# Holographic entanglement entropy with Born–Infeld electrodynamics in higher dimensional AdS black hole spacetime

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## Abstract

We examine the entanglement entropy in higher dimensional holographic metal/superconductor model with Born–Infeld (BI) electrodynamics. We note that the entanglement entropy is still a powerful tool to probe the critical phase transition point and the order of the phase transition in higher dimensional AdS spacetime. Due to the presence of the BI electromagnetic field, the formation of the scalar condensation becomes harder. For both operators \(\langle \mathcal {O}_{+}\rangle \) and \(\langle \mathcal {O}_{-}\rangle \), we show that the entanglement entropy in the metal phase decreases as the BI factor increases, but in condensation phase the entanglement entropy increases monotonically for stronger nonlinearity of BI electromagnetic field. Furthermore, we also study the influence of the width of the subsystem on the holographic entanglement entropy and observe that with the increase of the width the entanglement entropy increases.

## 1 Introduction

The entanglement entropy serves as a key quantity to measure how the subsystem and its complement are correlated [1]. In strongly coupled system, the entanglement entropy is expected to be a useful tool to keep track of the degree of freedom while other traditional methods might not be available. However, the calculation of entanglement entropy is usually a not easy task except for the case in \(1+1\) dimensions. In the spirit of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [2, 3, 4], Ryu and Takayanagi have provided a holographic proposal to compute the entanglement entropy in [5, 6]. It states that the entanglement entropy of a d+1 dimensional CFT at strong coupling can be investigated from a weakly coupled gravity dual characterized by an asymptotically \(\hbox {AdS}_d+2\) spacetime. With this elegant approach, the holographic entanglement entropy is widely used to study properties of phase transitions in holographic superconductor models [7, 8, 9, 10, 11, 12]. The behavior of the entanglement entropy in metal/superconductor phase transition was studied in [13] and observed that the entanglement entropy is lower in the superconductor phase than in the normal phase. In the insulator/superconductor model, the non-monotonic behavior of the entanglement entropy was found in Ref. [14]. Then, the study of entanglement entropy was also extended to other various holographic superconductors applications [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

*b*is the BI coupling parameter. In the limit \(b\rightarrow 0\), the BI field will reduce to the Maxwell field. It is also to be noted that the higher order terms in the factor

*b*essentially correspond to the higher derivative corrections of the gauge fields [39, 40]. In the present work, we will study the properties of phase transitions of the holographic superconductor model by calculating the behaviors of the scalar operator and the entanglement entropy and see how the phase transition is affected by the BI electrodynamics and the width of the subsystem in higher dimensional AdS black hole spacetime.

The framework of this paper is as follows. In Sect. 2, we introduce the basic field equations of holographic superconductor model with BI electrodynamics in n-dimensional spacetime and study the properties of the phase transition by calculating the scalar operator. In Sect. 3, the behavior of the entanglement entropy in the holographic superconductor model are investigated in derail. In Sect. 4, we conclude our main results of this paper.

## 2 Holographic superconducting model with BI electrodynamics

### 2.1 Basic field equations

*R*and

*g*are, respectively, the Ricci scalar curvature and the determinant of the metric, \(\Lambda =-(d-1)(d-2)/2L^2\) is the cosmological constant,

*L*is the AdS radius, \(\psi \) represents a scalar field with charge

*q*and mass

*m*, and \(L_{BI}\) is the Lagrangian of the BI gauge field.

*r*, and \(16\pi G=1\) was used. For purpose of getting the solutions in superconducting phase where \(\psi (r)\ne 0\), we need to impose the boundary conditions. At the horizon \(r_+\), the regularity condition gives the boundary conditions [41]

### 2.2 Metal/superconductor phase transition

*T*for different BI parameter

*b*is shown in Fig. 1. For a given b, the left panel shows that as the temperature

*T*exceeds a critical value \(T_c\), the scalar field is vanishing and this can be identified as the metal phase. However, the condensation of the operators emerges as \(T<T_c\), which can be thought of as a superconductor phase. Near the phase transition point \(T_c\), the critical behavior is found to be \(\langle \mathcal {O}_{+}\rangle \varpropto (T_{c}-T)^{1/2}\), which means the phase transition is second order [45, 46, 47, 48]. And it is noted that the curves for the scalar operators has similar behaviors to the BCS theory for different BI factor, where the condensation goes to a constant at zero temperature. In the right panel, it can be seen that the condensation of the operator \(\langle \mathcal {O}_{-}\rangle \) with respect of the temperature T is similar to the case of the operator \(\langle \mathcal {O}_{+}\rangle \). As the temperature is fixed, we observe that with the increase of the BI parameter the condensation gaps increases for both scalar operators \(\langle \mathcal {O}_{+}\rangle \) and \(\langle \mathcal {O}_{-}\rangle \), which implies that the scalar hair can be formed harder when the nonlinearity in the BI electromagnetic field becomes stronger. Interestingly, the condensation gap of the operator \(\langle \mathcal {O}_{+}\rangle \) is larger than the one of the operator \(\langle \mathcal {O}_{-}\rangle \) as the BI factor changes in same value. That is to say, the effect of the BI factor on the condensation of operator \(\langle \mathcal {O}_{+}\rangle \) is more powerful than the one of operator \(\langle \mathcal {O}_{-}\rangle \). From Table 1, we find that the critical temperature \(T_c\) for the operators \(\langle \mathcal {O}_{+}\rangle \) and \(\langle \mathcal {O}_{-}\rangle \) decreases as the BI factor increases, which agrees well with the results obtained in Fig. 1.

The critical temperature \(T_c\) for the operators \(\langle \mathcal {O}_{+}\rangle \) and \(\langle \mathcal {O}_{-}\rangle \) for different BI parameter

| 0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|---|

\(\langle \mathcal {O}_{+}\rangle \) | 0.1900 | 0.1884 | 0.1837 | 0.1761 | 0.1662 | 0.1542 | 0.1410 | 0.1269 | 0.1127 |

\(\langle \mathcal {O}_{-}\rangle \) | 0.3782 | 0.3780 | 0.3774 | 0.3763 | 0.3749 | 0.3730 | 0.3708 | 0.3683 | 0.3654 |

## 3 Entanglement entropy in holographic phase transition

*A*is

*T*, the BI factor

*b*and the belt width \(\ell \).

### 3.1 Entanglement entropy for operator \(\langle \mathcal {O}_{+}\rangle \)

We show the behavior of the entanglement entropy of operator \(\langle \mathcal {O}_{+}\rangle \) with respects to the temperature *T* and BI factor b in Fig. 2 with the dimensionless quantities \(s/ \rho ^{\frac{1}{3}},\ \ell \rho ^{\frac{1}{3}}, T/\rho ^{\frac{1}{3}}\). The left panel shows the cases of \(\ell \rho ^{\frac{1}{3}}=1\) with various *T* and the right panel is the cases of \(T/\rho ^{\frac{1}{3}}=0.1\), \(\ell \rho ^{\frac{1}{3}}=1\) with various *b*. It can be seen from the left-hand diagram that the slop of the entanglement entropy at the phase transition points indicated by the vertical dotted lines is discontinuous but the value of the entanglement entropy is continuous. Which indicates some kind of new degrees of freedom like Cooper pair would emerge after the condensation and this phase transition can be regarded as the signature of the second order phase transition. With the increase of the BI factor *b* the critical temperature \(T_c\) of the phase transition decreases which means that the stronger BI electrodynamics correction makes the scalar hair harder to condense. Moreover, the entanglement entropy in the superconductor phases for different *b* denoted by the solid lines are lower than the ones in the normal phases represented by the dot-dashed lines and decreases monotonously as temperature decreases. That is to say, the scalar hair turns on at the critical temperature and the formation of Cooper pairs make the degrees of freedom decrease in the superconductor phase. As the factor *b* becomes lager entanglement entropy in the mental phase decreases. In the condensation phase, it can be seen from the right diagram that the entanglement entropy increases monotonically for bigger *b* and this behavior of entanglement entropy is quite different from the results discussed in four-dimensional spacetime [49], where the entanglement entropy first increases and forms a peak at the threshold then decreases monotonically with increase of the BI parameter.

To get further understanding of the influence of the width \(\ell \) on the entanglement entropy in the superconductor phase, we plot the corresponding results in Fig. 3. For a given *b*, we find that the entanglement entropy becomes smaller as the temperature *T* gets lower. With the growth of the belt width \(\ell \) the value of the entanglement entropy *s* increases.

### 3.2 Entanglement entropy for operator \(\langle \mathcal {O}_{-}\rangle \)

## 4 Conclusion

In this paper, we calculated the holographic metal/ superconductor phase transition with BI electrodynamics in higher dimensional AdS spacetime. We first explored the properties of the phase transition of this system by analyzing the behaviors of the scalar operator and found that the value of the critical temperature \(T_c\) of the phase transition becomes smaller as the BI factor *b* increases. This means that the BI correction to the usual Maxwell field hinders the formation of the scalar hair. By comparing the effects of the BI parameter on the condensation, we observer that the effect of the BI factor on the condensation of operator \(\langle \mathcal {O}_{+}\rangle \) is more powerful than the one of operator \(\langle \mathcal {O}_{-}\rangle \). Instead of the probe limit, when taking the full-backreaction into consideration, we observed that the curves for two scalar operators has similar behaviors to the BCS theory for different BI factor. In addition, to further study the properties of the phase transition, we used holographic methods to explored the behavior of the entanglement entropy in the holographic model. For both operators \(\langle \mathcal {O}_{+}\rangle \) and \(\langle \mathcal {O}_{-}\rangle \) , the critical temperature \(T_c\) of the phase transition obtained from the behaviors of entanglement entropy is the same as the threshold temperature obtained from the behaviors of the scalar operator. And the discontinuous slop of the entanglement entropy at the critical temperature \(T_c\) corresponds to second order phase transition in this physical system. Consequently, the entanglement entropy is indeed a good probe to study the properties of the phase transition, and that its behavior can indicate not only the appearance, but also the order of the phase transition. Furthermore, the entanglement entropy of operator \(<\mathcal {O}_{+}>\) in the condensation phase increases monotonically for bigger *b* and this behavior of entanglement entropy is quite different from the results discussed in four-dimensional spacetime [49], in which the result of the entanglement entropy first increases and forms a peak at the threshold then decreases monotonically with increase of the BI parameter.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant nos. 11665015, 11875025; 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.S. Ryu, Y. Hatsugai, Entanglement entropy and the Berry phase in the solid state. Phys. Rev. B
**73**, 245115 (2006). arXiv:cond-mat/0601237 ADSCrossRefGoogle Scholar - 2.J.M. Maldacena, The large N limit of superconformal field theories andsupergravity Adv. Theor. Math. Phys.
**2**, 231–252 (1998). arXiv:hep-th/9711200 ADSCrossRefGoogle Scholar - 3.S.S. Gubser, I.R. Klebanov, Gauge theory correlators from noncritical string theory. A. M. Polyakov Phys. Lett. B
**428**, 105–114 (1998). arXiv:hep-th/9802109 ADSCrossRefGoogle Scholar - 4.E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys.
**2**, 253–291 (1998). arXiv:hep-th/9802150 ADSMathSciNetCrossRefGoogle Scholar - 5.S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett.
**96**, 181602 (2006)ADSMathSciNetCrossRefGoogle Scholar - 6.S. Ryu, T. Takayanagi, Aspects of holographic entanglement entropy. JHEP
**0608**, 045 (2006)ADSMathSciNetCrossRefGoogle Scholar - 7.D.V. Fursaev, Proof of the holographic formula for entanglement entropy. JHEP
**0609**, 018 (2006)ADSMathSciNetCrossRefGoogle Scholar - 8.T. Hirata, T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy. JHEP
**0702**, 042 (2007)ADSMathSciNetCrossRefGoogle Scholar - 9.T. Nishioka, T. Takayanagi, AdS bubbles, entropy and closed string tachyons. JHEP
**0701**, 090 (2007)ADSMathSciNetCrossRefGoogle Scholar - 10.I.R. Klebanov, D. Kutasov, A. Murugan, Entanglement as a probe of confinement. Nucl. Phys. B
**796**, 274 (2008)ADSMathSciNetCrossRefGoogle Scholar - 11.R.C. Myers, A. Singh, Comments on holographic entanglement entropy and RG flows. JHEP
**1204**, 122 (2012)ADSMathSciNetCrossRefGoogle Scholar - 12.A. Pakman, A. Parnachev, Topological entanglement entropy and holography. JHEP
**0807**, 097 (2008)ADSMathSciNetCrossRefGoogle Scholar - 13.T. Albash, C.V. Johnson, Holographic studies of entanglement entropy in superconductors. JHEP
**05**, 079 (2012)ADSCrossRefGoogle Scholar - 14.R.-G. Cai, S. He, L. Li, Y.-L. Zhang, Holographic entanglement entropy in insulator/superconductor transition. JHEP
**1207**, 088 (2012)ADSCrossRefGoogle Scholar - 15.R.-G. Cai, S. He, L. Li, L.-F. Li, Entanglement entropy and Wilson loop in Stckelberg holographic insulator/superconductor model. JHEP
**1210**, 107 (2012)ADSCrossRefGoogle Scholar - 16.R.-G. Cai, L. Li, L.-F. Li, S. Ru-Keng, Entanglement entropy in holographic P-wave superconductor/insulator model. JHEP
**1306**, 063 (2013)ADSCrossRefGoogle Scholar - 17.J. de Boer, M. Kulaxizi, A. Parnachev, Holographic entanglement entropy in Lovelock gravities. JHEP
**1107**, 109 (2011)ADSMathSciNetCrossRefGoogle Scholar - 18.L.-Y. Hung, R.C. Myers, M. Smolkin, On holographic entanglement entropy and higher curvature gravity. JHEP
**1104**, 025 (2011)ADSCrossRefGoogle Scholar - 19.N. Ogawa, T. Takayanagi, Higher derivative corrections to holographic entanglement entropy for AdS solitons. JHEP
**1110**, 147 (2011)ADSMathSciNetCrossRefGoogle Scholar - 20.W. Yao, J. Jing, Holographic entanglement entropy in insulator/superconductor transition with Born–Infeld electrodynamics. JHEP
**05**, 058 (2014)ADSCrossRefGoogle Scholar - 21.X. Dong, Holographic entanglement entropy for general higher derivative gravity. JHEP
**01**, 044 (2014)ADSCrossRefGoogle Scholar - 22.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 - 23.Y. Peng, Holographic entanglement entropy in superconductor phase transition with dark matter sector. Phys. Lett. B
**750**, 420–426 (2015)ADSCrossRefGoogle Scholar - 24.X.-X. Zeng, H. Zhang, L.-F. Li, Phase transition of holographic entanglement entropy in massive gravity. Phys. Lett. B
**756**, 170 (2016)ADSCrossRefGoogle Scholar - 25.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 - 26.Y. Peng, G. Liu, Holographic entanglement entropy in two-order insulator/superconductor transitions. Phys. Lett. B
**767**, 330–335 (2017)ADSMathSciNetCrossRefGoogle Scholar - 27.A. Strominger, C. Vafa, Microscopic origin of the Bekenstein–Hawking entropy. Phys. Lett. B
**379**, 99 (1996)ADSMathSciNetCrossRefGoogle Scholar - 28.O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Large N field theories, large N field theories, string theory and gravity. Phys. Rep.
**323**, 183 (2000)ADSMathSciNetCrossRefGoogle Scholar - 29.R. Emparan, H.S. Reall, Black holes in higher dimensions. Living Rev. Relativ.
**11**, 6 (2008)ADSCrossRefGoogle Scholar - 30.M. Born, L. Infeld, Foundations of the new field theory. Proc. R. Soc. A
**144**, 425 (1934)ADSCrossRefGoogle Scholar - 31.G.W. Gibbons, D.A. Rasheed, Electric-magnetic duality rotations in non-linear electrodynamics. Nucl. Phys.
**454**, 185 (1995)ADSMathSciNetCrossRefGoogle Scholar - 32.B. Hoffmann, Gravitational and electromagnetic mass in the Born–Infeld electrodynamics. Phys. Rev.
**47**, 877 (1935)ADSCrossRefGoogle Scholar - 33.W. Heisenberg, H. Euler, Folgerungen aus der Diracschen Theorie des Positrons. Z. Phys.
**98**, 714 (1936)ADSCrossRefGoogle Scholar - 34.H.P. de Oliveira, Non-linear charged black holes. Class. Quant. Grav.
**11**, 1469 (1994)ADSCrossRefGoogle Scholar - 35.J.L. Jing, S.B. Chen, Holographic superconductors in the Born–Infeld electrodynamics. Phys. Lett. B
**686**, 68 (2010)ADSCrossRefGoogle Scholar - 36.O. Miskovic, R. Olea, Conserved charges for black holes in Einstein–Gauss–Bonnet gravity coupled to nonlinear electrodynamics in AdS space. Phys. Rev. D
**83**, 024011 (2011)ADSCrossRefGoogle Scholar - 37.Y. Q. Liu, Y. Peng, B. Wang, Gauss–Bonnet holographic superconductors in Born–Infeld electrodynamics with backreactions. arXiv:1202.3586
- 38.W. Yao, J. Jing, Analytical study on holographic superconductors for Born–Infeld electrodynamics in Gauss–Bonnet gravity with backreactions. JHEP
**05**, 101 (2013)ADSMathSciNetCrossRefGoogle Scholar - 39.D. Roychowdhury, Effect of external magnetic field on holographic superconductors in presence of nonlinear corrections. Phys. Rev. D
**86**, 106009 (2012)ADSCrossRefGoogle Scholar - 40.D. Roychowdhury, AdS/CFT superconductors with Power Maxwell electrodynamics: reminiscent of the Meissner effect. Phys. Lett. B
**718**, 1089 (2013)ADSCrossRefGoogle Scholar - 41.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 - 42.P. Breitenlohner, D.Z. Freedman, Stability In gauged extended supergravity. Ann. Phys.
**144**, 249 (1982)ADSMathSciNetCrossRefGoogle Scholar - 43.P. Breitenlohner, D.Z. Freedman, Positive energy in anti-De Sitter backgrounds and gauged extended supergravity. Phys. Lett. B
**115**, 197 (1982)ADSMathSciNetCrossRefGoogle Scholar - 44.G.T. Horowitz, M.M. Roberts, Holographic superconductors with various condensates. Phys. Rev. D
**78**, 126008 (2008)ADSCrossRefGoogle Scholar - 45.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Building a holographic superconductor. Phys. Rev. Lett.
**101**, 031601 (2008)ADSCrossRefGoogle Scholar - 46.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Holographic superconductors. J. High Energy Phys.
**12**, 015 (2008)ADSMathSciNetCrossRefGoogle Scholar - 47.Q. Pan, B. Wang, E. Papantonopoulos, J. de Oliveira, A.B. Pavan, Holographic superconductors with various condensates in Einstein–Gauss–Bonnet gravity. Phys. Rev. D
**81**, 106007 (2010)ADSCrossRefGoogle Scholar - 48.Y. Peng, Y. Liu, A general holographic metal/superconductor phase transition mode. JHEP
**02**, 082 (2015)ADSCrossRefGoogle Scholar - 49.W. Yao, J. Jing, Holographic entanglement entropy in metal/superconductor phase transition with Born–Infeld electrodynamics. Nucl. Phys. B
**889**, 109 (2014)ADSMathSciNetCrossRefGoogle Scholar

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