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We reexamine the charged AdS domain wall solution to the Einstein-Abelian-Higgs model proposed by Gubser et al. as holographic superconductors at quantum critical points and comment on their statement about the uniqueness of gravity solutions. We generalize their explorations from 3 + 1 dimensions to arbitrary n + 1 Ds and find that the *n + 1 ≥ *5D charged AdS domain walls are unstable against electric perturbations.

The charged AdS domain walls are spaces interpolating two copies of anti-de Sitter space, one of which preserves the abelian gauge symmetry while the other one breaks it. The two sides of the domain walls have different AdS radius. 0Similar neutral domain walls (see [1,2] for references), are believed to have been formed in the early universe. also be observed in condensed matter systems, including in superconductors [3-6]. In references [7,8], S. S. Gubser and collaborators proposed that the quantum critical behavior and the emergent relativistic conformal symmetry in superfluids or superconductivities in strongly coupled gauge theories can be described by charged AdS domain wall solutions in several Einstein-Abelian-Higgs models. These works are mainly concerned with 3 + 1 dimension gravity theories and provide solutions they think be uniquely determined by the scalar field potential form and double boundary conditions.

The purpose of this paper is to generalize these discussions to arbitrary space-time dimensions and study properties of the charged AdS domain-wall. We will first in Section 2 point out that the double boundary conditions quotient by [7,8] are questionable and probably exclude the existence of solution families to the relevant dynamic equations. We then in Section 3 provide a new ansatz for the AdS domain walls and the corresponding equations of motion. The solution families in both 3 and 4 dimensions are provided also in this Section. While in Section 5 we study the electric perturbations to the solution and calculate the related electric-transporting coefficients. The last Section contains our main conclusions.

This Section has two purposes. The first is to provide basic ingredients to study the charged AdS domain walls. The second is to discuss the questionable aspects of reference [7,8] about the uniqueness of charged AdS domain wall solutions to the relevant equation of motion. The “uniqueness” of this two references states that, for an Einstein-Abelian-Higgs model system with given potential form such the following (1) and (2), when specifying the ultraviolet asymptotical behavior of the scalar field, the domain wall solution to the system is unique. Our analysis in the following is mainly from technical aspects. For readers who are more willing to catch physics by intuitions, we provide here a reason supporting First let us provide the basic ingredients of charged AdS-domain wall studyings. Taking the model of reference [

where we have adapted the dimension from 3 to arbitrary and used instead of reference [

and require that in the infrared limit sits on the global minimal of, i.e. and,. While the ultraviolet limit of the solution is constrained by the dual field theory to be some specific AdS featured. Reference [

Of these equations, the last one is looked as a constraint and the former 4 second order differential equations are solved numerically. The authors state that among the 8 integration constants: 1) one is used up by the constraint (9), 2) six are used up by the infrared-boundary conditions, i.e. as, (a)(b), (c), (d), (e), (f)3) the last one is determined by ultraviolet boundary condition that. So the solution to Equations (5)-(9) is unique, as long as the ultraviolet scaling dimension of the operator is fixed. There is no solution family parameterized by.

Here comes our standpoint against unique but supporting family of solutions. We have two reasons for our standpoint. The first is from physical analogue. Just as pointed out by reference [

scalar field varies from to. While all these things are implemented by changing the parameter of the previous paragraph without changing the asymptotical behavior(equivalence of the conformal dimension of) of the scalar field. This implies that is a freely tunable parameters instead of fixed number determined by the form of scalar field potentials and the conformal dimension of the corresponding field theory operators. This is our first, and probably the most strong reason supporting family instead of unique solutions to the Einstein-Abelian-Higgs model under the AdS-domain wall ansatz.

Our second reason is from technique analysis. In the counting of integration constants consummation of reference [

with being set arbitrarily. The fact that conditions (a) and (f) are equivalent implies that we cannot write the boundary conditions as

The setting method (1) accepts that is an arbitrary integration constant, while the setting method (1) introduces no any tunable integration constant and try to determine the value of through consistences of the double boundary value problems. We do not know if the author of reference [

For further supporting the above reasonings, we simultaneously solved the double boundary value problem of [

Summarizing reasons in this section, we conclude that, given the form of scalar field potentials and the ultraviolet conformal dimension of the corresponding operator, there is still a family of charged AdS-domain wall solutions. The members in this family are distinguished from each other by their charged density or wall-thickness. Numerically, it is the parameter that determines this features.

In this Section, we introduce a new ansatz for the AdS domain wall and more directly construct the family of solutions. The new ansatz has the advantage of reducing the order of differential equations which follows from minimizing the action of the system,

As long as has different values in the and limits, this will be domain walls interpolating between two AdS-spaces characterized by and. This ansatz can more explicitly express the asymptotical AdS features of the geometry. By rescaling coordinate, we can always set, then the value of will be determined by the equations of motion. As varies from the infrared region to the ultraviolet region, the speed of light in the two regions will change naturally. By the ansatz (14), the equation of motion reads

We checked that in these 5 equations, the last one can be derived out from its four predecessors only by differentiations and combinations. So it is not independent and can be looked as a constraint completely. Comparing with the equation of motion under the ansatz of reference [7,8], among our four other independent equations, two are first order, while the other two are second order. So essentially have only 6 equivalent first order equations, while the reference [7,8] need to solve 8 equivalent first order equations. Obviously, 6 first order differential equations need, and only needs 6 boundary conditions.

The exponent indices and involved in these expressions are determined from the infrared limit of Equations (18) and (19). Note Equation (22) contains information on two aspects, and as, so it should be counted as two boundary conditions. The same is true for Equation (23). The above equations of motion and boundary condition obviously defines a oneparameter family of solutions featured by either or (our choice is setting while let to feature solutions).

Although, the above boundary conditions only specifies the IR behavior of the solution, in the UV limit, as long as, i.e. as long as approaches the meta-stable point of the potential, the fate of other fields, including the asymptotical behavior of itself, are destined. It can be easily proven that

.

So the key question is, if for various solutions in the family we declared previously, goes to zero in the ultraviolet limit. Numerics tell us that, this is indeed the case, see

The charged AdS domain walls can also carry variable charge densities, but probably fixed charge/mass ratios.

It is worth pointing out that, the difference between heights of different domain walls’ electrostatic potentials, see

If we use these charged AdS domain wall systems as models of holographic superconductors, then one possible explanation is that, domain walls with different corresponding materials with different charge densities which implement superconductions, i.e. the density of superconductive electrons. The existence of charged domain wall families provides very good examples for the Criticality Paring Conjecture of [

Let us in this Section consider the electric perturbations of the charged AdS domain wall solutions. This consideration will give us information on two aspects. One is the stability of the domain wall configuration itself, the other is the transportation properties of the dual field system.

Including responses of the background metric on gauge field perturbations, we can write all the perturbed field as follows,

From the linearized Maxwell equation and Einstein Equation, we can derive out that, see reference [

Expanding the solution in the ultraviolet limit in the form

and using the AdS/CFT dictionary which says that, is proportional to the perturbing source while to the response i.e. currents in the CFT, we directly get the electric conductivity

Imposing infalling boundary conditions in the infrared limit region and solve Equation (30), we will get the relation directly.

By totally the same method of [7,13], we can verify the scaling law of [

When considering the effects of on this scaling law, we find that it only modulates the proportional constant in the above relation—makes it proportional to

. This means

The last proportionality is also easy to understand, because in the equation governing’s evolution, always appears as a whole. By reference [

One of the motivations leading references [7,8] to declare the uniqueness of charged domain wall solutions is, they hope to use this model as a description for the quantum criticalities observed in the high superconductors [14,15] which occurs only at one optimized doping rate. If the charged domain wall is non-unique, then one must suspect the reasonability of doing this. However, just as the previous Section of this work indicates, in a family of charged AdS domain wall solutions, differences between various members are only their charge densities, featured by the parameter. The observations (34) tell us that, changing this charge density does not affect the power law feature of the optical conductivity. This implies that, although the quantum phase transition in high superconductors is triggered by optimizing the doping rate of materials, it is not implemented through the changing of superconductive charges’ density. In other words, all members in a family of charged AdS domain walls can be used as holographic models of quantum critical superconductors. The only thing worthy

of noticing is that, quantum critical superconductions could occur in materials carrying different superconductive charge densities.

Calculating the conductivity of general dimensional charged AdS domain wall, we will see that the results are drastically different from those of case, see

where “” denotes derivatives with respect to. The general solution to this equation reads

where. By asymptotic expressions of the Bessel function, we know

Obviously, in the n ≥ 4 case, the perturbation does not converge. Instead it diverges in the form

as we follow down deep into the infrared region. This divergence of the perturbation in the deep infrared region obviously implies that, the charged domain walls are unstable. In the dual field theory, this means that the infrared fixed point (conformal) is unstable. From the gravity side, we know this instability neither depends on the hight of the domain wall measured by or, nor on the charge of the domain wall measured by or. It is completely determined by the dimension of the wall. Although strange, we think this is an interesting result and possibly not being noticed by earlier researchers.

Two conclusions of this work are worth emphasizing in this section. The first is, given the scalar fields’ potential form and its ultraviolet scalings, there is still a domain wall solution family to the relevant equations of motion. Different members of this family carry different charge density but probably fixed charge/mass ratios. Due to differences between the charge densities, different members in this domain wall family have different relative hight of electrostatic potentials in the ultraviolet and infrared region. In the dual field theory, this corresponds to different chemical potential or conserving charge densities. All these things are very similar to the case of extremal RN black holes, whose charge/mass ratio is fixed but the amount of charge or mass each-self is tunable. The second is, the higher dimensional charged AdS domain wall is perturbatively unstable. Our work uncovers a more closer similarity between the charged AdS-domain wall and the extremal Riessner-Nordström black branes. That is, the charged AdS-domain wall can be changed into neutral ones just as the extremal AdSRN black brane can be changed into simple AdS spaces by reducing their charge and mass density simultaneously.