Holographic entanglement entropy in metal/superconductor phase transition with Born-Infeld electrodynamics

We investigate the holographic entanglement entropy in the metal/superconductor phase transition for the Born-Infeld electrodynamics with full backreaction and note that the entropy is a good probe to study the properties of the phase transition. For the operator $<\mathcal{O}_{-}>$, we find that the entanglement entropy decreases (or increases) with the increase of the Born-Infeld parameter $b$ in the metal (or superconducting) phase. For the operator $<\mathcal{O}_{+}>$, we observe that, with the increase of the Born-Infeld parameter, the entanglement entropy in the metal phase decreases monotonously but the entropy in the superconducting phase first increases and forms a peak at some threshold $b_{T}$, then decreases continuously. Moreover, the value of $b_{T}$ becomes smaller as the width of the subsystem $A$ decreases.


Introduction
As the holographic principle provides a new insight into the investigation of strongly interacting condensed matter systems [1,2], there are a lot of works applying the anti-de Sitter/conformal field theory (AdS/CFT) duality [3][4][5] to condensed matter physics and in particular to superconductivity [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. It states that the bulk AdS black hole becomes unstable and scalar hair condenses as one tunes the temperature for black hole. As a matter of fact, in order to understand the influences of the 1/N or 1/λ (λ is the 't Hooft coupling) corrections on the holographic superconductors, the higher derivative corrections to the gauge field should be taken into consideration. For the high order correction related to the gauge field, one of the important nonlinear electromagnetic theories is Born-Infeld electrodynamics [23][24][25][26][27][28]. As is well known, the Born-Infeld electrodynamics, which was proposed in 1934 by Born and Infeld to avoid the infinite self-energies for charged point particles arising in Maxwell theory [23], displays good physical properties including the absence of shock waves and birefringence. It was also found that the Born-Infeld electrodynamics is the only possible nonlinear version of electrodynamics that is invariant under electromagnetic duality transformation [24]. Jing and Chen observed that the Born-Infeld coupling parameter makes it harder for the scalar condensation to form [29]. Then, the analytic study of properties of holographic superconductors in Born-Infeld electrodynamics was presented in Ref. [30]. In this paper, we would like to investigate the phase transition in the Born-Infeld electrodynamics with full backreaction of the matter fields electrodynamics on the AdS black hole geometry.
Recently, according to the AdS/CFT duality, Ryu and Takayanagi [31,32] have presented a proposal to compute the entanglement entropy of conformal field theories (CFTs) from the minimal area surface in gravity side. Since this proposal provides a simple and elegant way to calculate the entanglement entropy of a strongly coupled system which has a gravity dual, the holographic entanglement entropy is widely used to study various properties of holographic superconductors at low temperatures [33][34][35][36][37][38][39][40][41]. The entanglement entropy in the metal/superconductor system was studied in Ref. [42]. It was shown that the entanglement entropy in superconducting phase is always less than the one in the metal case and the entropy as a function of temperature is found to have a discontinuous slop at the transition temperature T c in the case of the second order phase transition. Ref. [43] considered the case with higher derivative corrections and studied the holographic entanglement entropy in Gauss-Bonnet gravity. Ref. [44] studied the holographic entanglement entropy for general higher derivative gravity and proposed a general formula for calculating the entanglement entropy in theories dual to higher derivative gravity. Then, the holographic entanglement entropy in the insulator/superconductor model was studied in Refs. [45][46][47][48] and it turned out that the entanglement entropy is a good probe to investigate the holographic phase transition. Furthermore, Kuang et al. examined the properties of the entanglement entropy in the four-dimensional AdS black hole and found that the entanglement entropy can be considered as a probe of the proximity effect of a superconducting system by using the gauge/gravity duality in a fully backreacted gravity system [49]. More recently, the entanglement entropy of general Stückelberg models both in AdS soliton and AdS black hole backgrounds with full backreaction was studied in Ref. [50]. However, the models mentioned above are all in the frame of Maxwell electromagnetic theory. It is of interest to investigate holographic entanglement entropy in the nonlinear electromagnetic generalization. We have studied in Ref. [51] the holographic entanglement entropy in the insulator/superconductor phase transition for the Born-Infeld electrodynamics, and found that the entanglement entropy increases with the increase of the Born-Infeld factor in the superconductor phase and the critical width of confinement/deconfinement phase transition is dependent of the Born-Infeld parameter. As a further study along this line, in this paper we will extend the previous works to investigate the entanglement entropy in metal/superconductor phase transition with Born-Infeld electrodynamics. We find that the entanglement entropy decreases with the increase of the Born-Infeld parameter b for both operators in the metal phase. However, In the superconducting phase, the entanglement entropy of operator O − increases with the increase of Born-Infeld factor. Interestingly, for the operator O + , the effect of the Born-Infeld parameter on the entanglement entropy is non-monotonic and the value of the threshold b T is related to the belt width of subsystem A.
The framework of this paper is as follows. In Section 2, we will derive the equations of motions and introduce the boundary conditions of the holographic model. In Section 3, we study the phase transition with Born-Infeld electrodynamics in AdS black hole spacetime. In Section 4, we calculate the holographic entanglement entropy in AdS black hole gravity with Born-Infeld electrodynamics. In Section 5, we conclude our main results of this paper.

Equations of motion and boundary conditions
The action for the gravity and Born-Infeld electromagnetic field coupling with a charged scalar field is described by where g is the determinant of the metric, ψ represents a scalar field with charge q and mass m, A is the gauge field, F μν is the strength of the Born-Infeld electrodynamic field F = dA, and b is the Born-Infeld coupling parameter.
In the limit b → 0, the Born-Infeld field will reduce to the Maxwell field.
To study the holographic entanglement entropy in Born-Infeld electrodynamics, we will take the full backreaction into account. The metric for the planar black hole can be taken as The Hawking temperature of this black hole is where r + is the black hole horizon. We consider the electromagnetic field and the scalar field in the forms Then, the equations of motion from the variation of the action with respect to the matter and metric can be obtained as where a prime denotes the derivative with respect to r, and 16πG = 1 was used. For this system, the useful scaling symmetries are Using the scaling symmetries (9), we can take r + = 1. Further employing the symmetries (10), we can let L = 1.
At the horizon r + , the regularity condition gives the boundary conditions And at the asymptotic AdS boundary (r → ∞), the asymptotic behaviors of the solutions are where μ and ρ are interpreted as the chemical potential and charge density in the dual field theory, and the exponent ± is defined by Notice that, provided − is larger than the unitarity bound, both ψ − and ψ + can be normalizable. According to the AdS/CFT correspondence, they correspond to the vacuum expectation values ψ − = O − , ψ + = O + of an operator O dual to the scalar field [52,53]. In the following calculation, we impose boundary condition that either ψ − or ψ + vanishes.

Phase transition with Born-Infeld electrodynamics
In this section, we will concretely study the phase transition for Born-Infeld electrodynamics with full backreaction in the 4-dimensional AdS black hole spacetime. For the normal phase, the metric becomes the Reissner-Nordström AdS black hole as the Born-Infeld factor approaches to zero. Thus, we have However, if the Born-Infeld factor is not equal to zero, the solution is the Born-Infeld AdS black hole.
For purpose of getting the solutions in superconducting phase where ψ(r) = 0, we can introduce a new variable z = r + /r. Then, the equations of motion can be rewritten as where the prime now denotes the derivative with respect to z. Using the shooting method, we can solve the equations of motion numerically and then discuss the effects of the Born-Infeld parameter b on the condensation of the scalar operators. In this paper, we set m 2 = −2 and q = 1. Since there are scaling symmetries described by Eq. (9) for the equations of motion, the following quantities can be rescaled as In Fig. 1, we plot the behaviors of condensate with the changes of the temperature and the Born-Infeld parameter in the dimensionless quantities O − / √ ρ, √ O + /ρ and T / √ ρ. From the left plot of Fig. 1, we can see that if the temperature T > T c , there is no condensation and this can be thought of as the metal phase. However, when the temperature decreases to be lower than the critical value T c , the condensation of the operator O − emerges and this corresponds to a superconducting phase. It should be noted that the value of the critical temperature T c becomes smaller as the Born-Infeld factor b increases, which means that the Born-Infeld correction to the usual Maxwell field makes the scalar hair harder to form in the full-backreaction model. For the operator O + (right plot), we also find that the critical temperature T c decreases with the increase of the parameter b. Our result is the same as the result in Ref. [53] as the factor b approaches to zero.

Holographic entanglement entropy of the holographic model
After obtaining the solutions to the metal phase and superconducting phase for the Born-Infeld electrodynamics in the AdS black hole geometry with full backreaction, we are now ready to study the behavior of holographic entanglement entropy in the Born-Infeld electrodynamics.
The entanglement entropy in conformal field theories can be calculated from the area of minimal surface in AdS spaces [31,32], and its formula is given by the "area law" where G N is the Newton constant in the Einstein gravity on the AdS space, S A is the entanglement entropy for the subsystem A which can be chosen arbitrarily, γ A is the minimal area surface in the bulk which ends on the boundary of A. We consider the entanglement entropy for a straight geometry which is described by − l 2 ≤ x ≤ l 2 and − R 2 < y < R 2 (R → ∞), where is defined as the size of region A. The holographic surface γ A starts from r = 1 at x = 2 , extends into the bulk until it reaches r = r * , then returns back to the AdS boundary r = 1 at x = − 2 . Thus, the induced metric on γ A can be obtained as follows By using the proposal given by Eq. (20), the entanglement entropy in the strip geometry is with where z * satisfies the condition dz dx | z * = 0 with z = 1 r . The term 1/ in Eq. (22) is divergent, while the term s is a finite term which is physically important. Under the scaling symmetries of Eq. (9), we can rescale the and s as Therefore, in the following calculation we can use these dimensionless quantities We now show the behavior of the holographic entanglement entropy of the operator O − or O + with respect to the temperature T , the Born-Infeld factor b and the belt width , respectively.

Holographic entanglement entropy of the operator O −
The behavior of the entanglement entropy of the operator O − is shown in Fig. 2 in which the dot-dashed lines describe the normal phases and the solid ones show the superconducting phases. It can be seen from the figure that the entanglement entropy presents a discontinuous change at a critical temperatures T c denoted by vertical dashed lines for different strengths of the Born-Infeld parameter b. The discontinuous change of the entanglement entropy indicates the phase transition point from the normal state to the superconducting state and the value of the T c becomes smaller with the increase of the Born-Infeld factor b. Which indicates that the holographic entanglement entropy is a good probe to study the properties of the phase transition. The figure also shows that the entanglement entropy decreases as the Born-Infeld parameter increases for the normal phase, but increases as the Born-Infeld parameter increases for the superconducting state. Moreover, the entanglement entropy in superconducting phase is less than the one in the normal case and drops as the temperature decreases, and this property holds for different values of the parameter b. This behavior of the entanglement entropy is due to the fact that the metal phase can be thought of as the one filled with free charge carriers, such as electrons. The condensate turns on at the critical temperature and the free charge carriers are continuously condensed to Cooper pairs as temperature decreases. Therefore, the formation of Cooper pairs make the degrees of freedom decrease in the superconducting phase. The entanglement entropy as a function of the parameter b for different in the superconducting phase is shown in Fig. 3. For fixed belt width , the entanglement entropy becomes smaller as the Born-Infeld factor b decreases. This is because that the condensation becomes stronger with higher condensation gap for smaller parameter b at low temperature so that the number of Cooper pairs is increased, which results in less degree of freedom. On the other hand, for a given Born-Infeld factor b, with the decrease of belt width the entanglement entropy also decreases.

Holographic entanglement entropy of the operator O +
The behavior of the entanglement entropy of the operator O + as a function of temperature T and the Born-Infeld factor b is described by Fig. 4. It is shown that the critical temperature T c of the phase transition decreases as the Born-Infeld factor b increases. That is to say, the stronger Born-Infeld electrodynamics correction makes the scalar condensation harder to form. We also find that the change of the entanglement entropy is discontinuous at T c and the entanglement entropy in the hair phase is less than the one in the normal phase. Interestingly, in the superconducting phase, the dependence of the entanglement entropy on the Born-Infeld factor b  is non-monotonic. In the left plot of Fig. 4, we find that the entanglement entropy increases with the increase of the factor b when b < b T . However, in the right plot of Fig. 4, we can see that the entanglement entropy decreases with the increase of the factor b when b > b T .
To further illustrate the effect of the Born-Infeld factor b on the entanglement entropy of the operator O + in the superconducting phase, we plot the entanglement entropy of the operator O + as a function of Born-Infeld factor b for different widths with T / √ ρ = 0.010 in Fig. 5. Obviously, with the increase of the factor b, the entanglement entropy first rises and arrives at its maximum as b = b T , then decreases monotonously. This process implies that there is some kind

Summary
We studied the behaviors of the holographic entanglement entropy in the metal/superconductor phase transition for the Born-Infeld electrodynamics with full backreaction. By calculating the entanglement entropy of the system, we noted that the critical temperature of the condensation for the operators O − and O + becomes smaller with the increase of the Born-Infeld parameter b, which implies that the Born-Infeld factor makes the scalar condensation harder to form. Both for the operators O − and O + , we found that the entanglement entropy in the superconducting phase is less than the one in the normal phase, and drops as the temperature decreases for the fixed parameter b and belt width . This is due to the fact that the formation of Cooper pairs makes the degrees of freedom decrease in the hair phase. For a given temperature, we observed that the entanglement entropy of the operator O − in the metal (or superconducting) phase decreases (or increases) with the increase of the Born-Infeld factor b for the fixed . Interestingly, the influence of the Born-Infeld factor b on the entanglement entropy of the operator O + is nontrivial. The entanglement entropy of the operator O + in the metal phase decreases with the increase of the Born-Infeld parameter. However, with the increase of the Born-Infeld parameter b for the fixed belt width of subsystem A, the entanglement entropy of the operator O + in the superconducting phase first increases and reaches the maximum at some threshold b T , then decreases monotonously. The threshold b T becomes smaller as the width of the subsystem A decreases. This process implies that there is some kind of the significant reorganization of the degrees of freedom which should be further studied in the future.