# Spatially Modulated Vacua in a Lorentz-invariant Scalar Field Theory

## TL;DR

This paper explores spatially modulated vacua in relativistic scalar field theories, extending the Nambu-Goldstone theorem to higher derivative models and demonstrating the emergence of NG bosons without canonical kinetic terms.

## Contribution

It introduces an adaptation of the Nambu-Goldstone theorem for higher derivative theories and demonstrates the existence of modulated vacua with NG bosons in a simple relativistic model.

## Key findings

- NG bosons appear without quadratic derivatives in modulated vacua.
- Stable phase modulation states are demonstrated in the model.
- Higher derivative terms enable novel symmetry-breaking phenomena.

## Abstract

Spatial modulation has been studied for a long time in condensed matter, nuclear matter and quark matter, so far in non-relativistic field theories. In this paper, spatially modulated vacua at zero temperature and zero density are studied in relativistic field theories. We first propose an adaptation of the Nambu-Goldstone theorem to higher derivative theories under the assumption of the absence of ghosts: when a global symmetry is spontaneously broken due to vacuum expectation values of space-time derivatives of fields, a Nambu-Goldstone (NG) boson appears without a canonical kinetic (quadratic derivative) term with a quartic derivative term in the modulated direction while a Higgs boson appears with a canonical kinetic term. We demonstrate this in a simple model allowing (meta)stable modulated vacuum of a phase modulation (Fulde-Ferrell state), where an NG mode associated with spontaneously broken translational and $U(1)$ symmetries appears.