Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 2 / a / Solution

Under spatial inversion the electric polarization, a polar vector, changes from $\mathbf{P}$ to $-\mathbf{P}$. Neumann's principle requires an equilibrium property of a phase retaining inversion symmetry to be unchanged. Thus $\mathbf{P}=-\mathbf{P}$, forcing $\mathbf{P}=0$: a centrosymmetric phase cannot itself be ferroelectric. A transition can, of course, remove that symmetry.

For an inversion-symmetric parent phase the Landau theory free energy satisfies $F(\mathbf{P})=F(-\mathbf{P})$. Every polynomial term of odd total degree changes sign and must have zero coefficient. Restricting to one fixed direction gives an even scalar expansion in $P$.