# Holographic insulator/superconductor transition with exponential nonlinear electrodynamics probed by entanglement entropy

- 152 Downloads

## Abstract

From the viewpoint of holography, we study the behaviors of the entanglement entropy in insulator/superconductor transition with exponential nonlinear electrodynamics (ENE). We find that the entanglement entropy is a good probe to the properties of the holographic phase transition. Both in the half space and the belt space, the non-monotonic behavior of the entanglement entropy in superconducting phase versus the chemical potential is general in this model. Furthermore, the behavior of the entanglement entropy for the strip geometry shows that the confinement/deconfinement phase transition appears in both insulator and superconductor phases. And the critical width of the confinement/deconfinement phase transition depends on the chemical potential and the exponential coupling term. More interestingly, the behaviors of the entanglement entropy in their corresponding insulator phases are independent of the exponential coupling factor but depends on the width of the subsystem \(\mathcal {A}\).

## 1 Introduction

As a strong-week duality, the anti-de Sitter/conformal field theories (AdS/CFT) correspondence [1, 2, 3] establishes a dual relationship between the \((d-1)\) dimensional strongly interacting theories on the boundary and the *d* dimensional weekly coupled gravity theories in the bulk. Based on this novel idea, the AdS/CFT correspondence have received considerable interest in modeling strongly coupled physics, in particular the construction of the holographic superconductor, might shed some light on the problem of understanding the mechanism of the high temperature superconductors in condensed matter physics [4, 5, 6, 7, 8, 9]. Such holographic superconductor models are interesting since they exhibit many characteristic properties shared by real superconductor. In recent years, the studies on the holographic superconductors in AdS spacetime have received a lot of attentions [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

In 1930’s Born and Infeld [49] introduced the theory of nonlinear electrodynamics to avoid the infinite self energies for charged point particles arising in Maxwell theory. The ENE theory, as a extended Born–Infeld-like nonlinear electrodynamics, was introduced by Hendi [50, 51]. It’s Lagrangian density is \(L=\frac{1}{4b^2}\left[ e^{-b^2 F^2}-1\right] \) with \(F^2=F^{\mu \nu }F_{\mu \nu }\). When the ENE factor \(b\rightarrow 0\), the Lagrangian will reduce to the Maxwell case. Compared to the Born–Infeld nonlinear electrodynamics (BINE) [52, 53, 54, 55], the ENE displays different effect on the electric potential and temperature for the same parameters and its singularity is much weaker than the Einstein–Maxwell theory [56, 57, 58]. Recently, this theory has applications in several branches of physics being particularly interesting in systems where the ENE is minimally coupled with gravitation as in the cases of charged black holes [59, 60, 61, 62, 63, 64] and cosmology [65, 66, 67].

Consequently, it is of great interest to investigate the holographic entanglement entropy in AdS spacetime by considering the exponential form of nonlinear electrodynamics. In our previous work [68], we have investigated the effects of the ENE sector on the holographic entanglement entropy in metal/superconductor phase transition. As a further step along this line, in this paper, we will further study the properties of phase transitions by calculating the behaviors of the scalar operator and the entanglement entropy in holographic insulator/superconductor model with ENE.

The paper is organized as follows. In the next section, we will derive the equations of motions and give the boundary conditions of the holographic model in AdS soliton spacetime. Then in Sect. 3, we will study the properties of holographic phase transition by examining the scalar operator. In Sect. 4, we will calculate the holographic entanglement entropy in insulator/superconductor transition with ENE. Finally, Sect. 5 is devoted to conclusions.

## 2 Equations of motion and boundary conditions

*g*is the determinant of the metric,

*L*is the radius of AdS spacetime,

*q*and

*m*are respectively the charge and the mass of the scalar field, \(F=F_{\mu \nu }F^{\mu \nu }\) here \(F_{\mu \nu }\) is the electromagnetic field tensor. The Einstein equation derived from the above action becomes

*r*. For the sake of integrating the field equations from the tip of the soliton out to the infinity for this system, we need to specify the asymptotic behavior both at the tip and the infinity. At the tip \((r=r_0)\), the above equations can be Taylor expand in the form [12]

## 3 Insulator/superconductor phase transition

In this section, we want to study of the phase transition in the five-dimensional AdS soliton background with ENE field. In order to obtain the solutions in the complicated model and ensure the validity of the results, we here concentrate on the case in the weak effects of ENE field and study its influences on the properties of the holographic phase transition. From above discussion, for given \(m^2, q, \psi {(r_0)}\), we can solve the equations of motion by choosing \(\phi (r_0)\) as a shooting parameter. Considering the BF bound [69, 70], we choose \(m^2=-\frac{15}{4}, q=2\) in this paper. Then, \(\psi _-\) can either be identified as an expectation value or a source of the operator \(\mathcal {O}\) of the dual superconductor. In the following calculation, we will consider \(\psi _-\) as the source of the operator and use the \(\psi _{+}=<\mathcal {O}_{+}>\) to describe the phase transition in the dual CFT.

Here, we plot pictures to display the explicit dependence of the chemical potential for operator \(\langle \hat{O}_+\rangle \) and charge density \(\rho \) on the ENE factor *b*. It can be seen from the Fig. 1 that there is a phase transition at the critical chemical potential \(\mu _c\) and its value is independent of the ENE factor b which is shown in the right-hand panel. That is to say, the ENE has no effect on the critical potential of the holographic phase transition for this physical model. When \(\mu <\mu _c\), the system is described by the AdS soliton solution itself which indicates a insulator phase turns on. When the chemical potential is bigger than the critical value \(\mu _c\) the condensation of the operator emerges, which means the AdS soliton reaches a superconductor phase. It is interesting to find that the effect of the ENE factor *b* on the operator in the condensate phase is not trivial. With the increase of the strength of the ENE, the value of the scalar operator becomes bigger. In Fig. 2, we note that the charge density \(\rho \) in the superconductor phase drops when the ENE parameter becomes lower and the insulator/superconductor phase transition here is typically the second order in this case.

## 4 Holographic entanglement entropy

*b*on the entanglement entropy. Since the choice of the subsystem \(\mathcal {A}\) is arbitrary, we can define infinite entanglement entropy correspondingly. For concreteness, we investigate the holographic entanglement entropy of dual field with a half space and a belt geometry in the AdS boundary, respectively.

### 4.1 Holographic entanglement entropy for half geometry

*s*is independent of the UV cutoff and \(s=-1\) corresponds to the pure AdS soliton. As the aim of requiring the lower bound of the integral is still equal to unit, we define a useful dimensionless coordinate in the form

*s*is a finite term, so it is physically important. According to the scaling symmetry (17), we here choose the following scale invariants to explore physics in the entanglement entropy

*s*.

*s*with respect of chemical potential \(\mu \) and the ENE factor

*b*in the half geometry. It can be seen from the figure that the entanglement entropy is continuous but its slop has a discontinuous change at the critical phase transition point \(\mu _c\). Which indicates some kind of new degree of freedom like the Cooper pair would emerge after the condensation. Furthermore, the discontinuous change of the entanglement entropy at \(\mu _c\) signals that the phase transition here is the second order transition. With the increase of the ENE factor, the value of \(\mu _c\) dose not change. Which means the ENE parameter has no effect on the critical point of the phase transition. Before the phase transition, the entanglement entropy is a constant as we change the parameters

*b*and \(\mu \) which can be interpreted as the insulator phase. After the phase transition, for a given

*b*, the entanglement entropy in the superconductor phase first increases and then decreases monotonously for larger \(\mu \). And the value of the entanglement entropy becomes lager as we choose a lager ENE parameter for a given \(\mu \). When the factor \(b\rightarrow 0\), the ENE field will reduce to the Maxwell field and our results are consistent with the one discussed in Ref. [34].

### 4.2 Holographic entanglement entropy for strip geometry

In the Following calculation, we are interested in a more nontrivial geometry which is a strip shape for region \(\mathcal {A}\). We assume that the strip shape with a finite width \(\ell \) along the x direction, along the \(\eta \) direction with a period \(\Gamma \), but infinitely extending in *y* direction. The holographic dual surface \(\gamma _\mathcal {A}\) defined as a codimension three surface is \(t=0,\ \ x=x(r),\ \ -\frac{R}{2}<y<\frac{R}{2}\ (R\rightarrow \infty ),\ \ 0\le \chi \le \Gamma \).

*z*coordinate is given by

*x*direction is symmetry, the corresponding Hamiltonian is conserved. Therefore, the equation of motion for the minimal surface from Eq. (24) can be deduced as following

*S*can be easily calculated in the form

*s*versus the width \(\ell \) of the subsystem \(\mathcal {A}\) and the ENE factor

*b*with the dimensionless quantities \(s\Gamma ^2, \mu \Gamma , \ell \Gamma ^{-1}\) and

*b*. We find that the discontinuous solutions represented the horizontal dotted lines in the figure is independent of the width \(\ell \) but its value of the entanglement entropy with a smaller

*b*is smaller. The connected configuration denoted by the solid lines has two solutions. Specifically, the so-called confinement/deconfinement phase transition [29, 30, 31] emerges as we change the width \(\ell \) and the critical value \(\ell _{c}\) indicated by the vertical dotted lines becomes bigger with the increase of the parameter

*b*. For a fixed

*b*, in the deconfinement phase where \(\ell <\ell _c\), considering the physical entropy determined by the choice of the lowest one, the entanglement entropy comes from the connected surface and the lowest branch in the figure is finally favored. However, the physical entanglement entropy in confinement phase where \(\ell >\ell _c\) is dominated by the discontinuous surface and has nothing to do with the factor \(\ell \). Thus, there exists four phases in the dual boundary field theory, including the insulator phase, superconductor phase and their corresponding confinement/deconfinement phases. When the parameter \(\ell \) is fixed, we observe that the entanglement entropy increases as we choose a bigger ENE factor both in the confinement and deconfinement superconducting phases.

Interestingly, the entanglement entropy with respect to the chemical potential \(\mu \) as one fixes the ENE factor *b* or the width \(\ell \) is presented in Fig. 5. At the insulator/superconductor phase transition point \(\mu =\mu _c\), we also find that the jump of the slop of the entanglement entropy indicates that the system undergoes the second order phase transition. Both in the confinement and deconfinement superconducting phases where \(\mu >\mu _c\), we can see that the behavior of the entanglement entropy as a function of the chemical potential is non-monotonic and similar to the case in the half geometry which we have discussed above. As the chemical potential is fixed, the value the entanglement entropy becomes smaller when the factors *b* and \(\ell \) become lower. More specifically, the effect of the ENE factor on the entanglement entropy is weaker than the width of the subsystem \(\mathcal {A}\). In their corresponding insulator phases where \(\mu <\mu _c\), we observe that the value of the entanglement entropy does not change as we alter the parameter *b* for a given \(\mu \). On the other hand, with the increase of the width the entanglement entropy increases. That is to say, the behavior of the entanglement entropy in the corresponding insulator phases is independent of the ENE factor but depends on the width of the subsystem \(\mathcal {A}\).

### 4.3 Phase diagram

In Fig. 6, the insulator phase and the superconductor phase are separated by the green vertical dashed line and the phase boundary between the confinement phase and the deconfinement phase is separated by the red horizontal dashed line and the solid curve. Therefore, the phases characterized by the parameters \(\mu \) and \(\ell \) contain the insulator phase, superconductor phase and their corresponding confinement/deconfinement phases. It can be clearly seen from the figure that the critical width \(\ell _c\) of the confinement/deconfinement phase transition in the insulator phase is independent of the ENE parameter. In the superconductor phase, however, the critical width \(\ell _c\) increases with the increase of the ENE factor. To further study, we observe that the critical width \(\ell _c\) has a non-monotonic change as the chemical potential becomes bigger. Concretely, the entanglement entropy first increases beyond the cusp at the certain chemical potential, reaches to a maximum and decreases to a minimum, and then approaches a plateau at very large \(\mu \).

## 5 Summary

We have studied the properties of phase transitions by calculating the behaviors of the scalar operator and the entanglement entropy in holographic insulator/superconductor model with ENE. On the basis of the behaviors of the scalar operator in this holographic model, we find that there is a insulator/superconductor transition at the critical chemical potential point and the effect of the ENE factor on the scalar condensation is quite different from those observed in the holographic metal/superconductor transition with ENE field model [68]. Specifically, in the holographic insulator/superconductor system the ENE factor does not have any effect on the critical chemical potential of the transition. These conclusions can also be understood from the behavior of the entanglement entropy. From the Fig. 3, the discontinuity of the slop of the entanglement entropy in half space at the critical chemical potential point signals some kind of new degree of freedom like the Cooper pair would emerge after the condensation and indicates the order of associated phase transition in the system. In Fig. 5, we observed the behavior of the entanglement entropy with respect chemical potential in strip geometry at the insulator/superconductor transition point is similar to the half case. That is to say, the entanglement entropy is indeed a good tool to search for the phase transition point.

In the superconducting phase, compared to the phenomenon observed in the scalar operator, the entanglement entropy versus the chemical potential displays more rich behaviors. Both in the half space and the belt space, the non-monotonic behavior of the entanglement entropy versus the chemical potential is general in this model as the ENE parameter is fixed. For a given chemical potential, the value the entanglement entropy becomes smaller when the ENE factor or the with becomes lower. In the insulator phase, however, the behavior of entanglement entropy is independent of the ENE parameter.

Interestingly, considering the effect of the belt width on the entanglement entropy, we obtained that the confinement/deconfinement phase transition appears in both insulator and superconductor phases and the complete phase diagram of the entanglement entropy with a straight geometry is presented in Fig. 6. It is shown that the critical width of the confinement/deconfinement phase transition depends on the chemical potential and the ENE term.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11665015, 11475061; Guizhou Provincial Science and Technology Planning Project of China under Grant No. qiankehejichu[2016]1134; The talent recruitment program of Liupanshui normal university of China under Grant No. LPSSYKYJJ201508.

## References

- 1.J.M. Maldacena, The large N limit of superconformal field theories andsupergravity. Adv. Theor. Math. Phys.
**2**, 231–252 (1998). arXiv:hep-th/9711200 ADSMathSciNetCrossRefMATHGoogle Scholar - 2.S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from noncritical string theory. Phys. Lett. B
**428**, 105–114 (1998). arXiv:hep-th/9802109 ADSMathSciNetCrossRefMATHGoogle Scholar - 3.E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys.
**2**, 253–291 (1998). arXiv:hep-th/9802150 ADSMathSciNetCrossRefMATHGoogle Scholar - 4.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Building a holographic superconductor. Phys. Rev. Lett.
**101**, 031601 (2008)ADSCrossRefGoogle Scholar - 5.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Holographic superconductors. JHEP
**12**, 015 (2008)Google Scholar - 6.G.T. Horowitz, M.M. Roberts, Holographic superconductors with various condensates. Phys. Rev. D
**78**, 126008 (2008)ADSCrossRefGoogle Scholar - 7.S.A. Hartnoll, Lectures on holographic methods for condensed matter physics. Class. Quantum Gravity
**26**, 224002 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar - 8.G.T. Horowitz, Introduction to holographic superconductors. arXiv:1002.1722 [hep-th]
- 9.G.T. Horowitz, Benson way complete phase diagrams for a holographic superconductor/insulator system. JHEP
**1011**, 011 (2010)ADSCrossRefMATHGoogle Scholar - 10.J. Jing, S. Chen, Holographic superconductors in the Born-Infeld electrodynamics. Phys. Lett. B
**686**, 68 (2010)ADSCrossRefGoogle Scholar - 11.J. Jing, Q. Pan, S. Chen, Holographic superconductors with Power–Maxwell field. JHEP
**11**, 045 (2011)Google Scholar - 12.Y. Peng, Q. Pan, B. Wang, Various types of phase transitions in the AdS soliton background. Phys. Lett. B
**699**, 383–387 (2011)ADSCrossRefGoogle Scholar - 13.S. Gangopadhyay, D. Roychowdhury, Analytic study of properties of holographic superconductors in Born-Infeld electrodynamics. JHEP
**05**, 002 (2012)ADSCrossRefGoogle Scholar - 14.Z. Zhao, Q. Pan, S. Chen, J. Jing, Notes on holographic superconductor models with the nonlinear electrodynamics. Nucl. Phys. B
**98**, 871 [FS] (2013)Google Scholar - 15.S. Gangopadhyay, D. Roychowdhury, Analytic study of Gauss-Bonnet holographic superconductors in Born-Infeld electrodynamics. JHEP
**05**, 156 (2012)ADSCrossRefGoogle Scholar - 16.J. Jing, Q. Pan, S. Chen, Holographic superconductor/insulator transition with logarithmic electromagnetic field in Gauss-Bonnet gravity. Phys. Lett. B
**716**, 385 (2012)ADSCrossRefGoogle Scholar - 17.W. Yao, J. Jing, Analytical study on holographic superconductors for Born-Infeld electrodynamics in Gauss-Bonnet gravity with backreaction. JHEP
**05**, 101 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar - 18.S. Dey, A. Lala, Holographic s-wave condensation and Meissner-like effect in Gauss-Bonnet gravity with various non-linear corrections. Ann. Phys.
**354**, 165–182 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar - 19.C. Lai, Q. Pan, J. Jing, Y. Wang, On analytical study of holographic superconductors with Born-Infeld electrodynamics. Phys. Lett. B
**749**, 437 (2015)ADSCrossRefMATHGoogle Scholar - 20.H.R. Salahi, A. Sheykhi, A. Montakhab, Effects of backreaction on Power-Maxwell holographic superconductors in Gauss-Bonnet gravity. Eur. Phys. J. C
**76**, 575 (2016)ADSCrossRefMATHGoogle Scholar - 21.Z. Sherkatghanad, B. Mirza, F.L. Dezaki, Exponential nonlinear electrodynamics and backreaction effects on Holographic superconductor in the Lifshitz black hole background. Int. J. Mod. Phys. D
**26**, 1750175 (2017)MathSciNetCrossRefGoogle Scholar - 22.A. Sheykhi, F. Shamsi, Holographic superconductors with logarithmic nonlinear electrodynamics in an external magnetic field. Int. J. Theor. Phys.
**56**, 916 (2017)CrossRefMATHGoogle Scholar - 23.A. Sheykhi, F. Shaker, Effects of backreaction and exponential nonlinear electrodynamics on the holographic superconductors. J. Mod. Phys. D
**26**, 1750050 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar - 24.A. Sheykhi, A. Ghazanfari, A. Dehyadegari, Holographic conductivity of holographic superconductors with higher order corrections. Eur. Phys. J. C
**78**, 159 (2018)ADSCrossRefGoogle Scholar - 25.S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett.
**96**, 181602 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar - 26.S. Ryu, T. Takayanagi, Aspects of holographic entanglement entropy. JHEP
**0608**, 045 (2006)Google Scholar - 27.D.V. Fursaev, Proof of the holographic formula for entanglement entropy. JHEP
**0609**, 018 (2006)ADSMathSciNetCrossRefGoogle Scholar - 28.T. Hirata, T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy. JHEP
**0702**, 042 (2007)ADSMathSciNetCrossRefGoogle Scholar - 29.T. Nishioka, T. Takayanagi, AdS bubbles, entropy and closed string tachyons. JHEP
**0701**, 090 (2007)ADSMathSciNetCrossRefGoogle Scholar - 30.I.R. Klebanov, D. Kutasov, A. Murugan, Entanglement as a probe of confinement. Nucl. Phys. B
**796**, 274 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar - 31.R.C. Myers, A. Singh, Comments on holographic entanglement entropy and RG flows. JHEP
**1204**, 122 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar - 32.A. Pakman, A. Parnachev, Topological entanglement entropy and holography. JHEP
**0807**, 097 (2008)Google Scholar - 33.T. Albash, C.V. Johnson, Holographic studies of entanglement entropy in superconductors. JHEP
**05**, 079 (2012)ADSCrossRefGoogle Scholar - 34.R.-G. Cai, S. He, L. Li, Y.-L. Zhang, Holographic entanglement entropy in insulator/superconductor transition. JHEP
**1207**, 088 (2012)ADSCrossRefGoogle Scholar - 35.R.-G. Cai, S. He, L. Li, L.-F. Li, Entanglement entropy and Wilson loop in Stuckelberg holographic insulator/superconductor model. JHEP
**1210**, 107 (2012)ADSCrossRefGoogle Scholar - 36.R.-G. Cai, L. Li, L.-F. Li, R.-K. Su, Entanglement entropy in holographic P-Wave superconductor/insulator model. JHEP
**1306**, 063 (2013)ADSCrossRefGoogle Scholar - 37.J. de Boer, M. Kulaxizi, A. Parnachev, Holographic entanglement entropy in lovelock gravities. JHEP
**1107**, 109 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar - 38.L.-Y. Hung, R.C. Myers, M. Smolkin, On holographic entanglement entropy and higher curvature gravity. JHEP
**1104**, 025 (2011)ADSCrossRefGoogle Scholar - 39.N. Ogawa, T. Takayanagi, Higher derivative corrections to holographic entanglement entropy for AdS solitons. JHEP
**1110**, 147 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar - 40.X. Dong, Holographic entanglement entropy for general higher derivative gravity. JHEP
**01**, 044 (2014)ADSCrossRefMATHGoogle Scholar - 41.X.-M. Kuang, E. Papantonopoulos, B. Wang, Entanglement entropy as a probe of the proximity effect in holographic superconductors. J. High Energy Phys.
**1405**, 130 (2014)ADSCrossRefGoogle Scholar - 42.Y. Peng, Holographic entanglement entropy in superconductor phase transition with dark matter sector. Phys. Lett. B
**750**, 420–426 (2015)ADSCrossRefMATHGoogle Scholar - 43.X.-X. Zeng, H. Zhang, L.-F. Li, Phase transition of holographic entanglement entropy in massive gravity. Phys. Lett.s B
**756**, 170 (2016)Google Scholar - 44.N.S. Mazhari, D. Momeni, R. Myrzakulov, H. Gholizade, M. Raza, Non-equilibrium phase and entanglement entropy in 2D holographic superconductors via Gauge-String duality. Can. J. Phys.
**10**, 94 (2016)Google Scholar - 45.Y. Peng, G. Liu, Holographic entanglement entropy in two-order insulator/superconductor transitions. Phys. Lett. B
**767**, 330–335 (2017)ADSMathSciNetCrossRefGoogle Scholar - 46.W. Yao, J. Jing, Holographic entanglement entropy in insulator/superconductor transition with Born-Infeld electrodynamics. JHEP
**05**, 058 (2014)ADSCrossRefMATHGoogle Scholar - 47.W. Yao, J. Jing, Holographic entanglement entropy in metal/superconductor phase transition with Born-Infeld electrodynamics. Nucl. Phys. B
**889**, 109 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar - 48.Y. Liu, Y. Gong, B. Wang, Non-equilibrium condensation process in holographic superconductor with nonlinear electrodynamics. JHEP
**02**, 116 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 49.M. Born, L. Infeld, Foundations of the new field theory. Proc. R. Soc. A
**144**, 425 (1934)ADSCrossRefMATHGoogle Scholar - 50.S.H. Hendi, Asymptotic charged BTZ black hole solutions. J. High Energy Phys.
**03**, 065 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar - 51.S.H. Hendi, A. Sheykhi, Charged rotating black string in gravitating nonlinear electromagnetic fields. Phys. Rev. D
**88**, 044044 (2013)ADSCrossRefGoogle Scholar - 52.B. Hoffmann, Gravitational and electromagnetic mass in the Born-Infeld electrodynamics. Phys. Rev.
**47**, 877 (1935)ADSCrossRefMATHGoogle Scholar - 53.W. Heisenberg, H. Euler, Folgerungen aus der Diracschen Theorie des Positrons. Z. Phys.
**98**, 714 (1936)ADSCrossRefMATHGoogle Scholar - 54.H.P. de Oliveira, Non-linear charged black holes. Class. Quantum Gravity
**11**, 1469 (1994)ADSCrossRefGoogle Scholar - 55.G.W. Gibbons, D.A. Rasheed, Electric-magnetic duality rotations in non-linear electrodynamics. Nucl. Phys.
**454**, 185 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar - 56.S.H. Hendi, Asymptotic Reissner-Nordstrom black holes. Ann. Phys. (N.Y.) 333, 282 (2013)Google Scholar
- 57.S. H. Hendi, Thermodynamic properties of asymptotically Reissner-Nordstrom black holes. Ann. Phys.
**346**, 42–50 (2014)Google Scholar - 58.S.H. Hendi, A. Sheykhi, M. Sepehri Rad, K. Matsuno, Slowly rotating dilatonic black holes with exponential form of nonlinear electrodynamics. Gen. Relati. Gravit.
**47**, 117 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar - 59.A. Sheykhi, S. Hajkhalili, Dilaton black holes coupled to nonlinear electrodynamic field. Phys. Rev. D
**89**, 104019 (2014)ADSCrossRefGoogle Scholar - 60.A. Sheykhi, A. Kazemi, Higher dimensional dilaton black holes in the presence of exponential nonlinear electrodynamics. Phys. Rev. D
**90**, 044028 (2014)ADSCrossRefGoogle Scholar - 61.M. Kord Zangeneh, A. Dehyadegari, A. Sheykhi, M.H. Dehghani, Thermodynamics and gauge/gravity duality for Lifshitz black holes in the presence of exponential electrodynamics. JHEP
**1603**, 037 (2016)MathSciNetCrossRefMATHGoogle Scholar - 62.S.I. Kruglov, Corrections to Reissner-Nordstrom black hole solution due to exponential nonlinear electrodynamics. Europhys. Lett.
**115**, 60006 (2016)ADSCrossRefGoogle Scholar - 63.S.I. Kruglov, Black hole as a magnetic monopole within exponential nonlinear electrodynamics. Ann. Phys.
**378**, 59 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar - 64.S. Hajkhalili, A. Sheykhi, Asymptotically (A)dS dilaton black holes with nonlinear electrodynamics. arXiv:1801.05697
- 65.V.A. De Lorenci, R. Klippert, M. Novello, J.M. Salim, Nonlinear electrodynamics and FRW cosmology. Phys. Rev. D
**65**, 063501 (2002)ADSCrossRefGoogle Scholar - 66.M. Novello, S.E. Perez Bergliaffa, J. Salim, Nonlinear electrodynamics and the acceleration of the Universe. Phys. Rev. D
**69**, 127301 (2004)ADSCrossRefGoogle Scholar - 67.M. Novello, A.N. Araujo, J.M. Salim, The cosmological origins of nonlinear electrodynamics. Int. J. Mod. Phys. A
**24**, 5639 (2009)ADSCrossRefGoogle Scholar - 68.W. Yao, J. Jing, Holographic entanglement entropy in metal/superconductor phase transition with exponential nonlinear electrodynamics. Phys. Lett. B
**759**, 533 (2016)ADSCrossRefMATHGoogle Scholar - 69.P. Breitenlohner, D.Z. Freedman, Stability in gauged extended supergravity. Ann. Phys.
**144**, 249 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar - 70.P. Breitenlohner, D.Z. Freedman, Positive energy in anti-de sitter backgrounds and gauged extended supergravity. Phys. Lett. B
**115**, 197 (1982)ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}