Phases of holographic superconductors with broken translational symmetry

We consider holographic superconductors in a broad class of massive gravity backgrounds. These theories provide a holographic description of a superconductor with broken translational symmetry. Such models exhibit a rich phase structure: depending on the values of the temperature and the disorder strength the boundary system can be in superconducting, normal metallic or normal pseudoinsulating phases. Furthermore the system supports interesting collective excitation of the charge carriers, which appears in the normal phase, persists in the superconducting phase, but eventually gets destroyed by the superconducting condensate. We also show the possibility of building a phase diagram of a system with the superconducting phase occupying a dome-shaped region on the temperature-disorder plane.


Introduction
Holographic superconductors have recently received a new wave of attention. It originated from several attempts [1][2][3][4][5][6] to provide a holographic description of systems which resemble more of the real-world superconductors. One of the essential features of the original holographic superconductor proposal of [7,8] is that it describes the system which exists in two states: a superconducting state which has a nonvanishing charge condensate, and a normal state which is a perfect conductor. As a direct conseguence, already in the normal phase the static electric response, namely the DC conductivity (ω = 0), is infinite. This is a straightforward consequence of the translational invariance of the boundary field theory, which leads to the fact that the charge carriers do not dissipate their momentum, and accelerate freely under an applied external electric field. Therefore one is motivated to introduce momentum dissipation into the holographic framework, breaking the translational invariance of the dual field theory. It is definitely interesting to construct a holographic superconductor on top of such dissipative backgrounds which is indeed going to have a finite DC conductivity in the normal phase, clearly distinguishable from the infinite one in the superconducting phase.
One efficient method to implement such a feature relies on the possibility of breaking diffeomorphism invariance in the bulk via giving the graviton a mass, as it has been proposed in [9]. It is very convenient to recast these Lorentz symmetry violating massive gravity theories into a covariant form introducing the Stueckelberg fields, namely the extra degrees of freedom appearing as a consequence of breaking of the diffeomorphism symmetry (see [10] for more details).
In the context of applied holography this construction was analyzed for the first time in [11] where momentum dissipation in the field theory was achieved by switching on neutral scalar operators depending linearly on the spatial coordinates of the boundary. These scalar fields on the boundary source the neutral scalar fields in the bulk. The resulting bulk system describes a holographic dual of the field theory with broken translational symmetry. Such a system possesses a finite DC conductivity [11] 1 .
The original idea of [11] has been put in a broader context in [16], where the most general form for the Lagrangian of the neutral scalars has been introduced 2 . This Lagrangian is weakly constrained by the consistency conditions in the bulk, which avoid ghost excitations and gradient instabilities [16]. It turns out that imposing physical consistency of the theory still leaves enough freedom to construct models, which exhibit new non-trivial features.
To be more specific, one can build models which possess the following attractive properties. The first one is an increase of conductivity as a function of temperature, for temperatures lower than a certain critical value T 0 , This property bears a resemblance to an insulating behavior, with the population of the conducting energy band depleting upon lowering the temperature. Still, it awaits a better understanding, because of an essentially non-vanishing value of the DC conductivity at zero temperature. We refer to the state (1.1) as pseudo-insulating. The second new feature of the model is an appearance of an extra structure in the optical conductivity. For temperatures lower than a certain critical value T , there appears a peak in the optical conductivity, signaling a new long-lived collective propagating excitation of the charge carriers 3 .
This paper is based on the idea to generalize the construction of [4,5] to the more generic effective models for momentum dissipative systems, proposed in [16]. The main questions which we aim to answer are the following: 1. Can one construct a model of holographic superconductor which is separated by the lines of the second order phase transition from the normal metallic phase and the normal pseudo-insulating phase (1.1)?
2. Does the peak in the optical conductivity of [16] continue to exist in the superconducting phase 4 ?
We have found that the answers are: 1. Yes, by combining the idea of [16] with the setting of a holographic superconductor one can obtain a system with a rich phase diagram where three different phases are present: superconductor, metal, and pseudo-insulator.
2. The peak in optical conductivity continues to exist in the superconducting phase, as the temperature is lowered below a critical temperature T c of the superconducting phase transition. However, at a certain temperature T = T the peak disappears.
Furthermore, we attempt to construct a holographic dual to a real doped superconductor (see, e.g., [20]). The main feature which we will attempt to implement holographically is an enclosing of a superconducting phase on the doping-temperature phase plane by a dome-shaped line of a phase transition (see figure 1). The most successful result would be to have a superconducting dome, separated from an insulating normal state at smaller values of the doping parameter, and a metal normal phase at larger values of the doping parameter. In figure 1 we provide a schematic sketch of what we would like to approach, the phase diagram for High-Tc superconductors.
We will demonstrate that implementing the momentum dissipation models of [16] in the holographic superconductor framework can indeed lead to a superconducting dome, located between pseudo-insulating and metallic phases. However, it appears that such models are too restricted to describe superconducting dome with realistic critical temperature of the superconducting phase transition. We have found that the critical temperature of the dome T c (α), where α is the magnitude of the translational symmetry breaking, is bounded from above by a small number (in units of charge density), of the order of 10 −8 . This makes the numerical calculation at finite temperature hopeless.
Nevertheless, at zero temperature it is possible to have analytical control of the SC instability through the BF bound reasonings and show the existence of a superconducting dome. In the Discussion section 8 we provide a few ideas to generalize our model, which might be useful to obtain a superconducting dome with a reasonably higher values of the critical temperature.
We will be considering charged black brane backgrounds with the neutral scalar fields having vacuum profiles, depending linearly on the spatial coordinates 5 : This configuration (1.2) breaks translational symmetry (and Lorentz invariance) of the boundary field theory but keeps untouched energy conservation. Within this choice we are going to retain homogeneity and rotational invariance. It would be interesting to reproduce the same sort of computations in an anisotropic setup as in [6]. Besides the parameter α, describing the magnitude of the translational symmetry breaking, we will also introduce another parameter m, which will be primarily important in the models with non-linear action for the neutral scalars (1.2). We consider the system at a finite charge density, which corresponds holographically to the time-like component of the U (1) gauge field having a non-trivial radial profile in the bulk, A t (u). The charged scalar ψ is dual to the condensate O of charge carriers. When the v.e.v. of the condensate is non-vanishing, O = 0, the system is in a superconducting phase. This corresponds holographically to a non-trivial configuration ψ(u) in the bulk, with the vanishing source coefficient of the near-boundary expansion of the ψ(u) [7,8].
We will study various superconducting systems, distinguished by the choice of the Lagrangian V (X) for the neutral scalar fields, where and L is the radius of AdS. In this paper we will be mostly interested in the following models: The model (1.4) gives the simplest way to describe the fields φ I and has been proposed in [11].
We will argue that already in the simple case of (1.4) it is possible to have a superconducting dome. We will demonstrate this analytically at zero temperature. Interestingly, the dome is achieved for the scaling dimension ∆ and the charge q of the scalar ψ, restricted to the small vicinity of the "dome" point, which we have found to be (∆ d , q d ) = (2.74, 0.6) . (1.7) In this case the superconducting dome exists in the middle of a normal metallic phase (the model (1.4) does not allow an insulating phase).
We will show that the model with the non-linear Lagrangian (1.5) also possesses the superconducting dome near the point (1.7). In this case it is possible to engineer a model where the dome is separated from metallic phase at larger values of the translational symmetry breaking parameter m, and from a pseudo-insulating phase at smaller values of m. This situation is the closest one to the actual real phase diagram for High-Tc superconductors. We are not aware of other holographic constructions giving this result in this kind of set-up. It is of course pending to improve the model to get a real insulating phase; we leave this issue for future work.
To support our statement about the superconducting dome with such a small critical temperature T c , we will calculate numerically the dependence of the critical temperature for the models (1.4), (1.5), on the scaling dimension ∆ and the charge q. We will show that as the (∆, q) approach the dome point (1.7), the critical temperature quickly declines.
The rest of this paper is organized as follows. In the next Section 2 we set up the model which we will be studying in this paper. We consider the general Lagrangian V (X) for the massless neutral scalar fields. In Section 3 we review the properties of the normal phase solution. In Section 4 we study the conditions for its instability towards formation of a non-trivial profile of scalar hair. From the field theory point of view this corresponds to a superconducting phase transition. In Section 5 we focus on the features of the broken phase, the condensate and the grand potential, demonstrating explicitly the second order phase transition at T = T c . In Section 6 we study the optical conductivity in the normal and superconducting phases. In Section 7 we describe the way to construct a superconducting dome in the middle of a metallic phase, for the model (1.4), and between pseudo-insulating and metallic phases, for the (1.5). We discuss our results in Section 8. Appendix A contains further details about the calculations of the condensate and the grand potential. Appendix B is dedicated to derivation of the on-shell action for bulk fluctuations, which are holographically dual to current and momentum operators on the boundary.

Action and equations of motion
The total action of our model is : where we have denoted the Einstein-Maxwell terms I 1 , the neutral scalar terms I 2 , and the charged scalar terms I 3 ;

2)
We have introduced an extra coupling κ, between the charged scalar ψ and the neutral scalars φ I . In this paper we will be mostly considering κ = 0, and comment on the models with non-vanishing κ in the discussion section 8. We have defined We denote D µ ψ = (∂ µ − i q A µ )ψ to be the standard covariant derivative of the scalar ψ with the charge q. We fix the cosmological constant to be Λ = −3/L 2 . In this paper we will consider 4-dimensional bulk, d = 3. The equations of motion following from the action I read 6 : where the dot stands for a derivative w.r.t. X, (2.5)

Background
We consider the following black brane ansatz for the background: The φ I scalars have profiles linear in the spatial coordinates x, y of the boundary. They effectively describe momentum dissipation mechanisms in the boundary field theory, making the DC conductivity of the theory finite [11]. We will take ψ to be real-valued, since due to the u component of Maxwell equations the phase of the complex field ψ is a constant. We are looking for charged black brane solutions with a scalar hair where u h is the position of the horizon, and the boundary is located at u = 0. We allow for non-trivial χ(u) because we want to have in general a non-trivial ψ(u). If ψ = 0, then χ = 0. The resulting equations of motion read: The Hawking temperature of the black brane (2.6) is given by: (2.11) Using eqs. (2.7)-(2.10), the temperature can be written as: with all the fields evaluated at the horizon u h .

Normal phase
In the case of a non-trivial condensate ψ(u) it is in general impossible to solve the background equations of motion (2.7)-(2.10) analytically. However, when ψ(u) = 0, the solution is known [16].
From now on we will fix the coupling κ to zero, The resulting normal phase background is given by: Due to (2.12) the temperature in the normal state reads: All the features of this normal phase solution are going to be reviewed in detail in the following section.

Normal phase features
As suggested in [16], for models with a specific choice of the Lagrangian V (X), the solution exhibits various interesting properties. Using the membrane paradigm the DC part (ω = 0) of the optical conductivity can be computed analytically [14] and for a generic Lagrangian V (X) it is given by [16]: .
The DC conductivity consists of two parts: which is a generic holographic feature The first one σ pair is due to pair creation in the background, and it is present even at zero charge density [25]. It corresponds exactly to the probe limit result. It is temperature independent, and therefore is always present (unless we introduce a dilaton field) as an offset in the value of σ DC , leading to σ DC (T = 0) = 0. The second term σ dissipation is really the one dealing with dissipative mechanism, and it can be thought as the strongly coupled analogue of the Drude formula for the conductivity. In the limit of zero translational symmetry breaking parameter m, this second term gives rise to the infinite DC conductivity, typical for backgrounds preserving translational symmetry, such as the AdS Reissner-Nordstrom black brane case. Due to the freedom of choice of the Lagrangian V (X) this solution can be either a metal or a pseudo-insulator and can provide a transition between the two phases (see figure 2). The pseudo-insulator phase is characterized by the conductivity, declining at smaller temperatures, dσ/dT > 0, for T < T 0 , but reaching a non-vanishing value at T = 0 (which is the reason why we are not calling it an insulating phase) 7 . The transition between the two phases is provided by the existence of a maximum in the DC conductivity as a function of temperature (see figure 2), at T = T 0 , which gives a clear separation between two different regimes: The temperature T 0 at which the metal-insulator transition happens can be obtained analytically, solving the following equation: The metal-insulator transition in the behavior of the DC conductivity is related to a non-trivial structure in the optical conductivity, namely a weight transfer from a Drude peak into a localized new peak in the mid-infrared regime (see figure 2). This feature corresponds to an emerging collective propagating excitation of the charge carriers, whose nature is not completely clear yet. The phase diagram of this normal phase is already rich and can give insights towards the interpretation about the various ingredients introduced into the model. In the case of the linear Lagrangian, which goes back to the original model [4], the parameters m and α are combined into m α, which can be interpreted as the strength of translational symmetry breaking. From the dual field theory point of view this is thought to be related to some sort of homogeneously distributed density of impurities, representing the doping of the material.
In the case of a more general V (X), the m parameter keeps this kind of interpretation while the α one represents the strength of interactions of the neutral scalar sector. This reasoning is confirmed by the study of the phase diagrams of the system (figure 3) which makes evident the difference between the two parameters. Indeed, while the m parameter, which we are going to interpret as the doping of our High-Tc superconductor, enhances the metallic phase, the α one clearly reduces the mobility of the electronic sector driving the system towards the pseudo-insulating phase.

Superconducting instability
In this section we will describe the instability conditions for the normal phase towards the development of a non-trivial profile of the charged scalar field. This allows one to determine a line of the second order superconducting phase transition, T c (α) (or T c (m) for the model (1.5)), in the boundary field theory, with broken translational symmetry.
We start by considering the system at zero temperature, which we are able to study analytically. Then we proceed to studying the normal phase at a finite temperature. Upon lowering the temperature, at a certain critical value T = T c , the normal phase becomes unstable. This is the point of a superconducting phase transition. We construct numerically T c as a function of the parameters ∆, q, α (or m), for the models with various V (X).

Zero-temperature instability
In the case of T = 0 the normal phase geometry interpolates between the AdS 4 in the ultra-violet and the AdS 2 × R 2 in the infra-red. We can apply the known analytical calculation to study the stability of the normal phase towards formation of a non-trivial profile of the scalar ψ [28]. Due to eq. (2.10), the effective mass M ef f of the scalar ψ is given by: Notice that at the boundary the mass of the scalar is just M 2 but at the horizon it gets an additional contribution. This is because near the horizon we have: at zero temperature. Due to (2.16), and the zero temperature T = 0 condition, with the temperature given by (2.17), we obtain: The normal phase is unstable towards formation of the scalar hair, if M ef f violates the BF stability bound in the AdS 2 , namely: In (4.4) we have denoted the AdS 2 radius as L 2 8 : Combining (4.1), (4.3), (4.5), the IR instability condition (4.4) finally reads 9 : where we have defined the function D as: In the usual RN case we have f (u h ) = 12 , and we find the usual L 2 2 = L 2 6 in d = 3. 9 This formula agrees with [5] in the case of V (X) = X 2 m 2 and κ = 0. For the practical calculations we will solve the equation T = 0, see (2.17), for the value of u h , giving the position of the horizon of the extremal black brane, We will measure all the dimensional quantities in units of ρ; both for zero temperature and finite-temperature instability analyses the ρ can be scaled out.
In figure 4 we plot the IR instability region on the (∆, q) plane, for the model 1, (1.4), with α = 2, as well as a few contour lines of the constant critical temperature. In figure 5 we plot the IR instability region and several T c = const curves on the (∆, q) plane, for the model 2, (1.5), with α = 0.25, m = 4. Analogous plot for ordinary holographic superconductor can be found in [28]. Plot in the case of the linear V (X) model first appeared in [5].

Finite-temperature instability
Consider the system at large temperature in a normal phase, which exists in a superconducting phase at low temperatures. Therefore as we decrease the temperature, at certain critical value T c the superconducting phase transition occurs. If T c is non-vanishing, then for T < T c the system is in a superconducting phase, with a non-trivial scalar condensate ψ(u).
Recall that near the boundary the scalar field with mass M : behaves asymptotically as:  Figure 8. Critical temperature as a function of α for the potential V (X) = X/2m 2 + β X 5 /2m 2 for different choices β. All the curves have a runaway behavior at α → ∞, and only the shape depends on the value of β.
where ψ 1 is the leading term, identified as the source in the standard quantization.
To find the value of T c we can look for an instability of the normal phase towards formation of the scalar field profile [28,29]. Near the second order phase transition point T = T c the value of ψ is small, and therefore one can neglect its backreaction on the geometry. The SC instability can be detected by looking at the motion of the QNMs of ψ in the complex plane. To be more specific, it corresponds to a QNM going to the upper half of a complex plane. Exactly at critical temperature we have a static mode at the origin of the complex plane, ω = 0, and the source at the boundary vanishes, ψ 1 = 0. In the next section we will solve numerically the equations (2.7)-(2.10) for the whole background, and confirm this explicitly.
The scalar field is described by eq. (2.10), which in the normal phase becomes: where f (u) is given by (2.16). To determine the critical temperature T c we need to find the highest temperature, at which there exists a solution to eq. (4.11), satisfying the ψ 1 = 0 condition. In this case for T < T c the system is in a superconducting state, with a non-vanishing condensate ψ 2 .
We are interested in the phases of the models (1.4)-(1.6) on the temperaturedoping plane. In figure 6 we plot T c (α) for the model 1, (1.4), and the model 3, (1.6), with N = 1/2, 2, 3, for different values of the charge q. It is clear that when the power N in the potential V is higher, the critical temperature for the SC phase transition is smaller. One interesting behavior, which still lacks an interpretation, is the non-monotonic behavior of T c as a function of α, which was already observed in the original model [4,5] and still persists in more generic setups.
In figure 7 we plot T c (∆) for q = 0.6 and α = 1 for the model 1, (1.4), and the model 3, (1.6), with N = 1/2, 2, 3. The T c (∆) curves explicitly show that the critical temperature quickly declines as ∆ approaches the border of the IR instability region. It is further underlined how higher powers/non-linearities in the potential lead to deeper supression of the critical temperature.
We also plot T c (α) for the generalized model 2, (1.5), in figure 8 for various amounts of non-linearity β X 5 , showing again the same behavior of suppression of the superconducting phase at larger β.

Broken phase and phase diagram
In this section we study superconducting phase and construct the phase diagram on the (m, T ) plane of the model (1.5). We will confirm existence of the second order phase transition between normal and superconducting phases by solving four equations (2.7)-(2.10) for the fully backreacted background. Knowing the near-boundary asymptotic behavior of this solution, one can determine the grand potential of the superconducting phase, and compare it with the grand potential of the normal phase to corroborate the phase transition at T = T c .
Running the numerical procedure described in details in appendix A, we were able to construct the condensate ψ 2 /ρ ∆/2 as a function of temperature T /ρ 1/2 . In figure 9 we provide the plot of the condensate, for the model (1.5) with ∆ = 2, q = 1, α u h = 0.5 (α in units of entropy density), m L = 1. There we also plot the grand potential for the broken and normal phases which confirms the superconducting transition at T = T c .
The holographic prescription for the calculation of the grand potential is: where S E is a Euclidean on-shell action of the bulk theory. After some computations showed in details in appendix A we obtain: where we have also used the area-law expression for the entropy: The grand potential is finally given by: where V is a volume of spatial region. In conclusion we obtain (denotingρ = ρL 2 ) the expected thermodynamic relation: where the energy density of the system is given by: We now have enough information to construct the full phase diagram of the nonlinear model (1.5). In figure 10 we plot the phase diagram of the model (1.5) with ∆ = 3, q = 4, α = 0.7 (in units of ρ = 1). We see that the superconducting region can be connected smoothly to both a metallic phase and a pseudo-insulating phase.

Optical conductivity
Our main aim in this section is to see whether the non-trivial structure in the optical conductivity (see figure 2), pointed out for the model (1.5) in the normal phase [16], persists to exist in the superconducting phase.

Fluctuation equations
In order to compute the optical conductivity, we study the fluctuations on top of the charged black brane background with spatially-dependent neutral scalars, as follows: Further in this section we consider homogenous case k = 0, for which it is consistent to put all the fluctuations, besides (6.1), to zero. In this section we also put L = 1. The equations for the perturbations read 10 : One can eliminate h tx from the second equation (6.3) right away, and substitute it into equations for a x and ξ [11]. It is then convenient to perform the following redefinition: (6.5) and reduce the problem to a 2x2 system: which in the normal phase agrees with the equations, derived in [16].

Superconducting phase
In order to extract the optical conductivity of the system we first derive the on-shell action for the fluctuations. We leave the technical steps for appendix B while here we just quote the result: where ζ = Z/u and we have defined the matrix M to be: and expanded the fluctuations near the boundary u = 0 as: a x (u, ω) = a (1) x (ω) + a (2) x (ω) u , (6.10) We solve two coupled fluctuation equations (6.6), (6.7) numerically, for two independent sets of initial conditions which satisfy the infalling behavior near the horizon [30] 11 . Due to linearity of the fluctuation equations (6.6), (6.7), the precise choice of the two sets of initial conditions is not important, and one can check that correlation matrix does not depend on it. For example, let us choose: . (6.14)  Figure 11. The AC conductivity for the model (1.5) with α = √ 2 (in units of 1/u h ), m 2 L 2 = 0.025, q = 4, and ∆ = 2. Black line is at the temperature, slightly below the corresponding critical temperature T c /ρ 1/2 0.16, and matches the result of the normal phase calculation at T = T c . Red, blue, orange and green lines are for T /ρ 1/2 = 0.15, 0.12, 0.09, 0.06, respectively. Notice that as we decrease the temperature, between blue and orange line, the peak in the imaginary part of the AC conductivity disappears. We call the corresponding critical temperature T /ρ 1/2 0.1. We also provide the condensate as a function of temperature and mark the points where we calculated the AC conductivity.
Near the boundary the fields behave as: and we can assemble the matrices of leading and subleading coefficients: We collect the entries of the matrices (6.16) by integrating the equations for the fluctuations numerically and extracting the asymptotic behavior using (6.15). Knowing (6.16) and (6.9), we can finally calculate the correlation matrix: Finally from the correlation matrix (6.17), it is straightforward to find the AC conductivity in superconducting phase, using the Kubo formula: In figure 11 we plot the AC conductivity for the model (1.5) with the non-linear Lagrangian for the neutral scalars, for ∆ = 2, q = 4, α u h = √ 2, m 2 L 2 = 0.025. We consider various values of temperature running from the normal phase to the superconducting phase. The AC conductivity for the normal phase of the model (1.5) has first appeared in [16], where it has been shown that (temperatures are measured in units of square root of charge density) 1. For T > T 0 (T 0 0.46 (for the considered model) the system exhibits a metallic behavior, dσ DC /dT < 0.
3. For T < T < T 0 , where T 0.35 (for the considered model) the non-trivial structure in the AC conductivity appears. To be more precise a mid-infrared peak shows up signaling a weight transfer mechanism and an emerging collective degree of freedom.
These properties are illustrated in figure 3. We checked that the sum rules for the optical conductivity are satisfied in both normal and broken phases.
After we couple this model of [16], with the potential (1.5) for the neutral scalars, to the superconducting sector, more features appear. For the choice of parameters ∆ = 2, q = 4, α u h = √ 2, m 2 L 2 = 0.025 we continue to enumerate what happens as we decrease the temperature: 4. At T c 0.16 (for the considered model) the second order phase transition occurs. The system lives in a superconducting phase, when T < T c .

At T = T
0.1 (for the considered model) the peak in the imaginary part of the AC conductivity disappears. The peak in the real part of the AC conductivity in superconducting phase gets smaller as the temperature is lowered and eventually disappears.
These properties can be seen in figure 11. We will comment more on these features in discussion section 8. It would be very interesting to find the QNM excitations of the system in both normal and broken phase to have complete control on its transport properties and its collective excitations. We leave this topic for future studies.

Dome of superconductivity
In this section we describe how to construct a superconducting dome, by tuning the parameters of the model (1.5) with the non-linear Lagrangian for the neutral scalars. In nature, High-Tc superconductors exhibit a dome of superconductivity (see figure 1) between insulating and metallic normal phases. Due to limitations of our system we cannot construct an actual insulator, however the non-linear model (1.5) still allows to distinguish between two qualitatively different states of the normal phase, (3.3) and (3.4).
The first observation is that when we decouple the translational-symmetry breaking sector of the neutral scalar fields, by setting m = 0, we restore the framework of an ordinary holographic superconductor. Therefore, in order to confine the superconducting phase inside a dome, we need to make sure that the ordinary holographic superconductor exists in the normal state at any temperature. The way to achieve this is to make sure that the parameters ∆ and q are such that the normal phase at T = 0 is stable. That is, we should have D > 0, where D is given by (4.7), with κ = 0 and m = 0. The T = 0 IR stability condition therefore reduces to a well-known inequality, which reads: superconducting dome, is to restore a superconductor at a finite value of m = m 1 , and then make sure that there is another value m 2 > m 1 , such that the system at m > m 2 is again in a normal phase. The procedure to search for the parameters which lead to the superconducting dome is the following. For the chosen value of α we plot D = 0 curves one the (∆, q) plane, with the D given by (4.7), parametrized by various values of m. We search for the points (∆, q) of intersection of two curves, corresponding to two different values of m. These values of m can be the boundaries of the dome region at T = 0. We then verify this explicitly by plotting the D for given α, ∆, q. In figure 12 we plot the D = 0 curves for α = 0.25, on the (∆, q) plane, and demonstrate explicitly that the dome requirement restricts us to consider a small sub-region on the (∆, q) plane. In figure 13 we repeat this for α = 0.5, and also plot the corresponding phase diagram. The superconducting phase is bounded from above by a small critical temperature, and is represented on the graph by a red interval.
We have found that the requirement of having an interval of superconductivity [m 1 , m 2 ] at T = 0 is rather restrictive 12 . We have found that on order to achieve the 'dome' at a vanishing temperature we need to tune ∆ and q to a small subregion of the region (7.1), centered around the point For such ∆ and q we can engineer a model which, at T = 0, exists in a normal pseudoinsulating phase for m ∈ [0, m 1 ], in a superconducting phase for m ∈ [m 1 , m 2 ], and a normal metallic phase for m > m 2 . The next step to construct the superconducting dome is to study the phase structure of the system at finite temperature. To determine the boundary of the superconducting region, that is the line of the second oder superconducting phase transition, we can start in the normal phase, at larger values of temperature, and determine when it becomes unstable towards formation of the scalar hair. This procedure has been reviewed in Subsection 4.2.
However, the point (7.2) is very close to the boundary of the T = 0 infrared instability region of the model (1.5). This behavior is rather generic and leads to the conclusion that the height size of the dome is very limited, the T c is very small and not accessible through stable numerical analysis. Another way of realizing this issue relies on noticing that the BF bound is very mildly violated in the dome region such that the instability is very soft.
We have repeated a similar dome analyses for the model (1.4), with the parameter A = α m playing the role of the doping. Interestingly enough, we have found that again for ∆ and q tuned to a small vicinity of the point (7.2) we obtain a superconducting dome. This time, however, the normal phase can only be metallic. The superconducting dome at T = 0 is an interval [A 1 , A 2 ], existing between two regions of normal metallic state, at A ∈ [0, A 1 ] and A > A 2 . The critical temperature is bounded from above by a small number, and we did not access the finite-temperature superconducting state. We plot our results for the dome in model (1.4) in figure 14.
This analysis shows that the existence of a superconducting dome region is a rather generic feature of these models, independent of the choice of the potential. In the next section we discuss the possible ways to alleviate the problem of the flatness of the dome. This seems to require introduction of an extra elements to our holographic system.

Discussion
In this paper we considered a holographic superconductor with broken translational symmetry, continuing the research, initiated in [4,5]. To break the translational symmetry we used the known technique [11], coupling our system to the sector of massless neutral scalar fields, depending linearly on the spatial coordinates. We studied the standard Lagrangian for these neutral scalars, as well as its non-linear generalization, proposed in [16]. We have constructed models, exhibiting the following non-trivial new features: 1. The Holographic superconductor in the non-linear Lagrangian model has a rich phase diagram on the temperature-doping plane. In particular the superconducting phase is separated from the normal pseudo-insulating phase and the normal metallic phase by the line of second order phase transition as shown in figure 10.
2. In the same model the optical conductivity exhibits a non-trivial emerging structure, signaling a collective excitation of the charge carriers localized in the mid-infrared. This has been observed in [16] in the normal phase of the same model, for temperatures, lower than a certain critical value. In this pa-per we have demonstrated that this structure persists in the superconducting phase. Eventually it gets destroyed by the charge condensate. This suggests a possible competition between the superconducting mechanism and the momentum dissipating one. In particular it seems clear that a large superfluid density completely screens this collective excitation which in a sense gets eaten by the large condensate. We are not aware of real superconducting system supporting a collective localized excitation like the one we see. It would be nice to see if other holographic models, providing translational symmetry breaking, support the same property. In this direction it would be very interesting to study the QNM structure of the system as initiated in [32].
3. We performed a complete analysis of the behavior of the critical (superconducting) temperature as a function of the various parameters of our model. In particular we studied the curious non-monotonic behavour of T c as a function of the graviton mass m, which was already observed in [4,5]. Our results suggest that this feature persists for generic Lagrangian for neutral scalars. We do not have any clear explanation of the big mass regime where T c actually increases with the strength of translational symmetry breaking. It is even tempting to doubt the model in that regime, reminiscing the following known issues: for large momentum dissipation it seems that the energy density of the dual field theory at zero charge density gets negative [32]; the diffusion bounds for the model are unrestricted from below and the diffusion constants go to zero in that limit [33].
A very similar behavior has been observed in holographic SC with helical lattices [6] and with disorder [2]. It would be interesting to further analyze the universality and the meaning of this feature.
It would be interesting to improve the model, so that pseudo-insulating phase is replaced by an actual insulating phase. This will make the phase diagram more resembling such of an actual high-Tc superconductor. This could be easily achieved introducing a dilaton field into the model. Another interesting open problem is to increase the critical temperature of the superconductor, enclosed by a dome-shaped line. One of the ways to accomplish this might be realized by an inclusion of a non-trivial coupling κ between the charged scalar condensate, and the neutral scalars.
A further interesting question is to look at universal properties of these large class of effective toy models such as the accomplishment of Homes' Law following [35]. We leave these interesting questions for future investigation. checked explicitly that the results are stable towards changing . We have the freedom of choice of the initial conditions ψ(u h ), A t (u h ), and χ(u h ). The freedom of choice of χ(u h ) is spurious, due to the time scaling symmetry, as we discuss below.
The values of ψ(u h ) and A t (u h ) are fixed by the requirement of having a fixed temperature T /ρ 1/2 and zero source ψ 1 = 0, see (4.10). Both the charge density ρ, in units of which me measure the temperature, and the source ψ 1 are determined by the near-boundary behavior of the numerical solution, with the gauge field behaving as: In practical calculation we do the following. Suppose the temperature is sufficiently small, so that the system is in a superconducting phase. We know that increasing the temperature will decrease the condensate, ψ 2 /ρ ∆/2 , until finally at the critical temperature T c the condensate is zero. At that point ψ(u h ) = 0, that is, we do not have the solution with vanishing source and non-trivial profile of ψ(u) in the bulk. Therefore we can start at ψ(u h ) = 0, and take gradually incrementing values of ψ(u h ). For each value of ψ(u h ) we search for A t (u h ), such that ψ 1 = 0. For an example of this kind of result see figure 15. Finally for the given pair (ψ(u h ), A t (u h )) we calculate numerically T /ρ 1/2 , ψ 2 /ρ ∆/2 as shown for example in figure 9.

Scaling symmetry
The equations of motion (2.7)-(2.10) are invariant under the scaling symmetry: where a is a parameter of the symmetry transformation. The temperature, chemical potential, and the charge density therefore transform as: The scaling symmetry (A.3) allows one to fix u h = 1. If u h is not fixed to one, then we should substitute u −2 hÃ t (u h ) as the initial condition for the flux at the horizon. We have checked explicitly that the results are invariant under change of u h .

Time scaling symmetry
The equations of motion (2.7)-(2.10) are invariant under the time scaling symmetry: where b is a parameter of the symmetry transformation. We can use the time scaling symmetry (A.5) to fix χ(0) = 0 at the boundary. This is necessary, so that the speed of light in the boundary field theory is equal to one. To achieve this, we impose the initial conditions on χ to be χ(u h ) + 2 log b, and on the flux to beÃ t (u h )/b. We fix χ(u h ) once and for all. We have demonstrated explicitly that the result is independent of the choice of χ(u h ).
After fixing χ(u h ), for the givenÃ t (u h ), we integrate numerically the equations of motion, with b = 1. We then impose b = e −χ(0)/2 , where χ(0) is determined numerically. For this b we impose the initial conditions χ(u h ) + 2 log b,Ã t (u h )/b and integrate the equations of motion again. This time, due to the time scaling symmetry (A.5), we have χ(0) = 0. We have verified this explicitly.
Running the described numerical procedure we were able to construct the condensate ψ 2 /ρ ∆/2 as a function of temperature T /ρ 1/2 . In figure 9 we provide the plot of the condensate, for the model (1.5) with ∆ = 2, q = 1, α u h = 0.5 (α in units of entropy density), m L = 1. For the same parameters we also plot the initial conditions (−A t (u h ), ψ(u h )), which we imposed, to enable the vanishing source ψ 1 = 0 in figure 15.

A.2 Grand potential
Here we provide intermediate steps for calculation of the grand potential.
The holographic prescription for the calculation of the grand potential is: where S E is a Euclidean on-shell action of the bulk theory. This should be supplemented with the boundary Gibbons-Hawking term, and the counter-terms 13 . The resulting action reads: where the boundary Gibbons-Hawking term is given by: where is a UV cutoff, h ab the pullback metric on the boundary and K ab the extrinsic curvature 14 . The counter-term action I c.t. is a sum of gravitational, scalar and axion fields counter-terms [4,36]: It is convenient to evaluate the following Lagrangian on shell: to get: After a straightforward calculation we obtain: Notice that which B(u H ) = 0. Therefore the full on-shell action is given by: To proceed with the calculation, we need to be able to evaluate the counter-term action (A.10) and the B( ) term of (A.14). We need to know the near-boundary behavior of the fields. That is given by 15 : 14 It is defined by where n µ is the unit vector normal to the boundary. 15 Note that this is true only if the potential reads V (X) = X + X n1 + X n2 + ... where the smallest power is always equal to one.
We are interested in the systems with vanishing source of the charged scalar, ψ 1 = 0. By solving equations of motion near the boundary, we obtain: Combining all the results together, we arrive at the final expression for the on-shell action:

B On-shell action for fluctuations
The calculation of the on-shell action for fluctuations is similar to the one for the grand potential performed in appendix A. The total action is a sum of the total bulk action (2.1), the Gibbon-Hawking (GH) term on the boundary (A.8), and the counter-term action (A.10): We evaluate the action (B.1) on the ansatz: φ y = α y , ψ = ψ(u) , and collect O( 2 ) terms, which describe dynamics of the fluctuations h tx , a x , ζ. The O( ) terms vanish due to equations of motion, satisfied by the background fields f , A t , φ x,y , ψ, and the O( 0 ) terms are contributions to the grand potential for the background. The GH term vanishes at the horizon. Therefore: We obtain: 4u 4 f 2 −2f e χ 2uh tx (u, t) u 3 f A t ∂ u a x (u, t) + 2q 2 uA t ψ 2 a x (u, t) + 4f ∂ u h tx (u, t) + 2αL 2 m 2 u∂ t ξ(u, t)V + u 2 − u 2 (∂ t a x (u, t)) 2 + f (∂ t h tx (u, t)) 2 + 2L 2 m 2 (∂ t ξ(u, t)) 2V + h tx (u, t) 2 (−2 (uf + 3) + f u 2 ψ 2 + 2uχ − 6 + L 2 2m 2 V − α 2 u 2 V + M 2 ψ 2 − u 2 e 2χ h tx (u, t) 2 u 2 A 2 t + 2q 2 A 2 t ψ 2 − 2u 4 f 3 (∂ u a x (u, t)) 2 − 4q 2 u 2 f 2 ψ 2 a x (u, t) 2 − 4L 2 m 2 u 2 f 3 (∂ u ξ(u, t)) 2V (B.4) To proceed, we integrate the a 2 x , h 2 tx , ξ 2 terms by parts, and substitute expressions for a x , h tx , ξ from the corresponding fluctuation equations. We need to keep track of the boundary terms. Then let us go to the momentum space. As a result we arrive atĨ f = B f , where: 2u 3 e χ h tx (u, −ω) u 3 A t a x (u, ω) − uh tx (u, ω) + 4h tx (u, ω) + uf u 2 a x (u, −ω)a x (u, ω) + 2L 2 m 2 ξ(u, −ω)ξ (u, ω)V , (B.5) where prime, as before, stands for a derivative w.r.t. u. The counter-term action (A.10) for the ansatz (B.2) is given by: where again not all the coefficients of expansion are independent, and in fact: (B.14) where we have kept ξ (1) , for brevity (but keep in mind it is not an independent expansion coefficient, due to (B.10)). It is convenient to replace ξ → Z, so that we are dealing with two fields, (a x , Z), which have the same near-boundary expansion, at least up to the first two orders. Due to (6.5), we obtain: which near the boundary becomes: We can represent I f tot in the form, convenient for calculation of correlation matrix: x (−ω) Z (2) (−ω) + · · · , (B.17) where dots denote ξ (1) terms. We cannot extract ξ (1) by solving system of equations for (a x , Z), because Z ∼ ξ . So we assume that ξ (1) is a constant of integration, which we fix to be: which is the choice enabling a diagonal matrix M . Let us rescale the fluctuation fields (this is a symmetry transformation of fluctuation equations): The corresponding matrix is: For the purpose of finding AC conductivity we only need the (a x , a x ) component of the correlation matrix.