Independent Goldstone modes for translations and shift symmetry from a real modulated scalar

We study a massive real scalar field that breaks translation symmetry dynamically. Higher-gradient terms favor 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 also break 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 reobtain and generalize some standard results for one-dimensional lattices. Eventually, we prove stability against geometric deformations extending existing analyses for elastic media to the higher-derivative cases.