1 Introduction

The Bardeen–Cooper–Schrieffer (BCS) theory [1, 2] is a successful microscopic theory to describe the superconductivity discovered in 1911 [3], in which the condensate is understood from Cooper pairs of electrons bound together by phonons. The electron-phonon interactions are much weaker, however in high temperature superconductors and the mechanism behind high temperature superconductivity is thus believed beyond the framework considered in BCS theory. Many scientists have been trying to reveal the basic principle behind them. But until now it still has no satisfied microscopic description for the high temperature superconductivity.

Alternatively, AdS/CFT correspondence [4,5,6,7], or more generally referring to holography, provides an effective description for the high temperature superconductors and also opens up a new window for studying the mechanism of the high temperature superconductivity. The main idea of holography is that a strongly coupled quantum field theory can be mapped to a higher-dimensional bulk gravity system. Based on this idea, a pioneering holographic superconductor model is proposed by Hartnoll, Herzog, and Horowitz (HHH) [8, 9]. This model exists two states: the superconducting state with a non-vanishing charge condensate and the normal state of a perfect conductor. It means that there exists an infinite electric direct current (DC) conductivity even in the normal state due to the translational invariance of the system. To construct a more realistic superconductor model, it is important to break the translational invariance in the holographic framework such that we have a momentum dissipation system.

There are several ways to break the translational invariance in the bulk gravity. The most brutal way is to introduce a spatially-dependent source in the dual boundary theory. It has been implemented by a spatially-periodic real scalar source or chemical potential [10,11,12,13,14]. They are usually referred to as the scalar lattice and the ionic lattice, respectively. In this method, such spatially-dependent sources typically lead to the inhomogeneity of the dynamic fields in the bulk, which generate a set of complicated coupled partial differential equations (PDEs). Though conceptually clear, the application of this framework is limited by numerical technology. This is because the accuracy of solving PDE numerically depends heavily on the temperature of the background. Thus, it is extremely difficult to explore the lattice effects at the extremely low temperature [15].

To bypass the technical complexity of solving the PDEs but capture the important aspect of the momentum dissipation, people develop a much simpler but elegant mechanism to break the translation symmetry but retain the homogeneity of the background geometry, for which we only need solve the ordinary differential equations (ODEs). So far, several models have been proposed to implement this mechanism, including holographic Q-lattices [16,17,18,19], helical lattices [20] and axions model [21,22,23,24,25,26,27,28,29,30,31,32,33]. Holographic Q-lattice model implements the breaking of the translational symmetry by the global phase of the complex scalar field. Holographic helical lattice model possesses the non-Abelian Bianchi VII\(_0\) symmetry, which results in the translational symmetry breaking but holds homogeneous background geometry. The simplest model is the so-called holographic axions model [21], which breaks the translational symmetry by a pair of linearly spatial-dependent scalar fields. Then, guiding in the spirit of effective holographic low energy theories, it is natural and interesting to explore the effect of higher-derivative terms of axion fields [34,35,36,37]. The higher-derivative effect turns out to have a heavy impact on the lower bound of charge diffusion [34] but have no effect on its upper bound [37]. Such higher-derivative term has no impact on the bound of energy diffusion [34]. It was also found that the lower bound of DC conductivity of the usual axions model [21] can be violated in the holographic effective theory with higher-derivative terms [35, 38], such that we have vanishing DC conductivity at zero temperature, which provides a framework to model a more realistic insulating state. In this holographic effective framework, the spontaneous symmetry breaking (SSB), which is associated with the gapless excitations called Goldstone modes in the low energy description, and the so-called pseudo-SSB associated with the pseudo-Goldstone mode, where the SSB dominates over the explicit symmetry breaking (ESB), can be implemented [36, 39,40,41,42,43,44]. We would like to point out that the holographic massive gravity studied in [45] also belongs to the framework of holographic effective axionic model. For the detailed discussions, please refer to [46]. In addition, we can also achieve the momentum dissipation in the dual boundary field theory by introducing a higher-derivative interaction term between the U(1) gauge field and the scalar field, which spontaneously generates a spatially dependent profile of the scalar field [47, 48].

The holographic superconductor models with broken translations have also been constructed and studied [49,50,51,52,53,54,55,56,57,58]. In Ref. [49], a pioneering work of holographic lattice superconductor model implemented by a periodic potential was set up. They reproduced some qualitative features of some cuprates, including the superconducting energy gap and the power law fall-off at the intermediate frequency. The holographic Q-lattice superconductor was also built in [50, 57]. Their result demonstrates that the condensate of the scalar field would be suppressed by the lattice effects and thus leads to a lower critical temperature. In particular, they found that if the normal state is a deep insulating phase, the condensation never happens for small charge of the scalar field. Moreover in holographic superconductor with axion fields [51], the authors found the existence of a new type of superconductor induced by the strength of momentum relaxation even at chemical potential being zero. It means that there exists a new “pairing” mechanism of particles and antiparticles interacting with the strength of momentum relaxation. In Ref. [58], a more phenomenologically relevant holographic superconducting model of momentum dissipation was constructed based on Gubser-Rocha model whose ground state entropy vanishes [59]. It was also shown that some universal properties of high \(T_c\) superconductors, including linear-T resistivity near \(T_c\) and Homes’ law, are observed [58]. In addition, a superconducting dome-shaped region is implemented on the temperature-doping phase diagram in holographic axion superconductor proposed in [56].

In this paper, we shall construct an effective holographic superconductor with broken translations, for which the higher derivative terms of axion fields are introduced. We intend to study the alternating current (AC) conductivity over the normal state and the superconducting properties of this effective holographic superconductor. The plan of this work is as follows: In Sect. 2, we construct the effective holographic model with broken translations. In Sect. 3, we study the conductivity in the normal state. The superconducting properties are explored in Sect. 4. In Sect. 5, we conclude.

2 Holographic framework

In this section, we construct an effective holographic superconductor model with broken translations. It includes the following key ingredients: the metric \(g_{\mu \nu }\), U(1) gauge field \(A_{\mu }\), the complex scalar field \(\psi \) and two axionic fields \(X^{I}\) (\(I=x\,,y\)) with x, y being spatial coordinates. The superconducting phase is supported by the complex scalar field \(\psi \), which can be defined by \(\psi = \chi e^{i\theta }\) with \(\chi \) being the real scalar field and \(\theta \) being the Stückelberg field. A pair of spatial linear dependent axionic fields result in the breaking of the translation symmetry.

Including all the ingredients, we write down the total action of the model as

$$\begin{aligned} S= & {} \int d^4x\sqrt{-g}\Big (R+6+{\mathcal {L}}_{M}+{\mathcal {L}}_{\chi }+{\mathcal {L}}_{X}\Big ), \end{aligned}$$
(1)
$$\begin{aligned} {\mathcal {L}}_{M}= & {} -\frac{Z(\chi )}{4}F^{2}, \end{aligned}$$
(2)
$$\begin{aligned} {\mathcal {L}}_{\chi }= & {} -\frac{1}{2}(\partial _{\mu } \chi )^{2} -H(\chi )(\partial _{\mu }\theta -q A_{\mu })^{2}-V_{\mathrm{int}}(\chi ), \end{aligned}$$
(3)
$$\begin{aligned} {\mathcal {L}}_{X}= & {} -\frac{J(\chi )}{4}{{\mathrm{Tr}}}[XF^{2}]-V(X). \end{aligned}$$
(4)

\(F_{\mu \nu }=\nabla _{\mu }A_{\nu }-\nabla _{\nu }A_{\mu }\) is the field strength of gauge field \(A_{\mu }\). The coupling function of Maxwell field strength takes the form as

$$\begin{aligned} Z(\chi )=1+\frac{a\chi ^{2}}{2}. \end{aligned}$$
(5)

We define \(X\equiv {{\mathrm{Tr}}}[X^{\mu }\,_{\nu }]\) with

$$\begin{aligned} X^{\mu }\,_{\nu }=\frac{1}{2}\partial ^{\mu }X^{I}\partial _{\nu }X^{I}, \end{aligned}$$
(6)

where \(I=x,y\). The potential V(X) in the Lagrangian density \({\mathcal {L}}_{X}\) is

$$\begin{aligned} V(X)=X\,, \end{aligned}$$
(7)

which is just the simplest axion term proposed in Ref. [21]. As in Refs. [34,35,36], we introduce a higher derivative term of axion fields coupled with the gauge field asFootnote 1

$$\begin{aligned} {{\mathrm{Tr}}}[XF^2]\equiv X^{\mu }\,_{\nu }F^{\nu }\,_{\rho }F^{\rho }\,_{\mu }. \end{aligned}$$
(8)

The coupling coefficient \(J(\chi )\) takes the form

$$\begin{aligned} J(\chi )=\alpha _{1}+\frac{\alpha _{2}\chi ^{2}}{2}. \end{aligned}$$
(9)

In the normal state, the setup is closely similar to that in [36]. The term \(V(X)=X\) and the gauge-axion term play the role of ESB and SSB, respectively. In our setup, \(\alpha _{1}\) controls the SSB, while \(\alpha \) controls both the ESB and SSB. However, in the setup of [36], the ESB and SSB can be independently controlled. We shall further discuss the roles of the term \(V(X)=X\) and the gauge-axion term in next section.

\({\mathcal {L}}_{\chi }\) is the Lagrangian density supporting the superconducting phase transition. The coupling function \(H(\chi )\) and the potential \(V_{\mathrm{int}}(\chi )\) are given by

$$\begin{aligned} H(\chi )=\frac{n\chi ^{2}}{2},\quad V_{int}(\chi )=\frac{M^{2}\chi ^{2}}{2}, \end{aligned}$$
(10)

where M is the mass of the scalar field \(\psi \). Without loss of generality, we choose the gauge \(\theta =0\) in what follows.

From the action above, we derive the covariant form of the equations of motion:

$$\begin{aligned}&\nabla _{\mu }\Bigg [ZF^{\mu \nu }-\frac{J}{2}\Big ((XF)^{\mu \nu }-(XF)^{\nu \mu }\Big )\Bigg ]\nonumber \\&\quad \quad -2Hq^{2}A^{\mu }=0, \end{aligned}$$
(11)
$$\begin{aligned}&\nabla _{\mu }\nabla ^{\mu }\chi -\partial _{\chi }Hq^{2}A^2-\frac{\partial _{\chi }Z}{4}F^{2}\nonumber \\&\quad \quad -\frac{\partial _{\chi }J}{4}{{\mathrm{Tr}}}[XF^{2}]-\partial _{\chi }V_{int}=0, \end{aligned}$$
(12)
$$\begin{aligned}&\nabla _{\mu }\Big [\nabla ^{\mu }\phi ^{I}+\frac{J}{4}(F^{2})^{\mu }\,_{\nu }\nabla ^{\nu }\phi ^{I}\Big ]=0, \end{aligned}$$
(13)

and

$$\begin{aligned}&R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R-3g_{\mu \nu }-\frac{1}{2}\nabla _{\mu }\phi \nabla _{\nu }\phi -\frac{1}{2}\nabla _{\mu }\chi \nabla _{\nu }\chi \nonumber \\&\quad -Hq^{2}A_{\mu }A_{\nu } -\frac{Z}{2}F_{\mu \rho }F_{\nu }\,^{\rho }-\frac{J}{4}\Big (\frac{1}{2}\nabla _{(\mu |}\phi ^I\nabla _{\sigma }\phi ^I(F^2)^{\sigma }\,_{|\nu )}\nonumber \\&\quad +F_{(\mu |\sigma }(FX)^{\sigma }\,_{|\nu )}+F_{(\mu |\sigma }(XF)^{\sigma }\,_{|\nu )}\Big )\nonumber \\&\quad -\frac{1}{2}g_{\mu \nu } ({\mathcal {L}}_{M}+{\mathcal {L}}_{\chi }+{\mathcal {L}}_{X})=0, \end{aligned}$$
(14)

where the symmetry brackets above mean \(A_{(\mu \nu )} = (A_{\mu \nu } + A_{\nu \mu })/2\).

To solve the above equations, we take the following ansatz

$$\begin{aligned} ds^2= & {} {1\over u^2}\left[ -(1-u)p(u)U_{1}dt^2+\frac{du^2}{(1-u)p(u)U_{1}} \right. \nonumber \\&\left. +U_{2} dx^2+U_{2} dy^2 \right] , \nonumber \\ A_{t}= & {} \mu (1-u)b(u),\quad \chi =u^{3-\Delta }\phi , \nonumber \\ X^{x}= & {} \alpha x,\quad X^{y}=\alpha y, \end{aligned}$$
(15)

with \(p(u)=1+u+u^2-\mu ^2u^3/4\) and \(\Delta =3/2\pm (9/4+M^2)^{1/2}\). \(u=1\) is the black hole horizon while the AdS boundary locates at \(u=0\). \(\alpha \) is a constant denoting the strength of broken translations. \(U_{1}\) and \(U_{2}\) are just the function of the radial coordinate u. We impose the boundary condition \(U_{1}(1)=1\) at the horizon so that we have the temperature of the dual system as

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

Through this paper we take \(M^2=-2\) such that \(\Delta =2\). Then for given coupling parameters a, \(\alpha _1\), \(\alpha _2\) and n, this holographic system is depicted by the dimensionless Hawking temperature \(T/\mu \) and the strength of broken translations \(\alpha /\mu \). For convenience, we abbreviate the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\) in the following.

3 Conductivity in normal phase

Obviously, in the case of \(\chi =0\), the action describes the non-superconducting phase. In the case of \(\chi =0\), the black hole solution can be analytically worked out. It is found that the gauge-axion coupling has no effect on the background solution and this exact background solution is just the Reissner–Nordström-AdS (RN-AdS) black hole solution with axions [21]. Here for numerical convenience, we still take the ansatz (15) to solve the background of the normal state.

Before proceeding, we would like to further explain the roles of the two terms in the Lagrangian density \({\mathcal {L}}_X\) (Eq. (4)). To this end, we assume that the axionic fields \(X^I\) depend on the full coordinates \(x^{\mu }\). If we only consider the usual axion term, i.e., the second term of \({\mathcal {L}}_X\), the asymptotic behavior of \(X^I\) near the AdS boundary is [36, 44]

$$\begin{aligned} X^I=X_I^{(0)}(t,x^i)+X_I^{(3)}(t,x^i)u^3+\cdots . \end{aligned}$$
(17)

Following the standard quantization, \(X_I^{(0)}(t,x^i)\) is the source of the dual operator and \(X_I^{(3)}(t,x^i)\) is interpreted as the expectation value. The special profile of axionic fields \(X^I=\alpha x^i\) means the source also has the form \(X_I^{(0)}(t,x^i)=\alpha x^i\) in the dual field theory. It indicates that the breaking of the symmetry is explicit.

If only the first term of \({\mathcal {L}}_X\) survives, the axionic fields near the AdS boundary behave like [36, 44]

$$\begin{aligned} X^I=X_I^{(-1)}(t,x^i)u^{-1}+X_I^{(0)}(t,x^i)+\cdots . \end{aligned}$$
(18)

Now, the subleading term is the second one in the above expansion and should be interpreted as the expectation value \(\left\langle {\mathcal {O}}^I\right\rangle \sim \alpha \delta ^I_ix^i\) with a vanishing source [36]. It means that a SSB appears, which is associated with the gapless excitations called Goldstone modes in the low energy description.

If the first term dominates over the second term in the Lagrangian density \({\mathcal {L}}_X\), the system shall be weak explicit symmetry breaking, which leads to a slightly gapped excitation and means the emergence of the pseudo-Goldstone mode [44]. It is obvious that in our setup, \(\alpha _{1}\) controls the SSB, while \(\alpha \) controls both the ESB and SSB.

Next, we shall study the DC and AC conductivities over the normal state, respectively.

3.1 DC conductivity

Though the gauge-axion coupling has no effect on the background solution, it enters into the perturbative equations leading to strong impact on the transports, which has been seen in the properties of DC transports [34,35,36]. For completeness, we shall give a brief review on DC conductivity over the normal state in this subsection.

Based on “membrane paradigm” [62, 63] (see also [64,65,66]), we can analytically work out DC conductivity determined by the background data at the horizon. According to the pivotal point of this method, we construct a radially conserved current \({\mathcal {J}}\) connecting the horizon and the boundary. At this stage, we can calculate DC conductivity as

$$\begin{aligned} \sigma _{DC}=-\frac{(\alpha ^{2}\mu ^{2}\alpha _{1}-4U_{2}(1))(\alpha ^{2}\mu ^{2}+\mu ^{2}b^{2}(1)U_{2}(1))}{\alpha ^{2}\mu ^{2}(4+\alpha _{1}\mu ^{2}b^{2}(1))U_{2}(1)}.\nonumber \\ \end{aligned}$$
(19)

Obviously, different from the simplest axionic model [21], DC conductivity from the higher-derivative corrections depends on the temperature T and also the broken translations strength \(\alpha \). Notice that the requirement of the positive definiteness of the conductivity imposes a constraint on the coupling parameter \(\alpha _1\) as \(0\le \alpha _1\le 2/3\) [35].

Fig. 1
figure 1

The DC conductivity with the \(\alpha _{1}\) coupling turned on by setting \(T=0\) while \(\alpha \) vary and by setting \(\alpha =\{0.5,1,1.2\}\) while T vary. Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image

We plot DC conductivity with the coupling parameter \(\alpha _{1}\) turned on in the top left in Fig. 1. The plot is exhibited to demonstrate DC conductivity as the function of \(\alpha \) at zero temperature. It is easy to find that at zero temperature, when the coupling parameter \(\alpha _{1}\) increases, DC conductivity decreases with \(\alpha \) increasing. In particular, for \(\alpha _{1}=2/3\), the DC conductivity tends to zero with \(\alpha \) increasing, which violates the DC conductivity bound in the simplest holographic axion model [21]. Notice that in the limit of \(\alpha \rightarrow 0\), due to the restoration of the violation of the translational symmetry, the DC conductivity tends to \(\alpha _1\)-independent infinity.

In holographic system, the temperature behavior of DC conductivity can well describe some characteristics of the system. In particular, in many holographic references [16,17,18,19,20, 38, 66,67,68,69,70,71,72,73,74], people often adopt the so-called operational definition to identify the metallic phase and insulating phase, i.e.,

  • Metallic phase: \(\partial _T\sigma _{DC}<0\).

  • Insulating phase: \(\partial _T\sigma _{DC}>0\).

  • Critical point (line): \(\partial _T\sigma _{DC}=0\).

Here we also study the temperature behaviors of DC conductivity, which are shown in the rest plots in Fig. 1. When the higher-derivative term is absent, i.e., \(\alpha _1=0\), \(\sigma _{DC}\) is independent of the temperature. Once the coupling parameter \(\alpha _1\) is turned on, DC conductivity decreases with T decreasing. In terms of the operational definition of phase of the holographic system described above, this holographic system exhibits the insulating behavior. But we note that unless \(\alpha _1\) takes the bounded value \(\alpha _1=2/3\) and \(\alpha \) tends to infinity, DC conductivity cannot vanish in the limit of zero temperature. For the fixed \(\alpha \), \(\sigma _{DC}\) decreases with \(\alpha _1\) increasing.

3.2 AC conductivity

In this subsection, we numerically calculate AC conductivity in the normal state. Since our system is homogeneous and isotropy, it is enough to turn on the following consistent linear perturbation

$$\begin{aligned} \delta A_x(t,u,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega }{(2\pi )^3}e^{-i \omega t}\delta b_x(u),\nonumber \\ \delta g_{tx}(t,u,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega }{(2\pi )^3}e^{-i \omega t}u^{-2}h_{tx}(u),\nonumber \\ \delta X^x(t,u,x^i)= & {} \int ^{+\infty }_{-\infty }\frac{d\omega }{(2\pi )^3}e^{-i \omega t}\delta \phi ^x (u). \end{aligned}$$
(20)

Thus we shall have three ordinary differential equations for \(h_{tx}, \delta b_x, \delta \phi ^x\). It turns out that near the UV boundary \((u\rightarrow 0)\), the asymptotic behavior of the Maxwell field falls in the following form

$$\begin{aligned} \delta b_x = b^{(0)}+b^{(1)}u+\cdots . \end{aligned}$$
(21)

According to holographic dictionary, we read off the conductivity as

$$\begin{aligned} \sigma (\omega )=-\frac{ib^{(1)}}{\omega b^{(0)}}. \end{aligned}$$
(22)

Then, we shall impose the ingoing boundary condition at the horizon to solve the perturbative system and read off the conductivity as the function of the frequency in terms of Eq. (22).

Fig. 2
figure 2

AC conductivity in the normal state at \(T= 0.015\) with different \(\alpha \) and \(\alpha _{1}\). The red dots correspond to DC conductivity obtained from formula (19). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image

Figure 2 exhibits the real and imaginary parts of AC conductivity in the normal state at \(T= 0.015\). At the high frequency regime, \(\omega \gg \mu \), the conductivity tends towards a constant determined by ultraviolet (UV) CFT fixed point. The interesting physics lies at the low frequency region.

Before proceeding, we present some comments on the AC conductivity behavior at low frequency without gauge-axion coupling (\(\alpha _{1}=0\)) [21], where the translation symmetry is explicitly broken. For this case, we observe that AC conductivity in real part displays a standard Drude peak at low frequency when the momentum dissipation is weak, i.e., \(\alpha \) is small. It implies that the weak momentum dissipation drives the current into a coherent state. Theoretically, it is because the total momentum of the system is approximately conserved, for which we can implement a perturbative expansion in the small momentum relaxation rate within the memory matrix formalism [75,76,77,78]. With the strength of the momentum dissipation increasing, the Drude peak gradually reduces and we observe a transition from coherent phase to incoherent phase (see the blue curves in Fig. 2, also see Refs. [23, 79]). In fact, because we must include the contribution at subleading order in the relaxation rate when the momentum dissipation is strong, the behavior of AC conductivity at low frequency should be depicted by a modified holographic formula, first derived in [80] and generalized to a more general holographic theory [81], instead of the standard Drude formula or even the modified hydrodynamic results.

However, when the gauge-axion coupling term \(\alpha _1\) \(Tr[XF^{2}]\) in the Lagrangian density Eq. (4), which controls the SSB, is introduced. AC conductivity exhibits some interesting behaviors. Recalling that in this model the strength of broken translations \(\alpha \) controls both the ESB and SSB. From the first and second rows in Fig. 2, we see that a peak emerges in the mid-IR in the AC conductivity for some \(\alpha \) and \(\alpha _{1}\). The peak moves to higher energy with \(\alpha _1\) or \(\alpha \) increasing (the first and second rows in Fig. 2). As discussed at the beginning of this section (also refer to Refs. [36, 44]), this peak attributes to the emergence of the pseudo-Goldstone mode, which is the result of the competition of SSB and ESB. Notice that in the setup of Ref. [36], they can separately control the strength of SSB and ESB to make the ESB is weak compared with the SSB such that the peak in min-IR is more pronounced. However, when the ESB dominates over the SSB (\(\alpha _1\) small or \(\alpha \) too large), the peak in min-IR disappears (see the third row in Fig. 2).

4 Superconducting phase

In this section, we turn to study the superconducting properties of this holographic effective theory with gauge-axion coupling. Some works have studied the superconducting properties of holographic axion model [51, 52, 56, 82, 83]. Here, we shall mainly focus on the effect of gauge-axion coupling.

4.1 Condensation

Fig. 3
figure 3

The condensation \(<O_2>\) as a function of temperature for given a (from left to right \(a=-0.1\), \(a=0\) and \(a=1\)) and various values of \(\alpha \). We have set \(q=2\) and \(n=1\). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image
Fig. 4
figure 4

The condensation \(<O_2>\) as a function of temperature for given \(\alpha \) (from left to right \(\alpha =0.5\), \(\alpha =1\) and \(\alpha =5\)) and various values of a. We have set \(q=2\) and \(n=1\). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image

Under the ansatz (15), we numerically solve the equations of motion (11)–(14) to explore the properties of the condensation. Again, it is easy to find that the gauge-axion coupling term \(J \mathrm{Tr}[XF^{2}]\) has no any effect on the background solution even in the superconducting phase. Only three parameters: the strength of broken translations \(\alpha \), the gauge coupling parameters a and n, exert their influence on the condensation. Notice that from the Lagrangian density \({\mathcal {L}}_{\chi }\), i.e., Eq. (3), we see that the relevant quantity is the product of q and n. At the same time, since the increase of charge makes the condensation easier, which had been widely studied in previous references for example [49, 50], we shall set \(q=2\) and \(n=1\) through this paper without loss of generality.

In the unit of the critical temperature we plot the condensation \(<O_2>\) as a function of the temperature by fixing the gauge coupling parameter a or by fixing the strength of broken translations \(\alpha \), respectively. When we fix the gauge coupling parameter a, the stronger the strength of broken translations, the expectation value of the condensation becomes much larger and thus the critical temperature becomes lower (Fig. 3), which had also been observed in many works for example [50, 51, 53]. Therefore, we conclude that the condensation becomes harder with the strength of broken translations increasing even in presence of the gauge coupling. We expect that the condensation is violated as the broken translations becomes stronger beyond some critical values.

And then, we fix the strength of broken translations \(\alpha \) and vary the gauge coupling parameter a to see the effects of gauge coupling on the condensation. From Fig. 4, we see that with a increasing, the expectation value of the condensate becomes much smaller. It indicates that the condensation becomes easier and the critical temperature becomes higher.

4.2 Superconducting conductivity

To calculate AC conductivity in the superconducting phase, we can follow the same procedure in the normal state outlined in Sect. 3.2. Different from the condensation, there are four theory parameters (\(\alpha \), a, \(\alpha _{1}\), \(\alpha _{2}\)) affecting the superconducting conductivity. In what follows, we shall explore the effects of these coupling parameters on the conductivity step by step.

Fig. 5
figure 5

The conductivities \(\sigma \) as a function of the frequency for different \(\alpha \) fixing \(a=0\), \(\alpha _{1}=0\), and \(\alpha _{2}=0\). The left panels are for the real part and the right panels are for the imaginary one. Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image
Fig. 6
figure 6

The critical behavior of the conductivity near the critical temperature in the low frequency region for the different \(\alpha \). We have turned off other coupling parameters. Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

Full size image

We first study the effects of the momentum dissipation on AC conductivity. To this end, we turn off the other coupling parameters. In Fig. 5, we plot the real and imaginary parts of AC conductivity as a function of frequency for various \(\alpha \). We also show the evolution of the conductivity with the temperature from normal phase to superconducting phase. We observe that once the system enters into the superconducting phase, the imaginary part of the conductivity climbs up rapidly and tends to infinity in the limit of \(\omega =0\). According to the Kramers-Kronig (KK) relation, a pole emerges in \(Im[\sigma ]\) implying a corresponding delta function in \(Re[\sigma ]\), which is one of the hallmarks of genuine superconductor. In order to exhibit such sudden change more distinctly, we plot the conductivity in low frequency region with the temperature dropping through the critical point in Fig. 6. Unlike the abrupt changing behavior of the imaginary part, the real part of the conductivity only changes slightly near the critical temperature. The real part of optical conductivity at low frequency goes down with the temperature decreasing and finally vanishes at extremal low temperature. In addition, we also work out the normal fluid density \(n_{n}\) defined by

$$\begin{aligned} n_{n}=\lim _{\omega \rightarrow 0}Re[\sigma (\omega )]. \end{aligned}$$
(23)

which depicts the contribution from the normal, non-superconducting component on the DC conductivity in the superconducting phase. The results are showed in Table 1. It is found that the values of \(n_n\) are still finite when the system just enters the superconducting phase, while \(n_n\) goes down with the temperature decreasing and vanishes at extremal low temperature. Therefore, our holographic model also resembles a two-fluid model as the standard holographic superconductor model [8, 9, 84], the holographic Q-lattice superconductor [50], and the holographic superconductor from higher derivative theory [85, 86].

Table 1 The normal fluid density \(n_n\) in the superconducting phase for the different temperature with different \(\alpha \). Here we turned off other coupling parameters and the dimensionless quantity \(\{\alpha /\mu \}\) is abbreviated as \(\{\alpha \}\)
Full size table
Fig. 7
figure 7

The behavior of conductivity with varying a for different \(\alpha \) in the superconducting state. Here we fix the temperature \(T/T_{c}\approx 0.2\). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

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Fig. 8
figure 8

The real part of conductivity as the function of the frequency at \(T/T_{c}\approx 0.2\). Left plot: different \(\alpha _{1}\) for \(\alpha =0.5\), \(a=0\) and \(\alpha _2=0\). Right plot: different \(\alpha _{2}\) for \(\alpha =0.5\), \(a=0\) and \(\alpha _1=0\). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

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Fig. 9
figure 9

The real part of conductivity as the function of the frequency with various \(\alpha _2\) for fixed \(\alpha =0.5\) and \(a=0\) at \(T/T_{c}\approx 0.2\). Left plot is for \(\alpha _1=1/3\) and right plot for \(\alpha _1=2/3\). Here we have abbreviated the two dimensionless quantities \(\{T/\mu ,\alpha /\mu \}\) to \(\{T,\alpha \}\)

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Furthermore, we study the combined effect of the strength of broken translations \(\alpha \) and the gauge coupling parameter a in the superconducting state. Here we have fixed the remaining parameters (\(\alpha _{1}\) and \(\alpha _{2}\)) to be zero. In Fig. 7, we show both real and imaginary parts of conductivity with different a and \(\alpha \). At low temperature (\(T/T_{c}\approx 0.2\)), the real part of the conductivity tends to zero in the limit of \(\omega \rightarrow 0\) (see the dashed red lines in Fig. 7 and also see Fig. 5). It implies that at this temperature, most of the normal components of the electron fluid have formed the superfluid component. For negative a (\(a=-0.1\)), the case is similar to that of \(a=0\). However, for positive a (\(a=1\)), the DC conductivity starts to rise up. It means that we need further cool the system to drive the normal component of the electron fluid forming the superfluid component.

We also explore the impact of gauge-axion coupling parameters (\(\alpha _{1}\) and \(\alpha _{2}\)) on the optical conductivity in the superconducting state. The results are exhibited in Figs. 8 and 9. In order to avoid the effect of gauge parameter a on the conductivity, here we fix \(a=0\). In addition, we also set \(\alpha =0.5\) without loss of generality. We summarize the main effects of the gauge-axion coupling as what follows:

  • Increasing the gauge-axion coupling \(\alpha _1\), it is easier to drive the normal component of the electron fluid forming the superfluid component and the superconducting energy gap becomes evident (left plot in Fig. 8). We also observe that the peak in the mid-IR in the normal state is smoothed out when the system enters into the superconducting phase.

  • To see the effect from the coupling \(\alpha _2\), which is a coupling among gauge field, axion fields and complex scalar field being responsible for condensation, we turn off \(\alpha _{1}\). The low frequency behaviors of conductivity are exhibited in the right plot in Fig. 8. For negative \(\alpha _{2}\), DC conductivity rises up with the absolute value of \(\alpha _{2}\) increasing and then the conductivity forms a small peak at low frequency for large absolute value of \(\alpha _{2}\). For positive \(\alpha _2\), we see that with \(\alpha _{2}\) increasing, DC conductivity goes down at first and then rises up exhibiting a non-linear change. When \(\alpha _2\) rises up to a large value, the conductivity also sprouts up a small peak at low frequency. Based on these observations, it is obvious that the role \(\alpha _{2}\) playing is completely different from that of \(\alpha _{1}\). We attribute this difference to the effect of condensation field \(\chi \). Furthermore, we would like to point out that the mechanism of this small peak emerging at low frequency deserving further pursuit.

  • We further study the combined effects of the coupling \(\alpha _{1}\) and \(\alpha _{2}\), and also the competition between them. Fig. 9 shows the real part of conductivity with various \(\alpha _{2}\) for fixed \(\alpha _{1}=1/3\) (left plot) and \(\alpha _{1}=2/3\) (right plot). It is clear that for negative \(\alpha _{2}\), DC conductivity is suppressed with the increase of \(\alpha _{1}\).

  • However, for positive \(\alpha _2\), the effect of the coupling term \(\alpha _{2}\) dominates over that of the coupling term \(\alpha _{1}\). From Fig. 9, we see that once the coupling terms \(\alpha _1\) and \(\alpha _{2}\) are both turned on, a peak emerges at low frequency. Recalling that for \(\alpha _1=0\), a gap is evident at low frequency even for \(\alpha _{2}=0.3\). With \(\alpha _1\) increasing, the peak at low frequency becomes more evident.

5 Conclusion

In this paper, we construct a holographic effective superconducting theory including some coupling terms among the gauge field, axion fields and complex scalar field. In the normal state, this holographic effective superconducting theory reduces to the so-called \({\mathcal {J}}\) model studied in Refs. [35, 36]. In the normal state, though the gauge coupling term \(\alpha _1\mathrm{Tr}[XF^{2}]\) has no effect on the background solution, it enters into the perturbative equations affecting AC conductivity. An important properties is that a peak emerges in the mid-IR in the AC conductivity for some \(\alpha \) and \(\alpha _1\). It is attributed to the emergence of the pseudo-Goldstone mode, which is the consequence of the competition of SSB and ESB. However, when the ESB dominates over the SSB, the peak disappears.

In the superconducting phase, we first study the properties of the condensation. Since the gauge-axion coupling term \(J\mathrm{Tr}[XF^{2}]\) has no effect on the black hole solution, there are only two theoretical parameter considered in our model (the strength of broken translations \(\alpha \) and the gauge coupling parameter a) exerting their influences on the condensation. It is easy to find that with the parameter \(\alpha \) increasing, the condensation becomes harder for fixed parameter a, while fixed parameter \(\alpha \) with the parameter a increasing, the condensation becomes easier. Such an opposite effect means a competitive relationship between \(\alpha \) and a. Similarly, the competitive behavior can be observed in superconducting conductivity.

Then we also study the properties of AC conductivity in the superconducting phase. We respectively explore the effects of coupling parameters (\(\alpha \), a, \(\alpha _{1}\), \(\alpha _{2}\)) in our model on the conductivity. First, we explore the effect of momentum dissipation on AC conductivity. The features of the conductivity with temperature from normal phase to superconducting phase resemble a two-fluid model as the standard holographic superconductor model [8, 9, 84], and the holographic Q-lattice superconductor [50], and the holographic superconductor from higher derivative theory [85, 86]. Then for the influence of the gauge coupling a, we find that a plays a key role in driving the normal component of the electron fluid forming the superfluid component. Especially, for positive a, it is harder to implement the transition from the normal component of the electron fluid forming the superfluid component. In the end, we explore the effect of the gauge-axion coupling. We find that the coupling parameter \(\alpha _{1}\) can drive the electron superfluid component forming which is similar with the effect of negative a. The increasing of \(\alpha _{1}\) leads to a more evident gap in AC conductivity. In addition, we find that the coupling of \(\alpha _{2}\) plays a completely different role from that of \(\alpha _1\), which is attributed to the effect of condensation field \(\chi \).