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Holography of Electrically and Magnetically Charged Black Branes

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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 [19–23]. 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 [24–33]. 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 [1–5]. 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 [6,9–18]. 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. [37–44]. (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 electri- cally 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) 35,45–54]. 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 [15,34,52–54]. 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.
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