Gapped Dilatons in Scale Invariant Superfluids

PHYSICAL REVIEW D 102, 076011 (2020)

Gapped dilatons in scale invariant superfluids

† ‡ Riccardo Argurio,1,* Carlos Hoyos ,2, Daniele Musso ,3, and Daniel Naegels 1,§ 1Physique Théorique et Mathématique and International Solvay Institutes, Université Libre de Bruxelles, C.P. 231, B-1050 Brussels, Belgium 2Department of Physics and Instituto de Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Universidad de Oviedo, c/ Federico García Lorca 18, E-33007 Oviedo, Spain 3Departamento de Física de Partículas and Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Inovalabs Digital S.L. (TECHEYE), E-36202 Vigo, Spain

(Received 9 July 2020; accepted 25 September 2020; published 19 October 2020)

We study a paradigmatic model in field theory where a global Uð1Þ and scale symmetries are jointly and spontaneously broken. At zero density the model has a noncompact flat direction, which at finite density needs to be slightly lifted. The resulting low-energy spectrum is composed by a standard gapless Uð1Þ Nambu-Goldstone mode and a light dilaton whose gap is determined by the chemical potential and corrected by the couplings. Even though Uð1Þ and scale symmetries commute, there is a mixing between the Uð1Þ Nambu-Goldstone and the dilaton that is crucial to recover the expected dynamics of a conformal fluid and leads to a phonon propagating at the speed of sound. The results rely solely on an accurate study of the Ward-Takahashi identities and are checked against standard fluctuation computations. We extend our results to a boosted superfluid, and comment the relevance of our findings to condensed matter applications.

DOI: 10.1103/PhysRevD.102.076011

I. INTRODUCTION extract properties of large charge operators of a CFT via the state-operator correspondence [1–4]. Scale invariance plays a special role in many-body and Whenever an internal symmetry is spontaneously broken high-energy physics. It underlies the emergence of univer- in a relativistic system, one expects to encounter gapless sality in many instances, such as critical phenomena, excitations in the form of Nambu-Goldstone (NG) modes Landau-Fermi liquids or cold atoms at unitarity, to name [5–7], one for each broken symmetry. If instead the a few. Scale transformations are a symmetry either at very breaking involves spacetime symmetries, the counting of low or very high energies compared to the intrinsic scales. modes becomes more complicated [8–11], yet the presence In most cases they represent only an approximate symmetry of a Nambu-Goldstone mode associated to scale invariance, valid in a restricted regime, requiring typically a certain commonly known as dilaton, is still a possibility. Similarly, degree of fine-tuning in the interactions, the thermody- the counting of NG modes deviates from the standard namic variables, the external parameters, or the support of Goldstone theorem expectation when the system is not additional symmetries. When a scale invariant system is Lorentz invariant [10,12–15].1 considered at non-zero particle number or at finite charge Even in relativistic systems, the presence of a nonzero density, scale symmetry is spontaneously broken; such charge density breaks the boosts spontaneously, thus the breaking is directly relevant to characterize the dynamics of NG modes may show some features similar to those the system mentioned above but it can also be useful to emerging in nonrelativistic systems. In the case of several internal symmetries, there can be additional gapped modes besides the gapless NG modes [17–22]. More specifically, *[email protected] in the presence of a chemical potential μ for a conserved † [email protected] ‡ charge Q, gapped modes emerge when the effective [email protected] Hamiltonian2 H˜ ¼ H − μQ does not commute with the §[email protected] broken generators. The gap is fixed by group theory Published by the American Physical Society under the terms of considerations and is proportional to the chemical potential. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1For a recent review on NG counting rules we refer to [16]. and DOI. Funded by SCOAP3. 2We are proceeding in analogy to [19,20,22,23].

2470-0010=2020=102(7)=076011(13) 076011-1 Published by the American Physical Society ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020)

In general, there can also be additional modes whose gap, ground states, the theory needs to have a moduli space of although proportional to μ, is not protected by symmetry vacua in addition to scale invariance: these are flat [15,21]. Such analysis was generalized in [20] to cases directions in the potential, supposing we refer to a field where μQ in the effective Hamiltonian is replaced by some theory with a Lagrangian.5 other deformation involving a symmetry generator, for Maybe contrary to expectations, the situation at finite instance the magnetic field times the spin in a ferromagnet density is similar despite the fact that the energy density is ˜ H ¼ H − gBzSz. nonvanishing. If the ground state is homogeneous and The breaking of scale invariance presents some similar- isotropic, the expectation value of the components of the ities with the story above due to the fact that the generator energy-momentum tensor correspond to constant energy of dilatations D does not commute with the Hamiltonian density and pressure ½D; H¼iH.3 Accordingly, the commutator with the 00 effective Hamiltonian is hT i¼ε; hTiji¼pδij: ð1:3Þ

½H;˜ D¼−iH: ð1:1Þ Scale invariance implies that the expectation value of the trace of the energy-momentum tensor will vanish h μ i¼0 ε ¼ For simplicity let us assume that Q is the generator of an T μ , which fixes the equation of state dp, where Abelian Uð1Þ symmetry. If this symmetry is spontaneously d is the number of spatial dimensions. In addition, we have broken, the ground state is not an eigenstate of Q. However, the usual relation between thermodynamic potentials at ε þ ¼ μρ ρ ¼h 0i ð1Þ it must be by definition an eigenstate of H˜ , so time zero temperature, p , where J is the U translations generated by H are spontaneously broken charge density. Combining the two, the energy density of ε ¼ ð þ 1Þμρ too. In fact time translations and the Uð1Þ symmetry are the scale invariant theory is d= d . At finite broken to a diagonal subgroup and there is just a single NG density the relevant quantity is not the energy density, but mode associated to both generators. the free energy (density) given by the effective Hamiltonian 00 − μ 0 Equation (1.1) implies that there is a mixing between the T J . A scale transformation changes the expectation Uð1Þ NG and the dilaton, then—even though Q commutes of the effective energy density as follows: with H˜ —the state produced by the corresponding charge δhT00 − μJ0i ∼ h−i½D; T00i − μh−i½D; J0i density J0 applied to the vacuum at some initial time is not an eigenstate of time evolution. Borrowing an analogy from ¼ðd þ 1Þε − dμρ ¼ 0: ð1:4Þ high-energy physics, the “flavor” eigenstates defined by the symmetry generators are not aligned with the “mass” Then, under quite general assumptions, scale transforma- eigenstates. If the dilaton were not dynamical, or if it were tions do not shift the free energy of a finite density state in a integrated out, the mixing implied by (1.1) would be scale invariant theory and it is legitimate to discuss the manifested in the form of an inverse Higgs constraint. physics of a dilaton mode, at least at zero temperature. Although interesting, one might wonder whether it is The observation above does not imply directly the sensible to discuss the physics of a dilaton in the first place, existence of a gapless (or gapped) mode. In the absence since the energy density is in general nonzero at nonzero of a general argument that would allow us to fix the charge density. In that case, a scale transformation would properties of a dilaton mode, we study a concrete model of change the vacuum energy density (as determined by the spontaneous breaking of scale invariance at nonzero temporal component of the energy-momentum tensor Tμν) density. We restrict the analysis to a relativistic theory in by an amount proportional to itself 3 þ 1 dimensions, and keep the analysis classical. Such simple model is informative because it can be interpreted as δhT00i ∼ h−i½D; T00i ¼ ðd þ 1ÞhT00i; ð1:2Þ an effective action ala` Ginzburg-Landau for the order parameter. where d þ 1 is the number of spacetime dimensions. Both The principal highlights of the present study are two. On here and henceforth, we assume a relativistic theory, thus one side is the characterization of the dilaton dispersion there cannot be a NG mode associated to the spontaneous relation and particularly its gap. This concerns mainly the breaking of scale invariance unless hT00i¼0.4 This is quite effects of the chemical potential and its role in defining the restrictive. Since a gapless mode requires a degeneracy of effective low-energy spectrum. On the other side, we propose and check a method based uniquely on the study of Ward-Takahashi identities, that in our setup just corre- 3 ∂ ¼ Nevertheless, it is a conserved charge because tD H,so spond to classical conservation equations. its total time derivative in the Heisenberg picture vanishes. 4Note also that the combination of Lorentz invariance (which fixes the expectation value of the energy-momentum tensor to 5 μν μν A related discussion about fine-tuning the cosmological hT i¼Λη ) and the Ward-Takahashi identity for scale invari- constant to zero in order to have a flat dilatonic direction is μ 00 ance, hT μi¼0, fixes hT i¼0. contained in [24–27].

076011-2 GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) Z 1 1 1 The paper is structured as follows. Section II introduces 4 μ μ μ 2 2 S ¼ d x ∂μτ∂ τ þ ∂μρ∂ ρþ ∂μϑ∂ ϑ−6λv ρ : the model at zero density, where we emphasize the need for quad 2 2 2 flat directions in the potential. This condition is relaxed in ð2:6Þ Sec. III, where we study the model at nonzero density. In Sec. IV the analysis is extended to allow for non- ρ 12λ 2 τ ϑ zero superfluid velocity. Each section has a subsection We thus see that gets a mass v while and are dedicated to the analysis of the Ward-Takahashi identities, massless. We identify the latter two with the Goldstone bosons for broken scale invariance, the dilaton and, for together with a check of the latter method against standard ð1Þ ð1Þ Lagrangian computations for the fluctuations. We conclude broken U symmetry, the U NG. The dispersion the paper in Sec. V with further comments on the results, relations are trivially relativistic, since Lorentz symmetry their interpretation, their applications and possible is preserved. extensions. In order to study the low-energy modes about (2.4), one can alternatively rely entirely on symmetry considerations and, specifically, on the Ward-Takahashi identities. II. THE MODEL As we will show in the next subsection, such symmetry- Consider the standard Goldstone model for a global aware approach permits to obtain the equations of motion Uð1Þ symmetry in four spacetime dimensions for the low-energy modes in a direct way, which is usually more transparent than the standard Lagrangian study of the Z fluctuations. 4 μ 2 2 2 S ¼ d x½∂μψ∂ ψ − λðjψj − v Þ ; ð2:1Þ A. WARD-TAKAHASHI IDENTITIES where ψ is a scalar complex field charged under the Uð1Þ AND LOW-ENERGY MODES symmetry, which acts as ψ → eiαψ, while λ and v Model (2.2) features a conserved Uð1Þ current given by represent—respectively—a dimensionless and a dimen- μ sionful coupling. Given the presence of a dimensionful Jμ ¼ ið∂μψ ψ − ψ ∂μψÞ; ∂ Jμ ¼ 0; ð2:7Þ coupling, the model (2.1) does not enjoy scale invariance. We can nonetheless make it scale invariant if we replace v while the improved energy-momentum tensor is with a dynamical real scalar field ξ acting as a compensator: Z Tμν ¼ 2∂ðμψ ∂νÞψ þ ∂μξ∂νξ − ημνL 4 μ 1 μ 2 2 2 S ¼ d x ∂μψ∂ ψ þ ∂μξ∂ ξ − λðjψj − ξ Þ : ð2:2Þ 1 1 2 þ ðη ∂2 − ∂ ∂ Þ ξ2 þjψj2 ð Þ 3 μν μ ν 2 : 2:8 The equations of motion are given by This expression satisfies on-shell the following Ward- Takahashi identities6 ∂2ψ þ 2λðjψj2 − ξ2Þψ ¼ 0; ∂2ξ − 4λðjψj2 − ξ2Þξ ¼ 0; ð Þ μ μ 2:3 T½μν ¼ 0; ∂ Tμν ¼ 0;Tμ ¼ 0: ð2:9Þ and the generic stationary solution is We expand around the vacuum (2.4) by considering the fluctuation parametrization (2.5). Up to linear order in the ξ ¼ v; jψj2 ¼ v2: ð2:4Þ fields, the Uð1Þ current is given by pffiffiffi The space of solutions (2.4) has two moduli, ξ itself and the Jμ ≃ 2v∂μϑ; ð2:10Þ phase of ψ. Consider the fluctuations around (2.4), para- metrized as follows so that its conservation equation gives the equation of ð1Þ motion for the U NG mode ipϑffiffi pτffiffi ρ τ ρ ϑ ψ ¼ 2v 3v þ pffiffiffi ≃ þ pffiffiffi þ pffiffiffi þ pffiffiffi pffiffiffi e ve v i ; μ 2 6 3 6 2 0 ¼ ∂ Jμ ≃ 2v∂ ϑ: ð2:11Þ

pτffiffi ρ τ ρ ξ ¼ ve 3v − 2 pffiffiffi ≃ v þ pffiffiffi − 2 pffiffiffi ; ð2:5Þ 6 3 6 The energy-momentum tensor expanded to linear order is where τ, ρ and θ are real. The quadratic action for the 6The trace Ward-Takahashi identity requires the improvement fluctuations is given by introduced in (2.8).

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v 2 III. SPONTANEOUS SYMMETRY BREAKING Tμν ≃ pffiffiffi ðημν∂ − ∂μ∂νÞτ; ð2:12Þ 3 AT FINITE DENSITY In this section we depart from the Lorentz-invariant setup and the trace Ward-Takahashi identity yields the equation discussed above, by introducing a nonzero chemical of motion for the dilaton potential μ for the charge associated to the global Uð1Þ pffiffiffi symmetry. As we will see, we will still be able to identify 0 ¼ μ ≃ 3 ∂2τ ð Þ T μ v : 2:13 the dilaton and the Uð1Þ NG, though their dispersion relations will be modified in an interesting way. From (2.11) and (2.13) we can observe that we recover the We start with a scale-invariant theory defined by the two massless modes of (2.6). The Ward-Takahashi com- action putation, however, descends directly from symmetry argu- Z ments, being therefore more convenient (and easier) to 1 ¼ 4 ∂ ψ ∂μψ þ ∂ ξ∂μξ−λðjψj2 −ξ2Þ2 −λ0ðjψj2Þ2 apply, especially when dealing with models more compli- S d x μ 2 μ ; cated than (2.2). In particular, this approach allows to ð Þ identify immediately and without ambiguities the nature of 3:1 each Goldstone boson, simply by associating every (gap- whose potential corresponds to the extension already less) mode to the Ward-Takahashi identity that yields its introduced in (2.14). We are going to switch on a chemical equation of motion. potential μ for the Uð1Þ symmetry. As discussed before, at It is important to stress that the model (2.2) is fine-tuned. finite chemical potential, the ground state is no longer Indeed, (classical) scale invariance dictates that the poten- determined by the Hamiltonian H but by the effective tial should contain only quartic terms in the scalars, but the ˜ ¼ − μ ð1Þ fact that the potential is a perfect square constitutes a fine- Hamiltonian H H Q, where Q is the U charge tuning, specifically considered to the purpose of having a operator. As we will discuss, this modifies the effective flat direction. The latter is of course a necessary condition potential of the theory and allows the fields to acquire a for the presence of a low-energy dilaton mode. nonzero value. Notably, one can recover the zero chemical potential symmetry breaking case described by (2.2) by The simple argument is as follows. In such a relativistic μ λ0 setup, scale invariance implies the absence of any reference means of an appropriate limit for both and . The main result of the present section is to show that the dilatonic scale in the (effective) Lagrangian. If scale invariance is to μ λ0 be broken spontaneously by a vacuum expectation value mode acquires a gap, which depends on and . (VEV), then the latter must be arbitrary. Hence this VEV A nonzero chemical potential can be implemented by parametrizes a noncompact flat direction. Moreover the extracting a time-dependent phase from the complex field absence of any reference scale means that the flat direction ψ ¼ eiμtϕ; ψ ¼ e−iμtϕ: ð3:2Þ must also correspond to a vanishing vacuum energy. The particle which corresponds to moving along this flat The equations of motion then read direction is the dilaton. We conclude that any effective theory that aims at describing spontaneous scale symmetry 2 2 ∂ ϕ þ 2iμ∂0ϕ − μ ϕ þ ∂ϕ Vðjϕj; ξÞ¼0; breaking (among others), must allow for a noncompact flat 2 direction in its potential. ∂ ξ þ ∂ξVðjϕj; ξÞ¼0; ð3:3Þ For instance, if we added a generic term preserving scale invariance but breaking the exchange symmetry between where Vðjϕj; ξÞ ≡ Vðjψj; ξÞ is given by (2.14). Note that jψj and ξ, namely (without loss of generality) these equations can equivalently be obtained introducing (3.2) in (3.1), identifying a new effective potential 2 2 V ¼ λðjψj2 − ξ2Þ2 þ λ0ðjψj2Þ2; ð2:14Þ Vϕðjϕj; ξÞ¼Vðjϕj; ξÞ − μ jϕj and taking the variation with respect to ϕ , ξ. Although Vϕ is not the true potential 2 2 the equations extremizing the potential would become (indeed, the energy density is E ∼ Vðjϕj; ξÞþμ jϕj ), the extrema of Vϕ correspond to solutions of the equations of λψðjψj2 − ξ2Þ¼−λ0jψj2ψ; ð2:15Þ motion of the original action (3.1). We will show in the following that Vϕ determines the ground state for the ˜ λξðjψj2 − ξ2Þ¼0: ð2:16Þ effective Hamiltonian H.

Considering λ0 > 0 for V to be bounded from below, the A. Effective Hamiltonian and ground state only solution is ξ ¼ 0 ¼ ψ, i.e., the flat direction is In order to determine the effective Hamiltonian and the completely lifted, even though scale invariance is associated ground state we need to find expressions for the respected. Uð1Þ charge Q and Hamiltonian. We will use the usual

076011-4 GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) definitions in terms of the temporal components of the Since t00 determines the effective Hamiltonian (3.5), we see energy-momentum tensor Tμν and Uð1Þ current Jμ that the ground state will correspond to the minimum of Z Z the effective potential. The effective potential has three 8 3 3 extrema H ¼ d xT00;Q¼ d xJ0: ð3:4Þ μ2 ξ ¼ ϕ ¼ 0; ξ ¼ 0 jϕj2 ¼ 2 ¼ ; Then, the effective Hamiltonian at finite chemical potential ; v 2ðλ þ λ0Þ is determined by the temporal component of an effective 2 2 2 2 μ energy-momentum tensor tμν ξ ¼jϕj ¼ ¼ ð Þ v 2λ0 : 3:14 Z Z ˜ 3 3 H ¼ d xðT00 − μJ0Þ ≡ d xt00: ð3:5Þ Out of the three extrema (3.14), the first two are saddle points and only the last is a minimum, which is the true ground state of the system. Note that for the true minimum The Uð1Þ current can be written as follows to exist, and for Vϕ to be bounded from below, we need to 0 2 have λ > 0. In other words, we need to lift the flat direction J0 ¼ 2μjϕj þ j0;J¼ j ; ð3:6Þ i i that we had at μ ¼ 0 in order to have a minimum, and μ ≠ 0 where symmetry breaking, when . We now proceed to investigate the low-energy spectrum around this (degenerate) minimum. jμ ¼ ið∂μϕ ϕ − ϕ ∂μϕÞ: ð3:7Þ

Similarly, for the energy-momentum tensor7 B. Nambu-Goldstone dynamics from Ward-Takahashi identities T00 ¼ μJ0 þ t00; ð3:8Þ We perturb the fields around the ground state 2 2 2 μ2 ξ ¼jϕj ¼ v ¼ 0. We use the same parametrization ¼ ¼ μ þ ¼ μ þ ð Þ 2λ T0i Ti0 Ji t0i ji t0i; 3:9 as in (2.5), though adapted to the field ϕ ¼ þ δ ðμ − 2μ2jϕj2Þ¼ þ δ μ ð Þ Tij tij ij J0 tij ij j0; 3:10 ipϑffiffi pτffiffi 1 pτffiffi 2 ϕ ¼ e 2v ve 3v þ pffiffiffi ρ ; ξ ¼ ve 3v − pffiffiffi ρ: ð3:15Þ 6 6 where

As before, the kinetic terms are diagonal and canonically tμν ¼ 2∂ðμϕ ∂νÞϕ þ ∂μξ∂νξ − ημνLϕ ϑ τ ρ ϑ normalized for , and . We still identify as the 1 1 fluctuation of the phase of the condensate and τ as a þ ðη ∂2 − ∂ ∂ Þ ξ2 þjϕj2 ð Þ 3 μν μ ν 2 ; 3:11 fluctuation of its magnitude, while ρ corresponds to an orthogonal direction of increasing potential energy. For and μ ¼ 0, ϑ and τ are naturally associated to the Uð1Þ NG and dilaton, while ρ enters as a Higgs fluctuation. This simple 1 picture is a bit complicated when μ ≠ 0, as the would-be L ¼ ∂ ϕ∂μϕ þ ∂ ξ∂μξ − λðjϕj2 − ξ2Þ2 ϕ μ 2 μ Goldstones undergo some mixing and also a nonvanishing − λ0ðjϕj2Þ2 þ μ2jϕj2: ð3:12Þ gap for one linear combination. We will study this effect in some approximation here and in more detail in the next section. Notice that from (3.9) we have that T0i ¼ Ti0 implying that the Ward-Takahashi identities for boost transformations are When the perturbation (3.15) is introduced in the effective potential (3.13) and expanded to quadratic order, satisfied, so that the full Lorentz symmetry is still preserved ϑ in the presence of a nonvanishing chemical potential. one finds no term for and the following mass matrix for ðτ; ρÞ The effective potential for Lϕ is the one we had identified previously in the equations of motion (3.3) pffiffiffi 4v2 2λ0 2λ0 M ¼ pffiffiffi : ð3:16Þ 2 2 2 0 2 2 2 2 3 0 0 Vϕ ¼ λðjϕj − ξ Þ þ λ ðjϕj Þ − μ jϕj ; ð3:13Þ 2λ λ þ 9λ

In principle both perturbations are massive and mixed, but 7 0 The notations are such that the capital letters (T00 etc.) refer to in the limit λ ≪ λ in which there is an almost flat direction the dynamics of ðψ; ξÞ (and by extension, of ϕ) dictated by (3.1). The lowercase letters refer instead to the dynamics given by (3.12) which is not the Lagrangian for ðϕ; ξÞ but it shares the 8Note that these uniform and static solutions are extrema of the same potential. effective potential (3.13), but not of the energy (3.8).

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q in the original potential (2.14), the mixing becomes very π˜ ≃ ϑ˜ − isignðω=qÞ pffiffiffi τ˜; small and there is a large hierarchy between the mass of τ, 2μ 2 0 2 2 2 2 rffiffiffi mτ ∼ λ v ∼ μ , and the mass of ρ, mρ ∼ λv . In the 2 2 σ˜ ≃ τ˜ − ðω μÞ 1 þ q ϑ˜ ð Þ following we will assume that we are in this situation, in i 3sign = 24μ2 : 3:23 which case the Higgs fluctuation ρ can be set to zero in the low energy description to a good approximation. A few comments are in order. In the first place, the The dynamical equations for the remaining fluctuations dispersion relation of π in (3.22) is such that it moves at can be derived from the Ward-Takahashi identities. When the speed of sound as fixed by conformal invariance evaluated on shell the Uð1Þ current should be conserved 2 cs ¼ 1=3, i.e., it can be identified as a conformal superfluid and the trace of the energy momentum tensor should vanish phonon, while σ is the gapped dilaton. This identification is consistent with an effective field theory approach, see e.g., ∂ μ ¼ 0 μ ¼ 0 ð Þ μJ ;Tμ : 3:17 [4]. Note that the mixing is necessary for this to happen, otherwise the phonon would move at the speed of light due This gives two equations, which is sufficient to determine to relativistic invariance of the rest of the terms. The second the dynamics of ϑ and τ. The trace of the energy- observation is that the gap of σ is fixed by the chemical momentum tensor, to linear order in the fluctuations, is potential mσ ¼ 2μ, and independent of the couplings λ and λ0 rffiffiffi in this approximation. This is very reminiscent of the pffiffiffi 4 2 massive Goldstone bosons appearing when internal sym- μ ≃ 3 ∂2τ þ μ2τ − 2 μ∂ ϑ ð Þ T μ v 3 3 0 ; 3:18 metries are spontaneously broken in the presence of a chemical potential. A last observation is that because of the whereas the divergence of the current is mixing, it is no longer true that each Ward-Takahashi identity is tied to one specific mode. Indeed reexpressing τ rffiffiffi ϑ π σ pffiffiffi 2 and in terms of and , one can easily see that both fields ∂μ ≃ 2 ∂2ϑ þ 2 μ∂ τ ð Þ appear in both equations (3.20). Jμ v 3 0 : 3:19

C. Exact dispersion relations This translates into the set of coupled equations rffiffiffi The results obtained from the Ward-Takahashi identities 4 2 are easy to interpret physically but we had to introduce ∂2τ þ μ2τ − 2 μ∂ ϑ ≃ 0 3 3 0 ; several approximations to derive them, in particular we rffiffiffi used the hierarchy between the masses of the Higgs 2 2 fluctuation and the dilaton to freeze out the first. In order ∂ ϑ þ 2 μ∂0τ ≃ 0: ð3:20Þ 3 to go beyond this approximation we need to include the Higgs mode in the analysis, whose dynamics is not As suggested by the general analysis in the introduction, captured by the Ward-Takahashi identities. This can be the chemical potential introduces a mixing between the more simply done using the effective Lagrangian. Uð1Þ NG and the dilaton. The equations can be diagon- 2 2 2 μ2 Consider again the vacuum ξ ¼jϕj ¼ v ¼ 2λ0 and the alized using expansions in Fourier modes fluctuations (3.15) around it. The quadratic Lagrangian for

Z 3 the fluctuations is dωd q 0 τð 0 xÞ¼ −iωx þiq·xτ˜ðω qÞ x ; 4 e ; ; ð2πÞ 1 μ 1 μ 1 μ L ¼ ∂μρ∂ ρ þ ∂μϑ∂ ϑ þ ∂μτ∂ τ Z 3 quad 2 2 2 dωd q 0 rffiffiffi ϑð 0 xÞ¼ −iωx þiq·xϑ˜ðω qÞ ð Þ x ; 4 e ; : 3:21 2 2 2 pffiffiffi ð2πÞ þ 2 μτ∂ θ þ pffiffiffi μρ∂ θ − 2μ2τρ 3 t 3 t 3 Expanding at low momentum q2=μ2 ≪ 1, the equations 2 9λ þ λ0 − μ2τ2 − μ2 ρ2 ð Þ have solutions when the modes satisfy the dispersion 3 3λ0 : 3:24 relations By going to Fourier space we get q2 5 ω2 ≃ ; ω2 ≃ 4μ2 þ q2: ð3:22Þ 1 3 3 L ¼ Tð−ω − Þ ðω Þ ðω Þ ¼ðϑ ρ τÞ quad 2y ; q ·M ;q ·y ;q ;y ; ; ; Therefore, there is a gapless mode π and a gapped mode σ, ð3:25Þ which at low momentum correspond respectively to the combinations where

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0 pffiffi 1 ω2 − 2 p2ffiffi μω 2pffiffi2μω flat directions associated to them, of course this will always q i 3 i 3 B pffiffi C be the case for internal symmetries. B 2ð9λþλ0Þ C ¼ B − p2ffiffi μω ω2 − 2 − μ2 −2 2μ2 C M @ i 3 q 3λ0 3 A: pffiffi pffiffi − 2pffiffi2μω −2 2μ2 ω2 − 2 − 4μ2 IV. BOOSTED SUPERFLUID i 3 3 q 3 Since the chemical potential breaks Lorentz invariance, it ð Þ 3:26 is interesting to study the effect on the NG modes when the superfluid is set on motion relative to the frame determined Studying the zeros of the determinant of M, one finds one by the effective Hamiltonian induced by the chemical massless mode, the Uð1Þ NG boson, and two gapped potential, that one can identify as the “laboratory” frame. modes: We consider again (2.2) and introduce both a chemical potential and a superfluid velocity ω2j ¼ 0 1 q¼0 ; μ μ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ ¼ iμ0uμx ϕ ψ ¼ −iμ0uμx ϕ ð Þ 3μ2 2 e ; e : 4:1 ω2 j ¼ λ þ λ0 λ2 − λλ0 þ λ02 ð Þ 2;3 q¼0 0 : 3:27 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 3 ⃗ 2 Where uμ ¼ γð1; −βÞ, γ ¼ 1= 1 − jβj is a timelike four- μ ¼þ1 μ ¼ γμ Expanding for low momentum q and for λ0 ≪ λ, we get velocity uμu . The chemical potential is 0, and the time direction in the laboratory frame is x0. The 1 background plane wave (4.1) is the same as (3.2) seen by a ω2 ≃ 2 ð Þ 1 3 q ; 3:28 boosted observer, compared to the laboratory frame. Since (2.2) is Lorentz invariant, the dispersion relations for the λ λ0 2λ0 gapless low-energy modes can be obtained by boosting ω2 ≃ 6μ2 1 þ þ 1 þ 2 ð Þ 2 λ0 3λ 9λ q ; 3:29 those obtained from (3.1) (i.e., the case with just a chemical potential). For the sake of providing an explicit check, we 0 0 repeat the exercise of computing them directly through the 2 2 λ 5 2λ 2 ω3 ≃ 4μ 1 − þ − q : ð3:30Þ Ward-Takahashi identities and through the perturbative 3λ 3 9λ Lagrangian approach.

Comparing with the dispersion relations in (3.22),we A. Effective Hamiltonian and ground state observe that the speed of the phonon is not modified by corrections depending on λ0, while the mass of the gapped We proceed in a similar fashion to the case of zero dilaton is corrected, though mildly. Indeed, contrary to velocity. The Hamiltonian and the charge are still deter- massive NG bosons associated to internal symmetries, the mined by the energy-momentum tensor and the current as mass of the gapped dilaton is not protected by the in (3.4), and the effective Hamiltonian at nonzero chemical symmetry. potential by (3.5). Because of the boost, the expressions for 0 2 2 the current and the energy-momentum tensor are slightly For λ ≪ λ, ω1 and ω3 reduce to the dispersion relations obtained in (3.22) from the study of the Ward-Takahashi modified. identities, and we have a hierarchy between the two ¼ 2μ jϕj2 þ massive modes. Furthermore, in the limit Jμ 0 uμ jμ; 2 2 α Tμν ¼ 2μ0uμuνjϕj þμ0ðuμjν þuνjμÞ−ημνμ0u jα þtμνðμ0Þ: μ2 μ → 0 λ0 → 0 → 2 ð Þ ð4:2Þ ; with 2λ0 v ; 3:31 μ Where j and tμν take the same form as before (3.7) and we recover the masses (2.6) of the relativistic model (2.2) (3.11), replacing μ by μ0. Recalling that the chemical potential is μ ¼ μ0u0 ¼ μ0γ, the effective Hamiltonian is ω2j ¼ 0; 1 q¼0 Z Z 2 2 ω2j ¼0 ¼ 12v λ; 3 3 ⃗ ⃗ q H − μQ ¼ d xðT00 − μJ0Þ¼ d xðt00ðμ0Þ − μβ · jÞ: ω2j ¼ 0 ð Þ 3 q¼0 ; 3:32 ð4:3Þ and ω3 describes the massless dilaton. This suggests a ϕ connection between the corrections to the mass of the Since ji vanishes for constant , the extrema of the effective μ gapped dilaton at finite chemical potential and the lack of a potential are the same as before (3.14) replacing by the effective chemical potential in the rest frame of the fluid μ0. flat direction in the potential at zero chemical potential. The 2 2 2 2 μ0 masses of gapped NGs might be protected only if there are The ground state is thus ξ ¼jϕj ¼ v0 ¼ 2λ0.

076011-7 ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) rffiffiffi B. Nambu-Goldstone dynamics from 2 4 2 2 ∂βτ þ μ0τ − 2 μ0∂ 0 ϑ ≃ 0; Ward-Takahashi identities 3 3 xβ rffiffiffi We can use the same parametrization for perturbations of 2 2 the ground state as in (3.15), replacing v by v0. The same ∂βϑ þ 2 μ0∂ 0 τ ≃ 0: ð4:9Þ 3 xβ considerations about the mass hierarchy of τ and ρ apply, so in this analysis we will assume λ0 ≪ λ and freeze ρ. The These are the same as before (3.20), replacing μ by μ0.We dynamics of the low energy modes are determined by the introduce an expansion of the modes in the rest frame in conservation equations for the current and the energy- plane waves momentum tensor. For the boosted superfluid they take the form Z 3 dωβd qβ − ω 0þ q x τð 0 x Þ¼ i βxβ i β· β τ˜ðω q Þ xβ; β ð2πÞ4 e β; β ; μ ⃗ ⃗ 2 μ Z ∂ Jμ ¼ 2μð∂0 þ β · ∇Þjϕj þ ∂ jμ; 3 dωβd qβ − ω 0þ q x ϑð 0 x Þ¼ i βxβ i β· β ϑ˜ðω q Þ ð Þ μ 2 2 μ μ xβ; β 4 e β; β : 4:10 T μ ¼ 2μ0jϕj − 2μ0u jμ þ t μðμ0Þ ð2πÞ 2 2 ⃗ ⃗ μ ¼ 2μ jϕj − 2μðj0 þ β · jÞþt μðμ0Þ: ð4:4Þ 0 We recover the expected low momentum dispersion relations in the rest frame Therefore, we should just replace the terms with a single μ∂ → μ ¼ 5 time derivative by the material derivative 0 D0 ω2 ≃ c2q2; ω2 ≃ 4μ2 þ q2; ð4:11Þ ⃗ ⃗ β s β β 0 3 β μð∂0 þ β · ∇Þ and otherwise change μ by the effective μ0: 2 ¼ 1 3 rffiffiffi where cs = is the speed of sound of the scale invariant pffiffiffi 2 theory. These expressions can be translated to frequency ∂μ ≃ 2 ∂2ϑ þ 2 μ τ Jμ v0 3 D0 ; and momentum in the laboratory frame using that rffiffiffi pffiffiffi 4 2 μ ≃ 3 ∂2τ þ μ2τ − 2 μ ϑ ð Þ ω ¼ γðωβ þ βqβ3Þ;q3 ¼ γðqβ3 þ βωβÞ; T μ v0 3 0 3 D0 : 4:5 q1 ¼ qβ1;q2 ¼ qβ1: ð4:12Þ

From this, we obtain the equations Note that the dispersion relations (4.11) are valid for low j j ≪ jμ j rffiffiffi momentum in the rest frame of the fluid qβ 0 . For the 4 2 gapless modes they can be matched with a low momentum ∂2τ þ μ2τ − 2 μ ϑ ≃ 0 3 0 3 D0 ; expansion in the laboratory frame jqj ≪ jμj, however for rffiffiffi the gapped modes this is not possible, as for generic β, 2 2 q3 ∼ ωβ ∼ μ. Therefore, finding the dispersion relations of ∂ ϑ þ 2 μD0τ ≃ 0: ð4:6Þ 3 the gapped modes at low momentum in the laboratory frame requires solving (4.6) directly. The dispersion relation for the gapless mode can be more We classify the dispersion relations of the gapless modes easily found by noting that μ ¼ γμ0 and using comoving taking as reference the direction of the superfluid velocity coordinates. Taking β⃗ parallel to the x3 direction, we in the laboratory frame. The dispersion relations for the introduce longitudinal modes is c β 0 0 3 3 3 0 ω ¼ s ¼ ¼ 0 ð Þ x ¼ γðxβ þ βxβÞ;x¼ γðxβ þ βxβÞ; k q3;q1 q2 ; 4:13 1 βcs 1 1 2 2 x ¼ xβ;x¼ xβ: ð4:7Þ while the dispersion relation for the transverse modes is

Then 2 2 2 2 q1 þ q2 ω ¼ c ;q3 ¼ 0: ð4:14Þ ⊥ s γ2ð1 − β2c2Þ ∂ ∂ s ¼ γð∂ þ β∂ Þ ¼ γð∂ þ β∂ Þ ∂2 ¼ ∂2 0 0 3 ; 3 3 0 ; β: ∂xβ ∂xβ These expressions agree with the ones obtained by rela- jβj ð Þ tivistic addition of velocities. Note that for >cs both 4:8 (positive frequency) longitudinal modes (4.13) propagate in the same direction as the superfluid velocity. This is the The equations become reason why we expressed linearly the dispersion relations.

076011-8 GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020)

2 2 2 0 4 2 2 2 For the gapped modes the low momentum dispersion V ¼ λðjϕj − ξ Þ þ λ jϕj − ðμ − k3Þjϕj ; ð4:16Þ relations are

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 For stationary solutions, the situation is not very different 2 2 2 βq3 5þβ cs q3 2 ωk ¼ 2 1−β c μ− þ ; from the case with just μ, in fact one just needs to replace μ s 31−β2 2 2 2 2 5=2 μ cs 12γ ð1−β cs Þ 2 2 2 by ðμ − k3Þ¼μ0 in (3.13). Therefore, we consider the q1 ¼ q2 ¼ 0; solution 2 2 2 2 2 2 5−β 2 2 ω⊥ ¼ 4ð1−β c Þ μ þ ðq1 þq2Þ;q3 ¼ 0; 2 s 2 2 μ0 3ð1−β csÞ ξ2 ¼jϕj2 jϕj2 ¼ ð Þ ; 2λ0 : 4:17 ð4:15Þ The fluctuation around (4.17) are still given by (3.15) where again the longitudinal dispersion relation is μ0 v ¼ v0 ¼ pffiffiffiffiffi k3 ¼ βμ expressed linearly. The gap is reduced by the superfluid where however 2λ0. Writing , the quad- velocity, but in this approximation remains finite even in ratic Lagrangian for the fluctuations is the limit β → 1, where the condensate vanishes (i.e., at fixed μ). Note also that, at leading order for momenta in the 1 1 1 L ¼ ∂ ρ∂μρ þ ∂ ϑ∂μϑ þ ∂ τ∂μτ same direction of the flow, the frequency is reduced. quad 2 μ 2 μ 2 μ rffiffiffi 2 2 C. Exact dispersion relations þ 2 μτð∂ þ β∂ Þθ þ pffiffiffi μρð∂ þ β∂ Þθ 3 t 3 3 t 3 We now study the effects of including the Higgs 0 2 pffiffiffi 2 9λ þ λ0 fluctuation ρ, and the corrections for finite λ =λ. We thus 2 2 2 2 2 − 2μ0ρτ − μ0τ − μ0 ρ : ð4:18Þ resort to expanding the full Lagrangian. According to (4.1), 3 3 3λ0 we switch on a chemical potential μ ¼ μ0γ and a background wave vector k3 ¼ μ0γβ. The effective potential is In analogy to (3.25) and (3.26), by going to Fourier space now: we get the kinetic matrix:

0 pffiffi 1 ω2 − 2 p2ffiffi μðω − β Þ 2pffiffi2 μðω − β Þ q i 3 q3 i 3 q3 B pffiffi C B 2ð9λþλ0Þ C p2ffiffi 2 2 2 2 2 2 B −i μðω − βq Þ ω − q − 0 μ − μ C: ð : Þ @ 3 3 3λ 0 3 0 A 4 19 pffiffi pffiffi − 2pffiffi2 μðω − β Þ − 2 2 μ2 ω2 − 2 − 4 μ2 i 3 q3 3 0 q 3 0

From the determinant of (4.19), one can find the exact dispersion relations. First of all, setting the momenta q ¼ 0 one finds that there is a massless mode corresponding to the Uð1Þ NG boson and two gapped modes: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ω2j ¼ 0 ω2 j ¼ λμ2 þ λ0μ2ð1 − 2β2Þ λ2μ4 − λμ2λ0μ2ð1 − 2β2Þþλ02μ4ð1 − 2β2Þ2 ð Þ 1 q¼0 ; 2;3 q¼0 λ0 0 cs 0 3 0 cs cs ; 4:20 where cs ¼ 1=3 as before. Now, expanding at low frequencies and momenta, one can extract analytically the dispersion relation for the Uð1Þ NG mode:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ¼ cs ð2 β ð1 − β2Þ2 2 þð1 − β2Þð1 − 2β2Þð 2 þ 2ÞÞ ð Þ 1 2 2 cs q3 q3 cs q1 q2 : 4:21 1 − csβ

Notice that the above expression is independent of the ratio λ0=λ. Indeed one can check that in the longitudinal and transverse case, it reproduces correctly the expressions (4.13) and (4.14), respectively. For the massive modes, one has to expand the frequencies around the respective gaps. To first order in momenta and in λ0=λ, the dispersion relations for the gapped dilaton are:

076011-9 ARGURIO, HOYOS, MUSSO, and NAEGELS PHYS. REV. D 102, 076011 (2020) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 0 2 2 ðgappedÞ 2 2 λ 1 − csβ 2β λ 1 − cs β ω ¼ 2μ 1 − c β 1 − − 1 − q3 þ … k s 6λ 1 − β2 3ð1 − c2β2Þ 3λ 1 − β2 s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 2 0 2 2 ð Þ λ 1 − c β 1 5 − β λ q þ q ω gapped ¼ 2μ 1 − c2β2 1 − s þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 2 þ … ð4:22Þ ⊥ s 6λ 1 − β2 2 2 1 − 2β2 6λ μ 12 1 − cs β cs

For λ0=λ ¼ 0 they agree with the dispersion relations obtained from the Ward-Takahashi identities (4.15). Again, we observe that the gap receives corrections in λ0=λ, so it is not protected by the symmetry. Finally, we present the dispersion relations of the Higgs mode, to the same order:

rffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 0 ð Þ 6λ λ 1 − c β 2λ β ω heavy ¼ 1 − β2μ 1 þ s − q þ … k λ0 6λ 1 − β2 9λ 1 − β2 3 rffiffiffiffiffi rffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 2 0 2 2 ð Þ 6λ λ 1 − c β 1 λ q þ q ω heavy ¼ 1 − β2μ 1 þ s þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 þ … ð4:23Þ ⊥ λ0 6λ 1 − β2 2 1 − β2 6λ μ

Note that the β → 1 limit at fixed μ seems to be ill defined. symmetry breaking to the purpose of elucidating the However this is an artifact of the expansion. For instance, characteristics of the resulting low-energy dynamics. It inspecting (4.20) and noticing that in this limit μ0 → 0,we would be interesting to look for a gapped dilaton in a find that the gap of the dilaton actually goes to zero, while strongly coupled theory, by means of the holographic the gap of the Higgs mode stays finite, but scales with μ, duality. This might be achieved by combining holographic which might be slightly nonintuitive (recall that in this limit models with a gapless dilaton in Poincar´e invariant vacua there is no condensate). [31,32] (see also [33]) and models with type II and gapped NG modes [34,35]. V. SUMMARY AND DISCUSSION We first examined a relativistic field-theory model (2.2) The two main highlights of the present paper are these: in four spacetime dimensions where scale symmetry and a Uð1Þ (1) The analysis of the low-energy mode associated to global symmetry are concomitantly and spontane- spontaneously broken scale symmetry and the char- ously broken. The scale-invariant potential must have two flat directions which translate into two gapless NG modes, acterization of how its zero-temperature gap depends ð1Þ on the finite density. the dilaton and the U NG both relativistic and both (2) The description of a generic method based on Ward- propagating at the speed of light, (2.11) and (2.13). Takahashi identities alone to study the low-energy In order to realize the same symmetry-breaking pattern at modes of an effective field theory. finite density, the model must be stabilized by means of an The analysis pursued in the present paper indicates extra scale-invariant term (3.1) which lifts the dilatonic flat that the spontaneous breaking of the scale symmetry at direction without affecting the spontaneous nature of the zero temperature gives rise to a light dilatonic mode breaking. The resulting low-energy modes are nonetheless ð1Þ whose gap is directly proportional to the chemical altered: the U NG remains gapless but propagates at the potential. This generic expectation can be relevant for conformal speed of sound, like a superfluid phonon; the the low-energy content of zero-temperature systems dilaton acquires a gap of the order of the chemical potential where the chemical potential, too, is small with respect μ whose value is however not protected by symmetry to the UV cutoff of the effective description (related to (3.22). The dilaton is light compared to other gapped modes some other physical scale such as an external magnetic only when the coefficient of the term that lifts the flat field [28]). direction (3.1) is tuned to be very small, in that case we Ward-Takahashi identities in quantum field theory are observe that the dilaton gap becomes independent of the known to be a key tool for the study of symmetries, either couplings. when these are preserved or broken, and even when the Our results at nonzero density belong to the line of breaking is explicit [29,30]. The present paper stresses that research on gapped NG modes [15,19–21,36]. In this Ward-Takahashi identities alone provide a sufficient frame- context, a natural future perspective is to embed the present work to study the dispersion relations of the low-energy analysis into a systematic Maurer-Cartan effective frame- modes of an effective field theory, providing an alternative work, thus assessing its universality and possible —generally simpler—approach than the direct fluctuation generalizations. analysis at the level of the Lagrangian. The method is One interesting field of applications is provided by generic, but we applied it to the specific study of scale condensed matter. The presence of a wide critical

076011-10 GAPPED DILATONS IN SCALE INVARIANT SUPERFLUIDS PHYS. REV. D 102, 076011 (2020) region in the phase diagram is a characteristic shared by to as homogeneous because they do not yield any spacetime many—generally strongly correlated—systems, among modulation of the energy density,9 they however provide which the cuprates. The critical phase is associated to acoustic phonon modes. It is an interesting open question to interesting phenomena like bad and strange metallicity and study whether and how these phonons would coexist with a non-Fermi liquid behavior [37]. It is also often conjectured dilatonic mode [49]. The relevance of the question is to lie at the basis of the mechanism for high-temperature threefold: it relates to the counting problem of NG modes superconductivity, see for instance [38]. for spacetime symmetries [9,16]; it concerns condensed The defining property of such critical region is the matter systems where a critical scaling and the breaking of validity of simple scaling rules whose origin, however, translations are intertwined10; it provides insight regarding can involve complicated and often elusive dynamics related holographic models where scaling and translation sym- – 11 to the presence of a quantum critical point [39 41] or, more metries are broken together [51–54]. generally, to the presence of a scaling sector [38,42]. This is sometimes referred to as generic scale invariance [43] and can be assumed among the defining symmetries of an ACKNOWLEDGMENTS effective description. We are grateful to Tomas Brauner for useful comments. Another paradigmatic example is provided by cold atoms R. A. and C. H. want to thank Nordita for their hospitality at unitarity, where there is an emergent nonrelativistic during the program “Effective Theories of Quantum Phases conformal symmetry [44], known as Schroedinger sym- of Matter.” R. A. and D. N. acknowledge support by IISN- metry. Gapped NG modes are known to appear when the Belgium (convention 4.4503.15) and by the F. R. S.-FNRS Hamiltonian is deformed by some of the symmetry gen- under the “Excellence of Science" EOS be.h Project erators of the Schroedinger algebra [45]. An extension of our No. 30820817. R. A. is supported as a Research Director analysis, along the lines of [23], to systems with Galilean of the F. R. S.-FNRS (Belgium). C. H. has been partially rather than Lorentz invariance would be quite interesting. supported by the Spanish Ministerio de Ciencia, As another remark, still related to condensed matter but Innovacion y Universidades Grant No. PGC2018- in the context of standard metals, it is relevant to mention 096894-B-100 and by the Principado de Asturias through that the low-energy modes of our analysis would not Grant No. GRUPIN-IDI/2018/000174. destabilize a Landau-Fermi liquid coexisting with them. This can be appreciated by means of an extension of the 9An example of inhomogeneous breaking of spatial trans- results of [46] to dilatations, a symmetry which does not lations in field theory was studied in [48]. Studying the cubic commute with either spatial or temporal translations: one polynomial in ω2 associated to (3.26), one can exclude the can show that the linear interaction term between the presence of complex solutions. Similarly, a numerical study of fermionic quasiparticles and, respectively, the Uð1Þ NG (4.19) showed no hints of finite-momentum instabilities. Thus, and the dilaton are both vanishing. for the purpose of studying translation symmetry breaking, the models introduced in the main text need to be enriched and The model adopted here allows for generalizations in generalized. which the Uð1Þ symmetry is coupled to translations and the 10Such as in the region of the phase diagram of cuprates symmetry-breaking preserves only a linear combination of overlapping with the critical, strange metal phase and the so- called pseudogap phase [37,50]. the two [47]. This would realize a spatial version of the 11 μ ≠ 0 Merging the last two points, there is a current in the literature pattern described above when and only a diagonal addressing the critical breaking of translations relevant for the component of the product of internal Uð1Þ and time study of strongly correlated electron systems (specifically strange translations was preserved. Such breakings are referred and bad metals), see for instance [55–62].

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