Holography of electrically and magnetically charged black branes

We construct a new class of black brane solutions in Einstein-Maxwell-dilaton (EMD) theory, which is characterized by two parameters $a,b$. Based on the obtained solutions, we make detailed analysis on the ground state in zero temperature limit and find that for many cases it exhibits the behavior of vanishing entropy density. We also find that the linear-T resistivity can be realized in a large region of temperature for the specific case of $a^2=1/3$, which is the Gubser-Rocha model dual to a ground state with vanishing entropy density. Moreover, for $a=1$ we analytically construct the black brane which is magnetically charged by virtue of the electric-magnetic (EM) duality. Various transport coefficients are calculated and their temperature dependence are obtained in the high temperature region.


I. INTRODUCTION
AdS/CFT correspondence provides a new direction for the study of strongly correlated systems [1][2][3][4]. In particular, great progress has been made in modelling and understanding the anomalous scaling behavior of the strange metal phase (see [5] and references therein).
Among of them the linear-T resistivity and quadratic-T inverse Hall angle are two prominent properties of the strangle metal, which have been widely observed in normal states of high temperature superconductors as well as heavy fermion compounds near a quantum critical point, which is universal in a very wide range of temperature. By holography, the linear-T resistivity has firstly been explored in [6,7]. Then different scalings between Hall angle and resistivity have also been investigated in holographic framework [5,[8][9][10][11][12][13][14][15][16][17]. In particular, both the linear-T resistivity and quadratic-T inverse Hall angle can be simultaneously reproduced in some special holographic models [18][19][20][21][22].
Currently it is still challenging to achieve the anomalous scales of strange metal over a wide range of temperature in holographic approach. It may be limited by the renormalization group flow which is controlled by the specific bulk geometry subject to Einstein field equations, and the scaling behavior of the near horizon geometry of the background. Therefore, in this direction one usually has two ways to improve the understanding of the transport behavior of the dual system. One way is to consider more general backgrounds within the framework of Einstein's gravity theory. The other way is to introduce additional scaling anomaly which may be characterized by Lifshitz dynamical exponent and hyperscaling violating parameter. In the latter case, the construction of asymptotic hyperscaling-violating and Lifshitz solutions have largely improved the scaling analysis of the exotic behavior in the strange metal [23][24][25][26][27][28][29][30][31][32]. In this paper, we will focus on the former case, namely the holographic construction of new backgrounds within the framework of Einstein theory, without the involvement of scale anomaly. In this way the Einstein-Maxwell-Dilaton (EMD) theory provides a nice arena for the study of electric and magnetic transport phenomena in a strongly-coupled system. Previously a particular model constructed in EMD theory is the Gubser-Rocha solution which describes an electrically charged black brane [33]. It is featured by a vanishing entropy density at zero temperature 1 . This model exhibits lots of peculiar properties similar to those of the strangle metal, including the linear specific heat [33] and the linear resistivity at low temperature [35]. Also, as a typical model for holographic studies, the Gubser-Rocha solution has been extended in various circumstances, see e.g. [36][37][38][39][40][41][42][43].
In this paper, we intend to construct new backgrounds which are applicable for the study of both electric and magnetic transport properties in holographic approach, aiming to provide more comprehensive understanding on the anomalous behavior of strange metals.
We first analytically construct a new class of black brane solutions which are electrically charged in Einstein-Maxwell-dilaton (EMD) theory in Section II. In particular, we study the transport behavior in the dual system and find that the linear-T resistivity holds in a large range of temperature for a 2 = 1/3. Then by virtue of the electric-magnetic (EM) duality for a 2 = 1, we construct a dyonic black brane solution in Section IV and Appendix A. Various transport coefficients are derived, including the resistivity, Hall angle, magnetic resistance and Nernst coefficient. It is expected to provide a useful platform for the study of both electric and magnetic transport behavior in the holographic framework.

A. Electrically charged dilatonic black branes
We consider Einstein-Maxwell-Dilaton-Axion (EMDA) theory in four dimensional spacetimes with the following action The axionic fields ψ x , ψ y are added to break the translation invariant, which is responsible for the finite DC conductivity over a charged black hole background.
In this paper, we propose a more general form for the potential and obtain a class of general AdS black brane solutions. The potential V (φ) and the gauge coupling Z(φ) we choose here are where a and b are two free parameters in this EMDA theory. And then, we take the following ansatz Under the above setting, the EMDA theory (1), with the potential (2), has two branches of the asymptotic AdS charged black brane solutions for β = 1/a and β = a, which are Some remarks on these solutions are presented in what follows.
• The AdS boundary is located at z = 0. µ, q are the chemical potential and charge density of the dual boundary system, respectively.
• The parameter Λ shall be determined in terms of a, b by the horizon condition h(z + ) = 0 with z + being the position of horizon. Namely, only a, b are free parameters in this model.
• To obtain the thermodynamics of the background, one need follow the standard holographic renormalization approach. We would like to recommend article [51,54,55], in which the thermodynamics of the above model with b = k = 0 have been well studied 2 .
We would also like to point out that the above two branches of solutions can be related by a coordinate transformation. That is to say, the second branch of solutions can be obtained from the first one under the following coordinate transformation Therefore, at this moment, it is enough to consider the first branch with β = 1/a only. Now, in order to make the solutions (4) become a black brane background, one should consider the horizon condition h(z + ) = 0, restricting the parameter Λ in terms of a, b In addition, the derivative of h(z) with respect to z gives From now on, we shall set Λ > 0, b ≥ 0. When k = 0, q = 0, h(z) monotonically decreases from the boundary h(z = 0) = 1 to the horizon, which guarantees h(z) > 0 for the neutral black hrane background. When q = 0, we can require a 2 ≥ 1/3 to ensure h(z) > 0.
Then, the temperature and entropy density can be calculated straightforward as It is obvious that the temperature T > 0 when h(z) is monotonic. This is the key point to establish a zero temperature ground state with a zero entropy density, since if T = 0 can be attained at a finite z + , then, the entropy density above must be finite. On the contrary however, a positive T > 0 for all finite z + , may decreases and becomes zero as z + → 0.

B. Analysis on the ground state
The ground state with zero entropy is physically acceptable. However, such ground state in holographic model is rare in the present literatures. As we know, the only simple example is the Gubser-Rocha solution [33]. Now, with a more fruitful AdS background (4) at hand we give a detailed analysis, case by case, to find the ground state with a vanishing entropy density. The method has been illustrated in the end of the last subsection, namely, we shall check whether z + → ∞ gives T → 0 as well as s → 0.
1. Neutral black brane background for q = 0, k = 0 We first consider the neutral black brane case without the axionic field, q = 0, k = 0.
The horizon condition (7) and the temperature (9) reduce to b (1 + Λz + ) Then, we set Λ = Λ 0 /z + . When z + is varying, Λ 0 > 0 is a fixed parameter satisfying Accordingly, the temperature and the entropy density (10) becomes When z + is varying, we have a simple relation Both the s, T tends to zero as z + → ∞. Such neutral background admits a ground state with zero entropy density.

Simple charged black brane background for b = 0
Next, we study a simple charged black brane case with q = 0 but b = 0, where the general solution (4) reduces to The above solutions have been well studied in [14,[51][52][53]. Especially, when a 2 = 1/3, it reduces the famous Gubser-Rocha solution [33]. Here, we try to complete the zero temperature analysis. We will find that not only the Gubser-Rocha solution, i.e., a 2 = 1/3, but also the case of a 2 > 3, k = 0, are the vanishing entropy density background at the zero temperature.
We first consider the case without axion fields, namely k = 0 where the horizon condition (7) and the temperature (9) reduces to When a 2 < 1/3, T = 0 can be achieved at a finite position which is of course finite.
However, when a 2 ≥ 1/3, the story is totally different, because the temperature (18) is always positive. We then check the temperature behavior as z + → ∞.
In the limit of z + → ∞, the relation (17) forces that Λz + + 1 ≈ Λz + → ∞ and gives Then the temperature in (18) and the entropy density in (10) approximatively read as It is clear that for a 2 ≥ 1/3 the entropy density deceases to zero as z + → ∞. But the zero temperature and zero entropy density can be simultaneously achieved only for a 2 = 1/3 or a 2 > 3. On the contrary, no zero temperature exists for 3 ≥ a 2 > 1/3. Actually, as z + → ∞, it is a high temperature limit with T → ∞.
Next, we consider the case k = 0. The horizon condition (7) and the temperature (9) with b = 0 reduces to Then, taking the limit z + → ∞, the horizon condition (23) becomes Immediately, we find that one can not obtain an extremal black brane solution with axionic fields for a 2 ≥ 3. It is in contrast to the case without axionic fields. Therefore, we conclude that once the axionic fields are taken into account, the simple charge black brane solution with vanishing ground state entropy density can be achieved only for a 2 = 1/3.

3.
Special cases for a 2 = 1, a 2 = 1/3, a 2 = 3 So far, the discussion is based on the general potential (2). For the special cases a 2 = 1, a 2 = 1/3, a 2 = 3, it is convenient to write down the form of Φ in the potential (2) after taking the limit, which are separately given by The background solutions of h(z) in (4) also take the form as However, the special parameters would not change the formula of h (z) and hence the temperature formula. Namely, we can use the former expressions (8) and (9) directly and fix a 2 = 1, a 2 = 1/3, a 2 = 3 respectively.
It is easy to see that, for the neutral case q = 0, k = 0, a zero temperature background with vanishing entropy also exists for these special parameters. While, for q = 0, k = 0, due to the logarithm divergence as z + → ∞ in (27), the system do not have ground state with vanishing entropy. For the simple charge case b = 0, k = 0, special parameters have no influence on the previous discussion. We still have the zero entropy background at zero temperature for a 2 = 1/3 or a 2 > 3.

III. LINEAR-T RESISTIVITY
In this section, we consider the electric transport behavior of the dual system over the black brane geometry (4). Specifically, we calculate the DC conductivity with the interest in its dependence on the temperature. We find the Gubser-Rocha case exhibits a linear-T resistivity valid in a wide temperature, which coincides to the universal behaviors of the strange metal.
Using the standard holographic techniques, one can derive the DC conductivity as Usually the relation between σ and T is complicated, see (7) and (9). It is thus hard to catch the universal relation of the resistivity of the temperature. To simplify, we first try to take the limit z + → ∞. Recall that when b = 0, such limit is invalid for the divergence in the horizon conditions, see for instance (27). Thus, we only consider the simple charge black brane case with b = 0.
We find that for 1/3 < a 2 < 3, the resistivity decrease as 1/T 2 in high temperature limit, which is independent of a. While for a 2 = 1/3, this holographic system captures the important property of the strange metal, i.e., linear-T resistivity.
Next, we shall further study the linear-T resistivity for the case of a 2 = 1/3 and argue that it holds not only in the low temperature limit but also for a large temperature range.
Our key observation is that it is not necessary to have z + → ∞ to obtain a simplified expression for the temperature. Instead, one only need the condition Λz + 1, which we may call the deep horizon region. This condition can be guaranteed if we set a huge number for Λ but vary z + in an appropriate region, which allow us to observe the conductivity in a wide region of temperature. We present the details as follows.
First, when a 2 = 1/3, both the temperature and the conductivity reduce to Under the condition Λz + 1, the horizon condition (23) becomes Thus Λ is fixed by the charge density q and the axionic parameter k. As a consequence, both the temperature and the conductivity only vary with z + . And then, we obtain a linear-T resistivity law where Λ is determined by Eq. (31). We stress that although the requirement Λz + 1 is the key point to achieve the linear-T resistivity, it does not mean taking any temperature limit here. In fact, the present result is valid in a wide range of temperature and can be viewed as a successful achievement of the linear-T resistivity in the strange metal scales.
We further clarify this point in what follows.
First, to eliminate the scale symmetry of asymptotical AdS background, one should use the scale dimensionless quantities, which is equivalent to set q = 1 or k = 1. Second, to have Λz + 1, we may require Λ 1 as well as z + > 1. And then, from Eq. (31), we deduce that q ∼ Λ 2 1. Here we have chosen k = 1. Furthermore, under this condition the temperature becomes T ∼ q 1/4 / √ z + , which varies in the region T ∈ (0, q 1/4 ), and the conductivity becomes σ ∼ q 1/4 √ z + . Since q is fixed by setting Λ, we can immediately conclude that the linear-T resistivity behavior holds in a large temperature range. The plot in FIG.1 clearly shows such behavior.
Note added. As this work was being completed, we were informed from Chao Niu that they also find the linear-T resistivity behavior at high temperature region in [58].

IV. DYONIC DILATONIC BLACK BRANE AND ITS TRANSPORTS
The lack of exact dyonic solution in gravity theory prevents us from investigating the magnetic transport behavior of the dual system in an analytical manner. Fortunately, in EMDA theory (1), we are able to find such an analytical dyonic solution for the special case of a 2 = 1 by virtue of the electromagnetic self-duality. Here we just list the dyonic solutions as below. The detailed derivation can be found in Appendix A. Moreover, we point out that an AdS dyonic solution with b = 0 as well as k = 0 has previously been reported in [56] 3 .
In [51], by detailed analysis on the boundary condition it is argued that a dyonic solution may only exist at a = 1 ( ξ = 1 in their paper). Here, interestingly enough, we provide an interpretation for this fact from a different angle of view, namely the electromagnetic self-duality.
For a = 1, the EMDA theory with the potential (2) exists the dyonic black brane solution as the following form where B is a constant magnetic field. The horizon condition h(z + ) = 0 gives rise to The temperature of this black brane is Now, we turn to study the DC transports. Employing the standard techniques developed in [17,36,63], we obtain the thermoelectric conductivities over the dyonic black brane geometry (33) as where Z ≡ Z(φ)| z + , H ≡ H(φ)| z + and q are, respectively And then, we give the charge transport coefficients, the DC resistivity ρ dc and the thermopower S as Also the magnetic transport coefficients, including the Hall angle tan θ H , the Hall Lorentz ratio L H , the magneticresistance ρ B and Nernst coefficient ν are calculated explicitly as .
In order to simplify the expression of temperature, we will only consider the case b = 0 in the follows, in which we can take a large z + limit. The horizon condition (34) and the temperature (35) reduce to We are particularly interested in the transport behavior of this system in the high temperature limit, in which Λz + 1 such that 1 + Λz + ≈ Λz + . In this case, Eqs. (40) and (41) reduce to Next we consider the situation that the magnetic field B is small, then z + and Λ can be solved as In the above equations, we have expressed z + and Λ up to the second order of B. As a consequence, we give the charge and magnetic transport coefficients up to the first order as The characteristics of the transport behavior in the high temperature region are summarized as what follows.
• Both DC resistivity ρ dc and thermopower S decrease with 1/T 2 at high temperature, which implies the thermal transport is dominant over the electric and electrothermal transport.

V. CONCLUSIONS AND DISCUSSIONS
In this paper we have constructed a new class of charged black brane solutions in EMDA theory, which is characterized by two free parameters a, b, which could be viewed as the extension of various charged solutions with b = 0 in literature [14,33,[51][52][53].
For different a, b, the background exhibits distinct behavior in zero temperature limit. In the neutral background q = 0, the zero temperature ground state with zero entropy density always exists for any a, b, while in the simple charged case q = 0, b = 0, it depends on a. For a 2 < 1/3, the zero temperature can be achieved at finite horizon position. But the entropy density is finite as well at the zero temperature. It is interesting to notice that a ground state with vanishing entropy density is allowed for a 2 = 1/3, which has previously been obtained in Gubser-Rocha model, and for a 2 > 3. While for 3 > a 2 > 1/3, the zero temperature can not be achieved and the deep horzion z + 1 corresponds to a high temperature limit.
When the translation invariance is broken by adding axion fields, by contrast, the vanishing entropy density ground state can be achieved only for a 2 = 1/3. For this special case we have demonstrated that the dual system is characterized by a linear-T resistivity in a large range of temperature, reminiscent of the key feature of the strange metal. We have also obtained dyonic black brane by virtue of the EM duality for a = 1. The transport coefficients have been calculated and their temperature dependence have been analyzed in high temperature region. We expect the EM duality as a valuable strategy may be applicable to more general gravity theories such that more analytic solutions of dyonic black brane could be constructed, which should be helpful for us to investigate the magnetic transport behavior of the dual system by holography.
where G = dH with H a being the dual gauge field and abcd is the completely antisymmetric Levi-Civita tensor. And then, we can write down the dual one of EMD theory aŝ From Eqs. (A1) and (A2), it is easy to find that under the EM duality, the gauge coupling transforms as Z(φ) → 1/Z(φ), which also implies a weak-strong coupling duality. Especially, there is a correspondence between the electric field F tx of the original theory and the magnetic field G yz of its dual one. Therefore, by EM duality we could quickly obtain a purely magnetic solution of the dual EMD theory (A2). We demonstrate it as follows.

EM S-duality and dyonic black brane solution
In this subsection, we construct the dyonic black brane solution from the EMDA theory (1) for a 2 = 1, in which the theory (1) is S-duality, namely the charge and its dual magnetic solutions are both valid for the same action.