# A simple holographic model for spontaneous breaking of translational symmetry

## Abstract

It has been shown that holographic massive gravities can effectively realize spontaneous breaking of translational symmetry in homogenous manners. In this work, we consider a toy model of such category by adding a special gauge-axion coupling to the bulk action. Firstly, we identify the existence of spontaneous breaking of translations by the analysis on the UV expansion. In the absence of explicit breaking, the black hole solution is simply the same as the Reissner-Nodström(RN) black holes, regardless of the non-trivial profile of the bulk axions. The associated Goldstone modes exist only when the charge density is non-zero. Then, we investigate the optical conductivity both analytically as well as numercially. Our result perfectly agrees with that for a clean system, while the incoherent conductivity gets modified due to the symmetry breaking. The transverse Goldstone modes are dispersionless since the solution is dual to a *liquid* state. Finally, the effect of momentum relaxation to the transverse modes is considered. In this case, the would-be massless modes are pinned at certain frequency, which is another key difference from the unbroken states.

## 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

*r*that the AdS boundary is located at \(r=0\), in the asymptotic region the background solution behaves like

*U*(1) chemical potential and the charge density in the boundary theory.

^{1}

^{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

*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

*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.

## 3 Electric conductivity

*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

*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.

^{3}

*relativistic*system in the hydrodynamic limit can be written as [1]

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.

*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.

## 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.

*f*(

*r*) becomes

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.^{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].

## Footnotes

- 1.
In practice, switching off the canonical kinetic term of the scalar fields theory means that the theory becomes strongly coupled at a relatively low energy scale. However, strong coupling is ubiquitous in field theories describing the low energy dynamics of systems with spontaneously broken translational symmetry. That is to say the bulk EFT of our model should be valid only down to a certain radial scale. A parallel and detailed argument can be seen from [9]. To achieve the expansion (9), we have also supposed that \(A_t'(0)\sim const\ne 0\), i.e., \(\rho \ne 0\).

- 2.
See the holographic renormalization procedure in the Appendix.

- 3.
Peaks in the spectral function is associated with the low-lying quasi-normal modes on the complex plane. Then, the dispersion relation of the Goldstone modes can be identified by the motion of the peaks. We would like to thank M. Baggioli for pointing out this.

- 4.

## Notes

### Acknowledgements

We are particularly grateful to M. Baggioli for many stimulating discussions, sharing us the Mathematica code for the optical conductivity and reading a preliminary version of the draft. We also thank C. Niu for helpful discussions on the numerics. This work is supported by the Natural Science Foundation of China under Grant No.11775036, 11847313 and Fundamental Research Funds for the Central Universities No.DUT 16 RC(3)097. WJL also would like to thank Shanghai University and Shao-Feng Wu for the warm hospitality during the completion of this work.

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