Holography of electrically and magnetically charged black branes
Abstract
We construct a new class of black brane solutions in EinsteinMaxwelldilaton (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 linearT resistivity can be realized in a large region of temperature for the specific case of \(a^2=1/3\), which is the GubserRocha 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 electricmagnetic (EM) duality. Various transport coefficients are calculated and their temperature dependence are obtained in the high temperature region.
1 Introduction
AdS/CFT correspondence provides a new direction for the study of strongly correlated systems [1, 2, 3, 4, 5]. In particular, great progress has been made in modelling and understanding the anomalous scaling behavior of the strange metal phase (see [6] and references therein). Among of them the linearT resistivity and quadraticT 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 linearT resistivity has firstly been explored in [7, 8]. Then different scalings between Hall angle and resistivity have also been investigated in holographic framework [6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In particular, both the linearT resistivity and quadraticT inverse Hall angle can be simultaneously reproduced in some special holographic models [19, 20, 21, 22, 23].
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 hyperscalingviolating and Lifshitz solutions have largely improved the scaling analysis of the exotic behavior in the strange metal [24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. 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 EinsteinMaxwellDilaton (EMD) theory provides a nice arena for the study of electric and magnetic transport phenomena in a stronglycoupled system. Previously a particular model constructed in EMD theory is the GubserRocha solution which describes an electrically charged black brane [34]. 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 [34] and the linear resistivity at low temperature [36]. Also, as a typical model for holographic studies, the GubserRocha solution has been extended in various circumstances, see e.g. [37, 38, 39, 40, 41, 42, 43, 44].
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 EinsteinMaxwelldilaton (EMD) theory in Sect. 2. In particular, we study the transport behavior in the dual system and find that the linearT resistivity holds in a large range of temperature for \(a^2=1/3\). Then by virtue of the electricmagnetic (EM) duality for \(a^2=1\), we construct a dyonic black brane solution in Sect. 4 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.
2 Electrically charged dilatonic black branes
2.1 Electrically charged dilatonic black branes

The AdS boundary is located at \(z=0\). \(\mu ,q\) are the chemical potential and charge density of the dual boundary system, respectively.

The parameter \(\Lambda \) 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.

When \(b=0,a^2=1/3\), the solution of the case \(\beta =1/a\) reduces to the GubserRocha one [34]. When \(b=0\), the solution of the case \(\beta =1/a\) becomes the well established results in [11, 15, 53, 54]. When \(k=q=0,b\ne 0\), by redefining the parameters, the solution coincides to that in [58].

To obtain the thermodynamics of the background, one need follow the standard holographic renormalization approach. We would like to recommend article [52, 55, 56], in which the thermodynamics of the above model with \(b=0\) have been well studied.^{2}
2.2 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 GubserRocha solution [34]. 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 \(z_+\rightarrow \infty \) gives \(T\rightarrow 0\) as well as \(s\rightarrow 0\).
2.2.1 Neutral black brane background for \(q=0,k=0\)
2.2.2 Simple charged black brane background for \(b=0\)
However, when \(a^2\ge 1/3\), the story is totally different, because the temperature (18) is always positive. We then check the temperature behavior as \(z_+\rightarrow \infty \).
2.2.3 Special cases for \(a^2=1,a^2=1/3,a^2=3\)
When we do the zero temperature analysis, the only thing needing to change is the the horizon condition (7), which should be replaced by using the above results (27) and require \(h(a^2=1/3,z_+)=0,\,h(a^2=1,z_+)=0,\,h(a^2=3,z_+)=0\) for \(a^2=1,a^2=1/3,a^2=3\) 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 \(q=0,k\ne 0\), due to the logarithm divergence as \(z_+\rightarrow \infty \) 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 \(a^2>3\).
3 LinearT 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 GubserRocha case exhibits a linearT resistivity valid in a wide temperature, which coincides to the universal behaviors of the strange metal.
Now we turn to explore the relation between the resistivity and the temperature. Using Eqs. (33) and (30), we plot the resistivity as the function of the temperature in Fig. 1. Obviously, the resistivity linearly depends on the temperature. Especially, we can see that the linearT resistivity survives in a large range of temperature, which shall be further addressed in what follows.
\(k=10^4\)  \(z_+=10^{3}\)  \(z_+=10^{2}\)  \(z_+=10^2\)  \(k=10^5\)  \(z_+=10^{2}\)  \(z_+=10^2\) 

\(\Lambda \)  9000.00  9900.00  9999.99  \(\Lambda \)  99899.99  99999.99 
T  477.47  158.36  1.59  T  503.04  5.03 
\(\Lambda \)  \(\approx 10^4\)  \(\approx 10^4\)  \(\approx 10^4\)  \(\Lambda \)  \(\approx 10^5\)  \(\approx 10^5\) 
T  \(\approx 503.29\)  \(\approx 159.16\)  \(\approx 1.59\)  T  \(\approx 503.29\)  \(\approx 5.03\) 
Next, we shall address that the linearT resistivity survives in a large range of temperature. To this end, we present two examples as what follows (see Table 1). The exact value of \(\Lambda \) and the temperature T in Table 1 are calculated by the expressions in (30) and (33). The approximate results of the temperature are obtained in terms of the first equation in Eq. (34). Note that we have used the relation \(\Lambda \thickapprox k\). From the above table (the second and third rows), we confirm the result that when \(k\gg 1\), \(\Lambda \gg 1\), and \(\Lambda z_+\gg 1\), \(\Lambda \) is approximately a constant, i.e., \(\Lambda \thickapprox k\). In addition, the approximate values of the temperature are consistent with the exact ones, which means that the approximate linearT resistivity expression, namely Eq. (34), holds very well in the region of \(k\gg 1\), \(\Lambda \gg 1\), and \(\Lambda z_+\gg 1\). More importantly, from this table, we can see that the temperature T indeed varies in terms of \(z_+\), crossing a large range.
In summary, the linearT resistivity is achieved when \(\Lambda \approx k\gg 1\), which is a good approximation. It always happens if one considers a system with large parameter \(k\gg 1\). In our examples, \(T<160\) is the good regime of linearT resistivity for \(k=10^4\), while \(T<503\) is, at least, the good regime for \(k=10^5\). Thus, for the parameters satisfying the certain conditions, the linearT resistivity is achieved for a wide range of temperature.
Note added. As this work was being completed, we were informed from Chao Niu that they also find the linearT resistivity behavior at high temperature region in [59].
4 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 selfduality. 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 [57].^{3} In [52], by detailed analysis on the boundary condition it is argued that a dyonic solution may only exist at \(a=1\) ( \(\xi =1\) in their paper). Here, interestingly enough, we provide an interpretation for this fact from a different angle of view, namely the electromagnetic selfduality.

Both DC resistivity \(\rho _{dc}\) and thermopower S decrease with \(1/T^2\) at high temperature, which implies the thermal transport is dominant over the electric and electrothermal transport.

With the increase of temperature, the Hall angle \(1/\tan \theta _H\) and Hall Lorentz ratio decrease (see the plots in Fig. 2), while the magneticresistance \(\rho _{B}\) increases (see the left below plot in Fig. 2). The Nernst coefficient \(\nu \) becomes a constant in the limit of high temperature (see the right below plot in Fig. 2).
5 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 [15, 34, 52, 53, 54].
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 \(q\ne 0,b=0\), 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 GubserRocha model, and for \(a^2>3\). While for \(3> a^2>1/3\), the zero temperature can not be achieved and the deep horzion \(z_+\gg 1\) 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 linearT 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.
Footnotes
 1.
Another important holographic model with vanishing entropy density ground state is presented in [35], in which the black brane is numerically constructed and the near horizon geometry at zero temperature possesses Lifshitz symmetry.
 2.
We are very grateful to Astefanesei for drawing our attention to [58] as well as the correct holographic renormalization approach.
 3.
Notes
Acknowledgements
We are very grateful to Chao Niu for many useful discussions and comments on the manuscript. This work is supported by the Natural Science Foundation of China under Grant Nos. 11575195, 11775036, 11747038, 11875053, 11847313.
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