Temporally, Spatially or Light-like Modulated Vacua in Lorentz Invariant Theories

We study the vacuum structure of a class of Lorentz invariant field theories where the vacuum expectation values are not constant but are (phase) modulated. The vacua are classified into spatial, temporal, and light-like modulation types according to the patterns of spontaneous breaking of translational symmetry. The conditions for having temporal or light-like modulated vacua imply severe constraints on the models. We utilize the notion of generalized Nambu-Goldstone modes which appear in the modulated vacua. Finally, we examine fluctuation modes around these vacua and discuss their dynamics and the absence of ghosts.


Introduction
Finding the vacuum structure is one of the most fundamental issues in the course of understanding quantum field theories. The true (false) vacuum in field theories is defined by being the global (a local) minimum of the energy functional (Hamiltonian). For models that consist of a canonical kinetic term (quadratic with second-order spacetime derivative) with a potential V , this is given by a constant field configuration determined by a minimum of V . In this case, the constant vacuum expectation value (VEV) trivially satisfies the equation of motion and the constant field configurations automatically yields the minimum energy since varying a field costs gradient energy. Contrary to this, the situation can generally become more involved when we consider models that contain quartic or higherorder kinetic terms in addition to the canonical one, but only if not all terms appear in the energy functional with a positive sign. Such models have terms with more than two spacetime derivatives and are called higher-derivative models. In some cases, non-constant field configurations can be energetically favored and become the vacuum. Depending on the symmetries of the potential, the energy may be minimized by a field configuration that has non-vanishing derivatives in the vacuum (lowest-energy state). The symmetry that will play an important role in this paper is shift symmetry, which when possessed by the potential allows for the field to be non-constant. In this situation, the kinetic term and the higher-order derivative terms can be seen as a "potential" for the "velocity" of the field, that is, for a non-vanishing gradient of the field. As mentioned above, if not all terms enter the energy functional with a positive sign, the "velocity" that minimizes the energy may be a non-vanishing constant. Even if the canonical kinetic term has the usual positive sign in the energy functional, a constant VEV may be just a local vacuum, and instead a modulated vacuum becomes the global vacuum. In this paper we will study the situation where the "velocity" is the phase of a complex field, although that is not necessarily the only possibility in general.
In condensed matter physics as opposed to relativistic quantum field theory, the lowestenergy state is not the vacuum but is called the ground state. Modulated ground states were originally discovered in non-relativistic theories, i.e. theories without the Lorentz invariance, in the context of superconductors. In such theories, spatially-modulated ground states can easily be found. For example, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states in superconductors were proposed a long time ago [1,2]. For the Fulde-Ferrell (FF) state [1], the phase of a condensate in the ground state is spatially modulated along one spatial direction, while for the Larkin-Ovchinnikov (LO) [2], the amplitude of the condensate is modulated. The LO state is the ground state when an external magnetic field is applied [3] (see also ref. [4]), while the FF state can be the ground state when a magnetic field penetrating a superconducting ring is present (see e.g. ref. [5]). Recent experiments reveal that these FFLO states are realized in several physical systems [6][7][8][9]. In addition to condensed matter physics, the FFLO states appear in a variety of fields. In the context of QCD, spatially-modulated chiral condensates of the FF-type, called a dual chiral density wave or chiral spiral [10,11], and the LO-type, called a real kink crystal [12,13], were contemplated to appear in a certain region of the QCD phase diagram. In the Gross-Neveu or Nambu-Jona Lasino model in 1+1 dimensions, FFLO states for which both the phase and the amplitude are modulated, called twisted kink crystals [14,15], were discussed at finite temperature and density [16]. The related issues were also discussed in the context of the AdS/CFT correspondence [17][18][19][20].
On the other hand, temporal modulations have been also studied recently. The idea of the time-crystal [21,22] proposes a time-dependent ground state where the continuous translational symmetry along the time direction is spontaneously broken to a discrete subgroup. This is an analogue of the ordinary crystalline structure of matter in spatial directions, namely, a conventional crystal. Although the time-crystal cannot be realized in thermal equilibrium states [23], it has been reported to be experimentally realized in a non-equilibrium state [24,25]. A diffusive Nambu-Goldstone mode corresponding to spontaneously broken time-translation in a time-crystal was discussed in ref. [26]. A temporally modulated state was also studied in QCD [27].
In all the examples in the literature that we mentioned so far, i.e. in condensed matter physics or in QCD, Lorentz invariance is either absent or explicitly broken by finite density, temperature, and/or an external magnetic field. On the other hand, modulated vacua in Lorentz invariant setups had not been studied until recently [28,29]. In refs. [28,29], we realized such a possibility in a simple Lorentz invariant model with higher-derivative terms, by studying the vacuum structure possessing a non-constant VEV. The VEV that we analyzed is a function of a specific direction in three-dimensional space. Such kind of vacuum is called spatially modulated.
In this paper, we study modulated vacua in Lorentz invariant field theories. As a continuation of our previous studies on spatially-modulated vacua [28,29], we here focus on temporally and light-like modulated vacua. Nevertheless, it proves instructive to include the spatially-modulated case here so as to put them all on equal footing. A temporally modulated vacuum is characterized by the property that the time derivative of fields develop a VEV. If the solution we are looking for is timelike -due to the Lorentz invariance of the class of theories we are considering -we can always set the spatial derivatives to zero without loss of generality simply by a change of Lorentz frame. Similarly, the lightlike modulation is defined by a VEV for the derivative of fields on the light-cone. A nontrivial problem for time-dependent solutions is that the extrema of the Hamiltonian do not necessarily coincide with all solutions to the Lagrangian equation of motion. A subset of the solutions to the Lagrangian equations of motion, as we will show in this paper, do yield extrema of the corresponding Hamiltonian; these are the states we are looking for. In the class of theories we consider in this paper, we will furthermore assume that there is no dependence on the field, ϕ, itself in the Lagrangian and that the Lagrangian only depends on the first derivatives of the field, ∂ m ϕ.
In restricting to a Lagrangian that only depends (nonlinearly) on the first time derivative of the field, we sidestep the problem of the Ostrogradsky instability and related ghost [30,31]. As in the ordinary vacuum in field theories, once the fields develop nontrivial VEVs, some of the symmetries of the Lagrangian are spontaneously broken. In the modulated states in non-relativistic theories, there appears Nambu-Goldstone (NG) modes associated with broken symmetries [26,32,33]. We will show that there is an analogue of NG modes (generalized NG modes) in our models. The generalized NG modes correspond to the flat direction of the "potential" whose quadratic kinetic term disappears in the Lagrangian for the case of spatial modulation [28]. However, this turns out not to be the case for temporal or light-like modulation.
In this paper, we study the FF-type (phase) modulation ϕ ∼ exp(iωt). One may wonder that this modulation is rather usual in non-relativistic theories. For instance, (nonlinear) Schrödinger systems, i∂ t ϕ = ∇ϕ + V ϕ (i∂ t ϕ = ∇ϕ + g|ϕ| 2 ϕ), have states with a time dependence of the above type. Therefore, such a time dependence is not usually called a temporal modulation. Note, however, that it is not the case for Lorentz invariant (relativistic) theories discussed in this paper.
The organization of this paper is as follows. In the next section, we discuss the general conditions for modulated vacua. We classify the vacua into the spatial, temporal and lightlike types. In section 3, we introduce the notion of generalized NG modes and discuss the relation between them and the zero modes of the generalized mass matrix. In section 4, by focusing on a higher-derivative model, we show an example of the modulated vacua and the generalized NG modes. We also examine the dynamics of the fluctuation modes in the vacua and discuss the absence of ghosts. Section 5 is devoted to a summary and discussion.

General discussion on modulated vacua
Throughout this paper, we consider Lorentz-invariant theories of a single complex scalar field ϕ. We require that the theory does not suffer from the Ostrogradsky instability [30,31] which can be caused by fourth or higher order of spacetime derivative terms. For simplicity of constructing models, we assume that theories have shift symmetry of the fields and a global U (1) symmetry: with constant parameters c and θ. The absence of the Ostrogradsky instability then implies that the Lagrangian consists of ∂ m ϕ, ∂ mφ only: In general, vacua are defined as field configurations that minimize the energy density and they satisfy the equation of motion. The conventional vacua, which are characterized by constant VEVs, trivially satisfy these conditions. However, if we look for vacua where VEVs are not constant but generally spacetime dependent functions, the conditions are not trivially satisfied. In the following, we write down the general conditions for vacua in this theory and discuss modulated vacua.

Conditions for modulated vacua
Starting from the Lagrangian (2.2), the equation of motion is This is expressed in compact form as: Hereφ = ∂ 0 ϕ and so on and the 2 × 2 matrices L mn (m, n = 0, 1, 2, 3) are defined as . (2.5) Note that only the symmetric part L (mn) contributes to the equation of motion and L †00 = L 00 , L †0i = L i0 , L †ij = L ji . Any global or meta-stable vacuum of the theory satisfies the equation of motion and it should be a local minimum of the energy functional. The Hamiltonian (energy) density is defined by The Hamiltonian depends only on the spacetime derivative of fields. Therefore it can be considered as a potential for ∂ m ϕ. The extremal condition of the energy with respect to ∂ m ϕ is then and its complex conjugate. More explicitly, we have These together with their complex conjugates are expressed as (2.9) One finds that the conventional vacuum configuration which is characterized by a constant VEV trivially satisfies the equation of motion (2.4) and the extremal condition of the energy density (2.9). However as we will see in section 4, there can be nontrivial solutions to these equations (2.4), (2.9) where parts of ∂ m ϕ develop nonzero VEV ∂ m ϕ = 0. In order to examine whether a configuration that satisfies eqs. (2.4) and (2.9) is a stable vacuum, we consider the fluctuation φ in the configuration. We consider a shift in the field ϕ → ϕ + φ, or equivalently ∂ m ϕ → ∂ m ϕ + ∂ m φ. After the shift in the fields, the Hamiltonian is expanded as where the symbol | 0 means that they are evaluated in the vacuum and we have defined the 2 × 2 matrices M mn and the vector φ m as: More explicitly, we have In order to define the Hessian matrix of the energy functional, we rearrange the expression (2.10): where we have defined the vector Since M †mn = M nm , the 2d×2d = 8×8 matrix M is a Hermitian Hessian matrix. In order for a vacuum ∂ m ϕ = 0 to be stable, the generalized mass matrix needs to be positive semi-definite in the vacuum which we denote M| 0 ≥ 0. Once a vacuum ∂mϕ = 0, for fixedm, that satisfies eqs. (2.4), (2.8) and M| 0 ≥ 0 is found, we consider the dynamical (fluctuation) field φ around the vacuum. Then the Lagrangian for the dynamical field is Here L mn are the matrices defined in eq. (2.5) and L 0 is a constant.
In the next subsection, we examine nontrivial solutions to the conditions (2.4) and (2.9). Among other things, we focus on vacuum configurations where the VEVs are modulated with respect to spacetime coordinates. We first classify the modulated vacua and then clarify the conditions for the vacua.

Classification of modulated vacua
The modulated vacua are classified according to the patterns of spontaneous breaking of the global and Poincaré symmetries. When some components of ∂mϕ develop a nonzero VEV, they break translational symmetry along the xm-directions as well as the rotational symmetry in a plane spanned by the xm coordinate. The shift and global U (1) symmetries are also spontaneously broken. If the VEV is a Fulde-Ferrell (FF, the phase modulated) type, i.e. ∂mϕ ∝ e icxm with a constant c, then the symmetry breaking pattern is classified by the vector generators Pm associated with the translational symmetry along the xmdirection into three cases (1) space-like, (2) time-like and (3) light-like.
(1) When Pm is space-like, the vacuum is called spatially modulated. In this case, we can choosem = 1 without loss of generality. The symmetry is spontaneously broken Here S is the shift symmetry group, P 1 is the translational group generated by P 1 and diag. stands for the diagonal subgroup corresponding to the simultaneous transformation: and implies a semi-direct product.
(2) When Pm is time-like, the vacuum is called temporally modulated. In this case, we can choosem = 0 without loss of generality. In this vacuum, the symmetry is spontaneously broken as is the Poincaré group in the spatial directions.
(3) On the other hand, when Pm is light-like (null), we can choose Pm to be the light-cone directions P ± = P 0 ± P 1 . We call the vacuum light-like modulated. The breaking pattern is We will discuss a generalization of the NG modes associated with these symmetry breakings in section 3. In the following, we examine the conditions for each modulated vacuum.
Spatial modulation In the case of the spatial modulation, the VEV is characterized by Assuming the Ansatz (2.19), we examine the conditions (2.4) and (2.9). By this assumption, we look for static field configurations. We first demand the following conditions Note that | 0 means that the conditions are satisfied in the vacuum for any x i . If we impose the above conditions, the energy extremal conditions in eq. (2.9) are trivially satisfied. The equation of motion (2.4) is also trivially satisfied since ϕ is x 0 -independent and ∂L ∂(∂ i ϕ) = 0 for any x i . Indeed, for a static field configuration, we have the relation H = −L and the condition δH = 0 automatically implies δL = 0. In refs. [28,29] we solved the condition (2.20) and found a spatially modulated vacuum.
If the theory that we consider is Lorentz invariant, one notices that only the combinationsφ 2 ,φφ appear in the Lagrangian: where A, B are any Lorentz invariant terms. Then, one can show that the relations, always hold for the spatial modulation Ansatz (2.19). From eq. (2.13), we therefore find the relations In particular one can show that L 0i | 0 = L i0 | 0 = M 0i | 0 = 0 for the spatial modulation. The generalized mass matrix is given by This is completely block diagonal. The explicit expression of M depends on the models under consideration. For the vacuum (2.19), we require that M| 0 is positive semi-definite. Once a vacuum is found, the Lagrangian for the fluctuation in a spatially modulated vacuum becomes The matrices L mn | 0 govern the kinetic terms of the fluctuation modes. For the spatial modulation, the zero modes of M mn | 0 coincide with those of L mn | 0 . This issue will be discussed in section 3.

Temporal modulation
The temporal modulation is characterized by the VEV: In the case of temporal modulations, it is not always true that extrema of the energy are solutions to the equation of motion. In order to look for a configuration characterized by eq. (2.26), we demand the following conditions: Indeed, the second condition in eq. (2.27) allows a solution in Lorentz invariant models. This substantially implies L 0i | 0 = 0. Since the solution (2.28) implies that ∂ i ∂ j ϕ = 0 at any points, the conditions (2.4) and (2.9) necessary for modulated vacua are now satisfied. By the Ansatz (2.26), we have For Lorentz invariant models, we can show ∂L 0i in the case of temporal modulation but M 00 | 0 and M ij | 0 are not equal to L 00 | 0 , L ij | 0 , in general. Since L 00 | 0 = 0, the kinetic term of the fluctuation along the time-direction vanishes identically. This distinguishes the temporal modulation from the spatial one.
Light-like modulation We define the light-like modulation as a vacuum characterized by the VEV: Here ∂ϕ · ∂ϕ = η mn ∂ m ϕ∂ n ϕ and so on. The metric is η mn = diag(−1, 1, 1, 1). In order to look for the configuration (2.30), we assumė If we demand the following conditions: then, both the conditions (2.4), (2.9) for modulated vacua are satisfied. However, these conditions are generally too strong and we will find relaxed conditions in a concrete model of light-like modulation in section 4.

Generalized Nambu-Goldstone modes
In the previous section, we made a classification of modulated vacua according to the spontaneous breaking of symmetries. Along with the spontaneous symmetry breakings, we expect that NG modes appear in the spectrum. Particular emphasis is placed on the rotational symmetry. Since the rotations in the (x 1 , x 2 ), (x 1 , x 3 ) planes are not independent of the translation along the x 1 -direction [34], there are no independent NG modes associated with the spontaneous breaking of the rotational symmetries. Therefore the only relevant part of our interest is or light-like. The NG modes associated with these symmetry breakings are well-captured by the notion of the generalized NG modes in ref. [28]. In this section, we discuss a generalization of the NG theorem in our setup. This provides a relation between zero modes of the generalized mass matrix (2.16) and spontaneous symmetry breaking. Most of the analysis in this section has been already discussed in ref. [28]. We therefore provide a brief sketch of the notion in the following. We assume that the energy functional, H, depends on the spacetime derivative of scalar fields only : H(ϕ m ,φ m ). Here we have defined ϕ m ≡ ∂ m ϕ,φ m ≡ ∂ mφ . Vacua are defined as extrema of H with respect to ϕ m ,φ m : In these extrema, we assume ϕ m ,φ m have the following VEVs: Herem is a fixed spacetime direction. The VEV v m can depend on the spacetime coordinate in general. In that case, the vacuum spontaneously breaks spacetime symmetries. Consider the fluctuations φ,φ around the VEV. As we have discussed in the previous section, the expansion of the Hamiltonian around the vacua results in ( Φ is defined in eq. (2.15)) In order that the vacua are local minima, we require that M| 0 is positive semi-definite. The zero modes of M| 0 correspond to flat directions. Let us clarify the relation between the zero modes of M| 0 and symmetries of the theory. The fields ϕ m ,φ m transform as Here Q A are generators of the symmetry group and T A is a Hermitian matrix representation of the generators. As we mentioned before, a part of ϕ m ,φ m develop VEVs (3.2). Then the spontaneous breaking of symmetries by modulations is defined if there exist operators ϕm with vm = 0 such that then the symmetry corresponding to TÂ with TÂ v = 0 is preserved in the vacuum while T A such that T A v = 0 is spontaneously broken. The energy functional H is invariant under the following transformation: where ε A is an infinitesimal parameter. Namely, we have By differentiating this relation with respect to ϕ m ,φ m , we obtain If we consider these relations in the vacuum, the factor ∂H/∂ϕ m vanishes and we find This results in the following relation: Here we have defined 0| Φ|0 = V where Φ is defined in eq. (2.15). If T A V = 0, namely, T A = T A is a broken generator, this is an eigenvector of the generalized mass matrix M| 0 associated with zero eigenvalue 1 . We call these zero modes T A V the generalized NG modes. Note that since only parts of the component in V are nonzero, i.e., only the subsector that corresponds to the nonzero VEV directionsm are relevant to specify the zero modes.

Higher derivative scalar model
In this section, we demonstrate the existence of modulated vacua by focusing on a concrete model with Lorentz invariance, and study fluctuations in each case according to the classification of spatial, temporal and light-like modulation.

Spatial modulation
We first consider the spatially modulated vacuum in the model. As we will show in the following, the results in this subsection are essentially the same with the ones in ref. [28], but Lagrangian here is more general. In order to make the paper be self-contained, we rederive the results in more general setup (4.4). For the spatial modulation, we employ the Ansatz where ϕ 0 , c are real constants and the VEV is of the FF-type in the x 1 -direction. Following the general procedure discussed in section 2.2, we examine the spatially modulated vacuum. The Ansatz implies Then, we have We first study the condition (2.9). As discussed in section 2.2, the nontrivial conditions boil down to ∂L ∂(∂ i ϕ) = 0. The conditions for i = 2, 3 in eq. (2.20) are automatically satisfied. On the other hand, for i = 1, we have This determines the amplitude of the VEV: Next, we analyze the other condition, i.e. eq. (2.4). The matrices L mn for the spatial modulation are calculated as
Global aspects of the potential and stability of the vacuum Although we have confirmed that the spatially modulated configuration (4.20) is a solution to the equation of motion and the energy extremum condition, it is still unclear whether the configuration is a (global or local) minimum of the energy functional. As we will show below, this holds true in a specific region of the parameters k, α, β, λ, µ. We first study the global stability in the ∂ 1 ϕ-direction. The energy density for the Ansatz (4.20) is This is a function of X ≡ |∂ 1 ϕ| 2 ≥ 0: y = kX −(λ+µ)X 2 −(α+β)X 3 . From this expression the condition for the existence of a local minimum is determined by the discriminant condition of y = k − 2(λ + µ)X − 3(α + β)X 2 = 0: D = 4 (λ + µ) 2 + 3k(α + β) > 0. This is the condition that the energy has a local minimum at X = 0. When these conditions are satisfied, one finds that the solution (4.24) is a local minimum of the energy. Since the energy is given in the form one finds that if f (X) ≡ −(α + β)X 2 − (λ + µ)X + k = 0 has no solution, then, the local minimum (4.24) is meta-stable. If f (X) = 0 has one solution, then the vacuum (4.24) is degenerate with the trivial vacuum X = 0. If f (X) = 0 has two solutions, then (4.24) becomes the global minimum. In summary, we have the following conditions for the global structures of the "energy potential" E sp (see fig. 1): meta-stable : (λ + µ) 2 + 4k(α + β) < 0, degenerate : (λ + µ) 2 + 4k(α + β) = 0, global : (λ + µ) 2 + 4k(α + β) > 0. (4.30) We now discuss the stability of the vacuum including theφ, ∂ 2 ϕ, ∂ 3 ϕ-directions. The local stability is guaranteed by the condition M| 0 ≥ 0. As discussed in section 2.2, since the "off-diagonal parts" of L mn vanish andφ =φ = 0, the matrices M mn are simply given by eq. (2.23) and the generalized mass matrix is which is totally block diagonal. One can easily find eigenvalues of each sector. The eigenvalues of M 00 | 0 are where we have defined Υ ≡ 3k(α + β) + (λ + µ) 2 + λ + µ, (4.33) and the eigenvalues of M 11 | 0 are The eigenvalues of M 22 | 0 and M 33 | 0 are respectively. The eigenvalue B 1 = 0 corresponds to the expected zero mode. Later, we will confirm that B 1 = 0 is indeed the generalized NG mode and discuss the meaning of the other zero modes for C 1 = 0. One finds a parameter region for which all the nonzero eigenvalues become positive. For example, if we consider β = µ = 0, 3 the eigenvalues become One can easily show that all the nonzero eigenvalues A 2 , B 2 , C 2 are positive in the region α < 0, λ > 0 and λ 2 + 3αk > 0, and the generalized mass matrix M| 0 is positive semidefinite. Thus in this region the spatially modulated configuration (4.20) is a stable vacuum in the model (4.4).
Generalized NG modes Next, we examine the generalized NG modes in the spatially modulated vacuum (4.20). The field that has a nonzero VEV is ϕ 1 = ∂ 1 ϕ. The VEV is characterized by As discussed in section 2.2, the relevant symmetry breaking of the spatially modulated vacuum is The corresponding broken and unbroken generators are given in ref. [28] by respectively, where each generator acts on the VEV v = icϕ 0 e icx 1 as Then, the unbroken generator acts on the VEV as T ub v = 0. For the broken generator, we have On the other hand, the normalized eigenvector of M 11 | 0 associated with B 1 = 0 is found to be One then finds that is the generalized NG mode. A comment on the extra zero modes C 1 = 0 appearing in the i = 2, 3 sectors is in store. We have chosen the x 1 -direction as a specific direction of the spatial modulation. We have then found that the configuration ∂ 1 ϕ = 0 is a (local) minimum of the energy density and there is a flat direction corresponding to the zero modes of M 11 | 0 . However, since the model that we consider is Lorentz invariant, the spatial directions x 1 , x 2 , x 3 are equivalent due to the SO(3) ∈ SO(1, 3) rotational symmetry. Therefore there should be flat directions even in the ∂ 2 ϕ, ∂ 3 ϕ-directions in the energy density. In this sense, the zero modes corresponding to C 1 = 0 are accompanying NG modes to the legitimate generalized NG mode in x 1 -direction. On the other hand, the zero mode A 1 = 0 in eq. (4.36) is genuinely accidental since it is nonzero for general parameters and appears only for the specific choice of parameters µ = β = 0.
Lagrangian for the fluctuation modes We will now write down the Lagrangian for the fluctuation modes in the spatially modulated vacuum. The Lagrangian (2.17) becomes where we have ignored the irrelevant constant. The eigenvalues of L 00 are positive and those of L ii = −M ii (i = 1, 2, 3) are negative or zero in the appropriate parameter region. Therefore, the fluctuation fields have the correct signs of their canonical kinetic terms and we expect that there are no dynamical ghost modes in the Lagrangian. Since M 00 = L 00 , M ii = −L ii (no summation over i), the zero modes of L ij and M ij coincide. We therefore conclude that the kinetic term of the generalized NG mode vanishes for the spatially modulated vacuum. As mentioned above, the results in this subsection are essentially the same as those in ref. [28]. However, the situation drastically changes when we consider modulations in the temporal and the light-like directions, which we shall turn to next.

Temporal modulation
For the temporal modulation, we assume the Ansatz for the VEV: with real constants ϕ 0 , ω. Assuming this configuration, the matrices L mn can be evaluated as L 00 = k + 4(λ + µ)|φ| 2 − 9(α + β)|φ| 4 2φ 2 −3(α + β)|φ| 2 + (λ + µ) 2φ 2 −3(α + β)|φ| 2 + (λ + µ) k + 4(λ + µ)|φ| 2 − 9(α + β)|φ| 4 , and the others are zero. We also have the matrices Let us analyze the conditions (2.4), (2.9). Since ∂L ∂(∂ i ϕ) = 0 is trivially satisfied, these conditions reduce to Sinceφ = iωφ,φ = −iωφ, the above conditions can be rewritten as Hence, they are satisfied when L 00 = 0. This is consistent with condition (2.32) in the general discussion. The condition L 00 = 0 imposes that the two independent components of L 00 in eq. (4.45) vanish. This is possible when the parameter k is chosen as For this choice of parameters, the solution to the condition L 00 = 0 is . (4.51) Note that since k is the coefficient of the canonical (quadratic) kinetic term, it should be positive k > 0. Then we find α + β < 0 and λ + µ < 0 are necessary for the temporal modulation. Global aspects of the potential and stability of the vacuum The energy density for the Ansatz (4.44) is This is a function of X ≡ |φ| 2 ≥ 0: y = X k + 3(µ + λ)X − 5(α + β)X 2 . Besides the trivial vacuum X = 0, there are nontrivial extrema. The discriminant D for y = k + 6(λ + µ)X − 15(α + β)X 2 = 0 is This is always positive D > 0. Therefore, there is a local minimum in the (X = |φ| 2 )direction, as expected. The solutions to y = 0 are . (4.54) The former is a local maximum while the latter is the temporally modulated vacuum (4.51). In order to clarify the (meta)stability of the temporally modulated vacuum, we examine the zeros of f which is defined by y = Xf . Since the discriminant of f ≡ −5(α + β)X 2 + 3(µ + λ)X + k = 0 is always positive, the vacuum (4.51) is energetically favored compared with the trivial vacuum X = 0. Thus, it is the global minimum at least in the |φ|-direction (see fig. 2). We note that the situation is different from the spatial case. In order that the temporally modulated vacuum is allowed, the coefficient k of the canonical kinetic term should be related to those of the higher-derivative terms via the relation (4.50) and α + β < 0, λ + µ < 0. Therefore, the model is severely constrained and there are no possibilities for having metastable or degenerate vacua as in the spatial modulation case.
Since ∂L 0i ∂φ = 0, ∂L ij ∂φ = 0 (i = j) for the Ansatz (4.44), we have the mass matrices (4.46), (4.47). Hence, the generalized mass matrix is where k is given in eq. (4.50). The eigenvalues of M 00 are There is a zero mode and A 2 > 0 as expected. The eigenvalues of M 11 , M 22 , M 33 are (4.58) One can always choose the parameters so that B 2 > 0 by choosing α + β < 0, λ + µ < 0. 4 Generalized NG modes The symmetry breaking pattern is essentially the same with the spatial case: The corresponding broken and unbroken generators are given by respectively, where each generator acts on the VEV v 0 = φ = iωϕ 0 e iωx 0 as (4.60) Then, we have T ub v = 0, and On the other hand, the normalized eigenvector of M 00 associated with the zero eigenvalue A 1 = 0 is found to be One therefore finds that is a generalized NG mode. 4 Note that in the supersymmetric limit µ, β → 0, we have an extra zero mode B2 = 0, see ref. [29].
The zero modes corresponding to B 1 = 0 are accidental since the normalized eigenvectors of M 11 = M 22 = M 33 are Hence, the zero modes in M 11 | 0 = M 22 | 0 = M 33 | 0 do not correspond to eigenvectors proportional to the vector generated by the broken generator T b v.
Lagrangian for fluctuation modes For the temporal modulation, there are no direct correspondences between the fluctuation modes and the zero modes of M| 0 , i.e. the generalized NG modes. This is because the matrices L mn are different from the components of the generalized matrix M mn .
Since the matrix L 00 vanishes in the temporally modulated vacuum, there is no canonical kinetic term for the fluctuation in the temporal direction. Only the gradient kinetic term survives in the fluctuation dynamics. The Lagrangian (2.17) becomes, In order to clarify whether physically unacceptable modes appear or not, we need the eigenvalues of the matrices where k = − (λ+µ) 2 3(α+β) , |φ| 2 = λ+µ 3(α+β) . The eigenvalues/vectors are found to be We can always choose the parameters so that s 2 < 0 by obeying the conditions α + β < 0, λ + µ < 0. 5 Therefore we can exclude unphysical ghost modes in appropriate regions of parameters.

Light-like modulation
For the light-like modulation, we assume the VEV: This implies ∂ϕ · ∂ϕ| 0 = ∂ϕ · ∂φ| 0 = 0. (4.69) In the following, we can choose the + sign in eq. (4.24) without loss of generality. With the Ansatz (4.68), the matrices L mn can be evaluated as and the others are zero. We further obtain  (4.72) and the others are zero. The conditions (2.4), (2.9) reduce to L 00 φ ϕ = 0, (4.74) (4.75) Here we have used ∂L ∂(∂ 2 ϕ) = ∂L ∂(∂ 3 ϕ) = 0 by assuming the Ansatz. Since we have the condition (4.75) becomes Again, by the Ansatz we have |φ| 2 = |∂ 1 ϕ| 2 . Then, the condition (4.74) becomes k + 4µ|φ| 2 + 4λ|φ| 2 φ k + 4µ|φ| 2 + 4λ|φ| 2 φ = 0. For the light-like modulation, we haveφ = ∂ 1 ϕ. Therefore, the condition (4.73) becomes However, this condition is automatically satisfied since the matrix L 00 + L 01 + L 10 + L 11 is identically zero by explicit calculations. Therefore, the light-like modulation is possible when the parameters of the model satisfy eq. (4.79). Apparently, the condition (4.79) is weaker than the condition k = λ = µ = 0 determined by the general discussion in section 2.2. The light-like modulation requires k = µ + λ = 0 but α, β can be nonzero. Note that for the light-like modulation, the amplitude ϕ 0 is not determined but is a free parameter. This is different from the spatial and the temporal modulation cases. A comment is in order for the light-like modulation. The model seems to be a little bit strange since the canonical kinetic term disappears in the Lagrangian (4.4). Although the Lagrangian contains only the 4th and the 6th order derivative terms, we will show that the dynamical (fluctuation) field has quadratic kinetic term in the Lagrangian.
Global aspects of the potential and stability of the vacuum For the light-like modulation, the energy density becomes constant: Therefore, there is no "potential" when we consider the Ansatz (4.68). This is why the amplitude ϕ 0 is not determined. The generalized mass matrix is where each matrix is obtained from eq. (4.72) by setting k = µ + λ = 0 as The eigenvalues of the matrices M 22 = M 33 are In order that all the nonzero eigenvalues become positive, we must require that It is always possible to choose the parameters such that they satisfy the above conditions.
Generalized NG modes For the light-like modulation, the symmetry breaking pattern is U (1) × P 0 × P 1 → [U (1) × P ± ] diag where P ± represent the translational symmetry group along the light-cone directions x ± = x 0 ± x 1 . The corresponding broken and unbroken generators are given by respectively. In the following, we choose the + sign in P ± . The actions of these generators on the VEV φ = ∂ 1 ϕ = v LL = iωϕ 0 e iω(x 0 +x 1 ) are given by (4.89) Thus, we find T ub v = 0 and On the other hand, the zero eigenvalues A 1 = A 2 = 0 of M are degenerate and whose eigenvectors are It is clear that the following linear combination of u 1 and u 2 defines a zero mode of M: Hence, we find the generalized NG mode is a zero mode of M : We calculate the eigenvalues of the following matrix: L ≡ L 00 | 0 L 01 | 0 L 10 | 0 L 11 | 0 , (4.95) where L 00 | 0 = L 11 | 0 = 2λ |φ| 2φ2 ϕ 2 |φ| 2 , L 01 | 0 = L 10 | 0 = −L 00 | 0 . Here we have ωϕ 0 = |φ|. Since λ > 0, we have s 4 > 0. There is only one fluctuation mode whose kinetic term has a positive coefficient. It is, however, not obvious whether the mode is physical or unphysical. We can say that it is not an unstable mode at least in the parameter region given by eq. (4.87).

Summary and discussion
In this paper, we have investigated modulated vacua in Lorentz invariant scalar field theories. We have found the general conditions for modulated vacua. We also have discussed a generalization of the NG theorem and have clarified the relation between the zero modes of the generalized mass matrix and the generalized NG modes. According to the spontaneous symmetry breaking patterns, the modulated vacua have been classified into the spatial, temporal and light-like ones. For the spatial and temporal modulations, the diagonal component of the U (1) and translational symmetries along the x 1 or x 0 direction are preserved. For the light-like case, the diagonal component of the U (1) and the translation along the light-cone direction is preserved. We have presented the general procedure to analyze each modulated vacuum. We have then demonstrated these modulated vacua in a specific model where up to 6th-order derivative terms of a complex scalar field are included. The model is a generalization of the one considered in ref. [28] where the spatially modulated vacuum was analyzed. We have shown that there are models that allow for temporal and light-like modulated vacua. The global structure of the energy potential in the temporal case is severely restricted, namely, the temporally modulated state is a global vacuum at least in the |φ| direction. This is different from the spatially modulated case, for which there is a parameter region where the vacuum is meta-stable, degenerate or global minimum. We have found that the light-like modulation is allowed in the model where only the quartic and sextic terms remain. We have then identified the generalized NG modes in each modulation. We also have written down the Lagrangians of the fluctuation modes in each modulated vacuum. As discussed in the previous paper [28], the kinetic terms of the generalized NG modes disappear in the spatial modulation. The absence of ghosts is guaranteed due to the positive semi-definiteness of the generalized mass matrix M. In the temporal case, the kinetic term of the fluctuation along the time direction vanishes, but this does not correspond to the generalized NG mode in general. We have found that there is a possible parameter choice for which no ghosts occur in the temporal modulation. For the light-like case, the gradient kinetic term along the x 2 , x 3 -directions are absent while there is only one mode that has a nonzero kinetic term in the x 0 , x 1 -directions. There are no obvious conditions for the absence of ghosts. However, we have pointed out that there is a parameter region where no ghosts appear in the fluctuation.
In this paper, we have considered only the FF-type modulation, i.e. the modulation of the phase of fields. It would be interesting to study a possibility of the LO-type where the amplitude of the fields is modulated. This could be realized by including a potential term. We have studied only the quadratic order of the fluctuations. Even though, part of the fluctuation modes are absent in each modulated vacuum, there are possible kinetic terms at higher orders. It would be interesting to study the dynamics based on these modes.
The modulated vacua in a Lorentz invariant setup studied in this paper are relatively overlooked possibilities of vacuum structures. As an application of the present discussions, we will consider a supersymmetric extension of the present model, which is possible for β = µ = 0, and will investigate supersymmetry breaking in modulated vacua [29,35]. To this end, we can embed our model in higher-derivative chiral models formulated in refs. [36][37][38]. A possibility to use such supersymmetry breaking vacua as a hidden sector for phenomenological models is an interesting future direction.
A generalization of the NG theorem in the classical regime has been discussed in this paper. It would also be important to investigate quantum nature of the generalized NG modes.