Double-well elastic theory for twist-bend nematic phases

The ground state of twist-bend nematic liquid crystals is a heliconical molecular arrangement in which the
nematic director precesses uniformly about an axis, making a fixed angle with it. Both precession senses are
allowed in the ground state of these phases. When one of the two helicities is prescribed, a single helical nematic
phase emerges. A quadratic elastic theory is proposed here for each of these phases which features the same
elastic constants as the classical theory of the nematic phase, requiring all of them to be positive. To describe the
helix axis, it introduces an extra director field which becomes redundant for ordinary nematics. Putting together
helical nematics with opposite helicities, we reconstruct a twist-bend nematic, for which the quadratic elastic
energies of the two helical variants are combined in a nonconvex energy. States with minimal energy and opposite
helicities may come in contact along an interface belonging to a special