Abstract
Spatial modulation has been studied for a long time in condensed matter, nuclear matter and quark matter, where the manifest Lorentz invariance is lost due to the finite density/temperature effects and so on. In this paper, spatially modulated vacua at zero temperature and zero density are studied in Lorentz invariant 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.
Highlights
FFLO states, called twisted kink crystals, were studied in the chiral Gross–Neveu model in 1+1 dimensions [11– 13]
The effective theory is in general a function of ∂mφ. In this set-up we study an adaptation of the NG theorem to higher derivative theories, stating that when a global symmetry is spontaneously broken due to vacuum expectation values of space-time derivatives of fields, an NG boson appears without canonical kinetic terms with a quartic derivative term in the modulated direction, while a Higgs boson appears with a non-zero canonical kinetic term
We show that an analogue of the NG boson appears without canonical kinetic term with a quartic derivative term
Summary
FFLO states, called twisted kink crystals, were studied in the chiral Gross–Neveu model in 1+1 dimensions [11– 13] (see [14] for application to a superconductor). The Cooper pair is usually refers to the particle–particle condensates, the chiral condensation is related to the particle– antiparticle (or hole) pairing They were proposed in diquark condensations exhibiting color superconductivity in high density QCD (see [20,21] as a review) and were discussed in the context of the AdS/CFT correspondence [22– 25]. The effective theory is in general a function of ∂mφ (complemented by a potential term) In this set-up we study an adaptation of the NG theorem to higher derivative theories , stating that when a global symmetry is spontaneously broken due to vacuum expectation values of space-time derivatives of fields, an NG boson appears without canonical kinetic (quadratic derivative) terms with a quartic derivative term in the modulated direction, while a Higgs boson appears with a non-zero canonical kinetic term. Our model admits (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
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