lowest-rank faithful nematic order parameter is the symmetric traceless tensorwhich is unchanged by $\mathbf{n}\mapsto -\mathbf{n}$.