# Axionic charged black branes with arbitrary scalar nonminimal coupling

## Abstract

In this paper, we construct four-dimensional charged black branes of a nonminimally coupled and self-interacting scalar field. In addition to the scalar and Maxwell fields, the model involves two axionic fields homogeneously distributed along the two-dimensional planar base manifold providing in turn a simple mechanism of momentum dissipation. Interestingly enough, the horizon of the solution can be located at two different positions depending on the sign of the parameter associated to the axionic field, and in both cases there exists a wide range of values of the nonminimal coupling parameter yielding to physical acceptable solutions. For a negative parameter that sustains the axionic fields, the allowed nonminimal coupling parameters take discrete values and the solution turns out to be extremal since its has zero temperature. A complete analysis of the thermodynamic features of the solutions is also carried out. Finally, thanks to the mechanism of momentum dissipation, the holographic DC conductivities of the solutions are computed in terms of the black hole horizon data, and we analyze the effects of the nonminimal coupling parameter on these conductivities. For example, in the purely electric case, we notice that as long as the nonminimal coupling parameter takes the discrete values associated to the extremal solution, the DC conductivity vanishes identically reproducing in turn an insulator behavior. In the non extremal case, we point out the existence of a particular value of the nonminimal coupling parameter (which is greater than the conformal one in four dimensions) yielding an infinite conductivity; this is due to the fact that the translation invariance is restored at this point. Finally, in the dyonic case, we show that the conductivity matrix for the extremal solution has a Hall effect-like behavior.

## 1 Introduction

In the last decades, the ideas underlying the anti de-Sitter/conformal field theory (AdS/CFT) correspondence have been applied to get a better understanding of phenomena that occur in the condensed matter physics as the quantum Hall effect, the superconductivity or the superfluidity, see e.g. [1, 2]. As a significative example, we can mention the case of charged hairy black holes with a planar horizon that may be relevant to describe the behavior of unconventional superconductors [3]. In this scenario, the nonzero condensate behavior of the unconventional superconductors is mimicked by the existence of a hair at low temperature that must disappear in the high temperature regime [3]. Nevertheless, finding black holes with such features is a highly nontrivial problem that is rendered even more difficult by the various no-hair theorems with scalar fields existing in the current literature, see e.g. [4]. Fortunately, the precursor works of Refs. [5, 6] have established that scalar fields nonminimally coupled seem to be an excellent laboratory in order to escape the standard scalar no-hair theorems. Indeed, as shown independently in Refs. [5, 6], conformal scalar field nonminimally coupled to Einstein gravity can support black hole configuration with a nontrivial scalar field. These black hole solutions have been dubbed BBMB solution in the current literature. However, the BBMB solution suffers from some pathology essentially because of the divergence of the scalar field at the event horizon. This inconvenient makes its physical interpretation and the problem of its stability a subject of debate. A way of circumventing this pathology consists in introducing a cosmological constant whose effect is to precisely push the singularity behind the horizon, and as a direct consequence, the scalar field becomes well-defined at the event horizon [7]. It is important to stress that for the BBMB solution or its extensions with cosmological constant dubbed as the MTZ solution [7, 8], the parameter \(\xi \) that couples nonminimally the scalar field to the curvature is always the conformal one in four dimensions, namely \(\xi =\frac{1}{6}\), and the horizon topology of the black hole solutions is either spherical or hyperbolic depending on the sign of the cosmological constant. There also exist examples of black hole solutions with the conformal coupling \(\xi =\frac{1}{6}\) but with a potential term that breaks the conformal invariance of the matter source, see e.g. [9, 10]. Nevertheless, charged black holes with planar horizon topology for a scalar field nonminimally coupled to Einstein gravity with or without a cosmological constant are not known.^{1} It seems that extra matter source is needed in order to sustain planar charged black hole with a nonminimal scalar field. This intuition is based on the works done in [14, 15] where a planar version of the MTZ solution was rendered possible thanks to the introduction of two 3-forms that were originated from two Kalb-Raimond potentials. Interestingly enough, this construction was also extended for arbitrary nonminimal coupling in [16]. Very recently, it has also been shown that (charged) planar AdS black holes can arise as solutions of General Relativity with a source composed by a conformal scalar field together with two axionic fields depending linearly on the coordinates of the planar base manifold [17]. The existence of these planar black holes is mainly due to the presence of the axionic scalar fields which, in addition of inducing an extra scale, allow the planar solutions to develop an event horizon. More precisely, as proved in [17], the black hole mass is related to the parameter associated to the axionic fields, and hence these axionic (charged) black branes can be interpreted as extremals in the sense that all their Noetherian charges are fixed in term term of the axionic intensity parameter. Many other interesting features are inherent to the presence of axionic fields for planar solutions. Among others, axionic fields depending linearly on the coordinates of the planar base manifold provide a very simple mechanism of momentum dissipation [18]. From an holographic point of view, this feature has a certain interest since, as established in Refs. [19, 20], the computation of the DC conductivities can be uniquely expressed in term of the black hole horizon data. Mainly because of these results, the study of axionic black hole configurations in different contexts has considerably grow up the last time, see e.g. [21, 22, 23, 24, 25, 26, 27, 28].

In the present, we plan to extend the work done in the conformal situation [17] to the case of a four-dimensional scalar field with an arbitrary nonminimal coupling with two axionic fields. The model also involves a parameter *b* that enters in the scalar potential and in the function that minimally couples the scalar and the axionic fields, see below (2.3a–2.3b). The range of this extra parameter will be fixed by some reality conditions as well as by demanding the solutions to have positive entropy. Asymptotically AdS planar dyonic black hole solutions will be presented with axionic fields homogeneously distributed along the orthogonal planar coordinates of the base manifold for a priori any positive value of the nonminimal coupling parameter \(\xi \). However, the positive entropy condition will considerably reduce the range of the permissible values of the nonminimal coupling parameter. As in the conformal case [17], the solutions only contain an integration constant denoted by \(\omega \). Interestingly enough, the location of the event horizon can be at two different positions depending on the sign of the parameter \(\omega \). Moreover, for \(\omega <0\), the range of permissible values of the nonminimal coupling parameter is discrete, and the solution is shown to be extremal since its has zero temperature. Finally, the full DC conductivities associated to the charged black brane solutions will be computed following the recipes given in [19, 20]. One of our motivations is precisely to identify the impact on the conductivities of the nonminimal coupling parameter.

The plan of the paper is organized as follows. In the next section, we present the model which consists of the Einstein gravity action with a negative cosmological constant with a source given by a self-interacting and nonminimally scalar field coupled to two axionic fields. In Sect. 3, the asymptotically AdS planar dyonic solutions are presented. In Sect. 4, a detailed analysis of the thermodynamic properties of the solutions through the Hamiltonian method is provided allowing to identify correctly the mass of the dyonic solutions. In the next section, following the perturbative method presented in [19, 20], the full DC conductivities of the (non) extremal solutions are computed and the effects of the nonminimal coupling parameter are analyzed. The last section is devoted to our conclusions.

## 2 Model, field equations and black brane solutions

^{2}Note that the particular value \(\xi =\frac{1}{4}\) which corresponds to the limit \(n\rightarrow \infty \) will be treated separately. The potential \(U_b(\phi )\) and the coupling \(\varepsilon _b(\phi )\) associated to the axionic fields depend on a positive constant denoted by

*b*whose range will be conditioned by some reality conditions as shown below

Range of the permissible values of the nonminimal coupling parameter \(\xi \) depending on the sign of the axionic parameter \(\omega \) ensuring a real solution

Sign of \(\omega \) | horizon \(r_{+}\) | Permissible values of |
---|---|---|

\(\omega >0\) | \(r_{+}=\frac{3\omega }{\sqrt{12\kappa }}\) | \(n>1\) i.e. \(\xi \in \ ]0,\frac{1}{4}[\) \(b\in ]0,b_0[\) |

\(\omega >0\) | \(r_{+}=\frac{3\omega }{\sqrt{12\kappa }}\) | \(n<0\), i.e. \(\xi \in \ ]\frac{1}{4},\infty [,\quad b\in ]b_0, \infty [\) |

\(\omega <0\) | \(r_{+}=-\frac{\omega }{\sqrt{12\kappa }}\) | \(n=1-2k\), i.e. \(\xi =\frac{k}{2(2k-1)}\), \(k\in \mathbb {N}{\setminus }\{0\},\quad b\in ]b_0, \infty [\) |

*b*of the model must belong to the interval \(b\in \,]0, b_0[\) with

*n*to be an odd negative integer \(n=1-2k\) with \(k\in \mathbb {N}{\setminus }\{0\}\) or equivalently the nonminimal coupling parameters is forced to take the discrete values given by \(\xi =\frac{k}{2(2k-1)}\). All these details are summarized in the Table 1. It is also interesting to stress that for \(\omega <0\), even if the scalar field vanishes at the horizon, the expressions of the potential (2.3a) and the coupling (2.3b) remain finite once evaluated on the solution at the horizon, i.e.

*b*given by \(b=b_0\) where \(b_0\) is defined in (2.7).

*U*and a coupling \(\varepsilon \) free of any couplings that read

## 3 Thermodynamics of the solutions by means of the Hamiltonian method

*T*stands for the temperature which is fixed by requiring regularity at the horizon. The temperature of the non-extremal solution (2.5) reads

*r*ranges from the horizon to infinity, i.e. \(r \ge r_{+}\), and the planar coordinates both are assumed to belong to a compact set, that is \(x\in \Omega _x\) and \(y\in \Omega _y\) with \(\int dx\,dy=\Omega _x \Omega _y\). We also assume a specific ansatz for the scalar, axionic and electromagnetic fields, i.e. \(\phi = \phi (r)\), \(A_{\mu }dx^{\mu } = A_{\tau }(r)d\tau + A_{x}(y)dx + A_{y}(x)dy\), and \( \psi _1=\psi _{1}(x)\), and \(\psi _{2}= \psi _{2}(y)\). In doing so, the reduced Euclidean action takes the form

*B*is a boundary term that will be properly fixed below. Here

*p*is the conjugate momentum of \(A_{\tau }\), \(p = -\dfrac{ r^2}{N(r)}A_{\tau }(r)'\), and the reduced Hamiltonian \(\mathcal {H}\) is given by

*B*is fixed by requiring that the reduced action has an extremum, that is \(\delta I_{\tiny {Euc}} =0\), yielding to

*B*) imply that

*N*is a constant, and this latter can be chosen without any loss of generality to be \(N(r)=1\). On the other hand, Gauss law implies that \(p=cst= q_e\). For the axionic fields, we note that their contribution strongly depends on the integral \(\int _{r_{+}}^{r} \varepsilon _b(\phi (r))\ dr\), which in our case can be computed yielding to

^{3}Working in the grand canonical ensemble, where \(\beta \) and all the potentials are fixed, the boundary term at the infinity can be integrated as

*b*; this is because the constant \(\tilde{G}\) as defined by (3.7) must be strictly positive, and this leads to

*b*must then be reduced to \(b\in ]\frac{1}{4}b_0, b_0[\). For the other values of the nonminimal coupling parameter \(n<0\) or equivalently \(\xi >\frac{1}{4}\), the entropy of the solutions turns out to be always negative because of the reality condition (2.6). Concerning the mass, it is interesting to note that the mass is always positive for \(\eta =0\) as defined by (3.5) or for the conformal case \(n=3\). On the other hand, for \(n=2\), the positivity of the mass requires \(b>\frac{1}{2}b_0\), and hence our static solution with \(n=2\) can have positive entropy and mass by demanding the parameter

*b*to belong to the set \(b \in ]\frac{1}{2}b_0, b_0[\). Finally, for the remaining values for which \(\eta =1\), namely \(n=-1, -2,\ldots \), the solutions will always have negative mass and entropy.

Signs of the entropy and mass of the dyonic solutions whose reality conditions are fixed by Table 1 and where \(b_0=\left( \frac{n-1}{8n\kappa }\right) ^{\frac{1}{n-1}}\)

Sign of \(\omega \) | Permissible values of \(\xi \) | Signs of entropy and mass |
---|---|---|

\(\omega >0\) | \(n>1\) and \(n\not =2\) i.e. \(\xi \in \ ]0,\frac{1}{4}[{\setminus }\frac{1}{8}\) | \(\mathcal {S}>0\) and\(\,\mathcal {M}>0\), for \(b\in ]\frac{1}{4}b_0, b_0[\) |

\(\omega >0\) | \(n=2\) i.e. \(\xi =\frac{1}{8}\) | \(\mathcal {S}>0\) and\(\,\mathcal {M}>0\), for \(b\in ]\frac{1}{2}b_0, b_0[\) |

\(\omega >0\) | \(n<0\) i.e. \(\xi \in \ ]\frac{1}{4},\infty [\) | \(\mathcal {S}<0\) and\(\,\mathcal {M}<0\), for \(b>b_0\) |

\(\omega <0\) | \(n=1-2k\), i.e. \(\xi =\frac{k}{2(2k-1)}\), \(k\in \mathbb {N}{\setminus }\{0\}\) | \(\mathcal {S}>0\) and\(\,\mathcal {M}>0\), for \(b>b_0\) |

## 4 Holographic DC conductivities

*r*. To that end, we follow the prescription as described in Refs. [19, 20], and we first turn on the following relevant perturbations on the black brane solution (2.5)

*v*,

*r*) such that \(v = t + \int \frac{dr}{f(r)}\). In this case, the gauge field will be well-defined by demanding

*rx*and

*ry*components of the linearized Einstein equations, obtaining a system of equations for \(h_{tx}(r_{+})\) and \(h_{ty}(r_{+})\), whose solutions are given by

*b*and the nonminimal coupling parameter

*n*on the electrical DC conductivity. We also recall that the parameter

*b*is subjected to some reality conditions (cf. Table 1) corresponding to the mathematical range of

*b*but also to physical constraints that ensure the positivity of the mass and entropy (cf. Table 2) that we will refer as the physical range of

*b*.

Firstly, it is straightforward to prove that there exists a value of the nonminimal coupling parameter \(n_0\approx 3.4681\) corresponding to \(\xi _0\approx 0.1779\) (2.2) such that for each \(n>n_0\), there exists a precise value of the parameter *b* denoted by \(b_1\) with \(\frac{b_0}{4}<b_1<b_0\) (cf. Table 2) such that the DC conductivity is strictly positive for \(b\in ]b_1, b_0[\) and \(\sigma _{DC}\) becomes infinite at \((n, b_1)\). In other words, this means that for any coupling greater than \(n_0\), one can always choose a parameter \(b\in ]b_1, b_0[\) of the theory that yields to well-defined physical solutions, namely solutions with positive mass and entropy and having a positive conductivity. Additionally, for the choice \(b=b_1\), the DC conductivity becomes infinite at the point \((n, b_1)\) with \(n>n_0\). This is due to the fact that the minimal coupling function \(\varepsilon _b(\phi )\) evaluated at the horizon vanishes at this point. This is not surprising since in order to ensure the mechanism of momentum dissipation yielding to finite conductivity, the coupling function \(\varepsilon _b(\phi )\) must not vanish.

*b*and

*n*, we will plot the graphics of \(\sigma _{DC}\) in function of

*b*and

*n*. In Fig. 1, we plot the conductivity vs the parameter

*b*for two distinct values of the nonminimal coupling parameter, namely \(n=2<n_0\) and \(n=4>n_0\). The graphic given by Fig. 1a for \(n=2\) is in fact representative of all the cases \(1<n\le n_0\). One can see that for the physical range of

*b*, namely \(b\in ]\frac{1}{4}b_0, b_0[\) or \(b\in ]\frac{1}{2}b_0, b_0[\) for \(n=2\) (cf. Table 2), the conductivity is strictly positive and finite.

^{4}In fact, the positivity of the conductivity is always ensured even for the full mathematical range of

*b*, i.e. \(b\in ]0, b_0[\). On the other hand, the graphic represented in Fig. 1b for \(n=4\) will be similar for any value \(n>n_0\). In this case, as mentioned before, one can see the existence of a a vertical asymptote at \(b=b_1\) (for \(n=4\) we have \(b_1\approx 0.66534\)) where the conductivity becomes infinite. Nevertheless, in contrast with the previous case, the conductivity is positive only for \(b\ge b_1\), and hence part of the physical range, namely \(b\in ]\frac{1}{4}b_0, b_1[\) yields to negative conductivity. We also include a plot of the non extremal situation for \(n<0\) or equivalently \(\xi >\frac{1}{4}\), see Fig. 2. The mathematical range of

*b*is located at the right of the blue line while its physical range is at the left of the dotted line. One can see that for a mathematically well-defined solution, the electric conductivity is always positive even if it has negative mass and entropy. We can also appreciate the influence of the nonminimal coupling parameter on \(\sigma _{DC}\) by drawing the graphics of this latter in function of

*n*, see Fig. 3. In order to achieve this task, we must be careful with the election of

*b*since its range of permissible values depends explicitly on

*n*, see Tables 1 and 2 where we have defined \(b_0\) in (2.7). As explained before, one can see that for \(n\le n_0\), the electric conductivity is positive while for \(n>n_0\), one has \(\sigma _{DC}<0\). Finally, we would like to mention that as in the conformal case [17], the DC conductivity is temperature-independent. To be more precise, on one hand we have seen that the temperature depends on the axionic parameter \(\omega \). On the other hand, in order to have solutions that respect the AdS symmetry at the boundary, the axionic charge must be related with the electric and magnetic charges through the relation (2.6). It is important to stress that this particular relation is due to the fact that the fall off of the nominally coupled scalar modifies the asymptotic behavior of our solutions. Moreover this relation makes our thermodynamic analysis integrable. However, as one uses the relation (2.6) in the conductivity matrix, we notice that \(\omega \)-dependence disappears as it can be appreciated in the expression (4.7).

^{5}

*B*the orthogonal magnetic field). Indeed, in the AdS/CFT dictionary, \((3+1)\)-dimensional AdS dyonic black holes are conjectured to be dual to a \((2+1)\) CFT. In this picture, the electric bulk gauge field does not have a counterpart in the dual field theory but instead it fixes the electric charge density \(\rho \) to be proportional to the electric charge of the black hole, i.e. \(\rho \propto q_e\). On the other hand, the magnetic bulk gauge field is in correspondence with an external magnetic field in the CFT side with a field strength \(B\propto q_m\). Hence, the Hall conductivity in our case (4.9) is proportional to the ratio between the magnetic field and the electric charge density, i.e. \(\sigma _{xy}\propto \frac{B}{\rho }\). As a final comment, one can notice that in the purely electric case \(q_m=0\), the DC conductivity for the extremal solution vanishes identically reproducing in turn an insulator behavior.

## 5 Conclusion

Here, we have considered a self-interacting scalar field nonminimally coupled to the four-dimensional Einstein gravity with a negative cosmological constant. The matter source is also supplemented by the Maxwell action with two axionic fields minimally coupled to the scalar field. Our model is specified from the very beginning by two parameters that are the nonminimal coupling parameter denoted \(\xi \) or equivalently *n* (2.2) and the constant *b* that enters in the minimal coupling as well as in the potential. For this model, we have obtained dyonic planar black holes with axionic fields depending linearly on the coordinates of the planar base manifold. We have noticed that these charged solutions depend on a unique integration constant denoted by \(\omega \) and the horizon can be located at two different positions depending on the sign of \(\omega \). Surprisingly, for \(\omega <0\), the temperature of the solution vanishes identically and hence one can interpret the solution as an extremal black brane. We have also shown that some reality conditions (cf. Table 1) supplemented by the requirement of having solutions with positive entropy and mass restrict considerably the permissible values of the nonminimal coupling parameter and of the parameter *b*, see Table 2. For a positive \(\omega >0\), the set of physically acceptable values of the nonminimal coupling parameter is given by \(\xi \in ]0,\frac{1}{4}[\) while for the extremal solution corresponding to \(\omega <0\), only discrete values of the nonminimal coupling parameter given by \(\xi =\frac{k}{2(2k-1)}\) with \(k\in \mathbb {N}{\setminus }\{0\}\) yield to solutions with positive mass and entropy. These restrictions on the nonminimal coupling parameter are to be expected since, even for purely scalar field nonminimally coupled to Einstein gravity, black hole configurations have been shown to be ruled out for \(\xi <0\) and \(\xi \ge \frac{1}{2}\), see Ref. [31].

In the last part of this work, we have taken advantage of the momentum dissipation ensured by the axionic fields to compute the different conductivities by means of the recipes given in Refs. [19, 20]. Many interesting results can be highlighted from the study of the holographic DC conductivities inherent to these dyonic solutions. For the non extremal solutions, we have shown that for \(n\le n_0\approx 3.4681\) or equivalently \( \xi \le \xi _0\approx 0.1779\), the dyonic solutions always enjoy a positive conductivity for any mathematically permissible value of *b*. On the other hand, for \(n>n_0\), the positive conductivity condition restricts the interval of *b* to be \(]b_1,b_0[\) with \(b_1>\frac{1}{4}b_0\). In other words, this means that the physical solutions (in the sense of having positive mass and entropy) for \(n>n_0\) with \(b\in ]\frac{1}{4}b_0, b_1[\) will have a negative conductivity. Also, we have shown that for \(n>n_0\) there always exist a value of the parameter *b* denoted by \(b_1\) which restores the translation invariance, in the sense that \(\sigma _{DC}(n, b_1)\rightarrow \infty \). Finally, for the extremal solution, we have shown that the diagonal elements of the conductivity matrix precisely vanish and its off-diagonal elements are similar to those inherent to the Hall effect.

An interesting extension of our model will be to consider an additional *k*-essence term for the axionic part of the action and to analyze the effects on the conductivities of the nonminimal coupling parameter \(\xi \) conjugated with the *k*-essence parameter, see Ref. [24]. In the same lines a natural generalization of these solutions would be the extension to higher dimensional scenarios following the lines of [32].

Finally, it will be very interesting to explore more deeply some of the properties of our solutions such as the extremality, the perfect conductivity or the Hall effect-like behavior of the extremal solutions. With this respect, in Ref. [33], it was shown that the Reissner-Nordstrom at the extremal limit experiences a sort of Meissner effect in the sense that the magnetic flux lines are expelled. Hence, a work to be done consists precisely in investigating the extremal solutions found here can exhibit a kind of Meissner effect.

## Footnotes

- 1.
- 2.
A simple calculation shows that the range of values \(\xi <0\) is not compatible with the reality condition as defined by Eq. (2.6).

- 3.
We may remember that the values \(n=0\) and \(n=1\) were excluded from the very beginning, and for \(n=3\) the expression multiplying \(\eta \) vanishes identically.

- 4.
It is worth mentioning that the values yielding negative conductivities are those for which the minimal coupling function \(\varepsilon _b(\phi )\) evaluated at the horizon is negative. Hence, in order to have a kinetic axionic term with the right sign to avoid ghost problems, the values associated to negative conductivities must be ruled out.

- 5.
For simplicity, we will only consider the DC conductivities along the

*x*-coordinate without magnetic charge.

## Notes

### Acknowledgements

This work has been partially supported by grant FONDECYT 11170274 (A. C). We would like to thank especially Moises Bravo-Gaete for stimulating discussions and nice comments to improve the draft. One of us, M. H. would like to dedicate this work to the memory of his late friend and professor Christian Duval.

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