# Phonon and Shifton from a Real Modulated Scalar

## TL;DR

This paper investigates a scalar field model with spontaneous translation symmetry breaking, revealing phonon and shifton modes, and extends stability analysis to higher-derivative theories with implications for condensed matter analogies.

## Contribution

It introduces a scalar field model with modulated vacua that produce phonon and shifton modes, generalizing stability results to higher-derivative theories.

## Key findings

- Identification of phonon and shifton modes in modulated scalar fields
- Generalization of stability analysis for higher-derivative models
- Analogy to rotons in superfluid systems

## Abstract

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.