Abstract
We analyze the concomitant spontaneous breaking of translation and conformal symmetries by introducing in a CFT a complex scalar operator that acquires a spatially dependent expectation value. The model, inspired by the holographic Q-lattice, provides a privileged setup to study the emergence of phonons from a spontaneous translational symmetry breaking in a conformal field theory and offers valuable hints for the treatment of phonons in QFT at large. We first analyze the Ward identity structure by means of standard QFT techniques, considering both spontaneous and explicit symmetry breaking. Next, by implementing holographic renormalization, we show that the same set of Ward identities holds in the holographic Q-lattice. Eventually, relying on the holographic and QFT results, we study the correlators realizing the symmetry breaking pattern and how they encode information about the low-energy spectrum.
Highlights
The Ward identities state a priori that the would-be superfluid is described at low energy by the massless Goldstone mode associated to the broken U(1)
We perform our analysis in a generic quantum field theory (QFT) with the symmetry breaking pattern above, and complement it with a concrete holographic realization consisting in a Q-lattice model [5]
Let us remark once again that both conformal and translation symmetries are broken by the same mechanism, that is by the same operator acquiring a nontrivial vacuum expectation value
Summary
We shall verify that the generating functional obtained above encodes the Ward identities obtained in the QFT analysis of section 2 This follows from the fact that the generating functional is invariant under the relevant local symmetries, diffeomorphisms and local rescalings, acting as (3.26), (3.27). We have proven that the holographic on-shell action reproduces exactly the Ward identities that we computed with standard field theory techniques. This may not be a complete surprise, considering that Ward identities are just a reflection of the symmetries of the problem, and the fact that the holographic model is built precisely to encode those symmetries. Once we obtain a generating functional that is invariant under the relevant symmetries, the Ward identities are guaranteed to follow
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