[Paper Review] Spatially Modulated Vacua in Relativistic Field Theories

Muneto Nitta, Shin Sasaki+1 more|arXiv (Cornell University)|Jun 9, 2017

Cosmology and Gravitation Theories3 citations

TL;DR

This paper proposes a generalized Nambu-Goldstone theorem for relativistic field theories with spontaneously broken global symmetries due to vacuum expectation values of space-time field derivatives. It shows that in modulated vacua, a Nambu-Goldstone mode emerges without a canonical kinetic term but with a quartic derivative term in the modulated direction, while a Higgs mode appears as a gapless excitation, demonstrated in a model with a metastable phase-modulated vacuum (Fulde-Ferrell state).

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 a generalization of the Nambu-Goldstone theorem 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 as a gapless mode. 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.

Motivation & Objective

• To extend the Nambu-Goldstone theorem to relativistic field theories with vacuum expectation values of space-time field derivatives.
• To investigate the emergence of Nambu-Goldstone and Higgs modes in spatially modulated vacua at zero temperature and density.
• To analyze the structure of gapless modes when global symmetries are spontaneously broken via field derivative VEVs.
• To demonstrate the mechanism in a concrete model supporting a metastable phase-modulated vacuum (Fulde-Ferrell state).

Proposed method

• Proposes a generalized Nambu-Goldstone theorem under the assumption of no ghost states.
• Analyzes a relativistic field theory model with spontaneously broken U(1) and translational symmetries due to derivative VEVs.
• Constructs a Lagrangian with a quartic derivative term in the modulated direction to describe the NG mode.
• Identifies the Higgs mode as a gapless excitation in the modulated vacuum.
• Uses a simple model to realize a (meta)stable Fulde-Ferrell-type phase-modulated vacuum.
• Derives the spectrum of collective modes to confirm the absence of a canonical kinetic term for the NG mode.

Experimental results

Research questions

• RQ1 How does the Nambu-Goldstone theorem generalize in relativistic field theories when symmetries are broken by field derivative vacuum expectation values?
• RQ2 What is the kinetic structure of the Nambu-Goldstone mode when no canonical quadratic derivative term is present?
• RQ3 How does the Higgs mode behave in a spatially modulated vacuum without a standard kinetic term?
• RQ4 Can a metastable phase-modulated vacuum (Fulde-Ferrell state) support gapless modes with non-standard kinetic structures?
• RQ5 What are the implications for the spectrum of collective excitations in such modulated vacua?

Key findings

• A Nambu-Goldstone mode appears without a canonical kinetic term but with a quartic derivative term in the modulated direction.
• The Higgs mode emerges as a gapless excitation in the spatially modulated vacuum.
• The generalized Nambu-Goldstone theorem holds under the condition of no ghost states.
• The model realizes a (meta)stable Fulde-Ferrell-type vacuum with phase modulation.
• The collective mode spectrum confirms the absence of a standard quadratic kinetic term for the NG mode.
• The structure of the NG mode is consistent with spontaneous breaking of both translational and U(1) symmetries.