Holography of Electrically and Magnetically Charged Black Branes

Eur. Phys. J. C (2019) 79:195 https://doi.org/10.1140/epjc/s10052-019-6705-8

Regular Article - Theoretical Physics

Holography of electrically and magnetically charged black branes

Zhenhua Zhou1,a, Jian-Pin Wu2,4,b,YiLing3,4,5,c 1 School of Physics and Electronic Information, Yunnan Normal University, Kunming 650500, China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China 3 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 4 Shanghai Key Laboratory of High Temperature Superconductors, Shanghai 200444, China 5 School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China

Received: 3 December 2018 / Accepted: 19 February 2019 / Published online: 4 March 2019 © The Author(s) 2019

Abstract We construct a new class of black brane solu- ity and quadratic-T inverse Hall angle can be simultaneously tions in Einstein-Maxwell-dilaton (EMD) theory, which is reproduced in some special holographic models [19Ð23]. characterized by two parameters a, b. Based on the obtained Currently it is still challenging to achieve the anomalous solutions, we make detailed analysis on the ground state in scales of strange metal over a wide range of temperature zero temperature limit and find that for many cases it exhibits in holographic approach. It may be limited by the renor- the behavior of vanishing entropy density. We also find that malization group flow which is controlled by the specific the linear-T resistivity can be realized in a large region of bulk geometry subject to Einstein field equations, and the temperature for the specific case of a2 = 1/3, which is the scaling behavior of the near horizon geometry of the back- Gubser-Rocha model dual to a ground state with vanishing ground. Therefore, in this direction one usually has two ways entropy density. Moreover, for a = 1 we analytically con- to improve the understanding of the transport behavior of struct the black brane which is magnetically charged by virtue the dual system. One way is to consider more general back- of the electric-magnetic (EM) duality. Varioustransport coef- grounds within the framework of Einstein’s gravity theory. ficients are calculated and their temperature dependence are The other way is to introduce additional scaling anomaly obtained in the high temperature region. which may be characterized by Lifshitz dynamical exponent and hyperscaling violating parameter. In the latter case, the construction of asymptotic hyperscaling-violating and Lif- 1 Introduction shitz solutions have largely improved the scaling analysis of the exotic behavior in the strange metal [24Ð33]. In this paper, AdS/CFT correspondence provides a new direction for the we will focus on the former case, namely the holographic study of strongly correlated systems [1Ð5]. In particular, great construction of new backgrounds within the framework of progress has been made in modelling and understanding the Einstein theory, without the involvement of scale anomaly. anomalous scaling behavior of the strange metal phase (see In this way the Einstein-Maxwell-Dilaton (EMD) theory pro- [6] and references therein). Among of them the linear-T resis- vides a nice arena for the study of electric and magnetic trans- tivity and quadratic-T inverse Hall angle are two prominent port phenomena in a strongly-coupled system. Previously a properties of the strangle metal, which have been widely particular model constructed in EMD theory is the Gubser- observed in normal states of high temperature superconduc- Rocha solution which describes an electrically charged black tors as well as heavy fermion compounds near a quantum crit- brane [34]. It is featured by a vanishing entropy density at ical point, which is universal in a very wide range of temper- zero temperature.1 This model exhibits lots of peculiar prop- ature. By holography, the linear-T resistivity has firstly been erties similar to those of the strangle metal, including the explored in [7,8]. Then different scalings between Hall angle linear specific heat [34] and the linear resistivity at low tem- and resistivity have also been investigated in holographic perature [36]. Also, as a typical model for holographic stud- framework [6,9Ð18]. In particular, both the linear-T resistiv-

1 a e-mail: [email protected] Another important holographic model with vanishing entropy density ground state is presented in [35], in which the black brane is numer- b e-mail: [email protected] ically constructed and the near horizon geometry at zero temperature c e-mail: [email protected] possesses Lifshitz symmetry. 123 195 Page 2 of 12 Eur. Phys. J. C (2019) 79 :195

+ 2 2+ ies, the Gubser-Rocha solution has been extended in various 2b 1 a φ 3 2 − 1 φ − a 3 φ V1 = (e 2a − 1) (a e a + e 2a ), circumstances, see e.g. [37Ð44]. (1 + a2)2 (2c) In this paper, we intend to construct new backgrounds ⎛ ⎞ which are applicable for the study of both electric and mag- 3a2−1 φ a2−3 φ a2−1 φ e 2a − 1 e 2a − 1 e a − 1 netic transport properties in holographic approach, aiming to =−b ⎝ + − ⎠ , 2 − 2 − 2 − provide more comprehensive understanding on the anoma- 3a 1 a 3 a 1 lous behavior of strange metals. We first analytically con- (2d) struct a new class of black brane solutions which are electrically charged in Einstein-Maxwell-dilaton (EMD) theory in where a and b are two free parameters in this EMDA theory. Sect. 2. In particular, we study the transport behavior in the And then, we take the following ansatz dual system and find that the linear-T resistivity holds in a large range of temperature for a2 = 1/3. Then by virtue of 1 dz2 2β2 ds2 = − f (z)dt2 + +(1 + z) 1+β2 (dx2 + dy2) , the electric-magnetic (EM) duality for a2 = 1, we construct z2 f (z) a dyonic black brane solution in Sect. 4 and Appendix A. Var- (3a) ious transport coefficients are derived, including the resistiv- = ( ), φ = φ( ), ψ = ,ψ= , ity, Hall angle, magnetic resistance and Nernst coefficient. It A At z z x kx y ky (3b) is expected to provide a useful platform for the study of both Under the above setting, the EMDA theory (1), with the electric and magnetic transport behavior in the holographic potential (2), has two branches of the asymptotic AdS framework. charged black brane solutions for β = 1/a and β = a, which are Case 1: β = 1/a, (4a) 2 Electrically charged dilatonic black branes 2 f (z) = (1 + z) 1+a2 h(z), (4b) q2(1 + a2) − 4 2.1 Electrically charged dilatonic black branes h(z) = 1 − z3(1 + z) 1+a2 4 a2−3 We consider Einstein-Maxwell-Dilaton-Axion (EMDA) the- − k2z2(1 + z) 1+a2 ory in four dimensional spacetimes with the following action 2 2 3a −1 a −3 (1 + z) a2+1 − 1 (1 + z) a2+1 − 1 − b + √ (φ) 2 − 2 − = 4 − − Z ab − 1(∂φ)2 + (φ) 3a 1 a 3 SEMD d x g R FabF V 2 4 2 2a −2 (1 + z) a2+1 − 1 − (∂ψ )2 . − , (4c) I (1) a2 − 1 I =x,y 2a qz φ(z) = ln(z + 1), At (z) = μ − , 1 + a2 1 + z The axionic fields ψx ,ψy are added to break the translation (4d) invariant, which is responsible for the finite DC conductivity over a charged black hole background. Case 2: β = a, (5a)

The black brane solutions of EMDA theory and their holo- 2a2 graphic properties have been widely studied in [11,15,34, f (z) = (1 + z) 1+a2 h(z), (5b) 35,45Ð54]. Analytical background can provide a more con- q2(1 + a2) 1−3a2 h(z) = 1 + z3(1 + z) 1+a2 trollable pattern in studying the holographic characteristics. 4 To obtain an analytical black brane solution, it is crucial to 1−3a2 (φ) − k2z2(1 + z) 1+a2 choose an appropriate potential V . For some speciﬁc form ⎛ of V (φ), the analytical AdS black brane solutions have been −3a2+1 −a2+3 ⎜ (1 + z) a2+1 − 1 (1 + z) a2+1 − 1 worked out in [15,34,52Ð54]. In this paper, we propose a − b ⎝ + 2 − 2 − more general form for the potential and obtain a class of 3a 1 a 3 (φ) ⎞ general AdS black brane solutions. The potential V and −2a2+2 the gauge coupling Z(φ) we choose here are (1 + z) a2+1 − 1 ⎟ − ⎠ , (5c) a2 − 1

aφ Z(φ) = e , V = V0(1 + ) + V1, (2a) 2a φ(z) =− ln(z + 1), At = μ − qz. (5d) φ − aφ φ − aφ 1 + a2 6(a2e 2a + e 2 )2 − 2a2(e 2a − e 2 )2 V = , (2b) 0 (1 + a2)2 123 Eur. Phys. J. C (2019) 79 :195 Page 3 of 12 195 3 2 2 ( 2 − )( + ) 2 ( − 2) Some remarks on these solutions are presented in what + b z + k z 3a 1 1 z + k z 3 a . follows. 1 + a2 1 + a2 1 + a2 (8) • The AdS boundary is located at z = 0. μ, q are the chem- ical potential and charge density of the dual boundary system, respectively. From now on, we shall set >0, b ≥ 0. When k = 0, q = • The parameter shall be determined in terms of a, b 0, h(z) monotonically decreases from the boundary h(z = by the horizon condition h(z+) = 0 with z+ being the 0) = 1 to the horizon, which guarantees h(z)>0forthe position of horizon. Namely, only a, b are free parameters neutral black brane background. When q = 0, we can require in this model. a2 ≥ 1/3 to ensure h(z)>0. • When b = 0, a2 = 1/3, the solution of the case β = 1/a Then, the temperature and entropy density can be calcu- reduces to the Gubser-Rocha one [34]. When b = 0, lated straightforward as the solution of the case β = 1/a becomes the well estab- lished results in [11,15,53,54]. When k = q = 0, b = 0, by redefining the parameters, the solution coincides to 2 that in [58]. − 2 2 2 2 2 (1 + z+) 1+a2 q z+(3a − 1) q z+ • To obtain the thermodynamics of the background, one T = + 4π 4 (1 + z+) need follow the standard holographic renormalization 3 2 approach. We would like to recommend article [52,55, b z 2 +( 2 − )( + +) 2 +( − 2) + + + k z 3a 1 1 z + k z 3 a , 56], in which the thermodynamics of the above model 1 + a2 1 + a2 1 + a2 with b = 0 have been well studied.2 (9) 4π 2 We would also like to point out that the above two branches = ( + +) 1+a2 . s 2 1 z (10) of solutions can be related by a coordinate transformation. z+ That is to say, the second branch of solutions can be obtained from the first one under the following coordinate transformation It is obvious that the temperature T > 0 when h(z) is mono- z z → . (6) tonic. This is the key point to establish a zero temperature 1 − z ground state with a zero entropy density, since if T = 0 can Therefore, at this moment, it is enough to consider the first be attained at a finite z+, then, the entropy density above branch with β = 1/a only. must be finite. On the contrary however, a positive T > 0for Now, in order to make the solutions (4) become a black all finite z+, may decreases and becomes zero as z+ → 0. brane background, one should consider the horizon condition h(z+) = 0, which determines the location of the horizon in terms of and a, b as below 2.2 Analysis on the ground state

2( + 2) − 4 a2−3 q 1 a 3 2 2 2 2 1 − z+(1 + z+) 1+a − k z+(1 + z+) 1+a The ground state with zero entropy is physically acceptable. 4 ⎛ However, such ground state in holographic model is rare in 3a2−1 a2−3 (1 + z+) a2+1 − 1 (1 + z+) a2+1 − 1 the present literatures. As we know, the only simple example − b ⎝ + 3a2 − 1 a2 − 3 is the Gubser-Rocha solution [34]. Now, with a more fruitful ⎞ AdS background (4) at hand we give a detailed analysis, case 2a2−2 by case, to find the ground state with a vanishing entropy (1 + z+) a2+1 − 1 − ⎠ = 0. (7) density. The method has been illustrated in the end of the last a2 − 1 subsection, namely, we shall check whether z+ →∞gives T → 0aswellass → 0. In addition, the derivative of h(z) with respect to z gives 2 2 2 2 2 − 4 q z (3a − 1) q z = , = h (z) =−(1 + z) 1+a2 + 2.2.1 Neutral black brane background for q 0 k 0 4 (1 + z) We first consider the neutral black brane case without the = , = 2 We are very grateful to Astefanesei for drawing our attention to [58] axionic field, q 0 k 0. The horizon condition (7) and as well as the correct holographic renormalization approach. the temperature (9) reduce to 123 195 Page 4 of 12 Eur. Phys. J. C (2019) 79 :195 ⎛ 3a2−1 a2−3 of a2 > 3, k = 0, are the vanishing entropy density back- (1 + z+) a2+1 − 1 (1 + z+) a2+1 − 1 b ⎝ + ground at the zero temperature. 3a2 − 1 a2 − 3 We first consider the case without axion fields, namely ⎞ 2a2−2 k = 0 where the horizon condition (7) and the temperature (1 + z+) a2+1 − 1 − ⎠ = 1, (11) (9) reduces to a2 − 1 2( + 2) − 4 3 2 q 1 a 3 2 b z+ − 2 z+(1 + z+) 1+a = 1, (17) T = ( + z+) 1+a2 , 2 1 (12) 4 4π(1 + a ) 2 − 2 2 2 2 2 (1 + z+) 1+a2 q z+(3a − 1) q z+ T = + . Then, we set = 0/z+. When z+ is varying, 0 > 0isa 4π 4 (1 + z+) fixed parameter satisfying (18) ⎛ 3a2−1 a2−3 ( + ) a2+1 − ( + ) a2+1 − 2 < / = ⎝ 1 0 1 1 0 1 When a 1 3, T 0 can be achieved at a finite position b + 3a2 − 1 a2 − 3 3−a2 ⎞ 1 3 4 1+a2 2a2−2 = . 2+ z+ (19) (1 + 0) a 1 − 1 μ 4 1 − 3a2 − ⎠ = 1, (13) a2 − 1 The corresponding entropy density is

2− 2 a 1 Accordingly, the temperature and the entropy density (10) μ 4 1+a2 s = , (20) becomes 3π 1 − 3a2

b3 − 2 which is of course ﬁnite. = 0 ( + ) 1+a2 , T 1 0 2 ≥ / 4πz+(1 + a2) However, when a 1 3, the story is totally different, 4π 2 because the temperature (18) is always positive. We then = ( + ) 1+a2 . s 2 1 0 (14) check the temperature behavior as z+ →∞. z+ In the limit of z+ →∞, the relation (17) forces that z+ + 1 ≈ z+ →∞and gives When z+ is varying, we have a simple relation a2+1 3a2−1 3a2−1 ( + 2) 2 a2+5 ∝ 2. 1 a q a2+5 a2+5 s T (15) = z+ ∼ z+ →∞. (21) 4 Both the s, T tends to zero as z+ →∞. Such neutral back- Then the temperature in (18) and the entropy density in (10) ground admits a ground state with zero entropy density. approximatively read as

2 3−a2 = 2 − ( + 2) 2 a2+5 2.2.2 Simple charged black brane background for b 0 3a 1 1 a q a2+5 T = z+ , 4π(1 + a2) 4 Next, we study a simple charged black brane case with q = 0 2 2a2+2 ( + 2) 2 a2+5 − = 1 a q a2+5 but b 0, where the general solution (4) reduces to s = 4π z+ . (22) 4 2 f = (1 + z) 1+a2 h(z), It is clear that for a2 ≥ 1/3 the entropy density deceases to →∞ q2(1+a2) − 4 a2−3 zero as z+ . But the zero temperature and zero entropy h = − z3( +z) 1+a2 − k2z2( + z) 1+a2 , 2 1 1 1 density can be simultaneously achieved only for a = 1/3 4 2 (16a) or a > 3. On the contrary, no zero temperature exists for ≥ 2 > / →∞ qz 2a 3 a 1 3. Actually, as z+ , it is a high temperature At = μ − ,φ= ln(z + 1), (16b) limit with T →∞. 1 + z 1 + a2 Next, we consider the case k = 0. The horizon condition The above solutions have been well studied in [15,52Ð (7) and the temperature (9) with b = 0 reduces to 54]. Especially, when a2 = 1/3, it reduces the famous 2( + 2) − 4 a2−3 Gubser-Rocha solution [34]. Here, we try to complete the q 1 a 3 2 2 2 2 1 − z+(1 + z+) 1+a − k z+(1 + z+) 1+a zero temperature analysis. We will find that not only the 4 Gubser-Rocha solution, i.e., a2 = 1/3, but also the case = 0, (23) 123 Eur. Phys. J. C (2019) 79 :195 Page 5 of 12 195 2 − 2 2 2 2 2 1 (1 + z+) 1+a2 q z+(3a − 1) q z+ − b (z(z − 2) + 2log(1 + z)) , (27c) T = + 8 4π 4 (1 + z+) k2z+(3a2 − 1)(1 + z+) k2z+(3 − a2) However, the special parameters would not change the for- + + , (24) ( ) 1 + a2 1 + a2 mula of h z and hence the temperature formula. Namely, we can use the former expressions (8) and (9) directly and 2 2 2 Then, taking the limit z+ →∞, the horizon condition (23) fix a = 1, a = 1/3, a = 3 respectively. becomes When we do the zero temperature analysis, the only thing needing to change is the the horizon condition (7), which 2( + 2) −4 a2−3 q 1 a 3 2 2 2 2 1 − z+(z+) 1+a − k z+(z+) 1+a = 0. (25) should be replaced by using the above results (27) and require 4 h(a2 = 1/3, z+) = 0, h(a2 = 1, z+) = 0, h(a2 = Immediately, we find that one can not obtain an extremal 3, z+) = 0fora2 = 1, a2 = 1/3, a2 = 3 respectively. black brane solution with axionic fields for a2 ≥ 3. It is It is easy to see that, for the neutral case q = 0, k = 0, in contrast to the case without axionic fields. Therefore, we a zero temperature background with vanishing entropy also conclude that once the axionic fields are taken into account, exists for these special parameters. While, for q = 0, k = 0, the simple charge black brane solution with vanishing ground due to the logarithm divergence as z+ →∞in (27), the state entropy density can be achieved only for a2 = 1/3. system do not have ground state with vanishing entropy. For the simple charge case b = 0, k = 0, special parameters have 2.2.3 Special cases for a2 = 1, a2 = 1/3, a2 = 3 no influence on the previous discussion. We still have the zero entropy background at zero temperature for a2 = 1/3 So far, the discussion is based on the general potential (2). For or a2 > 3. the special cases a2 = 1, a2 = 1/3, a2 = 3, it is convenient to write down the form of in the potential (2) after taking the limit, which are separately given by 3 Linear-T resistivity ⎛√ √ √ ⎞ φ − 4 3 φ − − 2 3 φ − In this section, we consider the electric transport behavior of 2 ⎝ 3 3e 3 3 3e 3 3⎠ (a =1/3) =−b − + , the dual system over the black brane geometry (4). Specifi- 2 8 2 cally, we calculate the DC conductivity with the interest in its (26a) dependence on the temperature. We find the Gubser-Rocha (a2 = 1) =−b (sinh φ − φ) , (26b) case exhibits a linear-T resistivity valid in a wide temperature, which coincides to the universal behaviors of the strange −b √2φ √2φ √ (a2 = 3) = 3e 3 e 3 − 4 + 4 3φ + 9 , metal. 24 Before the discussion, we should scale the quantities by (26c) ,, , the charge density q. Namely, in this section√ z+ k√ T should be understood as the scaled ones qz+,/ q, The background solutions of h(z) in (4) also take the form √ √ k/ q, T/ q. Then, using the standard holographic tech- as niques, one can derive the DC conductivity as 1 2 2 2 = 2a z+ h a σ = (1 + z+) 1+a2 1 + . (28) 3 2k2(1 + z+)2 q2z3 k2z2 = 1 − − Usually the relation between σ and T is complicated, since 3(1 + z)3 (1 + z)2 one needs to first solve z+(T ), (T ) by the relations in Eqs. 3 3z(3z + 2) − b log(1 + z) − , (27a) (7) and (9), and then substitute them into the above expression ( + )2 4 8 z 1 of the conductivity. It is thus hard to catch a simple universal h(a2 = 1) relation of the resistivity of the temperature. To simplify, q2z3 k2z2 one need take some approximation. We firstly consider the = 1 − − →∞ 2(1 + z)2 (1 + z) ‘deep horizon limit’ z+ as appeared in lots of the literature. Recall that when b = 0, such limit is invalid for z(z + 2) − b − log(1 + z) , (27b) the divergence in the horizon conditions, see for instance 2z + 2 (27). Thus, we only consider the simple charge black brane 2 h(a = 3) case with b = 0. 2 3 In this limit z+ →∞, which corresponds to low tem- q z 2 2 = 1 − − k z 2 = / (1 + z) perature limit at a 1 3 or high temperature limit for 123 195 Page 6 of 12 Eur. Phys. J. C (2019) 79 :195

ρ Also, we can obtain an approximate expression of the 0.20 resistivity. When a2 = 1/3, both the temperature in (24) 0.15 and the conductivity in (28) reduce to 0.10 1 1 1 T = √ 2+ ,σ= z+ 1 + . 4π z+ 33 22k2 0.05 (33)

0.00 T , 0 50 100 150 200 250 300 When k 1 and 1 z+ 1, the above equation can be approximately expressed as Fig. 1 The linear-T resistivity behavior in a large temperature range. Here, we have set k = 104, which ensures k 1, 1andz+ 1 1 1 1 1 T = √ 2 + ≈ √ 2k + . 4π z+ 33 4π z+k 3k3

3 > a2 > 1/3, the expression of temperature in (24) can be Note that in the above equation, we have used the relations, largely simplified. Together with the relation (25), we have y → 1 and → k, which can be directly derived from the p condition of z+ 1. Obviously, T is not sensitive to , but mainly depends on z+. The situation for the conductivity σ ∼ 2, > 2 > / , T 3 a 1 3 (29a) is similar. Both T and σ change mainly with z+, while can σ ∼ 1/T, a2 = 1/3. (29b) be treated as a constant, i.e. = k. And then, with the use of the relation k 1, the temperature and the conductivity The result indicates that for 1/3 < a2 < 3, the resistivity can be further simplified as / 2 decrease as 1 T in high temperature limit, which is inde- 2 = / 1 k pendent of a. While for a 1 3, this holographic system T ≈ ,σ≈ z+k. (34) captures the important property of the strange metal, i.e., 2π z+ linear-T resistivity [8]. However, the behavior of the linear-T Immediately, we obtain a linear-T resistivity law resistivity only holds in low temperature limit in [8]. Here 2π we shall address that the linear-T resistivity can also holds ρ ≈ T. (35) for a large temperature range if 1aswellasz+ 1. k We firstly check whether such region exists. When a2 = Next, we shall address that the linear-T resistivity survives 1/3, the horizon condition (23) becomes in a large range of temperature. To this end, we present two examples as what follows (see Table 1). The exact value of 1 3 −3 2 2 −2 − z ( + z+) − k z ( + z+) = . and the temperature T in Table 1 are calculated by the 1 + 1 + 1 0 (30) 3 expressions in (30) and (33). The approximate results of the To be clear, we rewrite the condition as temperature are obtained in terms of the first equation in Eq. (34). Note that we have used the relation ≈ k.Fromthe 1 k2 above table (the second and third rows), we confirm the result 1 − y3 − y2 = 0 ⇔ 22 = k2 y2 + k4 y4 + 4y3/3, (31) 34 2 that when k 1, 1, and z+ 1, is approximately a constant, i.e., ≈ k. In addition, the approximate values −1 where y = (1 + 1/(z+)) . We can see that z+ 1 of the temperature are consistent with the exact ones, which leads to y > 1/2 and then means that the approximate linear-T resistivity expression, namely Eq. (34), holds very well in the region of k 1, 22 > k2/2 + k4/8 + 1/6 > k2/2. (32) 1, and z+ 1. More importantly, from this table, we can see that the temperature T indeed varies in terms of Thus, when k 1 the regime with 1,z+ 1 z+, crossing a large range. survives. In summary, the linear-T resistivity is achieved when Now we turn to explore the relation between the resistivity ≈ k 1, which is a good approximation. It always hap- and the temperature. Using Eqs. (33) and (30), we plot the pens if one considers a system with large parameter k 1. resistivity as the function of the temperature in Fig. 1.Obvi- In our examples, T < 160 is the good regime of linear-T ously, the resistivity linearly depends on the temperature. resistivity for k = 104, while T < 503 is, at least, the good Especially, we can see that the linear-T resistivity survives in regime for k = 105. Thus, for the parameters satisfying the a large range of temperature, which shall be further addressed certain conditions, the linear-T resistivity is achieved for a in what follows. wide range of temperature. 123 Eur. Phys. J. C (2019) 79 :195 Page 7 of 12 195

Table 1 The exact value of − − − k = 104 z+ = 10 3 z+ = 10 2 z+ = 102 k = 105 z+ = 10 2 z+ = 102 and the temperature T obtained by the expressions in (30)and 9000.00 9900.00 9999.99 99899.99 99999.99 (33), which is presented in the . . . . . second and third rows, and the T 477 47 158 36 1 59 T 503 04 5 03 approximate results obtained in ≈ 104 ≈ 104 ≈ 104 ≈ 105 ≈ 105 terms of the ﬁrst equation in Eq. T ≈ 503.29 ≈ 159.16 ≈ 1.59 T ≈ 503.29 ≈ 5.03 (34), which is presented in the fourthandﬁfthrows

Note added. As this work was being completed, we were where B is a constant magnetic field. The horizon condition informed from Chao Niu that they also find the linear-T resis- h(z+) = 0 gives rise to tivity behavior at high temperature region in [59]. 2 2 q 3 −2 B 3 −1 1 − z+(1 + z+) + z+(1 + z+) 2 2 2 2 −1 − k z (1 + z+) + 4 Dyonic dilatonic black brane and its Transports z+(z+ + 2) − b − log(1 + z+) = 0, (37) 2(1 + z+) The lack of exact dyonic solution in gravity theory prevents us from investigating the magnetic transport behavior of the dual The temperature of this black brane is system in an analytical manner. Fortunately, in EMDA theory −1 2 2 2 2 (1), we are able to find such an analytical dyonic solution for (1 + z+) q z+ q z+ 2 = T = + the special case of a 1 by virtue of the electromagnetic 4π 2 (1 + z+) self-duality. Here we just list the dyonic solutions as below. 2 2 + k z+(1 + z+) + k z+ The detailed derivation can be found in Appendix A. More- = 3 2 2 2 2 2 over, we point out that an AdS dyonic solution with b 0as b z+ B z+ B z+ = 3 + − − (1 + z+) , (38) well as k 0 has previously been reported in [57]. In [52], 2 2 by detailed analysis on the boundary condition it is argued = ξ = that a dyonic solution may only exist at a 1( 1in Now, we turn to study the DC transports. Employing the their paper). Here, interestingly enough, we provide an inter- standard techniques developed in [18,37,64], we obtain the pretation for this fact from a different angle of view, namely thermoelectric conductivities over the dyonic black brane the electromagnetic self-duality. geometry (36)as For a = 1, the EMDA theory with the potential (2)exists the dyonic black brane solution as the following form ( 2 + + 2 2) σ = H q HZ B Z , xx B2q2 + (B2 Z + H)2 Bq(q2 + 2HZ + B2 Z 2) 1 dz2 σ = , (39a) ds2 = − f (z)dt2 + + (1 + z)(dx2 + dy2) , xy B2q2 + (B2 Z + H)2 z2 f (z) Hsq (36a) α = , xx B2q2 + (B2 Z + H)2 φ = ln(z + 1), (36b) Bs(q2 + HZ + B2 Z 2) qz α = , (39b) A = μ − dt + Bxdy, (36c) xy B2q2 + (B2 Z + H)2 1 + z 2 ( 2 + ) f = (1 + z)h(z), (36d) κ = s T B Z H , xx B2q2 + (B2 Z + H)2 q2 B2 h = − z3( + z)−2 + z3( + z)−1 2 1 1 1 Bqs T 2 2 κ xy = , (39c) − 2 2 + ( 2 + )2 − k2z2(1 + z) 1 B q B Z H z(z + 2) Z ≡ Z(φ)| H ≡ H(φ)| q − b − log(1 + z) , (36e) where z+ , z+ and are, respectively 2(1 + z) 2k2(1 + z+) = ( + +), = , Z 1 z H 2 z+ (1 + z+)μ 3 We are very grateful to Gouteráux for drawing our attention to the q = . (40) work in [52,57]. z+ 123 195 Page 8 of 12 Eur. Phys. J. C (2019) 79 :195

And then, we give the charge transport coefficients, the DC 2 2 q 3 −2 B 3 −1 ρ 1 − z+(1 + z+) + z+(1 + z+) resistivity dc and the thermopower S as 2 2 − 2 2 ( + )−1 = , 2 k z+ 1 z+ 0 (43) ρ = 1 = 2k , dc 2 2 (41a) 4πT = (1 + z+) σxx(B = 0) (1 + z+)(2k + μ ) 2 3 2 α (B = 0) 4πμ 3 q z+ k z+ S = xx = . × − − − . 2 2 (41b) σxx(B = 0) z+(2k + μ ) z+ 1 + z+ 2(1 + z+)3 1 + z+ (44) Also the magnetic transport coefficients, including the Hall θ angle tan H , the Hall Lorentz ratio L H , the magneticresis- We are particularly interested in the transport behavior of tance ρ and Nernst coefficient ν are calculated explicitly as B this system in the high temperature limit, in which z+ 1 such that 1 + z+ ≈ z+. In this case, Eqs. (43) and (44) 2 2 2 2 reduce to σxy Bμz B z + 4k + μ tan θH = = , (42a) σxx 2k2 B2z2 + 2k2 + μ2 2 2 2 q k B 2 1 − z+ − z+ + z = 0, (45a) κ π 2 3 2 + xy 16 2 2 L H = = , (42b) 2 2 T σxy z2 B2z2 + 4k2 + μ2 q k 4πT = 2 − z+ − z+ . (45b) 2 2 2 23 ρxx − ρxx(B = 0) 2B k z ρB = = , ρxx(B = 0) μ2 B2z2 +4k2 + 4k4 +μ4 Next we consider the situation that the magnetic field B is (42c) small, then z+ and can be solved as 1 αxy ν = − S tan θ π 5 5 σ H = π + 2048 T 2 + O( 4), B xx 4 T B B (46a) π 2 2 2 + 2 2 4π B2z2 + 2k2 32 k T q = . (42d) 3 3 2k2 + μ2 B2z2 + 2k2 + μ2 128π T z+ = π 2 2 2 + 2 32 k T q In order to simplify the expression of temperature, we will π 7 7 π 2 2 2 + 2 262144 T 16 k T q 2 4 only consider the case b = 0 in the follows, in which we + B + O(B ). 32π 2k2T 2 + q2 4 can take a large z+ limit. The horizon condition (37) and the temperature (38) reduce to (46b)

In the above equations, we have expressed z+ and up to the second order of B. As a consequence, we give the charge and magnetic transport coefﬁcients up to the ﬁrst order as

k2 q ρdc = , S = , (47a) 16π 2T 2 8πT 2 π 2 2 π 2 2 2 + 2 16 qT 64 k T q 3 tan θH = B + O(B ), (47b) k2 32π 2k2T 2 + q2 2 16π 2k4 q2 L = + + O(B2), (47c) H π 2 2 2 + 2 π 2 4 64 k T q 64 T π 4 4 π 2 2 2 2 + π 4 4 4 − π 6 2 6 + 4 2 512 T 192 k q T 1024 3 k T 16 k T 3q ρB =−B 32π 2k2T 2 + q2 4 + O(B4), (47d) 2048π 5k2T 4 ν = + O(B2). (47e) 32π 2k2T 2 + q2 2

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Fig. 2 The transports as the Coth θ [ H] L function of temperature T in the H 0.0020 high temperature region. Here 100.4 we have set k = 1, q = 10 and 0.0015 B = 1 0.0010 100.2 0.0005

100 T 0.0000 T 10 50 100 20 40 60 80 100

ρH 0.8 6.28 0.6 6.27

0.4 6.26

0.2 6.25

0.0 T 6.24 T 20 40 60 80 100 20 40 60 80 100

The characteristics of the transport behavior in the high While for 3 > a2 > 1/3, the zero temperature can not be temperature region are summarized as what follows. achieved and the deep horzion z+ 1 corresponds to a high temperature limit. When the translation invariance is broken • Both DC resistivity ρdc and thermopower S decrease by adding axion fields, by contrast, the vanishing entropy with 1/T 2 at high temperature, which implies the thermal density ground state can be achieved only for a2 = 1/3. For transport is dominant over the electric and electrothermal this special case we have demonstrated that the dual system transport. is characterized by a linear-T resistivity in a large range of • With the increase of temperature, the Hall angle 1/ tan θH temperature, reminiscent of the key feature of the strange and Hall Lorentz ratio decrease (see the plots in Fig. 2), metal. We have also obtained dyonic black brane by virtue = while the magneticresistance ρB increases (see the left of the EM duality for a 1. The transport coefficients have below plot in Fig. 2). The Nernst coefficient ν becomes been calculated and their temperature dependence have been a constant in the limit of high temperature (see the right analyzed in high temperature region. We expect the EM dual- below plot in Fig. 2). ity 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 5 Conclusions and discussions for us to investigate the magnetic transport behavior of the dual system by holography. In this paper we have constructed a new class of charged black brane solutions in EMDA theory, which is character- Acknowledgements We are very grateful to Chao Niu for many useful , discussions and comments on the manuscript. This work is supported by ized by two free parameters a b, which could be viewed the Natural Science Foundation of China under Grant Nos. 11575195, as the extension of various charged solutions with b = 0in 11775036, 11747038, 11875053, 11847313. literature [15,34,52Ð54]. For different a, b, the background exhibits distinct behav- Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This paper is a ior in zero temperature limit. In the neutral background theoretical study, for which no data is deposited.] q = 0, the zero temperature ground state with zero entropy density always exists for any a, b, while in the simple charged Open Access This article is distributed under the terms of the Creative q = , b = a a2 < / Commons Attribution 4.0 International License (http://creativecomm case 0 0, it depends on .For 1 3, the zero ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, temperature can be achieved at finite horizon position. But and reproduction in any medium, provided you give appropriate credit the entropy density is finite as well at the zero temperature. to the original author(s) and the source, provide a link to the Creative It is interesting to notice that a ground state with vanishing Commons license, and indicate if changes were made. Funded by SCOAP3. entropy density is allowed for a2 = 1/3, which has previously been obtained in Gubser-Rocha model, and for a2 > 3. 123 195 Page 10 of 12 Eur. Phys. J. C (2019) 79 :195

Appendix A: Electrically, magnetically charged dilatonic One can obtain the following purely magnetic solution 2 2a2 black branes and EM duality 2 1 ˜ 2 dz 2 2 ds = − f (z)dt + + (1 + z) 1+a2 (dx + dy ) , z2 f˜(z) The well-known dyonic black brane solution is given in (A4a) Einstein-Maxwell theory, which enjoys S-duality. While, 2 ˜ 2a usually it is hard to obtain new analytical solutions for dyonic f (z) = (1 + z) 1+a2 h(z), (A4b) black brane. In this section, we shall study the electrically, B2(1 + a2) 1−3a2 1−3a2 h(z) = + z3( + z) 1+a2 − k2z2( + z) 1+a2 1 1 1 magnetically charged black brane solutions in EMD theory ⎛ 4 −3a2+1 −a2+3 (1) with the help of EM duality. (1 + z) a2+1 − 1 (1 + z) a2+1 − 1 − b ⎝ + 3a2 − 1 a2 − 3 1. Pure magnetic solution ⎞ −2a2+2 (1 + z) a2+1 − 1 − ⎠ , (A4c) We first construct the purely magnetic solution. For this pur- a2 − 1 pose, we introduce EM duality of the EMD theory (1). We 2a define the dual field strength G by the Hodge star operation φ = ln(z + 1), A˜ = Bxdy, (A4d) ab 1 + a2 (φ) Z cd Gab := abcd F , (A1) 2 Note that this magnetic solution has also been found in [52]. where G = dH with Ha being the dual gauge field and 2. EM S-duality and dyonic black brane solution abcd is the completely antisymmetric Levi-Civita tensor. And then, we can write down the dual one of EMD theory as ⎛ In this subsection, we construct the dyonic black brane solu- √ tion from the EMDA theory (1)fora2 = 1, in which the ˆ 4 ⎝ 1 ab 1 2 SEMD = d x −g R − GabG − (∂φ) 4Z(φ) 2 theory (1) is S-duality, namely the charge and its dual mag- ⎞ netic solutions are both valid for the same action. For this case, the potential becomes 2⎠ +V (φ) − (∂ψI ) . (A2) I =x,y V = V0(1 + ) + V1, (A5a) 1 φ − 1 φ 1 φ − 1 φ From Eqs. (A1) and (A2), it is easy to find that under the EM 3(e 2 + e 2 )2 − (e 2 − e 2 )2 (φ) → / (φ) V0 = , (A5b) duality, the gauge coupling transforms as Z 1 Z , 2 which also implies a weak-strong coupling duality. Espe- b φ −φ − φ V = (e − 1)3(e + e 2 ), (A5c) cially, there is a correspondence between the electric field 1 2 Ftx of the original theory and the magnetic field G yz of its =−b (sinh φ − φ) , (A5d) dual one. Therefore, by EM duality we could quickly obtain a purely magnetic solution of the dual EMD theory (A2). We the charged black brane solution (4) becomes demonstrate it as follows. Given an electrically charged solution for the action in 1 dz2 ds2 = − f (z)dt2 + + (1 + z)(dx2 + dy2) , (1) which has a gauge coupling with an exponential function z2 f (z) (φ) = aφ Z e , one can obtain a purely magnetic solution for (A6a) the dual action in (A2) with gauge coupling 1/Z(φ) = e−aφ = ( + ) ( ), by virtue of the EM-duality, namely replace q → B in (5). f 1 z h z (A6b) Next, we change a →−a in the magnetic solution. The final q2z3 k2z2 h(z) = 1 − − result is a solution for the action with gauge coupling Z(φ) 2(1 + z)2 (1 + z) ˜ and potential V (φ) = V (a →−a,φ), which is z(z + 2) ⎛ − b − log(1 + z) , (A6c) 2z + 2 ˆ √ eaφ 1 Sˆ = d4x −g ⎝R − F2 − (∂φ)2 + V˜ (φ) qz EMD A = A (z)dt = μ − dt,φ= (z + ). 4 2 t + ln 1 ⎞ 1 z (A6d) 2⎠ − (∂ψI ) . (A3) 2 I =x,y Since for a = 1, the theory is S-duality, we also has the magnetic black brane solution

f˜ = (1 + z)h(z), (A7a) 123 Eur. Phys. J. C (2019) 79 :195 Page 11 of 12 195

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