1 Introduction

In most of real-word materials, the translational symmetry in spatial dimensions are inevitably broken both spontaneously and explicitly (In this paper, we call them SSB and ESB for short.) due to the existence of periodic lattice, striped orders, impurities, defects, etc. For crystalline states, the Goldstone bosons associated with the SSB of translations are usually called transverse and longitudinal phonons, all of which have linear dispersion relations and propagate freely at certain speed in the clean materials. While, for liquid states with SSB of translations, there is only one longitudinal phonon since the shear stresses cannot be supported.

In strongly interacting electronic systems, Goldstone modes and electrons can be mightily coupled which gives rise to novel collective behaviors and exotic transport properties [1,2,3,4]. To have a deeper understanding on such patterns, building a framework beyond the conventional perturbative methods has already become an important mission in condensed matter physics.

Holographic duality provides a tractable approach to the physics of strong correlated systems by mapping the many-body problems to classical gravity problems. Recently, some holographic effective models for solid states that spontaneously break the translations have been constructed [6,7,8,9,10,11,12,13,14,15]. A common feature in these models is that the translations are spontaneously broken in a homogenous manner(For inhomogeneous realizations, one can refer to [16,17,18,19,20,21,22,23,24,25]). This significantly simplifies the calculations and makes it possible to dissect the key properties of the system, say, the transport, in analytic ways. Recently, the homogeneous realization of phonons and pseudo-phonons has also been investigated in the field theory side [26].

In this paper, we mainly consider a new simple holographic models which can realize the liquid states with SSB of the translations, by introducing a special gauge-axion coupling. In the absence of relaxation, it is found that the background metric as well as the gauge field are exactly the same as the Reissner-Nodström(RN) black holes, while the profile of the bulk axions plays the role of the scalar condensate that breaks the translations. On top of this, we investigate the imprints of the transverse Goldstone modes on the optical electric conductivity. The plan of this work is as follows: In Sect. 2, we construct the simplest holographic model with a gauge-axion coupling and explain how the SSB of translations can happen in this model via analyzing the UV expansion of the bulk fields. In Sect. 3, we compute the electric conductivity in the purely SSB pattern both analytically and numerically. In Sect. 4, we consider the pinned modes in the presence of relaxation. In Sect. 5, we conclude.

2 Goldstone modes by gauge-axion coupling

To break the translations in an isotropic and homogeneous manner, one needs to introduce massless axion fields with a bulk profile \(\phi ^I=k\,\delta ^I_i\, x^i\). Essentially, these scalar fields give the gravitons an effective mass, hence breaking the diffeomorphism invariance of the bulk theory [27]. In this way, we can in principle construct a series of general effective holographic models with higher derivative terms [28]. Here, we will focus only on the following simple model with a special gauge-axion coupling:

$$\begin{aligned} \mathcal {S}&= \int \,d^4x\,\sqrt{-g}\nonumber \\&\quad \times \left( \,R\,-\,2\,\Lambda \,-\,\lambda X\,-\,\frac{1}{4}\,F^2\,-\,\frac{\mathcal {J}}{4}\,Tr \left[ \mathcal {X}\,F^2\right] \,\right) \end{aligned}$$
(1)

where the U(1) gauge field \(F_{\mu \nu }\equiv \nabla _\mu A_\nu -\nabla _\nu A_\mu \),

$$\begin{aligned} Tr[\mathcal {X}\,F^2]\equiv & {} {\mathcal {X}^\mu }_\nu {F^{\nu }}_{\rho } {F^\rho }_{\mu },\quad \text {with} \; {\mathcal {X}^\mu }_\nu \nonumber \\= & {} \frac{1}{2}\sum _{I=x,y}\partial ^\mu \phi ^I \partial _\nu \phi ^I, \end{aligned}$$
(2)

and \(X\equiv Tr[\mathcal {X}]\). We require that \(\lambda \ge 0\) for avoiding the ghost problem and a necessary condition \(0\le \mathcal {J}\le 2/3\) for unitarity and causality [28]. For convenience, we set the cosmological constant \(\Lambda =-3\) which implies a normalized AdS radius. Then, if taking \(\mathcal {J}=0\), it reduces to the simplest linear axion model.

From the action above, the covariant form of the equations of motion are given by

$$\begin{aligned}&\nabla _\mu \left[ F^{\mu \nu }-\frac{\mathcal {J}}{2}\left( (\mathcal {X}F)^{\mu \nu } -(\mathcal {X}F)^{\nu \mu }\right) \right] =0, \end{aligned}$$
(3)
$$\begin{aligned}&\nabla _\mu \Big [\lambda \nabla ^\mu \phi ^I+\frac{\mathcal {J}}{4} {(F^2)^\mu }_\nu \nabla ^\nu \phi ^I\Big ]=0, \end{aligned}$$
(4)

and

$$\begin{aligned}&R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R-\frac{\lambda }{2} \nabla _\mu \phi ^I\nabla _\nu \phi ^I-\frac{1}{2}\left( 6-\frac{\lambda }{2}\nabla _\sigma \phi ^I \nabla ^\sigma \phi ^I\right) g_{\mu \nu }\nonumber \\&\quad =\frac{1}{2} \Big ({F_\mu }^{\sigma }F_{\nu \sigma }-\frac{1}{4}g_{\mu \nu }F_{\rho \sigma }F^{\rho \sigma } \Big ) +\frac{\mathcal {J}}{4}\Big (\frac{1}{2}\nabla _{(\mu |} \phi ^I\nabla _\sigma \phi ^I{(F^2)^{\sigma }}_{|\nu )} \nonumber \\&\qquad +F_{(\mu |\sigma }{(F\mathcal {X})^{\sigma }}_{|\nu )}+F_{(\mu |\sigma }{(\mathcal {X} F)^{\sigma }}_{|\nu )}-\frac{1}{2}g_{\mu \nu } Tr\left[ \mathcal {X}F^2\right] \Big ).\nonumber \\ \end{aligned}$$
(5)

As is known that this model has the following isotropic charged black hole solutions:

$$\begin{aligned} ds^2= & {} -D(r)\,dt^2+B(r)\,dr^2+C(r)\, dx^idx_i, \nonumber \\ A_\mu= & {} A_t (r)\,dt, \quad \ \phi ^I=(0,\,0,\,k\,x,\,k\,y), \end{aligned}$$
(6)

where \(i=2,3\) denotes the two spatial directions. Choosing such radial coordinate r that the AdS boundary is located at \(r=0\), in the asymptotic region the background solution behaves like

$$\begin{aligned} D(r)&=\frac{1}{r^2}\left( 1-d_{(3)}\,r^3+\cdots \right) ,\nonumber \\ B(r)&=\frac{1}{r^2}(1+d_{(3)}\,r^3+\cdots ),\nonumber \\ C(r)&=\frac{1}{r^2},\nonumber \\ A_t(r)&=\mu -\rho \, r+\cdots , \end{aligned}$$
(7)

where the coefficient \(d_{(3)}\) is associated with the energy density, \(\mu \) and \(\rho \) are the U(1) chemical potential and the charge density in the boundary theory.

To explain why the SSB of translations can be realized in the holographic model (2), we firstly explain what role the profile of the scalars \(\phi ^i=k\,x^i\) plays in the following two different cases. Without loss of generality, let us now assume that \(\phi ^I\) depend on the full coordinates \(x^\mu \). If we set \(\lambda \ne 0\) and \(\mathcal {J}=0\) in (4), the asymptotic behavior of \(\phi ^I\) near the UV boundary is

$$\begin{aligned} \phi ^I=\phi _{(0)}^I(t,x^i)+\phi _{(3)}^I(t,x^i)\,r^3+\cdots . \end{aligned}$$
(8)

Then, the \(r-\)independent term \(\phi _{(0)}^i(t,x^i)\) dominates the second one, and hence plays the role of an external source that breaks the translations in the standard quantization. Obviously, the profile \(\phi ^i=k\,x^i=\phi _{(0)}^i\) means that the ESB of translations happens. And this has already been widely investigated in previous holographic studies on momentum relaxation [29,30,31].

Conversely, if we instead set \(\lambda =0\) and \(\mathcal {J}\ne 0\), the scalars behave likeFootnote 1

$$\begin{aligned} \phi ^I=\frac{\phi _{(-1)}^I(t,x^i)}{r}+\phi _{(0)}^I(t,x^i)+\cdots . \end{aligned}$$
(9)

In this case, the \(r-\) independent term is subleading and corresponds to the expectation value of a dual operator \(\mathcal {O}^I\).Footnote 2 Now, \(\phi ^i=k\,x^i\) should be interpreted as the expectation value \(\langle \mathcal {O}^i\rangle \sim k\,x^i\) with a vanishing source, i.e. \(\phi _{(-1)}^i=0\). Since such a scalar condensate \(\langle \mathcal {O}^i\rangle \) is not uniform in \(x^i\), the translational symmetry is broken spontaneously. The Nambu-Goldstone theorem claims that there should exist gapless excitations in the low energy description which are called Goldstone modes. With \(\lambda =0\), the background solution is given by

$$\begin{aligned} \begin{aligned} D(r)&=\frac{f(r)}{r^2}=\frac{1}{r^4B(r)},\\ f(r)&=1\,-\,\frac{r^3}{r_h^3}\,-\,\frac{\mu ^2\,r^3}{4\,r_h} \left( 1\,-\,\frac{r}{r_h}\right) ,\\ C(r)&=\frac{1}{r^2}, \quad A_t(r)=\mu -\rho \,r, \end{aligned} \end{aligned}$$
(10)

where \(r_h\) denotes the location of the horizon. Note that, unlike other massive gravity models, the background metric as well the gauge field do not depend on k. While the scalars can still have the non-trivial profile \(\phi ^i=k x^i\). Requiring the gauge field to be regular on the horizon gets \(\mu =\rho \,r_h\). Finally, the Hawking temperature is given by

$$\begin{aligned} T=\frac{3}{4\pi r_h}-\frac{\mu ^2 r_h}{16\pi }, \end{aligned}$$
(11)

which is also the same as the RN black holes. If we perturb the fields around the non-trivial solution like \(\phi ^I=k x^i+\chi ^I\) and \(A_\mu =A_t+a_{\mu }\), the axion-gauge term can be expanded as

$$\begin{aligned} \mathcal {L}_{\chi }= & {} -\frac{\mathcal {J}r^4\rho ^2}{8}\left[ (\dot{\chi }^I)^2 +({\chi ^I}')^2\right] +\cdots \end{aligned}$$
(12)

where the dots represents the higher derivative terms associated with \(\chi ^I\) and \(a_{\mu }\) interactions. The leading term in the action is quadratic and the dynamics of the Goldstone modes in the dual boundary theory is encoded in the eom of \(\chi ^I\). Note that this story happens only for finite density cases which is the similar case as the gapless sliding modes of charge density waves [32], however in contrast to the acoustic phonons which do not carry U(1) charges. For zero density case, the scalars \(\chi ^i\) will be decoupled from the other fluctuating fields in the bulk and will not affect the charge transport. In the next section, we will investigate the electric conductivity in the clean case.

Fig. 1
figure 1

Left plot: \(\text {Im}(\sigma )\) as the function of \(\omega /\mu \) for different temperature. The inset plot shows the Drude weight, \(\chi _{JP}^2/\chi _{PP}\), as the function of \(T/\mu \). Blue line is the analytical result obtained from Eq. (19) and the red dots are the numerical result. Right plot: The incoherent conductivity \(\sigma _0\) as the function of \(T/\mu \). Blue line is the analytical result obtained from Eq. (20) and the red dots are the numerical result. Here, we have set \(\mathcal {J}=1/3\) and \(k/\mu =1\)

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3 Electric conductivity

We now turn to study small fluctuations around the background solution. We denote \(g_{\mu \nu }=\bar{g}_{\mu \nu }+\delta g_{\mu \nu }\), \(A_\mu =\bar{A}_{\mu }+\delta A_{\mu }\) and \(\phi ^i=\bar{\phi }^i+\chi ^i\), where the quantities with bars are evaluated on the background, and introduce the time-dependent perturbations as follows

$$\begin{aligned} \delta A_\mu (t,r,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega d^{2}p_i}{(2\pi )^3}e^{-i \omega t+ip_ix^i}a_\mu (r),\nonumber \\ \delta g_{\mu \nu }(t,r,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega d^{2}p_i}{(2\pi )^3}e^{-i \omega t+ip_ix^i}r^{-2}h_{\mu \nu }(r),\nonumber \\ \chi ^I(t,r,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega d^{2}p_i}{(2\pi )^3}e^{-i \omega t+ip_ix^i}\psi ^I(r). \end{aligned}$$
(13)

To derive the conductivity, we focus on the homogeneous vector modes, namely setting all the momenta \(p_i=0\). Since the system is isotropic, we only need to consider the x-component of the vector modes, namely \(a_x\), \(h_{tx}\), \(h_{rx}\) and \(\psi ^{x}\). The linearized Maxwell, scalar equation and Einstein equations read

$$\begin{aligned}&f a_x''-\frac{\mathcal {J}}{4} k^2 r^2 f a_x''-\frac{1}{4} \mathcal {J} k^2 r^2a_x' f'+a_x' f'\nonumber \\&\quad -\frac{1}{2} \mathcal {J} k^2 r f a_x'-\frac{\mathcal {J} k^2 r^2 \omega ^2 a_x}{4f}+\frac{\omega ^2a_x}{f}\nonumber \\&\quad +\frac{i}{4} \mathcal {J} k^2 \rho r^2 \omega h_{rx}-i \rho \omega h_{rx}+\frac{1}{4} \mathcal {J} k^2 \rho r^2 h_{tx}'-\rho h_{tx}'\nonumber \\&\quad +\frac{1}{4} \mathcal {J} k^2 \rho r h_{tx}'+\frac{1}{2} i \mathcal {J} k \rho r \omega \psi ^x=0, \end{aligned}$$
(14)
$$\begin{aligned}&f{\psi ^x}''-\frac{2 i k \omega a_x}{\rho r}-k h_{rx} f'+f' {\psi ^x}'-k fh_{rx}'\nonumber \\&\quad -\frac{2 k f h_{rx}}{r}-\frac{i k \omega h_{tx}}{f}+\frac{2 f {\psi ^x}'}{r}+\frac{\omega ^2 \psi ^x}{f}=0, \end{aligned}$$
(15)
$$\begin{aligned}&\quad -\frac{i \mathcal {J} k^2 \rho r^4 \omega a_{x}}{4 f}+\frac{i \rho r^2 \omega a_{x}}{f}+\frac{\omega ^2 h_{rx}}{f}-\frac{i \omega h_{tx}'}{f}\nonumber \\&\quad -\frac{\mathcal {J}}{4} k^2 \rho ^2 r^4 h_{rx}+\frac{1}{4} \mathcal {J} k\rho ^2 r^4 {\psi ^x} '=0, \end{aligned}$$
(16)
$$\begin{aligned}&f h_{tx}''+\frac{\mathcal {J}}{4} k^2 \rho r^4 f a_x'-\rho r^2 f a_x'+i \omega f h_{rx}'-\frac{2 i \omega f h_{rx}}{r}\nonumber \\&\quad -\frac{2 f h_{tx}'}{r}-\frac{\mathcal {J}}{4} k^2 \rho ^2 r^4 h_{tx}-\frac{i \mathcal {J}}{4} k \rho ^2 r^4 \omega {\psi ^x}=0, \end{aligned}$$
(17)

In this case, the mass of the spin-1 gravitons should be read as \({\mathcal {M}(r)}^2=\mathcal {J}\frac{k^2\rho ^2r^4}{4}\), which varies non-trivially along the radial direction [28]. In particular, its value in the UV and IR satisfies that

$$\begin{aligned} \mathcal {M}(0)=0,\,\,\,\,\text {and}\,\,\,\,\mathcal {M}(r_h)=\left( \mathcal {J}\frac{\pi k^2\mu ^2}{s}\right) ^{1/2}\sim \text {finite}.\nonumber \\ \end{aligned}$$
(18)

As is pointed in the previous holographic study, this is exactly a condition for realizing the gapless Goldstone bosons [8]. The optical conductivity can be achieved numerically by setting the infalling boundary conditions at the horizon and solve the linearized equations of motion in the bulk. Since the electric current is a vector operator, the conductivity is sensitive to the transverse Goldstone modes but cannot mix with the longitudinal component which is a scalar mode. Then, one can directly read the information about the transverse modes from the conductivity.Footnote 3

Without explicit breaking, the optical conductivity of a relativistic system in the hydrodynamic limit can be written as [1]

$$\begin{aligned} \sigma (\omega )\xrightarrow {\omega \rightarrow 0}\sigma _0+\frac{\chi _{JP}^2}{\chi _{PP}}\frac{i}{\omega } \end{aligned}$$
(19)

where \(\chi _{JP}=\rho \), \(\chi _{PP}=\epsilon +P\) in our case and the finite part \(\sigma _0\) is the incoherent conductivity that is theory-dependent and irrelevant to the momentum relaxation. Unlike the DC conductivity, the incoherent one can always be achieved via the membrane paradigm, hence is UV insensitive from the RG perspective. In our model, it can be obtained similarly as in [10, 11]:

$$\begin{aligned} \sigma _0=\left( \frac{sT}{sT+\mu \rho }\right) ^2\left( 1-\mathcal {J}\frac{\pi k^2}{s}\right) , \end{aligned}$$
(20)

where the thermodynamic relation \(sT+\mu \rho =\epsilon +P\) has been applied. Eq. (19) means that the real part of conductivity has a delta infinity at zero frequency in the purely SSB pattern due to the absence of momentum relaxation.

In Fig. 1, we show that the numeric result of our holographic model with \(\lambda =0\) agrees with (19) and (20) very well. This again indicates that the background profile of the scalars \({\bar{\phi }}^i=kx^i\) should be interpreted as breaking the translational symmetry spontaneously rather than explicitly. And the Goldstone modes contribute the correction term in (20) to the incoherent conductivity that is controlled by the parameters \(\mathcal {J}\) and k.

The Goldstone modes associated with the SSB of translations in crystals are usually called phonons. Nevertheless, the transverse gapless modes in this model are not phonon like. For transverse and longitudinal phonons, they have the linear dispersion relations \(\omega \sim v_{T,L}\,p\) with finite sound speeds \(v_{T,L}\). Setting \(p_y=p\ne 0\) and repeat the numerical calculations(To do so, we adopt the gauge invariant formulation of the bulk fields like in [33], and solve three coupled equations of the shear modes numerically.), we find that the infinite peak does not move away from \(\omega =0\) as the momentum p is increased. In Fig. 2, we show the imaginary part of the conductivity with the varying momentum.

Therefore, the Goldstone modes are not dispersive, which means there is no freely propagating transverse phonons. This can also be understood in another way: For phonons, their sound speeds are related to the shear modulus \(\mathcal {G}\) and the bulk modulus \(\mathcal {K}\) by [34]

$$\begin{aligned} v_T^2=\frac{\mathcal {G}}{\chi _{PP}},\,\,\,\,\, v_L^2=\frac{\mathcal {G}+\mathcal {K}}{\chi _{PP}}. \end{aligned}$$
(21)

One can however check that in our model the \(\mathcal {J}\) coupling does not contribute a finite mass term to the spin-2 gravitons \(\delta g_{xy}\). Then, we conclude that the shear viscosity obeys the KSS bound \(\eta =\frac{s}{4\pi }\) and the shear modulus \(\mathcal {G}=0\) which is the case for a strongly coupled fluid instead of a solid. First relation in (21) implies that \(v_T=0\), the transverse modes are not phonon like. This does not conflict with the common sense that there is no transverse phonons in a fluid due to the lack of shear stress. Therefore, this holographic model provides a low energy description for the gapless Goldstone modes coupled with a conformal fluid. In the next section, we will further study the optical conductivity in the presence of the explicit source that breaks the translations. The numeric result of the holographic model captures another key feature of the SSB of translations which is called pinning effect.

Fig. 2
figure 2

\(\text {Im}(\sigma )\) as the function of \(\omega /\mu \) for different momenta. Here, we have set \(\mathcal {J}=1/3\), \(k/\mu =1\) and \(T/\mu =1\). For all the cases, there is always an infinite pole standing still at zero frequency

Full size image
Fig. 3
figure 3

Left plot: \(\text {Re}(\sigma )\) as the function of \(\omega /\mu \) for different values of \(\lambda \). Right plot: The pinning frequency as the function of \(\lambda \). The solid lines are the fitting results which show that \(\omega _0/\mu \approx 0.272532 \,\lambda ^{0.125} \) in the region \(\lambda \in [5\times 10^{-4},3\times 10^{-2}]\). Here, we have fixed \(\mathcal {J}=1/3\), \(k/\mu =1\) and \(T/\mu =0.005\)

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Fig. 4
figure 4

Left plot: \(\text {Re}(\sigma )\) as the function of \(\omega _0/\mu \) for \(\lambda =0.1\) and various values of \(T/\mu \). It shows that \(d\sigma _{dc}/dT>0\). Right plot: The optical conductivity for \(\lambda =0.01\), \(T/\mu =0.005\) and various values of \(p/\mu \). Note that the pinned modes are dispersionless which is the same as the purely SSB pattern. However, there appears a gapless peak again for large values of \(p/\mu \). The underlying physics still remains to be revealed in future. In both plots, we have fixed \(\mathcal {J}=1/3\), \(k/\mu =1\)

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4 Pinning effect

Now, we consider how the peak of the goldstone bosons moves in the presence of ESB. According to the UV analysis in Sect. 2, such a pinning effect can be realized by setting a non-zero value of \(\lambda \), or the external source equivalently. In consequence, the infinite delta at zero frequency should be removed and there will be a sharp but finite peak at a certain frequency (called pinning frequency) in the optical conductivity.

When \(\lambda \ne 0\), the blackfactor f(r) becomes

$$\begin{aligned}&f(r)=1\,-\,\frac{r^3}{r_h^3}\,-\,\left( \frac{\lambda k^2 r^2}{2}+\frac{\mu ^2\,r^3}{4\,r_h}\right) \,\left( 1\,-\,\frac{r}{r_h}\right) . \end{aligned}$$
(22)

And the linearized equations of motion:

$$\begin{aligned}&f a_x''-\frac{\mathcal {J}}{4} k^2 r^2 f a_x''-\frac{1}{4} \mathcal {J} k^2 r^2a_x' f'+a_x' f'\nonumber \\&\quad -\frac{1}{2} \mathcal {J} k^2 r f a_x'-\frac{\mathcal {J} k^2 r^2 \omega ^2 a_x}{4f}+\frac{\omega ^2a_x}{f}\nonumber \\&\quad +\frac{i}{4} \mathcal {J} k^2 \rho r^2 \omega h_{rx}-i \rho \omega h_{rx}+\frac{1}{4} \mathcal {J} k^2 \rho r^2 h_{tx}'-\rho h_{tx}'\nonumber \\&\quad +\frac{1}{4} \mathcal {J} k^2 \rho r h_{tx}'+\frac{1}{2} i \mathcal {J} k \rho r \omega \psi ^x=0, \end{aligned}$$
(23)
$$\begin{aligned}&f{\psi ^x}''-\frac{2 i k \omega a_x}{\rho r}-k h_{rx} f'+f' {\psi ^x}'-k fh_{rx}'\nonumber \\&\quad -\frac{2 k f h_{rx}}{r}-\frac{i k \omega h_{tx}}{f}+\frac{2 f {\psi ^x}'}{r}+\frac{\omega ^2 \psi ^x}{f} -\frac{4\lambda k f' h_{rx}}{\mathcal {J} \rho ^2 r^4} \nonumber \\&\quad +\frac{4 \lambda f'{\psi ^x}'}{\mathcal {J} \rho ^2 r^4}-\frac{4\lambda k f h_{rx}'}{J \rho ^2 r^4}+\frac{8 \lambda k f h_{rx}}{J \rho ^2 r^5}-\frac{4 \lambda i k \omega h_{tx}}{\mathcal {J} \rho ^2 r^4 f}\nonumber \\&\quad -\frac{8 \lambda f {\psi ^x} '}{J \rho ^2r^5}+\frac{4 \lambda f{\psi ^x} ''}{J \rho ^2 r^4}+\frac{4 \lambda \omega ^2 {\psi ^x}}{J \rho ^2 r^4 f}=0, \end{aligned}$$
(24)
$$\begin{aligned}&\lambda k{\psi ^x}'-\lambda k^2h_{rx}-\frac{i \mathcal {J} k^2 \rho r^4 \omega a_{x}}{4 f}+\frac{i \rho r^2 \omega a_{x}}{f}\nonumber \\&\quad +\frac{\omega ^2 h_{rx}}{f}-\frac{i \omega h_{tx}'}{f}-\frac{\mathcal {J}}{4} k^2 \rho ^2 r^4 h_{rx}+\frac{1}{4} \mathcal {J} k\rho ^2 r^4 {\psi ^x} '=0,\nonumber \\ \end{aligned}$$
(25)
$$\begin{aligned}&f h_{tx}''+\frac{\mathcal {J}}{4} k^2 \rho r^4 f a_x'-\rho r^2 f a_x'+i \omega f h_{rx}'-\frac{2 i \omega f h_{rx}}{r}\nonumber \\&\quad -\frac{2 f h_{tx}'}{r}-\frac{\mathcal {J}}{4} k^2 \rho ^2 r^4 h_{tx}-\frac{i \mathcal {J}}{4} k \rho ^2 r^4 \omega {\psi ^x}\nonumber \\&\quad -\lambda k^2h_{tx}-i\lambda k\omega \psi ^{x}=0, \end{aligned}$$
(26)

From the equations, the effective mass of the spin-1 gravitons should be identified as \(\mathcal {M}(r)^2\equiv k^2\left( \lambda +\frac{\mathcal {J} \mu ^2 r^2}{4}\right) \). With \(\lambda \ne 0\), the DC conductivity can be directly computed by the membrane paradigm [28]

$$\begin{aligned} \sigma _{DC}=\left( 1-\mathcal {J}\frac{\pi k^2}{s}\right) +\left( 1-\mathcal {J}\frac{\pi k^2}{s}\right) ^2\frac{\mu ^2}{\mathcal {M}(r_h)^2},\,\,\,\, \end{aligned}$$
(27)

The gapped modes can be identified with the peaks in the optical conductivity in the two plots of Fig. 3. As is expected, the pinning frequency becomes higher as the increase of \(\lambda \), i.e, the rate of momentum relaxation. However, the relation between \(\omega _0/\mu \) and \(\lambda \) exhibits a peculiar scaling that is different from the Gell-Mann-Okubo relation. In addition, it is obvious to see from the left plot of Fig. 4 that the pinning effect introduces a mechanism of transition from a metallic state(when \(\mathcal {J}=0\)) to an insulating state, which is quite similar as what happens in a doped Mott insulator [35]. Even though, the commensurability effect is in general absent in the holographic systems with homogeneity [36].

The quantitive relation between \(\lambda \) and the relaxation rate can be in principle checked by a full analysis on the quasi-normal modes of the black hole like in [8, 37, 38], which is however not a target in this work. Now, we consider how the propagation of the transverse modes would change when we vary the value of \(\lambda \). We turn on \(p\ne 0\) and obtain the optical conductivity with finite momentum in the right plot of Fig. 4. The numerical result shows that the peak of gapped modes becomes milder as increasing the momentum. However, these modes are still dispersionless, in contrast to the solid holographic massive gravity model [8, 33].

5 Conclusion and outlook

In this paper, we introduce a simple holographic model that can realize both the spontaneous and explicit breaking of the translational symmetry in the dual field theory. In this model, the SSB is induced by a gauge-axion coupling \(\mathcal {J}Tr[\mathcal {X}F^2]\), while the ESB can be realized by turning on the linear axion term \(\lambda X\).

When we turn off the external scalar source by setting \(\lambda =0\), the condensate of the dual operators that breaks the translations should be identified as the bulk profile of the axions, via the UV analysis. In this case, the background metric and gauge field is the same as the RN black holes. And the dynamics of the transverse Goldstone modes is encoded in the eoms of the spatial components of axions. Our numeric result of electric conductivity matches with that of a fluid with SSB of translations. We then turn on the explicit source to see its pinning effect on the Goldstone modes. It is found that the pinning frequency becomes higher as we increase the value of \(\lambda \). And the gapped modes are still dispersionless.

In this short paper, the analysis on the bulk mode is lacking. In fluids, there exists longitudinal phonons whose speed is related to both of the shear and bulk modulus. Then, the second relation in (21) can be checked by studying the coupled spin-0 fluctuations, which is more complicated.Footnote 4 We will leave this for future work [40].

Our model can be generalized, including the higher derivative terms like \(\sum _{n=2}^{\infty }Tr[\mathcal {X}^nF^2]\). One can, however, check that such terms do not change the UV expansion (9), hence will not change the story a lot, albeit further modifications on the incoherent conductivity. One can also consider another class of gauge-axion couplings, for instance, \(K\,Tr[X]F^2 \) [28]. This term will change the background solution and gives the system a non-zero shear modulus when \(\rho \ne 0\). Then the dual system may be interpreted as some kind of “electronic crystals”, whose impacts on the transport or elastic properties are also worth studying. In [41, 42], a general framework has been developed for computing the holographic 2-point function and the corresponding conductivities dual to a broad class of Einstein-Maxwell-Axion-Dilaton theories. We can generalize this study to include the higher derivative axion terms and see what novel phenomena emerge. In future, we will study this issue following the line in [41, 42].