Transitions and modulated phases in centrosymmetric ferroelectrics: Mean-field and renormalization-group predictions

A variety of ferrodisplacive transitions have Landau expansions which involve terms proportional to $p^2$(∂p), i.e., three powers of the order parameter (p) and one gradient (∂). Such expansions have also been suggested for polymeric liquid crystals. We have studied a rotationally invariant, centrosymmetric ferroelectric in which the order parameter is a vector. The $p^2$∂p term leads to modulated phases when it is large enough. The modulated phase close to the uniform-modulated transition and the location of the transition are discussed as a function of dimensionless parameters in a simple model for the free energy. Modifications expected from dangerous irrelevant variables are discussed. The free energy is studied within the d=4-ε renormalization group. A number of fixed points (always including a stable fixed point) are found as a function of the number of components of the order parameter and the dimensionality of space. The fixed points and the renormalization-group flows are discussed. Finally, the nature of the transition when both mean-field theory and the renormalization group are taken into account is discussed.