Holographic model for ferromagnetic phase transition in the Lifshitz black hole with the nonlinear electrodynamics

Abstract We numerically investigate the holographic paramagnetism–ferromagnetism phase transition in the 4-dimensional Lifshitz spacetime in the presence of three kinds of typical Born–Infeld-like nonlinear electrodynamics. Concretely, in the probe limit, we thoroughly discuss the effects of the nonlinear parameter b and the dynamical exponent z on the critical temperature, magnetic moment and hysteresis loop. The results show that the exponential form of nonlinear electrodynamics correction makes the critical temperature smaller and the magnetic moment harder to form with the absent external field for a constant nonlinear parameter b comparing it with the logarithmic form of nonlinear electrodynamics and the Born–Infeld nonlinear electrodynamics, especially for the case of larger dynamical exponent z. Moreover, the increase of nonlinear parameter b (for the fixed z) or dynamical exponent z (for the fixed b) will result in extending the period of the external magnetic field. Particularly, the effect of the exponential form of nonlinear electrodynamics on the periodicity of hysteresis loop is more noteworthy.


Introduction
It is well known that weakly coupled superconductors can be described by the BCS theory of superconductor with great accuracy. However, progress in the modern condensed matter physics in the last few decades has made it clear that this microscopic theory fails in understanding the pairing mechanism in materials (like high-temperature cuprates) which are strongly coupled. Hence it forces people to find other alternative theory which is provided by the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence [1][2][3] to address issues in strongly interacting system. The gauge/gravity duality [1,3] shows that a (d+1)-dimensional weak gravity system corresponds to a conformal field theory sitting on the d-dimensional boundary of this spacetime. The authors of Ref. [4] introduced the first holographic s-wave superconductor can indeed be reproduced in the 4-dimensional Schwarzschild anti-de Sitter black hole coupled to the Maxwell complex scalar field in this simple model. And following this, a variety of the holographic dual models have been shown in Refs. [5][6][7][8][9][10][11][12][13][14][15][16].
Note that all the above mentioned gravity duals for the holographic superconductors have been carried out in the framework * Corresponding author. of conventional Maxwell electrodynamics related to the Einstein-Maxwell gravity in the bulk. It is naturally interesting to investigate the possibility of describing such holographic superconductor in a nonlinear scenario [17][18][19][20][21]. In this context, one of the most important nonlinear electromagnetic theories is the Born-Infeld electrodynamics [22] that was proposed to avoid the infinite self energy arising in the Maxwell theory. And the Born-Infeld theory remains invariant under electromagnetic duality. Therefore, considering Born-Infeld electrodynamics, Jing and Chen firstly introduced holographic dual model in Born-Infeld electrodynamics and observed that the nonlinear Born-Infeld parameter makes it harder for scalar condensation to form [23], decreases the critical temperature, and changes the condensation gap. Moreover, the dependence of the condensation gap and the critical temperature on the Born-Infeld scale parameter is similar to that on the Gauss-Bonnet term (it corresponds to the higher derivative correction to gravitation) in the holographic superconductor. Along this line, there has been accumulated interest to investigate various holographic dual models with the nonlinear correction [24][25][26][27][28][29][30][31][32]. In order to back up numerical calculations, Siopsis et al. improved the variational method of Sturm-Liouville (S-L) eigenvalue problem to analytically calculate the critical temperature and found that the analytical results [33][34][35][36][37] are in good agreement with the numerical findings. And then this method is generalized to study the holographic insulator/superconductor phase transition [38] types of nonlinear electrodynamics in the context of gravitational field have been introduced that are similar to the case of Born-Infeld nonlinear correction. Two well-known nonlinear Lagrangians for electrodynamics are logarithmic (LEN) [39,40] and exponential (ENE) [41,43,44] Lagrangian. By considering three types of typical nonlinear electrodynamics, the authors of Ref. [45] observed that the exponential form of nonlinear electrodynamics has stronger effect on the condensation formation and conductivity for the holographic conductors in the backgrounds of AdS black hole. However, in the AdS soliton background the critical chemical potentials are independent of the explicit form of the nonlinear electrodynamics, i.e., the ENE, BINE and LNE corrections do not effect on the critical potentials.
Recently, some efforts have been made to generalize the AdS/CFT correspondence to magnetism. The authors of Ref. [46] realized the holographic description of the paramagnetism-ferromagnetism phase transition in a dyonic Reissner-Nordström-AdS black brane. In that model, the magnetic moment is realized by condensation of a real antisymmetric tensor field which couples to the background gauge field strength in the bulk. In the case without external magnetic field, the time reversal symmetry is spontaneously broken and the spontaneous magnetization happens in low temperatures. The critical exponents are in agreement with the ones from mean field theory. In the case of nonzero magnetic field, the model realizes the hysteresis loop of single magnetic domain and the magnetic susceptibility satisfies the Curie-Weiss law. Obviously, this model in Ref. [46] gives a good starting point to explore more complicated magnetic phenomena and quantum phase transitions. Since then, a large number of the holographic dual models have been constructed and some interesting behaviors have been found, for review, see Refs. [47][48][49][50][51][52][53][54] and references therein.
So based on the research about holographic superconductor with the nonlinear electrodynamics, we have studied the effects of BI coupling parameter and the other two types of nonlinear electrodynamics on the paramagnetism-ferromagnetism phase transition [55,56] in the background of Schwarzschild black hole. Thus, we wonder how other backgrounds influence the paramagnetismferromagnetism phase transition, especially in the nonrelativistic spacetimes, for example, the Lifshitz spacetime, which is our motivation in this letter.
The structure of this work is as follows. In section 2, we introduce the basic field equations of holographic ferromagnetism model with ENE and LEN in the Lifshitz black hole which have not been studied as far as we know, and compare them with the BINE holographic paramagnetism model. In section 3 by the numerical method we obtain the critical temperature and study the magnetic moment in the presence of the three kinds of typical nonlinear electrodynamics and dynamical exponent. Magnetic susceptibility density and hysteresis loop will be shown in section 4. Finally the summary and some discussions are included in the last section.

Holographic model
In this letter, we consider the Lagrangian density consisting of a U(1) field A μ and a massive 2-form field M μν in (d + 2)dimensional spacetime as where dM is the exterior differential of 2-form field M μν , m 2 is the squared mass of 2-form field M μν being greater than zero (see Ref. [50] for detail), λ and J are two real model parameters with J < 0 for producing the spontaneous magnetization, λ 2 characterizes the back reaction of the 2-form field M μν to the background geometry and to the Maxwell field strength, and V (M) is a nonlinear potential of the 2-form field describing the self-interaction of the polarization tensor. In order to numerical calculation simplicity, we take the form of V (M) as follows, where * is the Hodge-star operator. As shown in Ref. [50], this potential shows a global minimum at some nonzero value of ρ.
Meanwhile, L(F ) is the Lagrangian of three classes of Born-Infeldlike nonlinear electrodynamics, i.e., Here F ≡ F μν F μν and F μν is the nonlinear electromagnetic tensor. The Lagrangian L(F ) approaches to F μν F μν when the nonlinear parameter b tends to zero. Note that the higher order terms in the parameter b essentially correspond to the higher derivative corrections of the Abelian gauge fields and carry more plentiful information. With the same value of b, we can discuss the differences among the three types of the holographic dual models with the nonlinear electrodynamics quantitatively. It should be noted that the horizon geometry of nonlinear charged black holes is closed to the horizon of uncharged (Schwarzschild) black hole solution for very large values of b [41], so in this case L(F ) can be neglected. By varying action (1), we can get the equations of motion for 2-form field and gauge field for the three cases In what follows, we will work in the probe limit and consider the background being a D = d + 2 dimensional Lifshitz black hole with finite temperature [42] where is the cosmological constant, ϕ is a massless scalar and B rt is an abelian gauge field strength. And z is the Lifshitz dynamical exponent representing the anisotropy of the spacetime. Evidently, choosing the dynamical exponent z to be one reduces the In order to study systematically the effects of the b on the holographic paramagnetism-ferromagnetism phase transition, we just consider the case of d = 4 and z = d. Meanwhile we take the following self-consistent ansatz with matter fields, here B is a constant magnetic field viewed as the external magnetic field in the boundary field theory. Thus nontrivial equations of motion read, which are the same form for the three types of nonlinear electrodynamics (5). For the gauge field φ, however, we obtain the following equations of motion here a prime denotes the derivative with respect to r. Obviously, Eqs. (11) and (12) reduce to the standard holographic paramagnetism-ferromagnetism phase transition models discussed in Ref. [50] when b → 0 and z = 1. In order to solve the nonlinear Eqs. (11) and (12) numerically, we should first solve the equation of φ and put it into Eq. (12) and get the equation of p. And then we need to seek the boundary condition for ρ, φ and p near the black hole horizon r → r + and at the spatial infinite r → ∞.
The regularity condition for ρ(r + ) at the horizon gives the boundary condition φ(r + ) = 0. Near the boundary r → ∞, the nonlinear equations give the following asymptotic solution for matter fields with ± = where ρ ± , μ and σ are all constants, and μ and σ are interpreted as the chemical potential and charge density in the dual field theory respectively. The coefficients ρ + and ρ − correspond to the source and vacuum expectation value of dual operator in the boundary field theory when B = 0. Therefore one should set ρ + = 0 since one wants the condensation to happen spontaneously below a critical temperature. When B = 0, the asymptotic behavior will be governed by external magnetic field B.

Spontaneous magnetization
In this letter we work in the grand canonical ensemble where the chemical potential μ will be fixed. And the expression of magnetic moment in the background of Lifshitz black hole will be changed as [54] Here, we take J = −1/8, m 2 = 1/8 and λ = 1/2 as a typical example, which can capture the basic features of the model. In other words, the other choices of the parameters will not qualitatively modify our results. Using the shooting method, we can solve Eqs. (11) and (12) numerically and then discuss the effects of the nonlinear electrodynamics and the dynamical exponent z on the magnetic moment. Here, note that the system has the following scaling symmetry: Thus once obtaining the solutions of Eqs. (11) and (12), we can use the scaling symmetry (16) to find out the corresponding solutions with the same chemical potential.
In Fig. 1 we present the magnetic moment with the LNE (left two panels), BINE (middle two panels) and ENE (right two panels) as a function of temperature for the case of d = 4 by varying the nonlinear parameter b. And the upper half plane and the lower half plane of Fig. 1 correspond to the choice of z = 1 and z = 3/2, respectively. It is found that the spontaneous condensate of ρ (corresponding to the magnetic moment) in the bulk in the absence of external magnetic field appears and has similar behaviors for different b and z when the temperature is lower than critical temperature T c . Meanwhile, by fitting this curve in the vicinity of critical temperature, we find that the phase transition is a second order one with behavior N ∝ √ 1 − T /T c for all cases calculated above.
The results are still consistent with ones in the mean field theory and have been shown in Table II. In other words, considering the three kinds of the Born-Infeld-like nonlinear electrodynamics, the holographic paramagnetism-ferromagnetism transition still exists in Lifshitz black hole spacetime. From the right two panels of Fig. 1, we observe that the increasing value of the nonlinear parameter b makes the magnetic moment smaller with the ENE for the fixed values of z, which is similar to the cases of BINE and LNE. It means that the magnetic moment is harder to form in the nonlinear electrodynamics, which agrees well with the results given in [55]. In Table I and Table II, we present the critical temperature T c and the behavior of these condensation curves near T ∼ T c . It is easy to find that as b increases the critical temperature decreases for each nonlinear electrodynamic with z = 1 or z = 3/2, which are exhibited in Fig. 2 and agree well with the finding in the Fig. 1. This behavior has been seen for the holographic superconductor in the background   [45]. Similarly, for a constant value of b and comparing the situation of z = 1 with z = 3/2, we see that the critical temperature T c will decrease with the increasing z, and the magnetic moment will become smaller for each nonlinear electrodynamic. At the same time, the dependence of the magnetic moment and the critical temperature on the nonlinear parameter is similar to that on the Gauss-Bonnet term in the holographic superconductor, i.e., the higher curvature corrections make condensation harder to form. Therefore, we conclude that the ENE, BINE and LNE corrections to usual Maxwell field and the curvature corrections share some similar features for the condensation of the massive 2-form field ρ.
On the other hand, comparing with the curves for the magnetic moment in the three types of the nonlinear electrodynamics considered here, we find that the value of magnetic moment with ENE is smaller than ones in the BINE and LNE cases for the fixed value of nonlinear parameter b (except the case of b = 0, i.e., the conventional Maxwell electrodynamics) and dynamical exponent z, which means that the magnetic moment is more difficult to develop in the exponential form of nonlinear electrodynamics. This is also in good agreement with the results shown in Table I and in Fig. 2, where the critical temperature T c for the condensate of ρ with the ENE is smaller than ones in the BINE and LNE cases for the fixed value of b, especially for the case of z = 3/2.

The response to the external magnetic field
Let us turn on the external field to examine the response to magnetic moment N. This can be described by magnetic susceptibility density χ , defined as In the high temperature region T > T c , the ferromagnetic material is in a paramagnetic phase whose magnetic moments are The upper half plane: z = 1. The lower half plane: z = 3/2. randomly distributed. So the susceptibility obeys the Curie-Weiss law where C and θ are two constants. And following Eqs. (16), the magnetic susceptibility χ satisfies the scaling transition χ → a z−2 χ . Note that a significant difference between the antiferromagnetism and paramagnetism can be seen from the magnetic susceptibility. In the paramagnetic phase of antiferromagnetic material and paramagnetic material, the magnetic susceptibility also obeys the Curie-Weiss law, but the constant θ in Eq. (18) is positive and zero, respectively. For the three types of nonlinear electrodynamics, Fig. 3 shows the magnetic susceptibility as a function of temperature by solving Eq. (17) with b = 0, 0.05, 0.08 and z = 3/2. In the paramagnetic phase for all cases considered here, we observe that the magnetic susceptibility increases when the temperature is lowered for the fixed nonlinearly parameter b. Moreover, the magnetic susceptibility satisfies the Curie-Weiss law of the ferromagnetism near the critical temperature whether b = 0 or not, at the same time, it has nothing to do with dynamical exponent z qualitatively. Concretely, the results have been presented in Table III and Table IV for the chosen model parameters (i.e., b = 0, 0.05, 0.08, and z = 1, 6/5, 3/2). We learn from the Fig. 3 and Table III that coefficient in front of T T c for 1 χ increases with the increasing b for the case of z = 1 but it decreases for the case of z > 1. If we take a fixed coupling parameter b, for example, b = 0.05, it also decreases with the increasing z. Meanwhile, we see that the absolute value of θ μ will decrease when the nonlinear parameter b increases for fixed z or as the dynamical exponent z increases for fixed b.
On the other hand, from Fig. 3 and Table III we can see the value of coefficient in front of T T c for 1 χ of the ENE is larger than that of BINE and LNE for the fixed value of b (except the case of b = 0, i.e., the usual Maxwell electrodynamics), especially for the case of z = 3/2. Comparing the cases of BINE and LNE, however, the absolute value of θ μ for ENE is smaller. In the plot of Fig. 4, we show the magnetic moment with respect to external field B in region of T < T c (i.e., T = 0.89T c ) with different parameter b.
And from each line in Fig. 4, we see that the magnetic moment is not single valued when the external magnetic field continuously changes between −B max and B max periodically. Thus a hysteresis loop in the single magnetic domain will be obtained and the nonlinear parameter b has an effect on it quantitatively. Along the horizontal direction (the magnetic moment has been taken a same value), one needs a larger external field as the nonlinear parameter b increases. In other words, the nonlinear electrodynamics makes the periodicity of hysteresis loop bigger which is different from the effect of Lifshitz dynamical exponent z on it whether in the presence of coupling parameter b [54] or not. Particularly, for the case of ENE, whose effect on the periodicity of hysteresis loop is more noteworthy. However, all the curves will overlap once the value of the magnetic field exceeds the maximum corresponding to the case of b = 0.14, which can be seen from Fig. 4.

Summary and discussion
In this letter, we have systematically investigated holographic paramagnetism-ferromagnetism phase transition in 4-dimension Lifshitz black hole spacetime in the presence of the three kinds of typical Born-Infeld-like nonlinear electrodynamics correction to the Maxwell electrodynamics, and obtained the effects of the nonlinear parameter b and dynamical exponent z on the holographic paramagnetism-ferromagnetism phase transition. Considering that   black hole background, we found it has stronger effects on critical temperature T c and the magnetic moment N, especially for the case of larger z. In the vicinity of the critical point, however, the behavior of the magnetic moment is always as (1 − T /T c ) 1/2 , which is independent of the explicit form of the nonlinear electrodynamics and the anisotropy of spacetime, i.e., the ENE, BINE, LNE correction terms and dynamical exponent do not have any effect on the relationship. Meanwhile, it is in agreement with the result from mean field theory. Moreover, in the presence of the external magnetic field, the inverse magnetic susceptibility as T ∼ T c behaves as C /(T + θ), (θ < 0) in all cases, which satisfies the Cure-Weiss law. Yet both the constant C for the case of z = 1 and the absolute value of θ for all cases of z decrease with the increasing nonlinear parameter b, and that the value of C will increase as z > 1. Furthermore, we have observed the hysteresis loop in the single magnetic domain when the external field continuously changes between the maximum and minimum values periodically with b. The increase of the nonlinear parameter b could result in extending the period of the external magnetic field. Especially, the effects of the exponential form of nonlinear electrodynamics and the larger dynamical exponent on the periodicity of hysteresis loop are more noteworthy. The reason for this phenomenon maybe attributed to the three forms of nonlinear electrodynamics. If we fix the gauge field F , the exponential form of nonlinear electrodynamics L(F ) change the fastest compared with the other two forms of nonlinear electrodynamics. In other words, the electrostatic repulsion for the case of exponential form of nonlinear electrodynamics is easier to overcome the gravitational attraction for the formation of the 2-form field. Hence, the condensation of 2-form field is harder to form. In this way, when we take the same value of nonlinear parameter b (except for the case of b 2 = 0), the exponential form of nonlinear electrodynamics makes the critical temperature T c and the magnetic moment N more smaller. It also illustrates that the exponential form of nonlinear electrodynamics makes the phase transition difficult to happen. That is to say, the magnetic moment N is not more likely to arrive at a certain value with the effect of exponential form of nonlinear electrodynamics. Therefore, one must need a bigger external magnetic field to realize the hysteresis loop, which is in accordance with the results in Fig. 4. Note that in this letter we just investigate the influences of the three kinds of nonlinear electrodynamics on paramagnetismferromagnetism phase transition in Lifshitz black hole. It would be of interest to generalize our study to holographic paramagnetismantiferromagnetism model and holographic paramagnetism phase transition with the other corrections, such as the Power-Maxwell correction and weyl correction. Work in this direction will be reported in the future.