Massive 2-form field and holographic ferromagnetic phase transition

Journal of High Energy Physics, Nov 2015

In this paper we investigate in some detail the holographic ferromagnetic phase transition in an AdS4 black brane background by introducing a massive 2-form field coupled to the Maxwell field strength in the bulk. In two probe limits, one is to neglect the back reaction of the 2-form field to the background geometry and to the Maxwell field, and the other to neglect the back reaction of both the Maxwell field and the 2-form field, we find that the spontaneous magnetization and the ferromagnetic phase transition always happen when the temperature gets low enough with similar critical behavior. We calculate the DC resistivity in a semi-analytical method in the second probe limit and find it behaves as the colossal magnetic resistance effect in some materials. In the case with the first probe limit, we obtain the off-shell free energy of the holographic model near the critical temperature and compare with the Ising-like model. We also study the back reaction effect and find that the phase transition is always second order. In addition, we find an analytical Reissner-Norström-like black brane solution in the Einstein-Maxwell-2-form field theory with a negative cosmological constant.

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Massive 2-form field and holographic ferromagnetic phase transition

Received: July Massive 2-form eld and holographic ferromagnetic phase transition Rong-Gen Cai 0 1 2 4 5 6 7 Run-Qiu Yang 0 1 2 4 5 6 7 Ya-Bo Wu 0 1 2 3 5 6 7 Cheng-Yuan Zhang 0 1 2 3 5 6 7 e ect 0 1 2 5 6 7 0 Dalian , 116029 , China 1 Beijing 100190 , China 2 Chinese Academy of Sciences 3 Department of Physics, Liaoning Normal University 4 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics 5 We also study the back reaction 6 nd it behaves 7 reaction of the 2-form In this paper we investigate in some detail the holographic ferromagnetic phase transition in an AdS4 black brane background by introducing a massive 2-form to the Maxwell eld strength in the bulk. In two probe limits, one is to neglect the back eld to the background geometry and to the Maxwell eld, and the other to neglect the back reaction of both the Maxwell eld and the 2-form that the spontaneous magnetization and the ferromagnetic phase transition always happen when the temperature gets low enough with similar critical behavior. We calculate the DC resistivity in a semi-analytical method in the second probe limit and as the colossal magnetic resistance e ect in some materials. In the case with the probe limit, we obtain the o -shell free energy of the holographic model near the critical temperature and compare with the Ising-like model. Holography and condensed matter physics (AdS/CMT); AdS-CFT Corre- 1 Introduction 2 The model 3 Ansatz and trivial solution 4 Probe limit in the case of Spontaneous magnetization and susceptibility Holographic Ginzburg-Landau formulation Spontaneous magnetization and susceptibility 5.2 DC conductivity in the ferromagnetic phase 6 Phase transition with back reaction 6.1 6.2 On-shell free energy Tensor hairy solutions and phase transition 7 Summary and discussion 5 Probe limit by neglecting the back reaction of all matter elds A Ginzburg-Landau theory for Ising universality class B Semi-analytic calculations near the critical temperature B.1 Spontaneous magnetization B.2 Susceptibility and hysteresis loop Introduction In recent years, a promising new route named AdS/CFT duality [1{4] provided a new viewpoint to understand gravity and strongly coupled or correlated phenomena in physics. This duality relates a gravity theory in a weakly curved (d + 1)-dimensional anti de Sitter (AdSd+1) spacetime to a strongly coupled d-dimensional eld theory living on its boundary. The AdS/CFT duality maps questions about strongly correlated many-body phenomena to solvable single- or few-body classical problems in a curved geometry, which opens a new window to solve the strongly correlated system in condensed matter physics and considerable progresses have been made since the duality was proposed. For instance, holographic superconductor/superfuild [5, 6] (for reviews, see [7{11]), holographic (non-)fermi liquid [12{14], holographic charge density wave and metal/insulator phase transition [15{17], and some systems far-from thermal equilibrium [18{22] have been studied intensively. So far, most of attentions about the duality application to condensed matter physics have been focused on the electronic properties of materials. In condensed matter physics, magnetism also plays an important role in materials including high temperature superconductors and heavy fermion metals in many strongly correlated electronic systems. The gauge/gravity duality provides an approach and perspective to understand these challenging problems. Though there exist a few works involving magnetism in holographic superconductor models, magnetism does not pay the central role. Yet the scarcity of models on magnetism is due to various technical challenges in holographic context. In a previous work [23], we rst proposed a new example of the application of the AdS/CFT correspondence by realizing the paramagnetism/ferromagnetism phase transition in a dyonic Reissner-Nordatrom-AdS black brane. This model also was extended to realize the paramagnetism/antiferromangnetism phase transition by introducing two real antisymmet elds with interaction between them and they are coupled to the background gauge eld strength [24]. In ref. [25], by combining the holographic p-wave superconductor model [26] and the holographic ferromagnetism model, we studied the coexistence and competition of ferromagnetism and p-wave superconductivity. It is found that the results depend on the self-interaction of magnetic moment of the complex vector eld and on which phase (superconductivity or ferromagnetism) appears rst. We noted that the ferromagnetic superconductivity has been also discussed in ref. [27] by introducing two SU(2) gauge elds in the bulk. In that model superconductivity and ferromagnetism happens simultaneously and the magnetic susceptibility is nite at the ferromagnetic critical temperature. Note that, however, the ferromagnetic p-wave superconductor in heavy fermion systems such as UCoGe, URhGe and UGe2, shows the two critical temperatures are di erent in general [28] and magnetic susceptibility diverges at ferromagnetic critical temperature [29]. In the previous works, the reasons of using an antisymmetric tensor eld to model ferromagnetic phase transition in the holographic setup are as follows. (1) On the viewpoint of symmetry breaking, ferromagnetic phase transition breaks the time reversal symmetry spontaneously in low temperature (if spatial dimension is more than 2, it also breaks spatial rotation symmetry), which in general is not associated with other symmetry breakings such as U(1), SU(2) and so on. Thus the dual operator does not carry U(1) or SU(2) charge, which implies that we need to use a real eld dual to such an operator. (2) From the point of view of covariance, magnetic eld is the spatial component of a SO(1,3) tensor F , so magnetic moment should be also a component of a tensor. This is very important when we construct a holographic model to describe magnetism of materials. In condensed matter physics, we usually say magnetic eld is a vector, because the system is of course non-relativistic.1 However, in holographic model, it more convenient to write down the theoretical model in its relativistic form as the boundary theory has Lorentz invariance. In this case, we should use electromagnetic 1Strictly speaking, magnetic eld and magnetic moment are two pseudo-vector elds rather than vector elds even in non-relativistic case, which play the central role in the phenomena involving spontaneously magnetic ordering. eld tensor F to replace the magnetic eld or electric eld. The magnetic moment N~ as the response to magnetic eld, with the electric dipole moment P~ as the response to electric eld, should be combined into a tensor eld, (E~ ; B~ ) ! F ; (N~ ; P~ ) ! M This replacement has many advantages when we are going to write down a theoretical model in a Lorentz invariant framework but has no any obstacle if we only care about its non-relativistic limit. By an antisymmetric tensor eld, we can use a very compact form to take magnetic moment and electric dipole moment into account in Lorentz invariant form. We can see later that magnetic moment is a pseudo-vector eld automatically in this setup. (3) Furthermore, according to the origin of magnetic moment, magnetism of material is controlled by the intrinsic magnetism of electrons which is obtained from the Lagrangian for electrons, Le = i( @= where A is the gauge potential, F is the gauge strength eld and j is current density which is de ned as j density !N is the spatial component of an antisymmetric tensor eld, @ ) . Then the magnetic moment N i = i in the boundary eld in a covariant manner needs an antisymmetric tensor and its spatial component corresponds to the magnetic moment. These considerations can be generalized into 2+1 dimensions, where magnetic moment should still be regarded as the spatial components of an antisymmetric tensor eld. Then according to the spirit of AdS/CFT, we can use an antisymmetric tensor eld in the bulk to dual such a tensor operator in the boundary. Recently, it has been found in ref. [30] that the original model in [23] contains a vector ghost and a new model has been proposed, there the exterior di erential is used to replace the covariant derivative in the kinetic term of the antisymmetric tensor eld. With such a simple modi cation it has been shown that there does not exist any ghost and causality violation does not appear in the new model, while the key results in the original model are kept qualitatively. While in ref. [30] we paid our main attention to the health of the model and showed that the spontaneous magnetization can happen when the temperature gets low enough, in this paper, we are going to study this model in some detail with some directions. Before going on, we mention here that in this paper we assume the physical phase is homogeneous and will not consider inhomogeneous phases. This assumption is just for simplifying our discussions in technology. The possibility whether the inhomogeneous phase could appear spontaneously in this model under the homogeneous boundary conditions is left to study This paper is organized as follows. In section 2, we will describe the holographic model. We will give our ansatz for matter eld and derive equations of motion in section 3. In this section, we will show that there exists an analytical black hole solution di erent from the AdS Reissner-Nordtrsom solution in this model and discuss associated properties of the solution. In sections 4 and 5, we will investigate the paramagnetism/ferromagnetism phase transition in two di erent probe limits. One is to neglect the back reaction of the 2-form eld to the black brane geometry and to the Maxwell eld, and the other to neglect the back reaction of both the Maxwell eld and the form eld. In the former case, we will also calculate the o -shell free energy of the holographic model near the phase transition point by using the Sturm-Liouville eigenvalue method and obtain a free energy form like the Ginzburg-Landau one, and make a comparison with the Ising-like universal class model. In the latter case, we calculate the DC resistivity in the ferromagnetic phase, which shows the behavior of the colossal magnetic resistance e ect in condensed matter physics. We study the full back reaction e ect in section 6 by solving the full equations of motion of the model and nd that the phase transition is always second order. The summary and some discussions are included in section 7. The model In this paper, the model we are considering is just Einstein theory with a negative cosmolog 3=L2, a U(1) eld A space-time. The ghost free action reads [30] and a massive 2-form eld M in 4-dimensional S = 1 Z L1 = R + L2 = the gravitational constant in the bulk, which will be set to be unity in the following. g is the determinant of the bulk metric g . dM is the exterior di erential of 2-form m2 6= 0 is the squared mass of 2-form and must be greater than zero, which will be explained shortly. and J are two real model parameters with J < 0 in order the magnetization to happen spontaneously. 2 characterizes the back reaction of the 2-form to the background geometry and to the Maxwell eld strength.2 V (M nonlinear potential of the 2-form eld. It describes the self-interaction of the polarization tensor, which should be expanded as the even power of M . In this model, we take the following form, ) = ( M )2 = [ (M ^ M )]2: 2Note that here can be also understood as the measurement of the coupling between the tensor eld and the Maxwell eld strength F by rescaling the tensor eld and the parameter J. = 0; = T : is the Hodge-star operator. The choice of nonlinear potential is not unique. We choose this form just for simplicity. As shown in ref. [30], this potential shows a global minimum at some nonzero value of . By varying action (2.1), we can get the equations of motion for the matter elds and gravitational eld as ) = F ; The energy-momentum tensor T In the AdS/CFT correspondence, a hairy black hole with appropriate boundary conditions can be explained as a condensed phase of the dual eld theory, while a black hole without hair is dual to a normal phase. Clearly when M vanishes, the model admits the AdS Reissner-Nordstrom (RN) black brane solution, which corresponds to the normal phase in the dual eld theory. When M appears, we will see that there exists an analytical black brane solution di ering from the AdS-RN solution. The new solution corresponds to the normal phase rather than the usual AdS RN solution. When we lower the temperature, the system exhibits an instability which triggers to break time reversal symmetry spontaneously as well as spatial rotation symmetry since the condensate will pick out one direction as special (if spatial dimension is more than 2) and the paramagnetism/ferromagnetism phase transition happens. Ansatz and trivial solution As we will discuss the full solution to the action (2.1) including the back reaction to the spacetime geometry, we start with the following ansatz for the metric ds2 = r2f (r)ea(r)dt2 + + r2(dx2 + dy2); and take the self-consistent ansatz for polarization eld and U(1) eld as p(r)dt ^ dr + (r)dx ^ dy; (r)dt + Bxdy; with some real functions f (r), a(r), (r), p(r),and (r). The bulk eld B is a constant magnetic eld, which can be regarded as external magnetic eld in the dual boundary m2 + 4Jp2e a where a prime denotes the derivative with respect to r. The rst three equations are for the polarization eld and Maxwell eld, and the last two are the two independent components of gravitational eld equations. In fact, there are three nonzero components of gravitational eld equations, but only two of them are independent due to the Bianchi identity. We are interested in the black brane con gurations which have a regular event horizon g A A being nite at the horizon. We require the regularity conditions at the horizon nite values at rh and admit a series expansion in terms of (r rh). Then, at the horizon, we have the following relations, = 0; 0 = 0; = 0; = 0; = 0; m2 + 2B + 4B2 theory. We will denote the position of the horizon as rh and the conformal boundary will be at r ! 1. Since we would like to study a dual theory with nite chemical potential or charge density accompanied by a U(1) symmetry, we turn on At in the bulk. According to gauge/gravity duality, the Hawking temperature of the black brane is identi ed with the temperature of boundary thermal state, which is given by T = (r2f (r))0ea(r)=2 r=rh S = 4 A = 4 rh2V2; and the thermal entropy S is given by the Bekenstein-Hawking entropy of the black brane Put the ansatz (3.1) and (3.2) into equations (2.3), the independent equations of mo = 0; 0 = 0 = f 0 = m2 + 4Jp2e a + B The black brane solution should be asymptotically AdS. Thus one has the following asymptotic solutions near the AdS boundary, +r(1+ )=2 + r(1 )=2 + f = 1 + a = a0 + p = = p constants. The Breitenlohner-Freedman (BF) bound requires m2 > asymptotic solution of . According to the AdS/CFT dictionary, up to a normalization, the coe cients are directly related to the chemical potential and charge density in the dual system, respectively. The magnetic moment density in the dual theory is de ned and f0 are all 14 according to the by the integration [23, 30] N = From the equation for in eqs. (3.5), we see that reversal transformation, which leads to the property of magnetic moment de ned in eq. (3.8) under the time reversal transformation such as N ! N . This is agreement with the fact that magnetic eld is pseudo-vector in 3+1 boundary dimensions or pseudo-scalar in 2+1 B under the time boundary dimensions. Note that there is an additional restriction on m2 when we treat the magnetic eld B as the source of . In order to keep damping, we have to impose boundary condition for the integration (3.8) will diverge or the equation for can not be linearized near the the constant + should be viewed as the source of the corresponding operator in the boundary eld theory, according to AdS/CFT duality. In order the symmetry to be broken spontaneously, one has to turn o the source term. In addition, in order to make condense in the case without external magnetic eld when the temperature is low enough, the model parameters should violate the BF bound near the horizon, whose geometry is an AdS2 for an extremal black brane. To see how this requirement restricts the parameters, we rst consider the solution 2 Z 1 ea=2 (r) = (1 rh=r); f (r) = 1 p(r) = ~ = 0 = 0; = 0; = 0: Then the temperature of the black brane is T = Compared with the usual planar AdS RN black brane solution [31], we see that 2 in the metric of usual AdS RN solution is just replaced by ~ in the new solution. Namely, the new solution has the same form as the planar AdS RN solution with the same Maxwell eld (r), but also with a nontrivial pro le of p(r) for the massive 2-form Then we can easily calculate the free energy and the charge density of the sys (T; ) = = (T; ) = = 2 1 We see that the properties of this black brane solution depends on the value of 1 nonzero U(1) gauge eld and polarization eld. To investigate the physical properties when to chemical potential, = 6rh 1 It is easy to see that eq. (3.13) is positive when 1 2=4m2 > 0. ary system to be chemical instability. To understand it, one can image that a box is submerged into the environment with xed temperature T and chemical potential equilibrium state. The energy and charge can exchange between the interior of the box and environment through the wall (see gure 1). Now because of thermal uctuation, the chemical potential in the box then is < , which leads to some charge coming into the box. However, because eqs. (3.13) is negative, adding the charge of the system will decrease the chemical potential of the system. Then we see that the chemical potential will decrease again, which will lead that more charges come into the box. So the charge inside the box will be more and more and the system is unstable. From eq. (3.13), we can the dual boundary system is chemical stable region only when temperature and chemical potential satisfy4 1 T < : If eq. (3.13) is positive, we see that adding charge will increase the chemical potential, so after some time, the system can be in equilibrium again. In fact, in the case of 3The total charge density can be also calculated from the second equation in (2.3), which gives the same value as the one in (3.12). This also con rms that the chemical potential of the system is given by . 4It is worth noting that this condition is only valid in grand canonical ensemble. In other ensembles, the stable condition is di erent in general. 2=4m2. we have to replace 2 by e. But in the case of 1 di erent from those of usual AdS RN black brane. For example, the temperature gets increased when we increase the chemical potential while xing the horizon radius and there 2=4m2 < 0, the properties are very does not exist zero temperature entropy. In this paper, we only focus on the situation of 1 2=4m2 > 0, which can give a chemical stable dual boundary system and zero temperature black hole solution whose IR geometry has the form of asymptotic AdS2 geometry. With this, we consider the possibility of spontaneous symmetry breaking of this system in low temperature. This can be analyzed by solving the following equation in the background (3.10), = 0: The condition of instability for in some low temperature is that the BF bound in AdS4 is retained but the BF bound in AdS2 is violated. Near the horizon for an extremal black brane, the geometry is asymptotic AdS2. In this region, the asymptotic solution for equation (3.9) is, +r (1 p1+4 m~2)=2 + r (1+p1+4 m~2)=2; m~ 2 = So the conditions for spontaneous condensate are, m~ 2 = One can immediately see that these inequalities have solution only when J < 0. Under the restriction of (3.17), there is a critical temperature, lower than which the begins to appear. Then the full coupled equations of motion do not admit an analytical solution. Therefore, we have to solve them numerically. We will use shooting method to solve equations (3.5). In order to nd the solutions for all the ve functions it into the equation of 00 and obtain an equation for p, we must impose suitable boundary 1. There are two kinds of scaling symmetries which are useful when we perform numerical computations: 4J 2=r4)p and then put r; (t; x; y) ! (t; x; y)= ; Under the above two scaling symmetries, we nally have four independent parameters fa(rh); (rh); p(rh)g and horizon radius rh at hand. In this paper, we will x the boundary chemical potential to be unitary. When these four parameters are given, we can integrate the equations out of the horizon to obtain the whole solutions. When we perform the the coupled di erential equations, one should use the second scaling symmetry (3.20) to x the chemical potential for each solution the same. By this method, we get one-parameter solution (r; T ) for T < Tc. And then we can get the behavior of magnetic moment density N with respect to temperature T . order to make action be invariant, we have to have . So when B = 0 but 6= 0 in the source free case, the time reversal symmetry is broken spontaneously. In order to see the main properties of the model, let us rst study the model in probe limit for simplicity. Here we may consider two kinds of probe limit. The rst one is to take the model parameter ! 0 as we did in ref. [23]. This case is just to neglect the back reaction of the massive 2-form eld to the black brane geometry and to the Maxwell eld, but to consider the e ect of the Maxwell eld to the background geometry. This probe limit corresponds to the case in such materials that their electromagnetic response properties are very weak compared with the external eld and have little e ects on the their transport properties. In other words, in this probe limit, the in uence of external eld on the materials is considered, but we neglect the back reaction of electromagnetic response to the external eld and structures of materials such as crystal structure or energy band. The other is to neglect all back reaction of matter elds including the Maxwell eld to the background geometry. In this probe limit, the interaction between the electromagnetic response and external eld is taken into account so that we can study how spontaneous magnetization in uences the electric transport, but they both have little in uence on the structures of materials. We will study these two cases in the following sections separately. Probe limit in the case of Spontaneous magnetization and susceptibility Let us rst investigate the spontaneous magnetization in the limit of limit, we neglect the back reaction of polarization eld to the gauge eld and background geometry. The background geometry and the Maxwell eld can be taken as (r) = (1 f (r) = 1 T = Note that the expression of p(r) can be solved directly. We put it into the equation of (r), = 0: This equation shows that will be spontaneously condensed below a critical temperature only when m2 > 0. When m2 > 0, near the critical temperature where is very small and therefore the term 2 can be neglected in (4.3), thus increasing the chemical potential5 will decrease the e ective mass at the horizon (note that J < 0), which leads that condensed spontaneously below a critical temperature.6 Furthermore, the restrictions on the parameters in the case of m~ 2 = m2 + 4J p2 r2 = 0: is very small, the nonlinear term of can be neglected in (4.3).To nd the critical temperature, we can solve the linearized equation of with initial condition 0 = m6 + 4J (3 4 T ) 4 T m4 + = 0 at r ! Without loss the generality, we take the initial value of at horizon to be unity, and treat T as the shooting parameter to match the source free condition. There will be many solutions for shooting parameters T , we choose the highest one as the critical temperature Tc. We nd the critical temperature Tc= ' 1:78 for the case of given model parameters. When the temperature is lower than the critical temperature Tc, in order to examine whether the polarization can make spontaneous magnetization when the external eld B = 0. We plot the value of + versus (rh) at the horizon in the left gure 2. Each curve in this plot corresponds to di erent temperature. This plot shows a typical example in the high and low temperature cases which correspond to the red and the 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 5This implies that the temperature is decreased in grand canonical ensemble [see (4.1)]. will get spontaneously condensed at a temperature higher than a critical temperature. This solution is unstable and we will not discuss it any more. J = 1=8. The critical temperature Tc= ' 1:78. intersecting point with horizontal axis except a trivial point at the origin, which corresponds to a trivial solution with hand, when the temperature is low enough, we nd that there exists a nontrivial solution system gets into a ferromagnetic phase. As a result, we see that the model indeed can give rise to a paramagnetism/ferromagnetism phase transition in the case without external magnetic eld. Furthermore, when the temperature is lower than the critical temperature Tc, we have magnetic moment with respect to temperature is shown in the right panel of gure 2. In addition, since the magnetic moment of the polarization eld obtains an expectation value, the time reversal symmetry is broken spontaneously. By tting this curve near the critical temperature, we nd that there is a square root behavior for the magnetic moment versus temperature, which is a typical behavior for a J = 1=8, we have N 2= 4 2 ' 4:910(1 T =Tc): Except for the magnetic moments which is one of the characteristic properties of ferromagnetic material, another remarkable one is the behavior of susceptibility density of the material in the external magnetic eld. The static susceptibility density is de ned by = lim 1.0005 1.001 1.0015 1.002 1.0025 1.003 critical temperature when m2 = J = 1=8. Here we set 2 2 = 1 for convenience. When we turn on the external magnetic eld B, the function is nonzero in any temperature. In order to compute the susceptibility density, we need to shoot for the boundary conditions with one parameter (rh) for equation (4.3) under the given external magnetic B and temperature T . From the de nition of , which involves only the behavior of B ! 0, so the computation can be simpli ed in the following way. In the case of T > Tc, considering the result = 0 when B = 0, we expect that is in the same order as B, then 2 is as the same order as B2. So in the case with weak magnetic eld, we can neglect nonlinear terms in equation (4.3). In that case, we just need to solve the \linearized" equation = 0: In gure 3, we show the magnetic susceptibility as a function of temperature and nd it satis es the Curie-Weiss law of ferromagnetism in the region of T ! Tc+. Concretely, for the chosen model parameters, we have So we can conclude that the dual system is in a paramagnetic phase in high temperature and ferromagnetic phase in low temperature. The model can describe a paramagnetism/ferromagnetism phase transition. We will see later that the exponents in eqs.(4.6) and (4.10) are exact and can be obtained by analytical methods. = N = ' 4:0499(T =Tc In action (2.1), two parameters appear in the Lagrangian. One is the mass of the polar eld, which corresponds to the conformal dimension of the dual operator by the standard dictionary. The other is the self-interaction coupling constant J . So a natural question is what the meaning of this constant is in the dual boundary theory? To answer this question, we need to compare our holographic GL free energy with the one in microcosmic theories. Unfortunately, in this bottom-up setup of the holographic model, we have no such microcosmic theories in strong correlated system. But, in general, a universal phenomenological theory can emerge from several di erent microcosmic theories. In this subsection, we will try to give a process about how to give out a quantitative interpretation if we know the microcosmic theory in the boundary. As an example to perform this process, we assume that some microcosmic theories could be approximately described by Ising-like universal class model. In this subsection, we also will build a holographic Ginzburg-Landau formulation for our holographic model. By it, we can understand the properties of our holographic model at critical point in the probe limit by analytical method. Since we are working in the probe limit, the geometry and external Maxwell eld are xed, this leads to a simpli cation to compute the partition function of the bulk theory. This is the reason that we consider this kind of probe limit here. Our method is to compute the e ective grand thermodynamic potential in both sides and to equate them (which are equivalent to compute the partition functions in both sides and to equate them). Then we can \read o " the meaning of J in the dual boundary theory. Now let us frist consider the gravity side of the holographic model to compute the grand thermodynamic potential. It is convenient to make a coordinate transformation by 4J 2z4)2f z2 = 0: As we will care about the behavior of T ! Tc, the value of will be a small quantity near the transition point. In this case, we can make a Taylor's expansion on the nonlinear term of in eq. (4.11) as, 4J 2z4)2 = Neglecting the high order terms, eq. (4.11) can be rewritten as Lb = Jef 3z8 Jef = q(z) = m2 + 32J 2 2=m6 < 0: = e Up to the order of 4, the part of polarization eld in action (2.1) can be written as, S(T; B; ) 2 Lb + B which is a function of T and B, but a functional of . The asymptotic solution for (4.13) is +z (1+ )=2 + z (1 )=2: potential or free energy in grand canonical ensemble (T; B; ) = e(T; B; )V2 = 2 Lb + B According to thermodynamic relationship, d (T; B) = V2N dB ) N= 2 = It seems that the magnetic moment should be, @ (T; B) N = N= 2 = However, in our previous papers, we de ned the magnetic moment as, (kT; k2B; k2 ) = k (T; B; ), which gives, sion (4.18) is not true. The reason is as follows. The relation (4.17) is on-shell, while the equation (4.16) is o -shell. In order to obtain the di erential relation of (T; B; ) with respect to (T; B; ), we need to use the Euler homogenous function theorem. We should rst note that under the scaling transformation z ! kz; (t; x; y) ! k(t; x; y), we have (T; B; ) = Submitting (4.16) into (4.20) and considering the on-shell condition = 0, we nd N= 2 = @ (T; B) This is just the de nition (4.19). Sturm-Liouville problem,7 which is the following ODE: The key step for computing the grand thermodynamic potential is to construct the Pb n = Lb n = n = 7The method is similar to the one used in ref. [32], but is completely di erent from the Sturm-Liouville (SL) eigenvalue method in ref. [33], there the precision depends on the trial function one chooses. with the boundary conditions: (b) At z ! 0+, we impose n(0) = 0. The weight function !(z) can be an arbitrary positive continuous function in the region of [0; zh]. From a practical point of view, we choose weight function such that the values of n will not in uence the asymptotic behaviors of equation (4.22) when z ! 0+. There k with an integer k > 2. Once note that the asymptotic solution when r ! 1 for equation (4.22) is, n = +z (1+ )=2 + z (1 )=2; be the Hilbert space of square integrable functions on [0; zh], i.e., L2([0; zh]; !(z); dz) = !(z)jh(z)j2dz < 1 with the inner product hh1; h2i = !(z)h1(z)h2(z)dz; and D be the subspace of L2([0; zh]; !(z); dz) that satis es the boundary conditions of (a) and (b), i.e., Then we can prove that Pb is the self-adjoint operator on D, i.e., 8h1; h2 2 D; hh1; Pbh2i = hPbh1; h2i: According to the properties of SL problem, the solutions of (4.22) form a function basis on D with which one can expand any functions belonging to D, i.e., with cn = h n; hi. h n; ki = nk; 8h 2 D; 9fcng R; h(z) = X cn n(z) Note that all the parameters in equation (4.22) depend on temperature, so do the eigenvalues n and eigenfunctions n. Indeed, the minimal eigenvalue, i.e., the rst eigenvalue 1 is the function of temperature. There is a critical temperature Tc, at which we have 1 = 0. We can nd that when T > Tc, 1 > 0. In order to show this, we take at temperature of T ' 2:00Tc(left give the eigenvalues n. Here Tc ' 1:7766 . eigenvalue n. We can see that all the eigenvalues are positive when T > Tc and when T < Tc. For any temperature, the equation has in nite eigenvalues and the smallest eigenvalue exists, which shows the system is stable. Let us now turn our attention to the free energy (4.16). For convenience, we will use into the case of xing chemical potential in the nal results. dose't need to be the solution of EoM (4.13). We can use the eigenfunction n to expand (r) and magnetic moment as, N = edz = 2 n=1 cn = e ndz; Nn = where cn and Nn are coe cients, de ned as, Here we have assumed that f ng is an unit base. Then the variational principle of (T; B; ) underlying the equations of motion, or nding a solution of EoM (4.13), is equivalent to (T; B; cn) with respect to cn's. this case, we have e(T; cn) = Jef z8 4=4i ; = P1 e(T; cn) = 8 4dz = 2 n=1 If T > Tc, then n > 0. Because of Jef < 0, we can nd that e(T; cn) 0. The minimization only when 1 < 0, i.e., T < Tc. This is just what we have obtained in the pervious section. When T ! Tc , we can set 1 = a0(T =Tc 1) with a0 > 0 and assume that the o -shell solution is dominated by the rst term in (4.30) only, i.e., jc1j 2 in (4.31). As a result, we have, dz 14z8 ' 2 a0(T =Tc with a1 = 14 R01 14z8dzjT =Tc > 0 and, Put (4.35) into (4.34), we can obtain, c1N1=2: e(T; cn) ' e(T; N ) ' 22aN02 (T =Tc We can see that this is just the Ginzburg-Landau (GL) theory of ferromagnetic model. To understand the meaning of the parameters appearing (4.36), we can compare it with some suitable microcosmic model. Unfortunately, a universal microcosmic model to describe spontaneously magnetization in strongly coupling system has not yet been built. One of classes of theory is so-called Ising universality, which is an approximate theoretical model appearing in variety of microcosmic models [34, 35]. Recently, ref. [36] shows that scale invariance implies conformal invariance for Ising universality class in all the dimensions, which implies some intrinsic connections with Ising universality class and conformal eld theory. Here we aren't going to investigate this topic deeply. As an example about how to use our GL-formulism in eq. (4.36), we here assume that the the spontaneous magnetization in the boundary theory can be described by Ising universality class model at least in the vicinity of critical temperature. The Hamiltonian for the Ising universality class model is, H = 1 X r0)!s (r) !s (r0) + X K[!s (r)2 where r; r0 are the positions of lattices, J (r r0) > 0 is the exchange integration of lattices r and r0, s(r) the z-component of spin at lattice r, K is a positive constant which corresponds to the deviation from the Ising model. When K ! 1, Hamiltonian (4.37) reduces to that of the usual Ising model. In this paper, the dual boundary is a (2+1)-dimensional spacetime, i.e., a lm system. compare with our holographic model, it is convenient to change Hamiltonian (4.37) into the one in continuous limit, H = 2 J l2RJ2 s(x)!r2s(x) (J + 2K)s(x)2 + Ks(x)4 ; them, which gives, we can obtain, where l is the lattice spacing and J = P r J (r), R2 = P r2J (r). The summations for J r J and RJ2 are only in one crystal lattice. We see that the Ising like Hamiltonian is a potential in the mean eld approximation in the high temperature limit reads, ms2 = s = temperature Tc = 4 ms2= s E. Here we have used secl to represent the classical value of s. We see that there is a critical When T > Tc, the thermodynamic equilibrium phase Now the grand thermodynamic potentials of the holographic model in the gravity side and in the boundary theory side are in hand, we can use the AdS/CFT duality to relate Here we should mention that the dual theory in the boundary being an Ising-like model is an assumption. With this equality, at the critical temperature, comparing (4.36) with (4.39), QF T = Note that in materials, the spontaneous magnetization is proportional to the expectation value of z-component of spin, i.e., N / secl. Following the de nition of Jef in eqs. (4.13) and interpret the parameter J in action (2.1) as the deviation from the standard Ising model in the boundary theory. Here some remarks are in order. First, the equality eq. (4.41) is exact according to the AdS/CFT correspondence. However, the expressions for the grand thermodynamic potentials in both sides are just some approximations. In gravity side, because we used the probe limit which neglects the back reaction of polarization eld to the background geometry and to external eld and we only computed the grand thermodynamic potential in the classical level which neglects the quantum correction. In the boundary side, we used mean eld expansion and only took the tree level of the quantum uctuation into account. So the relationship (4.42) is only approximately valid and our explanation for J is only a qualitative description. Second, the Ising-like model (4.37) can only describe the local moment system, which is usually suitable for insulator. For metal magnetic materials, we need to use di erent model. So what the meaning of J is in these materials needs to be considered in the future. With the grand thermodynamic potential in eq. (4.36), we can obtain the expression of magnetic moment in the ferromagnetic phase as, N= 2 = T =Tc)1=2: f (r) = 1 This con rmed the critical behavior obtained in the numerical calculations and the critical compare them with the numerical ones. We can also compute the grand thermodynamic potential when magnetic eld is nonzero and get the magnetic susceptibility and hysteresis loop. The details are given in appendix B. Probe limit by neglecting the back reaction of all matter elds Spontaneous magnetization and susceptibility In the probe limit by neglecting the back reaction of all matter elds including the Maxwell eld, the background geometry is just the planar AdS Schwarzschild black brane with, In the background, the equations for Maxwell eld read, Note that in this case, there does not exist the AdS2 geometry, the near horizon geometry of an extremal black brane with vanishing temperature. To have the spontaneous magnetization when the temperature is lowered, the restrictions (3.17) need to be reconsidered. To nd the restriction about the parameters, let us consider eqs. (5.3) in the high temperature region where vanishes and the solutions for and p are the same expressions shown in eqs. (3.10). We can read o the e ective mass square of at the horizon as, m2e = m2 + 4J p(rh)2 = m2 + m4r2 = m2 + Because of J < 0, the temperature term contributes a negative term into the e ective mass square, which is divergent when T ! 0. Thus we see that in the grand canonical ensemble, the instability always appears provided that the temperature is low enough. As a result, in this case, we need not the restrictions in (3.17). But, the parameters have to satisfy the 0 = 0; = 0: m2 + 4J p2 = 0; 2r4 + 16J 2 0 = 0; p = 0: We can directly obtain (r) = Rr1h dr(m2 4J 2=r4)p with (rh) = 0 and put it into the equation of 00, then the equation for p can be obtained. So in this probe limit, we need only solve the three equations of matter elds numerically, condition 2 < 4m2. To see this, let us recall the charge density in eq. (3.12). In this probe limit, we have, Then we reach, We see that it is positive only when 2 < 4m2. Now we consider the spontaneous magnetization in this probe limit. First, let us of probe limit, the polarization is a small quantity near the critical temperature, we can neglect the nonlinear terms of in the equations of and p. Then we can get (r) = rh=r) and p(r) = The equation of is just eq. (3.15). At the horizon, the initial conditions are, (T; ) = 2 1 = 2 1 4 T 3 : : 0 = (rh) = 1: the temperature is also xed. By adjusting the chemical potential, we shoot the boundary to transform our results into the case in grand canonical ensemble where the chemical potential is J = 1=8 = 1=2. The critical temperature is Tc= ' 1:7871. Similarly, we can plot the relationship between + and shooting parameter , in order to examine whether spontaneous condensation when T < Tc. We nd that the solution of source free always appears, which results in the spontaneous magnetization of the system, and breaks the time reversal symmetry in low temperatures. When the temperature is lower than the critical one Tc, we have to solve eq. (5.3) to get the solution of the order parameter , and then compute the value of magnetic moment N , of magnetic moment N as a function of temperature. We see that when the temperature is corresponds to a time reversal symmetry breaking spontaneously. In addition, let us stress here that if the boundary spatial dimension is three, the spatial rotational symmetry is also broken spontaneously, since a nonvanishing magnetic moment chooses a direction as a special. The numerical results show that this phase transition is a second order one with the behavior N / the mean eld theory description of the paramagnetism/ferromagnetism phase transition. Next we calculate the static magnetic susceptibility in this probe limit, de ned by eq. (4.7). Based on the previous analysis, the magnetic susceptibility is still obtained by solving eq. (4.8), and the only di erence is the form of f (r). Thus we can also set the magnetic eld B = 1 and get = 5:7(T =Tc 1), which satis es the Curie-Weiss law of ferromagnetism in the region of Te > Tc. Its inverse is shown in the right panel of gure 5. 1.005 1.01 1.015 1.02 1.025 1.03 1.035 havior of the inverse susceptibility density in the paramagnetic phase near the critical temperature. Here, we choose parameters as m2 = J = 1=8. The temperature Tc= ' 1:7871. DC conductivity in the ferromagnetic phase The electric transport is also an important property in the materials involving spontaneous magnetization. Now let us study how the DC conductivity is in uenced by spontaneous magnetization in this model. In order to simplify our computation in technology, we will work in the probe limit by neglecting back reactions of all the matter elds. This limit can give out the main features near the critical temperature. However, in the case of near zero temperature, we have to consider the model with full back reaction. We will consider this in the future. To compute the conductivity, we have to consider some perturbations for gauge eld with harmonically time varying electric eld. Due to the planar symmetry at the boundary, the conductivity is isotropic. Thus for simplicity, we just compute the conductivity along the x-direction. According to the dictionary of AdS/CFT, we consider the perturbation of polarization eld in the rst order of . As a result, we have to consider the perturbations for all the components of gauge eld and polarization eld. However, if we only care the conductivity in the low frequency limit, i.e., T ! ! 0, the problem can be simpli ed. In the low frequency limit, we only need turn on the three perturbations, Ax = ax(r)e i!t; Mrx = Crx(r)e i!t; Mty = Cty(r)e i!t; 4Jp Crx + O(!) = 0; 4rJ4pf mC2ty + O(!) = 0; and corresponding equations for the three perturbations in the low frequency limit read with p and determined by eqs. (5.3). Here O(!) is the terms with order of ! which can is zero at the horizon, which leads to that the limit of ! ! 0 is ambiguous. However, at the horizon, if we impose the ingoing conditions for Crx; Cty and ax, 2Crx=4)]0 + + O(!) = 0; Cty = Ct(y0) + Ct(y0)(r Crx = e i!r [Cr(x0) + Cr(x1)(r ax = e i!r [a(x0) + a(x1)(r At the AdS boundary with the source free condition, we have the following asymptotic Cty = Cty+r(1+ )=2+Cty r(1 )=2+ ; Crx = ; ax = ax++ ax + ductivity is given by, = lim !!0 i!ax+ As a holographic application of the membrane paradigm of black holes, we can directly obtain the DC conductivity from eqs. (5.9) using the method proposed by Iqbal and Liu in [37]. In fact, the transport coe cients in the dual eld theory can be obtained from the horizon geometry of the dual gravity in the low frequency limit. Applying this into U(1) gauge eld, this conclusion implies that the DC conductivity is given by the coe cient of the gauge eld kinetic term evaluated at the horizon. To see this, we assume that T > 0 and ! ! 0, then we can neglect all the terms of ! in eqs. (5.9). We rst note that, lim r2f (r)(a0x 2Crx=4) = r2 = (1 2=4m2)hJxi: Eq. (5.9c) shows that this quantity is conserved along the direction r. So at the horizon, using eqs. (5.9b) and (5.9c), we have, = r2f r=rh m2 = J = 1=8 and = 1=2. The critical temperature Tc= ' 1:7871. Right panel: temperature dependence of resistivity for various single crystals of La1 xSrxMnO3. Arrows indicate the Curie temperature. For more details, see ref. [38]. Combining eqs. (5.9a) and (5.9b) and considering the fact that Cty is regular at the horizon, Cty = at r ! rh . Thus we have from eqs. (5.15) and (5.14) that 2=4m2)hJxi = lim r2f a0x 1 4(m4 + 16J 2p2 2=r4) Now let us take the ingoing condition for ax at the horizon, which tells us that, nally we get, Here p0 and 0 are the initial values of p(r) and (r) at the horizon, which can be computed from eqs. (5.3). In the low frequency limit, eqs. (5.9) imply that the electric eld is constant, the horizon as, 2=4m2 1 4(m4 + 16J 2p20 20=rh4) r2f a0x = ax = hJxi = i!ax(rh) 4(m4 + 16J 2p20 20=rh4) function of temperature. We see that the resistivity shows a metallic behavior when the temperature is below the Curie temperature. With the appearance of ferromagnetism, DC resistivity decreases when the sample gets cooling, which shows in many interesting phenomena in condensed matter physics, especially in a class of manganese oxides which are widespread because of the discovery of colossal magnetoresistance (CMR) and receive a lot of interest both in theory and experiment [39, 40]. Note that this e ect has a completely di erent physical origin from the \giant" magnetoresistance observed in layered and clustered compounds. Over the past twenty years, CMR is among the main topics of study within the area of strongly correlated electron systems and its popularity is reaching the level comparable to that of the high-temperature superconducting cuprates. In the right panel of gure 6, we show the experimental data from a typical CMR material La1 xSrxMnO3 as an example. We see that our model gives a very similar behavior as the latter with x 0:175. Of course, we should mention here that there still exist some di erences between our model result and experimental data on CMR. In general, when T > Tc, the material shows a semiconductor or insulator behavior and the DC resistivity increases with cooling the sample. In our model, however, the DC resistivity is a constant when T > Tc. So this model only gives partial property of CMR when T < Tc. But this is an exciting and enlightening result, because it implies that this model can lead to a possibility to build a holographic CMR model and to investigate this typical and important strong correlated electrons system in the AdS/CFT setup. We are going to investigate this issue in the future.8 Phase transition with back reaction In previous sections, we have studied the spontaneous magnetization in two kinds of probe limit. However, probe limit may lead to some information lost. For example, in some holographic superconductor models, it will lead to the appearance of the rst or zeroth order phase transition when the strength of back reaction gets beyond some values (see refs. [26, 42, 43], for example). In addition, when the temperature is low enough, the probe limit may lose its validness. To have a complete phase diagram for the holographic model, we need go beyond the probe approximation and include the back reaction. On-shell free energy The model admits various solutions, in order to determine which phase is thermodynamically favored, we should calculate the free energy of the system for both normal phase and condensed phase. In gauge/gravity duality the grand potential of the boundary thermal state is identi ed with temperature T times the on-shell bulk action with Euclidean signature. Since we are considering a stationary problem, the Euclidean action is related to the Minkowski one by a minus sign as SE = where Lm = + 2L2 and g is the determinant of the metric. We rst show that, when evaluated on a solution, this action reduces to a simple surface term at the AdS 8A holographic realization of metal/insulator phase transition and the CMR has been recently studied boundary. From the symmetries of the solution (5) and (6), the yy component of the stress energy tensor only has a contribution from the terms proportional to the metric. Thus, the gravitational led equations imply that L2 = 2Ryy + 2T yy: The Euclidean action is then SE = dr (2r3f ea=2 f 0ea=2)0 2 )ea=2 action that contains the surface term at r SE = 2r3f ea=2 f 0ea=2jr=r1 2 )ea=2 As the rst item of eq. (6.4) diverges when r ! 1 and must be regulated. This counter terms we need to regulate the action are the standard ones (see for example [25]): Sc:t: = g1( 2K + 4=L) jr=r1 ; 1 is the induced metric on the boundary r = r 1 and K = g the extrinsic curvature (n is the outward pointing unit normal vector to the boundary). The summation SEonu-cslhideellan = SE + Sc:t: is now action becomes, after considering the asymptotical forms in (3.7), nite in the limit r ! 1. The regularized = T SEonu-cslhideellan=V2 = 2 )ea=2 Tensor hairy solutions and phase transition We are interesting in the black brane solutions with nontrivial space-space component the tensor eld. For this, we have to adopt the numerical method to nd such solutions. Without loss of generality, the location of rh can be xed to be unity in our numerical calculation. We are then left with two independent parameters f (rh); p(rh)g. By choosing p(rh) as the shooting parameter to match the source free condition at r ! 1, i.e., Other coe cients can be expressed in terms of those parameters. After solving the set of equations, we can calculate the spontaneous magnetization N and free energy density. With a xed m2, we scan a wide range of J and within the limitation eq. (3.17) in 3-dimensional plane in order to trace out the evolution of critical temperature Tc versus these parameters. Figure 7 plots the critical temperature Tc as a function of J and gure 8 that the critical temperature is weakly dependent on the parameter . Moreover, the smaller the value of m2, the larger the phase transition temperature Tc when the same value of J are considered. For each value of , the analytical black brane solution (3.10) always the surface is cut because that J and should satisfy the relation (3.17). in the case with model parameter m2 = J = 1=8. exists. However, for su ciently low temperature, we always nd additional solutions with which are thermodynamically favored. That is to say, for each value of we take, there is a phase transition occurring at a certain temperature Tc where the black brane developing a new \tensor hair" with nontrival becomes thermodynamically favored. In the dual eld theory side, it means that magnetic moment acquires a vacuum expectation value breaking the time reversal symmetry spontaneously. The left panel of gure 9 presents the condensate as a function of temperature for Tc. For small , the curve is similar to the case with the probe limit (compare the case of gets larger, the condensate increases. However, near the critical temperature, J = 1=8 for both plots. the square root behavior still holds as The free energy di erence of the condensed phase and the normal phase is expressed by , which is plotted in the right panel of gure 9. It is obvious that below the critical temperature Tc, the state with non-vanishing magnetic \tensor hair" is indeed thermodynamically favored over the normal phase because always is less than zero. Moreover, our numerical calculation indicates that the order of the phase transition is only second order, no matter how we increase the strength of the back reaction in the allowed paramdoes not appear in this model. Summary and discussion In this paper we have presented a holographic model to realize the paramagnetism/ferromagnetism phase transition in AdS black brane background by introducing a massive 2eld in the bulk. This 2-form eld couples to the background Maxwell eld strength and carries self interaction. The model admits a new analytical black brane solution with a non-trivial time-space component of the tensor eld. The properties of the black brane solution depend on the canonical ensemble unless the temperature and chemical potential satisfy some additional conditions. In that case, there is no corresponding extreme black hole, i.e., the horizon the space-time geometry is just the planar AdS Schwarzschild geometry but both the U(1) eld and 2-form eld do not vanish. totic AdS2 geometry near the horizon emerges when the temperature tends to zero. By this emergent AdS2 geometry, we obtained the condition under which the spontaneous symmetry breaking can happen. If the parameters satisfy the condition, time reversal symmetry will be broken spontaneously and the paramagnetism/ferromagnetism phase transition can happen when the temperature is lower than a critical value. In order to understand the properties of this holographic ferromagnetic phase transition, we investigated the paramagnetism/ferromagnetism phase transition in two kinds of probe limit and in the case with full back reaction, respectively. In the case of the rst kind of probe limit where the model parameters probe limit neglects the back reactions of the 2-form eld to the Maxwell eld and background geometry and makes it simple to study the behavior of spontaneous magnetization. In this probe limit, we computed the critical exponents by both numerical and analytical approaches, which are agreement with the mean eld results. In addition, we obtained a Ginzburg-Landau-like free energy near the critical temperature for the holographic model and in order to give a possible explanation for the model parameter in terms of microscopic theory in the boundary, we related the Ginzburg-Landau-like free energy in the bulk to the one for a Ising-like model. In the second kind of probe limit, we neglected all back reaction of matter elds to the background geometry but considered the interaction between the tensor eld and the eld. In this probe limit, we are able to study the in uence of spontaneous magnetization on the electric transport properties in a relatively simple way, where the background geometry is xed. We found that the critical exponents are the same as ones in the rst kind of probe limit. We also computed the DC resistivity in this probe limit. It was found that the DC resistivity is suppressed by spontaneous magnetization and shows a metallic behavior. This is very similar to the strong correlated phenomenon named CMR e ect found in the some manganites. Next we considered the case with full back reaction and solved the full equations of motion numerically. It was found that the free energy di erence between the condense phase and normal phase, , is zero at critical temperature and always negative when T < Tc. The phase transition is always a second order one as one increases the strength of the back reaction. Main calculations are made in 3 + 1 dimensions in this paper, but it can be easily extended into the higher dimensional case. In the latter case, the space rotation symmetry will be broken spontaneously when the phase transition happens. In addition, we can also generalize this model to the case with the Lifshitz symmetry in the bulk and study the in uences of the Lifshitz dynamical exponent z on the condensate both in the probe limit and in the case with the back reaction. In addition, in all the calculations in this paper, we assumed that the solution in the bulk or the phase at the boundary is homogeneous, which is a strong assumption. In fact, inhomogeneous solution may exist even in a model whose Lagrangian has translation symmetry and the boundary conditions are homogeneous (see ref. [17], for example). In many materials, the inhomogeneous phase can appear spontaneously in a chemical homogeneous materials. Therefore it is of great interest to study whether the inhomogeneity could appear spontaneously in this model. Although we focused on the ferromagnetic phase transition only in this paper, which has been understood well in condensed matter physics, it o ers a framework rather than only a speci c model, which can be regarded as a basic starting point to understand more complicated phenomenon involving spontaneous magnetization. As a result, there are various prospects to study and we expect more exciting results could be reported in the future. Acknowledgments This work was supported in part by the National Natural Science Foundation of China ( No.11375247 and No.11435006 ), National Natural Science Foundation of China (Grant Nos.11175077) and the Doctoral Program of Higher Education, Ministry of Education, China (Grant No. 20122136110002) Ginzburg-Landau theory for Ising universality class In this appendix, we will compute the grand thermodynamic potential of the Ising-like Hamiltonian (4.38). We can rewrite it into the form of 4 theory, In addition, the Euclidian action contains the linear term of , but this term contributes nothing and can be dropped. And, if we neglect the cubic and quartic order terms of , we can obtain the result in mean eld approximation by only taking the tree level of 9Here we assume the relativistic depression relation. Turn into the Lagrangian form9 and go to the case with Euclidian signature, we have the SE = 1 Z the quantum e ective potential, we take the mean eld expansion method. We split as usual the eld into classical background part and quantum uctuation part, The quadratic part of in the action is then controlled by a kinetic operator of the form, 22SsE [secl] = M 2(secl) = H = dx x ln 1 c0 = (3); c1 = E =2; Here E = 0:5772 function. In this limit we have, is the Euler constant and (3) = 1:202 is the value of Riemannfrom which we can obtain the expression in (4.39). Semi-analytic calculations near the critical temperature Spontaneous magnetization In this appendix, we will compute the values of N1, a0 and a1 appearing in section 4. With these we can get the coe cients in (4.43) and compare with the numerical results in the Let us rst compute N1 and a1. For this we have to rst nd the eigenfunction 1 the grand thermodynamic potential density in the mean eld approximation as, account. This is a good approximation if s and ssecl are small.10 Then it is easy to nd energy and can be neglected. Then the second term in the integral of (A.6) then can be dx x ln 1 can get a series expression by high temperature expansion, where x0 is treated as a small 2M 2(secl). The result of (A.7) does not admit a compact expression. We where x20 = previous sections. which is the solution of, at T = Tc with the conditions, + q(x) n = 0 1(1) = 1; 1+ = 0: 10In the vicinity of critical temperature, ssecl is always small. So in this region, the cubic term can always be neglected. However, s in general may not be a small quantity. In this case, loop-corrections must be considered. T /Tc − 1 −0.01 −0.005 0.005 0.01 −0.01 −0.005 T /Tc − 1 J = 1=8 and ! = z4(left) and ! = z3(right). For convenience, here we do not assume that f ng form an unit base. Thus we have, Pb T = here C1 is the normalization coe cient and N1 = 1dz; a1 = 1 Z 1 C12 = h 1; 1i = In order to compute a0, we need to solve equation (B.1) in the limit T ! Tc . To clarify as two examples. Then we t the relation 1 = a0(T =Tc nd a0. Figure 10 shows for k = 3. We have, N 2= c2 = T =Tc) ' a2(1 T =Tc): give di erent values for N1, a1 and a0, but the same value of magnetic moment N (up to a numerical error). The value of a0 can also be obtained directly by solving ODE (4.22). In the region near the critical temperature, we assum 1 = a0(T =Tc 1). Note that all quantities in (4.22) are solved, then the task to nd a0 becomes to solve a non-homogenous eigenvalue problem, At the AdS boundary, T has the same asymptotic behavior as (4.23), thus we can impose the boundary conditions as We nd that T 2 D. We then use the basis f ng to expand T , i.e., Furthermore we get, 1i = when we perform numerical computation. If we x rh = 1, the shooting parameter is chemical potential . The relation between temperature in grand canonical ensemble and chemical potential is given by, Thus the expression (B.12) can be rewritten as a0 = = c Here c is the critical chemical potential when we x rh = 1 and Tc = 34 c2 is the critical c temperature in grand canonical ensemble. Combining (B.3) and (B.14), we have, N 2= c2 4 = T =Tc): We can see that it is determined by the equation (B.1), but independent of the weight function! The expression (B.14) depends on the weight function, because it depends on C1. In order to check the formula (B.14), let us compute the values of a0 in the cases We see that, up to numerical errors, they are the same as what we have obtained by tting the curve what we obtained in the numerical calculation. Susceptibility and hysteresis loop The solution of equation (4.13) can be expressed as, 0 = Lb + B = X clLb l = X clCl 1 l! l Thus we can get the magnetic moment density as and the magnetic susceptibility cn = ; with n = N= 2 = = 2 = = lim = X cn n B X When T ! Tc+, we have 1 = a0(T =Tc in the summation of (B.21) and its inverse can be expressed as is dominated by the rst term = c = 2a0 (T =Tc In the case of m2 = J = 1=8, we have 2 1 to our numerical result 2 1 = c ' 4:0499(T =Tc = c ' 4:0520(T =Tc 1), which is very close 1) given in the numerical calculation. Consider (4.30), we can rewrite it to TTT >=< TTTccc x 10−3 Using the expansion expression (4.30), we have, assume that the rst term in (4.30) dominates only, i.e., jc1j 2 in (4.31), N= 2 = c1N1=2: Taking n = 1 in (B.25), we have, For a given temperature T ! Tc, we can combine (B.26) and (B.27) to obtain a relation between external magnetic eld B and magnetic moment density N . Figure 11 shows the J = 1=8. We see that it is very similar to what we have obtained in the previous work [23]. Finally, let us notice that in GL theory, the equation for magnetic moment density is, Tc)N + A2N 3 B = 0 with two positive coe cients A1 and A2. However, it is easy to see that the equation for N in our model is di erent from the usual form (B.28) from the GL theory, which can be obtained by combining (B.26) and (B.27) to eliminate c1. Namely, although our model gives the similar results near the critical temperature in GL theory, there exist some di erences between the holographic model and the GL theory even in the region near the critical temperature. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE]. string theory, Phys. Lett. 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Rong-Gen Cai, Run-Qiu Yang, Ya-Bo Wu, Cheng-Yuan Zhang. Massive 2-form field and holographic ferromagnetic phase transition, Journal of High Energy Physics, 2015, 21, DOI: 10.1007/JHEP11(2015)021