  It has been known that at high density, the local orientation of
banana-shaped molecules shows a spontaneously bent state, giving rise to
interesting liquid-crystalline phases such as splay-bend and twist-bend. This
spontaneous bend can be modelled theoretically by allowing the bend elastic
constant in the Frank elastic energy to become negative. Here we extend this
idea to polar banana and pizza-slice-shaped molecules which can also splay
spontaneously. By allowing both splay and bend elastic constants to be negative
we discovered two additional new liquid crystalline phases. In particular,
using renormalization group technique, we showed that the phase transition
belongs to a new constrained ferromagnet universality class.
