Holographic paramagnetic-ferromagnetic phase transition of Power-Maxwell-Gauss-Bonnet black holes
aa r X i v : . [ h e p - t h ] F e b Numerical study of Gauss-Bonnet holographic paramagnetic-ferromagnetic phasetransition with Power-Maxwell electrodynamics
B. Binaei Ghotbabadi,
1, 2
A. Sheykhi,
1, 3, 2, ∗ and G. H. Bordbar
1, 2, † Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, P. O. Box: 55134-441, Iran
Based on shooting method, we numerically investigate the properties of holographicparamagnetism-ferromagnetism phase transition in the presence of Gauss-Bonnet ( GB ) correctionson the gravity side. On the matter field side, however, we consider the effects of the Power-Maxwell( PM ) nonlinear electrodynamics on the phase transition of this system. For this purpose, we intro-duce a massive 2 − form coupled to PM field, and neglect the effects of 2 − form fields and gauge fieldon the background geometry. We observe that increasing the strength of both the power parameter q and GB coupling constant α decrease the critical temperature of the holographic model, and leadto the harder formation of magnetic moment in the black hole background. Interestingly, we findout that at low temperatures, the spontaneous magnetization and ferromagnetic phase transitionhappen in the absence of external magnetic field. In this case, the critical exponent for magneticmoment has the mean field value, 1 /
2, regardless of the values of q and α . In the presence of externalmagnetic field, however, the magnetic susceptibility satisfies the Cure-Weiss law. PACS numbers:
I. INTRODUCTION
The duality between gravity in an anti-de Siter (AdS) spacetime and a conformal field theory (CFT), knownas AdS/CFT correspondence, provides a powerful tool to investigate strongly coupled systems [1–5]. A significantapplications of this conjecture is investigation of the electronic properties of materials and magnetism [6–15]. Thegauge/gravity duality, which is a new approach for calculating the properties of superconductors using a dual classicalgravity, also provides a fascinating tool to shed the light on high temperature superconductors. Based on this theory,one can describe a superconductor using a dual classical gravity description. It has been shown that some propertiesof strongly coupled superconductors can be potentially described by classical general relativity living in one higherdimension, which is known as holographic superconductors [16]. The idea of holographic superconductor was initiatedby Hartnol, et. al. [16, 17]. They considered a four-dimensional Schwarzschild-AdS black hole coupled to a Maxwelland a scalar fields to construct a holographic s -wave superconductor. Based on their model, in order to describea holographic superconductor on the boundary, a transition from hairy black hole to a no hair black hole in thebulk for temperatures below and upper the critical value is required [16]. The appearance of hair corresponds to thespontaneous U (1) symmetry breaking [16]. This theory opened up a new perspective in condensed matter physicsto study the high temperature superconductors. During the past decade, the explorations on the holographic dualmodels have attracted a lot of attentions (see e.g. [18–32] and reference therein). The studies were also generalizedto investigate paramagnetic-ferromagnetic phase transition using the holographic description [33–43, 56]. The firstholographic paramagnetic-ferromagnetic phase transition model was a dyonic Reissner-Nordstrom-AdS black brane[33]. This model provides a starting point for exploration of more complicated magnetic phenomena and quantumphase transition, by considering a real antisymmetric tensor field which is coupled to the background gauge fieldstrength in the bulk. It was argued that the spontaneous magnetization which happens in the absence of an externalmagnetic field, can be realized as the paramagnetic-ferromagnetic phase transition. Most investigations on holographicparamagnetism-ferromagnetism phase transition have been carried out by considering the gauge field as a linearMaxwell field in Einstein gravity [34–41]. It is also of great interest to explore the effects of nonlinear electrodynamicson the properties of the holographic ferromagnetic-paramagnetic phase transition. These holographic setups havebeen widely studied in the presence of nonlinear electrodynamics, those involve more information than the usualMaxwell state [42, 43]. It has been observed that in the Schwarzschild AdS black hole background and in the absenceof external magnetic field, the higher nonlinear electrodynamics corrections make the magnetization harder to be ∗ Electronic address: [email protected] † Electronic address: [email protected] formed. Among various nonlinear extension of Maxwell electrodynamics, the PM nonlinear electrodynamics whichpreserves the conformally invariant feature in higher dimensions has received more attraction [44]. The conformallyinvariant PM action in ( n + 1)-dimensional may be written, I P M = Z d n +1 x √− g ( −F ) q , (1)where F = F µν F µν is the Maxwell invariant and q is the power parameter. Under conformal transformation g µν → Ω g µν and A µ → A µ , the above action remains invariant. The corresponding energy-momentum tensor given by T µν = 2 (cid:18) qF µρ F ρν F q − − g µν F q (cid:19) , (2)is traceless for 4 q = n + 1. Black hole solutions in the presence of PM electrodynamics have been constructed by manyauthors (see e.g. [45–48] and references therein). The properties of holographic superconductor with conformallyinvariant P M electrodynamics have been studied in Refs.[49–55]. Recently, we explored the effects of PM nonlinearelectrodynamics on the properties of holographic paramagnetic-ferromagnetic phase transition in the background ofSchwarzchild-AdS black hole [56]. We investigated how the PM electrodynamics influences the critical temperatureand magnetic moment. We found that the effects of P M field lead to the easier formation of magnetic moment athigher critical temperature. The studies were also generalized to other gravity theories. In the context of GB gravity,the phase transition of the holographic superconductors were explored in Refs.[57–65]. Their motivations are to studythe effects of higher order gravity corrections on the critical temperature of holographic superconductors. They foundthat when GB coefficients become larger, the condensation on the boundary field theory becomes harder to be formed.In our previous work [56] we have considered the effects of P M on paramagnetic-ferromagnetic phase transition inEinstein gravity. It it also interesting to examine the effects of this kind of nonlinear electrodynamics when the higherorder corrections on the gravity side such as GB terms, is taken into account. We would like to examine whetheror not the holographic paramagnetic-ferromagnetic phase transition still hold in the presence of higher order gravitycorrections. We shall apply the shooting method to numerically disclose the influences of both the higher order GB curvature correction terms, as well as the nonlinear P M electrodynamics on the holographic system.This paper is organized as follows. In section II, we introduce the action and basic field equations in the presenceof PM electrodynamics by considering the higher order GB curvature correction terms. In section III, we employ theshooting method to obtain numerically critical temperature and magnetic moment of the system. We also study themagnetic susceptibility density. We finish with closing remarks in the last section. II. HOLOGRAPHIC SET-UP
To investigate an ( n ) − dimensional holographic ferromagnetism model, we consider an ( n + 1)-dimensional actionof Einstein-Gauss-Bonnet gravity in AdS spaces which is coupled to a P M field as, S = 12 κ Z d n +1 x √− g h R −
2Λ + α R − R µν R µν + R µνρσ R µνρσ ) + L ( F ) + λ L i , (3)where κ = 8 πG with G is Newtonian gravitational constant, g is the determinant of metric and α is the GB coefficient.In the above action, R, R µν , R µνρσ are, respectively, Ricci scalar, Ricci tensor and Riemann curvature tensor. When α →
0, the above action reduces to the Einstein one. In addition Λ = − n ( n − / l is the cosmological constant of( n + 1)-dimensional AdS spacetime with radius l . Here L ( F ) = −F q /
4, where F = F µν F µν in which F µν = ∇ [ µ A µ ] and A µ is the gauge potential of U (1) gauge field and q is the power parameter of the P M field. In the limitingcase where q = 1, the P M
Lagrangian will reduce to the Maxwell case, L = − F µν F µν /
4, and the Einstein-Maxwelltheory is recovered. Besides, for q = ( n + 1) /
4, the energy-momentum tensor of the
P M
Lagrangian is traceless inall dimension and the theory is conformally invariant [44]. The L Lagrangian is defined as [37] L = −
112 ( dM ) − m M µν M µν − M µν F µν − J V ( M ) , where λ and J are two constants with J < λ characterizesthe back reaction of the two polarization field M µν and the Maxwell field strength on the background geometry. Inaddition, m is the mass of 2-form field M µν being greater than zero [37], and dM is the exterior differential 2-formfield M µν . The nonlinear potential of 2-form field M µν , V ( M µν ), describes the self interaction of polarization tensorwhich should be expanded as the even power of M µν . For simplicity, we take the following form for the potential, [38] V ( M ) = ( ∗ M µν M µν ) = [ ∗ ( M ∧ M )] , (4)where ∗ is the Hodge star operator. This potential shows a global minimum at some nonzero value of ρ [37]. Since weconsider the probe limit, the gauge and matter fields do not back react on the background metric. The line elementof the metric with flat horizon is given by [67] ds = − r f ( r ) dt + dr r f ( r ) + r n − X i =1 dx i , (5)with f ( r ) = 12 α " − s − α (cid:18) − r n + r n (cid:19) , (6)where r + is the positive real root of Eq. f ( r + ) = 0. At large distance where r → ∞ , the metric function reduces to f ( r ) ≈ α (cid:2) − √ − α (cid:3) , (7)Since f ( r ) should be a positive definite function, it implies 0 ≤ α ≤ / α = 1 / L eff as L = 2 α − √ − α . (8)The Hawking temperature associated with the black hole event horizon, which can be interpreted as the temperatureof CFT on the boundary, is given by [57] T = f ′ ( r + )4 π = nr + π , (9)Varying action (3) with respect to 2 − form field, M µν , and the gauge field, A ν , we arrive at the following field equations[56] 0 = ∇ τ ( dM ) τµν − m M µν − J ( ∗ M τσ M τσ )( ∗ M µν ) − F µν , (10)0 = ∇ µ (cid:18) qF µν ( F ) q − + λ M µν (cid:19) . (11)Our aim here is to investigate the effects of the power parameter q and the GB coefficient α on the holographicferromagnetic-paramagnetic phase transition. We adopt the following self-consistent ansatz for the matter and gaugefields [38] M µν = − p ( r ) dt ∧ dr + ρ ( r ) dx ∧ dy, (12) A µ = φ ( r ) dt + Bxdy, (13)where B is a uniform magnetic field which is considered as an external magnetic field of dual boundary field theory.Here ρ ( r ) and p ( r ) are the components of M µν , while φ ( r ) stands for the electric potential. Inserting the above ansatzinto Eqs. (10) and (11), lead to the following field equations,0 = ρ ′′ + ρ ′ (cid:20) f ′ f + n − r (cid:21) − ρr f (cid:2) m + 4 Jp (cid:3) + Br f , (cid:18) m − Jρ r (cid:19) p − φ ′ , (14)0 = φ ′′ + 2 φ ′ r " n − φ ′ + (cid:0) q − n +32 (cid:1) B r (2 q − φ ′ − B r + λ q q +1 (cid:18) p ′ + ( n − pr (cid:19) (cid:16) φ ′ − B r (cid:17) − q (2 q − φ ′ − B r , (a) α =0 (b) α =0.1 (c) α =0.2 FIG. 1: The behavior of magnetic moment N and the critical temperature with different values of power parameter q and GB coupling α = 0 , . . n + 1 =)5 − dimensions. Here we have taken m = 1 / J = − / (a) q =3/4 (b) q =1 (c) q =5/4 FIG. 2: The behavior of magnetic moment N and the critical temperature for different values of GB parameter α and powerparameter q in five dimensions. Here we have taken n = 4, m = 1 / J = − / where prime denotes the derivative with respect to r . Obviously, for Einstein gravity ( α →
0) the above equationsreduce to the corresponding equations in Ref. [37] in the Maxwell limit where q → n = 3. In order to solveEqs.(14) numerically, we should specify the boundary conditions for the fields. Imposing the regularity conditions atthe horizon( r = r + ), yields the following boundary conditions [33] φ ( r + ) = 0 , ρ ′ ( r + ) = ρ ( r + )( m + 4 Jp ) − B πT . (15)Since the behaviors of model functions are asymptotically AdS , thus we solve the field equations (14) near theboundary ( r → ∞ ). We find the asymptotic solutions as φ ( r ) ∼ µ − σ / (2 q − r ( n − q − − , p ( r ) ∼ σ / (2 q − ( n − q q − ) r ( n − q − ,ρ ( r ) ∼ ρ − r ∆ − + ρ + r ∆ + + Bm , (16)with ∆ ± = 12 (cid:20) ( n − ± q − m L + ( n − (cid:21) . (17)Based on the AdS/CFT correspondence, ρ + and ρ − are interpreted as the source and vacuum expectation value ofthe dual operator when B = 0, which plays the role of order parameter in the boundary theory. Moreover, µ and σ are regarded as the chemical potential and charge density of dual field theory, respectively. Therefore, condensationhappens spontaneously below a critical temperature when we set ρ + = 0. By considering B = 0, the asymptoticbehavior is governed by external magnetic field B . Since the boundary condition for the gauge field φ depends onthe power parameter q of the P M field, the values of q should be restricted to n − q − − >
0, which implies that thepower parameter q ranges as 1 / < q < n/
2. In the next section, we solve the field equations numerically and obtainthe physical properties of our holographic model.
TABLE I: Numerical results of T c /µ for different values of q and α , in five dimensions. q / / α = 0 3 . . . α = 0 . . . . α = 0 . . . . N with different values of power parameter q and GB coefficient α in five dimensions. q / / α = 0 2 . − T /T c ) / . − T /T c ) / . − T /T c ) / α = 0 . . − T /T c ) / . − T /T c ) / . − T /T c ) / α = 0 . . − T /T c ) / . − T /T c ) / . − T /T c ) / III. NUMERICAL CALCULATION FOR SPONTANEOUS MAGNETIZATION AND SUSCEPTIBILITY
In this paper, we work in the grand canonical ensemble where the chemical potential µ is a fixed quantity. We haveto solve Eq. (14) to get the solution of the order parameter , ρ and then compute the value of the magnetic moment N , which is defined as N = − λ Z ρ r n − dr. (18)Here we employ the shooting method [6] to numerically investigate the behavior of the holographic ferromagnetism-paramagnetic phase transition in GB gravity. In order to compare our results with Ref. [56], we choose n = 4. Thereis a good agreement between our results with corresponding cases of Ref. [56] when α →
0. Since different parameterswill give similar results, we choose m = − J = 1 / λ = 1 / z = r + /r instead of r which transforms the coordinate r to z . In terms of this newcoordinate, z = 0 and z = 1 correspond to the boundary ( r → ∞ ) and horizon ( r = r + ), respectively. Besides setting l to unity, we also set r + = 1 in the numerical calculation, for simplicity, which may be justified by virtue of the fieldequation symmetry, r → ar, f → a f, φ → aφ. First, we expand Eq.(14) near the horizon ( z = 1), ρ ≈ ρ (1) + ρ ′ (1)(1 − z ) + ρ ′′ − z ) + · · · , (19) φ ≈ φ ′ (1)(1 − z ) + φ ′′ − z ) + · · · . (20)In the above equations, we have imposed the boundary condition φ (1) = 0. In our numerical process, we find ρ (1), φ ′ (1) such that the desired values for boundary parameters in Eq. (16) are attained. At boundary, one can seteither ρ − or ρ + to zero as source, and find the value of the other one as the expectation value of order parameter h O i . We consider the cases of different P M parameter q and GB coefficient α in five spacetime dimensions. Wepresent our results in Fig. 1 when GB parameter is fixed for three different values of q . We also provide Fig. 2 byfixing the power parameter q , for studying the behavior of this system for three allowed values of GB parameter.These figures show the behavior of magnetic moment as a function of temperature for different choices of nonlinearityand GB parameters. Increasing both of these parameters makes decreasing for magnetization value. This is due tothe fact that the larger parameters α and q inhibit the ferromagnetic phase transition. It means that the magneticmoment is harder to be formed which is in a good agreement with similar works [42, 43]. This behavior has beenseen for the holographic superconductor in the Schwarzschild- AdS black hole, where the three types of nonlinearelectrodynamics make scalar condensation harder to be formed [66]. When the temperature is lower than T C , thespontaneous magnetization appears in the absence of external magnetic field. These numerical results show thatthe second order phase transition happen which its behavior obtain by fitting this curve ( N ∝ p − T /T C ). Theresults have been presented in Table II. We find that there is a square root behavior for the magnetic moment versustemperature. According to these results and the same as in Ref. [56], the GB parameter does not change the thecritical exponent(1 /
2) which is the same as that of the mean field theory. In other word, there is a holographicferromagnetic-paramagnetic phase transition by considering the
P M electrodynamics in the GB gravity similar to (a) q = 3 / , α =0 (b) q = 3 / , α =0.1 (c) q = 3 / , α =0.2(d) q = 1 , α =0 (e) q = 1 , α =0.1 (f) q = 1 , α =0.2(g) q = 5 / , α =0 (h) q = 5 / , α =0.1 (i) q = 5 / , α =0.2 FIG. 3: The behavior of the inverse of susceptibility density 1 /χ with respect to the critical temperature for different values of α and q in five dimensions. Here we have taken n = 4, m = 1 / J = − / the cases of nonlinear electrodynamics in Einstein gravity discussed in Refs. [42, 56]. Table I gives information aboutthe values of critical temperature based on µ for different values of GB parameter as well as power parameter q .Considering the GB coefficient leads to the same effect as the larger values of power parameter q on the criticaltemperatures. It can be found a decreasing for the critical temperature at stronger values of q and α , which presentthe magnetization harder to be formed, and it corresponds to a ferromagnetic phase in boundary field theory. Onthe other hand, increasing α in the allowed range can cause the transition to ferromagnetic phase to be harder forany values of the power parameter q . We see also from Table I that by increasing the power parameter q , the criticaltemperature T c decreases for fixed value of α . It means that the magnetic moment is harder to be formed and thephase transition is made harder in the Einstein- Gauss-Bonnet gravity. This behavior have been reported previouslyin Ref. [42] for nonlinear electrodynamics in Einstein gravity too.The behavior of susceptibility density of the materials in the external magnetic field is another remarkable propertiesof ferromagnetic material. In order to study the static susceptibility density of the ferromagnetic materials in the GB gravity, we follow the definition χ = lim B → ∂N∂B . (21)Here we turn on the external magnetic field B to examine the response of magnetic moment N . In the presence ofmagnetic field, the function ρ is nonzero at any temperature. The magnetic susceptibility obtained by solving Eq. (10)based on the previous analysis which one has been discussed in Ref. [37]. Fig.3 shows the behavior of susceptibilitydensity near the critical temperature in five dimensional GB gravity. We see that when the temperature decreases, χ increases for any allowed values of q and α . This is due to the fact that with the increasing these parameters, thesystem will become instable and then the ferromagnetic phase will be broken into the paramagnetic phase. In theregion of T → T + c , the susceptibility density satisfies the cure-Weiss law of ferromagnetism and the paramagneticphase happens. χ = CT + θ , T > T C , θ < , (22)where C and θ are two constants. The results have been presented in Table III. Obviously, we can see that thecoefficient in front of T /T c for 1 /χ increases, by decreasing the power parameter( q ) and GB coefficient( α ). It meansthat for the smaller values of these two parameters our system goes to stability. TABLE III: The magnetic susceptibility χ with different values of q and α . q / / α = 0 λ /χµ . T /T c + 211 . . T /T c + 57 . . T /T c + 47 . θ/µ . . . α = 0 . λ /χµ . T /T c + 122 . . T /T c + 34 . . T /T c + 28 . θ/µ . . . α = 0 . λ /χµ . T /T c + 78 . . T /T c + 24 . . T /T c + 20 . θ/µ . . . IV. CLOSING REMARKS
We have numerically investigated the behavior of a holographic ferromagnetic model with the
P M electrodynamicsbased on shooting method, by considering the higher order GB corrections terms on the gravity side of the action.On the gauge field side, however, we have considered the effects of PM nonlinear electrodynamics on the system.We have focused on 1 / < q < n/ GB coefficient make a decreasing for the critical temperature.Numerical calculations indicate that increasing the values of nonlinearity and GB parameters in different dimensionscan make the magnetization harder to be formed, because the increasing α and q always inhibit the ferromagneticphase transition. Increasing the effect of PM parameter in GB gravity leads to the same behavior as in case ofEinstein gravity [56]. We observed that the enhancement in GB parameter α causes the paramagnetic phase moredifficult to be appeared. These results are reflected in Figs.2. We find out that the magnetic moment behaves as(1 − T /T c ) / which is in agreement with the result of the mean field theory. Our investigation of critical exponentindicates that the critical exponent has the mean field value (1 / q , α and n .In the presence of external magnetic field, the inverse magnetic susceptibility near the critical point behaves as ( CT + θ )for all allowed values of the power parameter q and different values of the GB coupling α in different dimensions,and therefore it satisfies the Cure-Weiss law. The absolute value of θ increases by increasing the q . When T < T c the ferromagnetic phase happens, and for T > T c this model goes to the paramagnetic phase. As a result, our modelprovides a holographic description for the ferromagnetic-paramagnetic phase transition. Acknowledgments
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