Holographic ferromagnetism model with the Maxwell field strength corrections

Article history: Received 4 January 2019 Received in revised form 8 March 2019 Accepted 13 March 2019 Available online 19 March 2019 Editor: M. Cvetič


Introduction
There has been an immense amount of interest in studying strongly coupled system using one of the most fascinating developments in modern theoretical physics, the AdS (anti-de Sitter Spacetime)/CFT (Conformal Field Theory) correspondence [1][2][3][4]. The correspondence gives an exact duality between a gravity theory living in a (d+1)-dimensional AdS spacetime and a field theory having conformal invariance sitting on the d-dimensional boundary of this spacetime. It has been extensively applied in order to study various phenomena in condensed matter physics.
One important application of the duality is to study high-Tc superconductivity. For example, the holographic s-wave superconductor model known as Abelian-Higgs model was first implemented [5], in which the formation of a scalar field condensate near the horizon of a black hole is possible below a certain temperature for a charged AdS black hole coupled minimally to a complex scalar field. The condensation of the scalar breaks the U(1) symmetry of the system, mimicking the conductor-superconductor phase transition. Following [5], an overwhelming numbers of papers have appeared which try to investigate various properties of the holographic superconductors from different perspectives . Of course, except for the above mentioned models for the holographic superconductors in the frame of Maxwell electromagnetic theory, people also investigate holographic superconductors in the nonlinear electromagnetic generalization [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].
The other application of duality is to study ferromagnetism where the electron spins align to produce a magnetization, which breaks the time reversal symmetry spontaneously and happens in the ferromagnets at the Curie temperature T c (sometimes, it is even higher than the indoor temperature). The authors of Ref. [48] introduced the holographic paramagnetism-ferromagnetism model in a dyonic Reissner-Nordstrom-AdS black brane and observed that the properties of a (2+1)-dimensional ferromagnetism can indeed be realized in this simple model. However, there is a vector ghost in the above model by a more careful analysis. Hence, in Ref. [49], a modified Lagrangian density was put forward, which is ghost free and in which causality is well defined, and keeps all the significant results in the original model qualitatively. On the basis of Ref. [48], the model was further extended to realize a holographic model of paramagnetism/antiferromagnetism phase transition [50], a holographic insulator/metal phase transition [51] and the coexistence and competition of ferromagnetism and p-wave superconductivity [52]. Furthermore, other studies based on this model can be discovered, for example, the effect of backreaction on the holographic paramagnetism/ferromagnetism phase transition [53]. At present, the holographic duality has been applied to two dimensional magnetic systems [48,53,54] and three-dimensional magnetic systems [55]. Especially for the case of latter, it can describe the magnetic behaviors in low temperatures where the technology of spintronics is actively developed. in principle can provide a means to analyze phenomena involving magnetization and spin transport, and thus it can introduce new perspectives in the field of spintronics. Of course, except for the case of holographic magnetic models in the relativistic spacetimes, we investigate holographic paramagnetism-ferromagnetism phase transition in different dimensional Lifshitz black holes by means of numerical and semianalytical methods [56]. But as discussed in the holographic superconductors, the higher-order correction term appears in the Maxwell gauge field [29,57]. Thus, along this direction, in this paper, we will focus on the influence of the correction to the gauge field on the holographic ferromagnetism phase transition.
This paper is outlined as follows: in the next section, we introduce the basic field equations of holographic paramagnetism model with the higher correction to the gauge field in the background of four-dimensional AdS Schwarzschild black hole. In section 3 we obtain the critical temperature and study the magnetic moment by numerical shooting method. Magnetic susceptibility density and hysteresis loop will be shown in section 4. The last section is devoted to concluding remarks and some discussions.

Holographic model
In this paper, in order to grasp the influence of the correction to the gauge field on holographic model, we will turn off the curvature correction and study in a pure Einstein gravity background for simplicity. Following Ref. [49], below, we consider the Lagrangian density consisting of a U(1) field A μ and a massive 2-form field M μν in 4-dimensional spacetime. The ghost free action 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. [49] for details), λ and J are two real model parameters with J < 0 for producing the spontaneous magnetization, λ 2 character- where * is the Hodge-star operator. As shown in Ref. [49], this potential shows a global minimum at some nonzero value of ρ. L F 4 is determined by a quartic field strength correction to the usual Einstein-Maxwell field [58][59][60][61][62] with the real numbers c 1 and c 2 . When c 1 and c 2 are zero, it reduces to the model considered in [49]. Interestingly, just as is shown in the following discussion, the constant can be relaxed to the case of 2c 1 + c 2 = 0, where the model (1) reduces to the standard holographic paramagnetism/ferromagnetism phase transition [49]. By varying action (1), we can get the equations of motion (EoMs) for the matter fields as In what follows, we will work in the probe limit (i.e., neglecting the backreactions of the missive 2-form field to the background and Maxwell field, also including the Maxwell field to the background geometry. In this probe approximation, the interaction between the electromagnetic response and external field is taken into account so that in the future we can study how spontaneous magnetization influences the electric transport, but they both have little influence on the structures of materials) and the background is a 4-dimensional planner Schwarzschild-AdS black hole where the r + is the event horizon of the black hole and the Hawking temperature is In order to study systematically the effects of c 1 and c 2 on the holographic ferromagnetism model, we take the following self -consistent ansatz with matter fields, where B is a constant magnetic field viewed as the external magnetic field in the boundary field theory. Thus nontrivial equations of motion read, (1 here a prime denotes the derivative with respect to r. Obviously, the equation of motion for gauge field φ(r) is more complicated than that in usual Maxwell theory due to the presence of L F 4 correction (i.e., the influence of parameters c 1 and c 2 ). But it just corresponds to the case in the probe limit. If we consider the backreaction with F 4 corrections, we will obtain five equations, including three equations for the polarization field and Maxwell field, and two independent components of gravitational field equations. At this point, the equations are more difficult to be solved. However, once finding a reasonable solution for the equations beyond the probe approximation, we may get a complete phase diagram. This will be the main focuses for us in the future and we also look forward to obtain some exhilarating results. Meanwhile, it notes that the Maxwell equation in eqs. (9) can be integrated to give the conserved (i.e. Noether) electric charge. This can also directly be seen from eqs. (4), where the covariant divergence can be expressed as a total r derivative once the ansatz for the fields are used. This then admits a first integral which is just the conserved charge. In order to solve the nonlinear equations (9) numerically, we have to specify boundary conditions for the fields. At the horizon r → r + , we impose φ(r + ) = 0 to satisfy the finite form A μ , and ρ = − Thus once given the initial values of ρ and p at the horizon, one can get the solution of equations of motion (9). While near the boundary r → ∞, the linearized equations give the asymptotic solution for matter fields, here ± = 1 2 ± 1 2 √ 1 + 4m 2 and ρ ± , μ and σ are all constants.
According to gauge-gravity and the explanation for the source in Ref. [48], we treat ρ + as the source of the dual operator when B = 0, namely, ρ + = 0. When B = 0, the asymptotic behavior is governed by external magnetic field B. Here μ and σ are chemical potential and charge density of dual field theory, respectively. The Breitenlohner-Freedmeltaan (BF) bound requires m 2 > −1/4, and the mass squared m 2 of massive 2-form field has the lower bound. Within the BF bound condition, there does not exit the AdS 2 geometry and the near horizon geometry of an external black brane with vanishing temperature. To find the restriction to the parameters, let us consider Eqs. (9) in the high-temperature region where ρ vanishes, and we can read off the effective mass square of ρ at the horizon as Because J < 0, the temperature term contributes a negative term to the effective mass square, which is divergent when T → 0. It follows that, the instability appears provided that the temperature is low enough and produces the paramagnetism/ferromagnetism phase transition.

Spontaneous magnetization
In this letter we will work in the grand canonical ensemble where the chemical potential μ will be fixed and the system has the following scaling symmetry: r → βr, (t, x, y) → β −1 (t, x, y), T → β T , (13) Thus once obtaining the solutions of Eqs. (9), we can use the scaling symmetry (13) to find out the corresponding solutions with the same chemical potential. As a typical example, in what follows, we choose parameters as m 2 = − J = 1/8 and λ = 1/2 which can capture the basic features of the model. In other words, the other choices of the parameters will not qualitatively modify our results.
To study the spontaneous magnetization, we need set B = 0.
Since near the critical temperature, ρ is very small, the nonlinear term of ρ can be neglected in (9). To find the critical temperature, we can solve the Eqs. (9) with no nonlinear term of ρ and the initial condition is at the horizon r + = 1 and boundary condition ρ + = 0 at r → ∞. Without loss the generality, we take the initial value of ρ at horizon to be unity, and treat p as the shooting parameter to match the source free condition. As taken different parameters a (i.e., a = −0.1, 0, 0.1, 0.3), we will find the critical temperature T c /μ ≈ 1.90, 1.78, 1.68, 1.53. Here we have set a = 2c 1 + c 2 which can be used to describe the quartic field strength F 4 correction to the usual Maxwell field in the case without external magnetic field. While for the reason why only one parameter a is needed when B = 0, we can give a further explanation. The combination a = 2c 1 + c 2 is somewhat interesting because a = 0 corresponds to the case where the F 4 term can be written as ( * F μν F μν ) 2 . At the moment, the combination is essentially ( − → E · − → B ) and hence vanishes in the absence of an external magnetic field. In Table 1, we present the critical temperature T c for the magnetic moment and find that the critical temperature decreases as the a increases, which is exhibited in Fig. 1. It means the increasing value of nonlinear parameter makes the condensation harder to form. When the temperature is lower than the critical temperature T c , in order to examine whether the polarization field ρ can make spontaneous magnetization when the external magnetic field B = 0. We plot the value of ρ + versus ρ(r + ) at the horizon in the  Here, δ = 2 − + .
plot (a) to (c). These plots show a typical example in the high and low temperature cases which correspond to the red and black lines, respectively, while the case with the critical temperature is shown by the blue curve. In the case of high temperature, we see that the curve has no intersecting point with horizontal axis except a trivial point at the origin, which corresponds to a trivial solution with ρ = 0 and it describes the paramagnetic phase. On the other hand, when the temperature is lower enough, we find that exists a nontrivial solution which locates at ρ(r + ) = 0 whether the correction parameter a is zero or not. This solution breaks the time reversal symmetry and the system gets into a ferromagnetic phase. In addition, here, it is worth stressing that in 5-dimensional space-time the spatial rotational symmetry is also broken spontaneously, since a nonvanishing magnetic moment chooses a direction as a special. In order to compare the influence of parameter a on the intersecting point in the low temperature, we present it in the plot (d) in Fig. 2. It is found that the value of intersecting point is the biggest when a = −0.1. But anyway, as a result, we see that the model indeed can give raise to paramagnetism/ferromagnetism phase transition in the case without external magnetic field. When the temperature is lower than the critical temperature T c , we have to solve Eq. (9) to get the solution of the order parameter ρ and then computer the value of magnetic moment N, which is defined in 4-dimensional Schwarzschild black hole by  Fig. 3 shows the value of magnetic moment N as a function of temperature with various F 4 correction parameters a in 4-dimensional Schwarzschild black hole. It will be found that when the temperature is lower than critical temperature T c the nontrivial solution ρ = 0 and spontaneous magnetic moment appears in the absence of external magnetic field which is independent of parameter a. Meanwhile, the numerical results show that the phase transition is a second-order one with the behavior N ∝ √ 1 − T /T c in the vicinity of temperature for all cases calculated above. The result has been list in Table 1 and is still consistent with the one in the mean field theory. The behavior of the condensates for the massive 2-form field shows that the holographic paramagnetismferromagnetism phase transition still exists even we consider F 4 correction terms to the usual Maxwell electrodynamics. From the Fig. 3, we observe that the higher correction parameter a makes the magnetic moment smaller. It means that the condensation is harder to be formed when adding F 4 correction to the usual Maxwell field. In fact, the Table 1 shows that the critical temperature T c for the condensation decreases as the correction terms a increases, which agrees well with the finding in Fig. 3. This behavior is reminiscent of that seen for the holographic superconductor in the background of a Schwarzschild -AdS black hole, where the higher correction to the Maxwell field makes the condensation harder to form and changes the excepted relation in the gap frequency. However, in the soliton background, the correction to the Maxwell field does not influence the holographic superconductor and insulator phase transition [32].

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 randomly distributed. So susceptibility obeys the Curie-Weiss law  where C and θ are two constants. 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. (17) is positive and zero, respectively. Fig. 4 shows the magnetic susceptibility as a function of temperature. In the paramagnetic phase, we observe that the magnetic susceptibility increases when the temperature is lowered for the fixed F 4 correction parameter c 1 and c 2 . Moreover, the magnetic susceptibility satisfies the Curie-Weiss law of the ferromagnetism near the critical temperature. Concretely, the results have been presented in Table 2 for the chosen model parameters. It is easy to see that the coefficient in front of T /T c for 1/χ decreases with the increasing a (here a = 2c 1 + c 2 ), which meets with the discovery in Fig. 4. However, the absolute value of θ/μ will increase when the higher correction term takes a bigger constant. For the first three plots of Fig. 5, we show the magnetic moment with respect to external field B in the region of T < T c (i.e., T = 0.85T c ), T = T c and T > T c , and take the value of a as −0.01, 0, 0.01 from plot (a) to (c) respectively.
It shows that the magnetic moment is not single valued only in the case of T < T c when the external magnetic field continuously changes between −B max and B max periodically. Meanwhile, in order to compare the effect of different parameters a on the model, we present the case individually in plot (d) and a hysteresis loop in the single magnetic domain will be obtained. In addition, we find the higher correction term a just has an effect on hysteresis loop quantitatively. Along the horizontal direction (the magnetic moment has been taken a same value), one needs a smaller external field as the F 4 correction parameter increases. In other words, the higher correction term makes the periodicity of hysteresis loop smaller which is similar to the effect of Lifshitz dynamic exponent z on it. However, all the curves will overlap once the value of the magnetic field exceed the maximum that correspondings to the case of a = −0.1, which can be seen from Fig. 5.

Summary and discussion
In this letter we have studied holographic paramagnetism/ferromagnetism phase transition in the presence of a quartic field strength correction F 4 to Maxwell electrodynamics in 4-dimensional Schwarzschild -AdS black hole spacetime, and obtained the effect of the higher correction parameter a on the holographic paramagnetism/ferromagnetism phase transition. Adopting the probe limit, we have found that the improving of the F 4 correction term results in the decrease of critical temperature T c and magnetic moment in the case without external field. It means that the phase transition becomes difficult, and the magnetic moment is harder to form. This behavior is similar to that seen for the holographic superconductor in the background of a Schwarzschild -AdS black hole, where the F 4 correction term makes scalar condensation harder to form. In the vicinity of the critical point, however, the behavior of the magnetic moment is independent of F 4 correction term. It is always as (1 − T /T c ) 1/2 and 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 and the absolute value of θ increase with the increasing F 4 correction term. 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 F 4 correction. The increase of correction parameter a could result in shortening the period of the external magnetic field. Note that in this paper we just investigate the influence of F 4 correction term on paramagnetism/ferromagnetism phase transition by the numerical shooting method. It would be of interest to generalize our study to the analytic method. Work in this direction will be reported in the future.