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.
Highlights
(2) We consider models which enjoy a shift symmetry and cases where this is broken by a mass term and show that the spontaneous breaking of translations can be attained in both cases
Thereby, we show that shift symmetry is not a necessary ingredient to the purpose of describing phononic modes
To push further the comparison to discrete chains, if we look to a chain with several types of atoms, the internal oscillations within a unit cell correspond to optical modes. This might give a physical picture of the reason why we have a tower of optical modes in our continuous model
Summary
For about 100 years [1], translation symmetry breaking has been extensively studied, though its dynamical origin is often left aside and neglected. In typical condensed matter circumstances, there is a large hierarchy in energy between the physics of a crystal formation/ melting and its low-energy excitations. These latter determine the thermodynamic and linear response properties, which can usually be described by low-energy effective theories without considering the dynamical origin of the lattice. (2) We consider models which enjoy a shift symmetry and cases where this is broken by a mass term and show that the spontaneous breaking of translations can be attained in both cases. (3) In the shift-symmetric real scalar models that we analyze, the shift symmetry is spontaneously broken concomitantly with spatial translations. We argue that at low energy the Nambu-Goldstone modes associated with the two symmetries, e.g., the shifton and the phonon, are well separated in momentum space and can be regarded as effectively independent modes, in analogy to what happens for (superfluid) phonons and rotons in superfluid helium [2,3,4,5]
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