]>PLB30298S0370-2693(14)00444-410.1016/j.physletb.2014.06.039The AuthorsTheoryFig. 1Schematic figures for (a) nematic and (b) smectic-A phases in a liquid crystal system.Broken spacetime symmetries and elastic variablesTomoyaHayataabYoshimasaHidakab⁎aDepartment of Physics, The University of Tokyo, Tokyo 113-0031, JapanDepartment of PhysicsThe University of TokyoTokyo113-0031JapanbTheoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198, JapanTheoretical Research DivisionNishina CenterRIKENSaitama351-0198Japan⁎Corresponding author.Editor: J.-P. BlaizotAbstractWe discuss spontaneous breaking of continuum symmetries, whose generators do explicitly depend on the spacetime coordinates. We clarify the relation between broken symmetries and elastic variables at both zero and finite temperatures, and/or finite densities, and show the general counting rule that is model-independently determined by the symmetry breaking pattern. We apply it to three intriguing examples: rotational, conformal, and gauge symmetries.1IntroductionSymmetry is one of the most important concepts in physics. When a global symmetry group G is spontaneously broken into a subgroup H, the ground state infinitely degenerates. In other words, a uniform change associated with the continuum symmetry, characterized by a parameter π, does not cost energy. If π is slowly varying in space, the energy will increase with (∇π)2. Such a variable is called the elastic variable [1]. In the effective Lagrangian approach, π is identified as the coordinate of coset space G/H [2–5]. For example, in hadron physics, pions that mediate the strong force between nucleons can be identified as the elastic variables. In a crystal, it corresponds to a displacement from the position of the atom at equilibrium.The elastic variables couple to the charges associated with the broken symmetries (broken charges) as canonical variables, and form gapless propagating modes called the Nambu–Goldstone (NG) modes [6–8]. Such gapless modes play an important role at a low energy region. In particular, at low temperature, the number of them, their dispersions, and their damping rates are reflected in the equation of state, heat capacity, temperature dependence of the order parameter, and so on [9–14].In the case that the broken charge densities do not explicitly depend on the spacetime coordinates, the number of elastic variables NEV is equal to the number of broken symmetries (or broken charges) NBS; however, the number of NG modes NNG is not always equal to NEV or NBS. The relation between NNG and NBS(=NEV) is given by [15,16](1)NNG=NBS−12rank〈[Qˆa,Qˆb]〉, where Qˆa are the broken charges. The nonvanishing expectation value 〈[Qˆa,Qˆb]〉 implies that the broken charge densities are also the elastic variables, and form canonical pairs [17]. Such charge densities do not generate the independent NG modes, and thus the total number of NG modes is reduced by as many as the number of such pairs. In fact, the second term in Eq. (1) represents the number of the canonical pairs. The NG modes associated with nonvanishing 〈[Qˆa,Qˆb]〉 are classified as type-B, while the other NG modes are classified as type-A [15]. Their dispersions for type-A and type-B NG modes are linear and quadratic in momentum |k| unless fine-tuning [15,16]; therefore, type-A and type-B NG modes coincide with type-I and type-II NG modes classified by Nielsen and Chadha [18], in which the dispersions of type-I and type-II NG modes are odd and even powers of |k|, respectively.On the other hand, when broken charge densities do explicitly depend on the spacetime coordinates (we refer to the symmetries as spacetime symmetries), the situation is totally different. The counting rule of NG modes as in Eq. (1) is not always valid. A famous example is a crystalline order, in which space-translational and rotational symmetries are spontaneously broken [1]. The charges associated with the rotational symmetries are the angular momenta that explicitly depend on the space coordinates. There appear the three gapless phonons (NG modes) accompanied by the translational symmetries in three dimensions, but no NG modes for the rotational symmetries. To our knowledge, for spacetime symmetries, the general relation between broken generators, elastic variables, NG modes, and their dispersions has not been completely understood yet. It is a great theoretical challenge to generalize the counting rule of NG modes to cover the spontaneous breaking of spacetime symmetries.For Lorentz-invariant theories at zero temperature, the counting rule of NG modes for spontaneous breaking of spacetime symmetries was discussed by Low and Manoha [19], in which the condition that different broken symmetries do not generate the independent NG modes was shown. In the effective field theory approach, this can be understood by the so-called “inverse Higgs mechanism” [19–23]. For the nonrelativistic systems, Watanabe and Murayama proposed a criterion, “Noether constraints,” for the redundancy of the broken symmetries, using quantum operators and the vacuum state [24]. This criterion gives a sufficient condition and covers the redundancy of NG modes not only for spacetime symmetries but also for internal ones, although the applicability is limited at zero temperature.In this paper, as a first step towards constructing the counting rule of NG modes to cover the spacetime symmetries, we focus on the relation between elastic variables, corresponding to the flat directions of the free energy, and spontaneous breaking of spacetime symmetries at both zero and finite temperatures, and/or densities. We show the general counting rule of elastic variables that is model-independently determined by the symmetry breaking pattern.2Spontaneous symmetry breaking and elastic variables2.1(Non)translationally invariant chargeWe consider a system whose microscopic theory has translational symmetries. Charges of translations for time and space are the Hamiltonian Hˆ and the momentum Pˆi, respectively. We use indices with capital (A,B,…) and small (a,b,…) letter for charges that are generators of G and G/H, respectively. We also use the hat symbol to indicate quantum operators. In general, charges transform under spatial translation Tˆx as(2)TˆxQˆATˆx†=cAB(x)QˆB. The coefficients are given by cAB(x)=[exp(−iT¯kxk)]AB, where [T¯k]AB≡−ifkAB, with [Pˆk,QˆA]=ifkABQˆB. Einstein's convention on repeated indices is understood. Since Tˆx+x′=Tˆx′Tˆx, cAB(x+x′)=cAD(x′)cDB(x) and [T¯k,T¯l]=0 are satisfied. For Hermitian charges, fkAB and cAB(x) are real. If QˆA does not explicitly depend on the space coordinates, QˆA commutes with Pˆk, i.e., fkAB=0, and cAB(x)=δAB. We call the charge operator the translationally invariant charge. Conversely, QˆA that does explicitly depend on the space coordinates is called the non-translationally invariant charge. A typical example of non-translationally invariant charges is the angular momentum Lˆij, which transforms under Eq. (2) as(3)TˆxLˆijTˆx†=Lˆij−xiPˆj+xjPˆi.For later use, we define the expectation value of an arbitrary operator Oˆ as 〈Oˆ〉≡trρˆeqOˆ, where ρˆeq≡exp(−βHˆ)/trexp(−βHˆ) is the Gibbs distribution density operator with the inverse temperature β=1/T. For a nonzero chemical potential μ, one may replace Hˆ with Hˆ−μNˆ, where Nˆ is the number operator. The zero temperature theory is obtained by the T→0 limit.2.2Elastic variablesLet us discuss how many elastic variables appear when continuum symmetries are spontaneously broken. For this purpose, we employ the free energy at finite T (and μ) because the elastic variables correspond to the flat directions of the free energy. First, we define the thermodynamic potential as(4)W[J]≡−1βlntrexp[−βHˆ+β∫ddxϕˆi(x)Ji(x)], where d represents the spatial dimension, and ϕˆi(x) are local-Hermitian operators belonging to a linear representation, which may be either elementary or composite. At least, the set of ϕˆi(x) is chosen to contain one order parameter for each broken generator. We assume that ϕˆi(x) transform as ϕˆi(x)=Tˆxϕˆi(0)Tˆx†. Next, the free energy F[ϕ] is given by the Legendre transformation of W[J]:(5)F[ϕ]=W[J]−∫ddxJi(x)δW[J]δJi(x). The first-variational derivative of F[ϕ] with respect to ϕj(y) at J=0 gives the stationary condition, δF[ϕ]/δϕj(y)=0. The second-variational derivative at J=0 is equal to the inverse susceptibility,(6)χij(x,y)=δ2F[ϕ]δϕi(x)δϕj(y)|ϕ=〈ϕˆ〉J=0, which satisfies ∫ddwχik(x,w)χkj(w,y)=δijδ(d)(x−y), where χik(x,w)≡limJ→0δ〈ϕi(x)〉J/δJk(w) is the susceptibility. The subscript 〈⋯〉J denotes the thermal average in the external field J. When we identify χij(x,y) as an operator, we can write the eigenvalue equation as(7)∫ddyχij(x;y)ψj(n,k;y)=λn(k)ψi(n,k;x), where ψi(n,k,x)=ψi(n,k,x). The wave function satisfies the normalization condition,(8)∫ddxψi⁎(n,k;x)ψi(m,k′;x)=δnm(2π)d′δ(d′)(k−k′), where d′ is the dimension of the unbroken translations as explained below. The eigenvalue λn(k) is nonnegative because of convexity of the free energy. We assume that the translational symmetry is not completely broken at least one direction, which may be discrete. This is necessary to identify the eigenmode of unbroken translation, which is characterized by translational operator TˆR with a translation vector R. Under this translation, x→x+R, the eigenfunction satisfies ψi(n,k;x+R)=eik⋅Rψi(n,k;x), where k denotes a point in the first Brillouin zone. The eigenfunction of Eq. (7) with zero eigenvalue at k=0 corresponds to the flat direction of the free energy. Therefore, the number of elastic variables is given by the number of independent eigenfunctions of Eq. (7) with zero eigenvalues at k=0.Here, we consider symmetry of the free energy to find the eigenfunctions with the zero eigenvalue associated with spontaneous symmetry breaking. For charges satisfying [ρˆeq,QˆA]=0, the free energy satisfies(9)∫ddyδF[ϕ]δϕj(y)〈hˆAj(y)〉J=0, where hˆAi(x)≡i[QˆA,ϕˆi(x)]=i[TA]ijϕˆj(x), and the absence of quantum anomalies is assumed [25]. Differentiating Eq. (9) with respect to ϕi(x), we obtain(10)∫ddyδ2F[ϕ]δϕi(x)δϕj(y)〈hˆAj(y)〉J+∫ddyδF[ϕ]δϕj(y)δ〈hˆAj(y)〉Jδϕi(x)=0. At the stationary point and J=0, the second term vanishes, and thus we obtain(11)∫ddyχij(x,y)hAj(y)=0, using Eq. (6), where hAi(x)≡limJ→0〈hˆAi(x)〉J. When continuum symmetries are spontaneously broken, there exist nonvanishing order parameters hai(x), which are assumed to be linearly independent in the sense such that cahai(x)=0 if and only if ca=0. The index a runs from 1 to NBS. We assume there exists no other elastic variable that is not accompanied by the spontaneous symmetry breaking. Then, from Eq. (11), linear combinations of nonvanishing hai(x) are candidates of elastic variables. To be the eigenfunctions of Eq. (7), they should be eigenfunctions of unbroken spatial-translation. When the translational symmetry is not broken, for translationally invariant charges, hai(x) are constant, so that hai(x) are eigenfunctions of translations with k=0, whose number coincides with NBS, and NEV=NBS. However, for general cases, hai(x) are not always eigenfunctions of translations. To see this, let us consider an unbroken translation TˆR under which the thermal state is invariant. The order parameter transforms into(12)hai(x)=i〈[TˆRQˆaTˆR†,TˆRϕˆi(x)TˆR†]〉=icab(R)〈[Qˆb,ϕˆi(x+R)]〉=cab(R)hbi(x+R). If a linear combination of hai(x), fahai(x) is an eigenfunction of the unbroken translation with k=0, it satisfies11We note that this linear combination must satisfy the normalization condition Eq. (8).(13)fahai(x)=facab(R)hbi(x). In addition, with Aab(R)≡δab−cab(R), faAab(R)=0 must be satisfied for any translation vector R. Therefore, the number of independent-elastic variables is given by the dimension of the nontrivial solutions of faAab(R)=0, which equals the dimension of the kernel of Aba(R),(14)NEV=dimkerA(R), for arbitrary R. We note that the indices of Aab(R) run for broken generators. If the unbroken translation is continuum, the matrix A is understood as Aab(ϵ)=ϵkfkab with the infinitesimal-translation vector ϵk.This result does not mean that a non-translationally invariant charge does not generate any eigenfunctions. In the following, we show three examples in which broken non-translationally invariant charges do or do not generate the elastic variables, depending on the pattern of the symmetry breaking.3ExamplesIn this section, we apply our result to three systems with spontaneous breaking of non-translational symmetries: rotational, conformal, and gauge symmetries.3.1Nematic and smectic-A phases in a liquid crystalThe first example is a liquid crystal system, whose microscopic theory is invariant under rotational and translational transformations. We consider the nematic and smectic-A phases of the liquid crystal, shown in Fig. 1. In both phases, the space-rotational symmetry is spontaneously broken, O(3)→O(2), whose broken generators are Lxz and Lyz. The order parameter transforms under Eq. (2) as(15)〈[TˆRLˆizTˆR†,ϕˆk(x)]〉=〈[Lˆiz,ϕˆk(x)]〉−Ri〈[Pˆz,ϕˆk(x)]〉+Rz〈[Pˆi,ϕˆk(x)]〉. In the nematic phase (Fig. 1(a)), where the translation symmetry is not broken, the second and third terms in the right hand side of Eq. (15) vanish, and thus Aab(R)=0. In this case, NEV=NBS, and two non-translationally invariant charges Lˆxz and Lˆyz generate two elastic variables. In contrast, in the smectic-A phase (Fig. 1(b)), the continuum translational symmetry along the z direction is broken into the discrete one in addition to the rotational symmetry breaking, i.e., NBS=3. In this instance, APzPz(R)=APzLiz(R)=0 and ALizPz(R)=Ri. For each Ri, the dimension of A is equal to two. However, one of them depends on Ri, while the other does not. Thus, the dimension of A for arbitrary Ri is equal to one; there appears the only one elastic variable, which is associated with spontaneous breaking of translational symmetry [1].3.2Spontaneous breaking of conformal symmetryThe second example is the system with conformal symmetry. There are three types of non-translationally invariant charges Mˆμν, Dˆ, and Kˆμ, which are generators associated with Lorentz, scale, and special conformal transformations, respectively. The commutation relations between these charges and translational operators are(16)[Mˆμν,Pˆρ]=−i(ημρPˆν−ηνρPˆμ),(17)[Dˆ,Pˆμ]=−iPˆμ,(18)[Kˆμ,Pˆν]=−2i(ημνDˆ+Mˆμν). When a uniform condensation of a scalar field, 〈Φˆ(x)〉≠0, exists, the scale and special conformal symmetries are spontaneously broken; i〈[Dˆ,Φˆ(x)]〉≠0 and i〈[Kˆμ,Φˆ(x)]〉≠0. The number of broken symmetries is five in (3+1) dimensions. In this case, the system is still Lorentz invariant. Thanks to it, the elastic variables coincide the NG fields. The matrix elements are ADD(ϵ)=AKμKν(ϵ)=ADKμ(ϵ)=0, and AKμD(ϵ)=2ϵμ, so that dimkerA=1. Thus, the only one NG mode appears, which is associated with broken scale invariance. The NG mode corresponding to the special conformal symmetry does not appear [4,5,19,26–28].3.3Quantum electrodynamicsThe last example is quantum electrodynamics, in which the photons can be understood as NG modes in a covariant gauge at zero temperature [29–31]. We choose the gauge-fixing term as LGF=B(x)∂μAμ(x)+α(B(x))2/2, where B(x), Aμ(x), and α are the Nakanishi–Lautrup field, the gauge fields, and the gauge parameter, respectively. In this framework, again thanks to the Lorentz symmetry, the elastic variables are identical to the NG fields. In this gauge fixing condition, there are two charges associated with residual gauge symmetries, δAμ(x)=∂μθ(x) with θ(x)=aμxμ+b: Qˆμ and Qˆ. The commutation relations between these charges and the translational operators satisfy [Pˆν,Qˆ]=0 and [Pˆν,Qˆμ]=iηνμQˆ. Therefore, Qˆμ is non-translationally invariant charge. Furthermore, Qˆμ is always broken since the residual gauge transformation gives δAˆν=i[aμQˆμ+bQˆ,Aˆν(x)]=aν, so that i〈[Qˆμ,Aˆν(x)]〉=ημν. If Qˆ is not broken, dimkerA=4. Therefore, there appear four NG modes: Two transverse components of them correspond to the physical photons, while the longitudinal and scalar components correspond to unphysical NG modes that do not appear in the physical state. On the other hand, if Qˆ is spontaneously broken, i.e., if the vacuum is in the Higgs phase, NBS=5, AQQ(ϵ)=AQQμ(ϵ)=0 and AQμQ(ϵ)=ϵμ. We have dimkerA=1. This fact implies that the photons are no longer NG modes nor massless [29–31]. In addition, the NG mode associated with broken Qˆ becomes an unphysical mode. We note that this result depends on the choice of gauge fixing condition. For example, if one chooses Rξ gauge, which explicitly breaks the global U(1) symmetry, there is no massless NG mode, even in the unphysical sector, although the physical spectra are independent from the choice of gauge fixing condition.4Summary and discussionWe have discussed the relation between elastic variables and spontaneous breaking of continuum symmetries including spacetime ones at finite temperature. The elastic variables are given as the degrees of freedom corresponding to the flat directions of the free energy. The general counting rule of elastic variables is given by Eq. (14). Although we begun with the free energy with a set of local fields at finite temperature, the result does not depend on the choice of local fields, and only depends on the symmetry breaking pattern. Our result also works at zero temperature by taking the T→0 limit.For symmetry breaking, whose generator explicitly depends on the time coordinate, i.e., [QˆA,Hˆ]≠0, QˆA does not generate an elastic variable at finite temperature, because [ρˆeq,QA]≠0, and thus Eq. (11) is not satisfied. This is the case for Lorentz or Galilean boost, in which the order parameter is i〈[Mˆ0i,Tˆ0j(x)]〉=ηijh with the enthalpy h. In fact, the susceptibility of Tˆ0i(x) is proportional to h and does not diverge [32]. Note, however, that T0i(x) contains the acoustic phonon as the NG mode associated with broken boost symmetry, although T0i(x) is not elastic variable [33,34].In this paper, we only focused on the elastic variables, not NG modes, which should be distinguished. As it is discussed in Refs. [15–17], the elastic variables are not always independent in the sense of the canonical variables. This will also happen for spacetime symmetries if 〈[Qˆa,Qˆb]〉≠0.It should be remarked on the previous work by Low and Manohar [19], who discussed NG modes rather than elastic variables in Lorentz invariant theories at zero temperature. In their analysis, the fluctuation field is introduced as δϕi(x)≡ca(x)[Ta]ij〈ϕˆj(x)〉 with 〈[Qˆa,ϕˆi(x)]〉=[Ta]ij〈ϕˆj(x)〉. The nontrivial solution satisfying(19)iPkδϕi(x)=(∂kca(x)−cb(x)fkba)[Ta]ij〈ϕˆj(x)〉=0 does not generate gapless excitations [19,20]. Their result for constant ca reduces to our result for the elastic variable, when the unbroken translational symmetry is continuum. We emphasize that their counting rule for NG modes does not work at finite temperature and/or non-Lorentz invariant systems, in particular, when 〈[Qˆa,Qˆb]〉≠0.For dispersions of NG modes associated with spontaneous breaking of spacetime symmetries, they will not be robust; type-I and type-II do not coincide with type-A and type-B. For example, NG modes in a nematic phase of liquid crystal, which are type-A NG modes in the sense of 〈[Qˆa,Qˆb]〉=0, have the dispersions, ω=ak2+ibk2 with constants a and b [35]. The real and imaginary parts have the same order in k, while for spontaneous breaking of internal symmetries, the imaginary part of dispersion of the NG mode is smaller than the real part at small k [1,34]. This behavior can be understood as follows: The NG mode can be regarded as a propagating wave in which the elastic variable and the charge density are the canonical pair. Their equations of motion (the generalized Langevin equations) are formally derived from Mori's projection operator method [16,36], where the expectation value of commutation relation between operators plays a role of Poisson bracket. Since the charge density of the rotational symmetry Lˆ0ij(x)=xiTˆ0j(x)−xjTˆ0i(x) explicitly depends on the spatial coordinates xi that are not operators, one should employ Tˆ0i(x) as the pair operator of the elastic variable rather than Lˆ0ij(x). When only the rotational symmetry is spontaneously broken, the expectation value of commutation relation between the elastic variable and Lˆij0(x) does not vanish at momentum k=0, while that between the elastic variable and Tˆ0j(x) does. At small momentum, it is proportional to k because xi in Lˆ0ij turns to ∂ki in momentum space. Therefore, the additional k appears in the Poisson bracket between the elastic variable and Tˆ0i(x), which makes the dispersion of NG mode quadratic [34]. This suggests that the power of k in the real part of the dispersion relation is strongly related to the power of x in the non-translationally invariant charge density. In contrast, the imaginary part comes from the diffusion term in the Langevin equation for the charge density, which is obtained from the Kubo formula that does not contain the Poisson bracket. Thus, the power of k in the imaginary part for a type-A NG mode is the same as that for internal symmetry breaking, i.e., the order of k2 [34]. Note that for a type-B NG mode, the situation is different because the charge density itself is the elastic variable. For the internal symmetry breaking, the imaginary part behaves like k4 [34]. In general, the parameter a strongly depends on temperatures; at some temperature a vanishes, and thus, the mode is not propagating but overdamping [37].Another nontrivial example is a capillary wave or ripplon, which propagates along the phase boundary of a fluid, and has the fractional dispersion, ω∼k3/2 [38–40]. This dispersion belongs neither to type-I nor type-II NG modes in the Nielsen–Chadha classification [18]. In this case, the translation normal to the phase boundary of the fluid T0z is spontaneously broken, and the mass density ρ becomes the elastic variable. Here, we chose z axis as the normal direction of the phase boundary. Since the expectation value of the translation and the mass density is nonzero, the capillary wave is classified as a type-B NG mode. For internal symmetry breaking, the time derivative of charge density behaves like ∂0na∼∂i2nb, where na and nb are broken charge densities [15,16], and it leads to the quadratic dispersion ω2∼k4. On the other hand, for the capillary wave, the time derivative of ρ is given by its continuity equation as ∂0ρ=−∂iT0i∼−∂zT0z. The power of k in the equation of motion for ρ is reduced to linear from quadratic for a conventional type-B NG mode, while the equation of motion for T0z is the same as the conventional one, ∂0T0z∼∂i2ρ (i=x,y). As a result, we have ω2∼k3, and thus, the fractal dispersion relation is realized ω∼k3/2. It is interesting future problem to investigate the dispersion relations of NG modes associated with spontaneous breaking of spacetime symmetries.AcknowledgementsWe thank Y. Hirono, T. Kugo, Y. Tanizaki, and A. Yamamoto for useful discussions. T.H. was supported by JSPS Research Fellowships for Young Scientists (Grant No. 24008301). Y.H. was partially supported by JSPS KAKENHI Grants Nos. 24740184, 23340067. This work was also partially supported by RIKEN iTHES Project.References[1]P.M.ChaikinT.C.LubenskyPrinciples of Condensed Matter Physics2000Cambridge University Press[2]S.R.ColemanJ.WessB.ZuminoStructure of phenomenological Lagrangians. 1Phys. Rev.177196922392247[3]C.G.CallanJr.S.R.ColemanJ.WessB.ZuminoStructure of phenomenological Lagrangians. 2Phys. Rev.177196922472250[4]D.V.VolkovPhenomenological LagrangiansSov. J. Part. Nucl.419733[5]V.I.OgievetskyNonlinear realizations of internal and space–time symmetriesProc. of X-th Winter School of Theoretical Physics in Karpacz, vol. 11974117[6]Y.NambuG.Jona-LasinioDynamical model of elementary particles based on an analogy with superconductivity. 1Phys. Rev.1221961345358[7]J.GoldstoneField theories with superconductor solutionsNuovo Cimento191961154164[8]J.GoldstoneA.SalamS.WeinbergBroken symmetriesPhys. Rev.1271962965970[9]J.GasserH.LeutwylerChiral perturbation theory to one loopAnn. Phys.15811984142210[10]C.P.HofmannEffective analysis of the O(n) antiferromagnet: low-temperature expansion of the order parameterPhys. Rev. B601999406413[11]C.P.HofmannSpontaneous magnetization of the O(3) ferromagnet at low temperaturesPhys. Rev. B652002094430arXiv:cond-mat/0106492[12]J.O.AndersenEffective field theory for Goldstone bosons in nonrelativistic superfluidsarXiv:cond-mat/0209243[13]J.O.AndersenTheory of the weakly interacting Bose gasRev. Mod. Phys.762004599639[14]T.BraunerSpontaneous symmetry breaking and Nambu–Goldstone bosons in quantum many-body systemsSymmetry22010609657arXiv:1001.5212[15]H.WatanabeH.MurayamaUnified description of Nambu–Goldstone bosons without Lorentz invariancePhys. Rev. Lett.1082012251602arXiv:1203.0609[16]Y.HidakaCounting rule for Nambu–Goldstone modes in nonrelativistic systemsPhys. Rev. Lett.1102013091601arXiv:1203.1494[17]Y.NambuSpontaneous breaking of lie and current algebrasJ. Stat. Phys.1152004717[18]H.B.NielsenS.ChadhaOn how to count Goldstone bosonsNucl. Phys. B1051976445[19]I.LowA.V.ManoharSpontaneously broken spacetime symmetries and Goldstone's theoremPhys. Rev. Lett.882002101602arXiv:hep-th/0110285[20]E.IvanovV.OgievetskyThe inverse Higgs phenomenon in nonlinear realizationsTeor. Mat. Fiz.251975164177[21]A.NicolisR.PencoF.PiazzaR.A.RosenMore on gapped Goldstones at finite density: more gapped GoldstonesJ. High Energy Phys.13112013055arXiv:1306.1240[22]S.EndlichA.NicolisR.PencoPhys. Rev. D892014065006arXiv:1311.6491[23]T.BraunerH.WatanabeSpontaneous breaking of spacetime symmetries and the inverse Higgs effectPhys. Rev. D892014085004arXiv:1401.5596[24]H.WatanabeH.MurayamaRedundancies in Nambu–Goldstone bosonsPhys. Rev. Lett.1102013181601arXiv:1302.4800[25]S.WeinbergThe Quantum Theory of Fields, vol. II1996Cambridge University PressCambridge, UK[26]C.IshamA.SalamJ.StrathdeeSpontaneous breakdown of conformal symmetryPhys. Lett. B311970300302[27]S.ColemanAspects of Symmetry1985Cambridge University PressCambridge, UK[28]K.HigashijimaNambu–Goldstone theorem for conformal symmetryProceedings of XX International Colloquium on Group Theoretical Methods in Physics1994223228[29]R.FerrariL.PicassoSpontaneous breakdown in quantum electrodynamicsNucl. Phys. B311971316330[30]R.A.BrandtW.-C.NgGauge invariance and massPhys. Rev. D1019744198[31]T.KugoH.TeraoS.UeharaDynamical Gauge bosons and hidden local symmetriesProg. Theor. Phys. Suppl.851985122135[32]Y.MinamiY.HidakaRelativistic hydrodynamics from the projection operator methodPhys. Rev. E8722013023007arXiv:1210.1313[33]A.M.J.SchakelEffective field theory of ideal-fluid hydrodynamicsMod. Phys. Lett. B101996999arXiv:cond-mat/9607164[34]T. Hayata, Y. Hidaka, Y. Hirono, in preparation.[35]M.HosinoH.NakanoMolecular theory of hydrodynamic equations for nematic liquid crystalsProg. Theor. Phys.681982388401[36]H.MoriTransport, collective motion, and Brownian motionProg. Theor. Phys.3331965423455[37]P.G.de GennesJ.ProstThe Physics of Liquid Crystals1993Oxford University Press[38]L.D.LandauE.M.LifshitzFluid Mechanicssecond ed.1987Butterworth HeinemannOxford, UK[39]H.TakeuchiK.KasamatsuNambu–Goldstone modes in segregated Bose–Einstein condensatesPhys. Rev. A882013043612arXiv:1309.3224[40]H.WatanabeH.MurayamaNambu–Goldstone bosons with fractional-power dispersion relationsPhys. Rev. D892014101701arXiv:1403.3365