  We study a massive real scalar field that breaks translation symmetry
dynamically. Higher-gradient terms favour modulated configurations and neither
finite density nor temperature are needed. In the broken phase, the energy
density depends on the spatial position and the linear fluctuations show
phononic dispersion. We then study a related massless scalar model where the
modulated vacua break also the field shift symmetry and give rise to an
additional Nambu-Goldstone mode, the shifton. We discuss the independence of
the shifton and the phonon and draw an analogy to rotons in superfluids.
Proceeding from first-principles, we re-obtain and generalise some standard
results for one-dimensional lattices. Eventually, we prove stability against
geometric deformations extending existing analyses for elastic media to the
higher-derivatives cases.
