Holographic paramagnetic-ferromagnetic phase transition with Power-Maxwell electrodynamics

TL;DR

This paper investigates how Power-Maxwell nonlinear electrodynamics influences holographic paramagnetic-ferromagnetic phase transitions in Schwarzschild AdS black holes, revealing effects on magnetic moment formation, critical temperature, and susceptibility behavior.

Contribution

It introduces a massive 2-form coupled to Power-Maxwell fields to study nonlinear effects on magnetic phase transitions in holography, using numerical methods in the probe limit.

Findings

Increasing power parameter makes magnetic moment formation harder.

Critical temperature decreases with higher power parameter.

Magnetic susceptibility follows Curie-Weiss law in external magnetic fields.

Abstract

We explore the effects of Power-Maxwell nonlinear electrodynamics on the properties of holographic s-wave paramagnetic-ferromagnetic phase transition in the background of Schwarzchild Anti-de Sitter (AdS) black hole. For this purpose, we introduce a massive 2-form coupled to the Power-Maxwell field. We perform the numerical shooting method in the probe limit by assuming the Power-Maxwell and the 2-form fields do not back react on the background geometry. We observe that increasing the strength of the power parameter causes the formation of magnetic moment in the black hole background harder and critical temperature lower. In the absence of external magnetic field and at the low temperatures, the spontaneous magnetization and the ferromagnetic phase transition happen. In this case, the critical exponent for magnetic moment is always 1/2 which is in agreement with the result from the mean field theory. In the presence of external magnetic field, the magnetic susceptibility satisfies the Cure-Weiss law.

Tables

Table 1. Table 1: Numerical results of T c / μ subscript 𝑇 𝑐 𝜇 {T_{c}}/{\mu} for different values of α 𝛼 \alpha in 4D and 5D.

$\alpha$	$3/4$		$1$		$5/4$
$4D$	$3.6023$		$1.7870$		$-$
$5D$	$3.9388$		$2.4367$		$2.0103$

Table 2. Table 2: The magnetic moment N 𝑁 N with different values of α 𝛼 \alpha in 4D and 5D.

$\alpha$	$3/4$	$1$	$5/4$
$4D$	$6.0253{\left(1-T/Tc\right)}^{1/2}$	$2.9409{\left(1-T/Tc\right)}^{1/2}$	$-$
$5D$	$3.8213{\left(1-T/Tc\right)}^{1/2}$	$2.9810{\left(1-T/Tc\right)}^{1/2}$	$2.6211{\left(1-T/Tc\right)}^{1/2}$

Table 3. Table 3: The magnetic susceptibility χ 𝜒 \chi with different values of α 𝛼 \alpha .

	$\alpha$	$3/4$	$1$	$5/4$
$4D$	${\lambda }^2/\chi \mu$	$3.7667\left(T/Tc+0.0034\right)$	$3.9906\left(T/Tc-1\right)$	$-$
	$\theta /\mu$	$0.0124039$	$-1.7871$	$-$
$5D$	${\lambda }^2/\chi \mu$	$6.1814\left(T/Tc+0.0022\right)$	$3.7874\left(T/Tc+0.0222\right)$	$3.04777\left(T/Tc+0.0803\right)$
	$\theta /\mu$	$0.0089$	$0.05421$	$0.1614$

Equations

$$I$$

$$I_M=\frac{1}{8{\kappa }^2}\int d^{d}x\sqrt{-g}(-F{)}^{\alpha },$$

$$T_{\mu \nu }=2(\alpha F_{\mu \rho }F{}_{\nu }^{\rho }{F}^{\alpha -1}-\frac{1}{4}{g}_{\mu \nu }{F}^{\alpha }).$$

$$L_2=-\frac{1}{12}(dM{)}^2-\frac{m^2}{4}{M}_{\mu \nu }{M}^{\mu \nu }-\frac{1}{2}{M}^{\mu \nu }{F}_{\mu \nu }-\frac{J}{8}V(M),$$

$$V\left(M\right)={\left({}^{\ast }{M}_{\mu \nu }{M}^{\mu \nu }\right)}^2={[}^{\ast }(M\wedge M){]}^2,$$

$$0$$

$$d{s}^2=L^2\left(-r^{2}f(r)d{t}^2+\frac{d{r}^2}{r^{2}f(r)}+r^2\sum _{i=1}^{d}d{x}_i^2\right),$$

$$f(r)=1-{\left(\frac{r_{+}}{r}\right)}^{d-1},$$

$$T=\frac{f^{\prime }(r_{+})}{4\pi }=\frac{(d-1)r_{+}}{4\pi },$$

$$M_{\mu \nu }=-p(r)dt\wedge dr+\rho (r)dx\wedge dy,$$

$$A_{\mu }=\varphi (r)dt+Bxdy,$$

$$0$$

$$\varphi (r)\sim \mu -\frac{{\sigma }^{\frac{1}{2\alpha -1}}}{r^{\frac{(d-2)}{2\alpha -1}-1}},p(r)\sim \frac{\sigma }{m^2{r}^{(d-2)}},$$

$$\rho (r)\sim \frac{{\rho }_{-}}{r^{{\Delta }_{-}}}+\frac{{\rho }_{+}}{r^{{\Delta }_{+}}}+\frac{B}{m^2},$$

$${\Delta }_{\pm }=\frac{1}{2}\left[(5-d)\pm \sqrt{4m^2+16+(d-1)(d-9)}\right].$$

$$\sigma =-\frac{{\lambda }^2\mu }{4m^2}+\alpha \mu {\left(\frac{2B^2}{r^4}-\frac{2\mu {}^2}{r^4}\right)}^{\alpha -1}.$$

$$N=-{\lambda }^2\int \frac{\rho }{2r^{d-2}}dr.$$

$$r\to ar,f\to f,\varphi \to a\varphi ,$$

$$\rho$$

$$\varphi$$

$$\chi =\underset{B\to 0}{\lim }\frac{\partial N}{\partial B}.$$

$$\chi =\frac{C}{T+\theta },T>T_C,\theta <0,$$

Full text

I Introduction

The AdS/CFT duality provides a correspondence between a strongly coupled conformal field theory (CFT) in $d$-dimensions and a weakly coupled gravity theory in ($d+1$)-dimensional anti-de Sitter (AdS) spacetime 1 ; 2 ; 3 . Since it is a duality between two theories with different dimensions, it is commonly called holography. The idea of holography has been employed in the condensed matter physics to study the various phenomena such as superconductivity hartnoll ; 5 ; 6 ; 7 ; 8 . For describing the properties of low temperature superconductors, the BCS theory can work very well 9 ; 10 . The electronic properties of materials have been studied using the duality in strongly correlated systems. Recently, the magnetism also have been attracted the attentions about the duality application to the condensed matter physics. There are a few works in investigating the magnetism from the holographic superconductors point of view montull ; Donos ; albash ; m.pujo ; iqbal . An example is the holographic paramagnetic-ferromagnetic phase transition in a dyonic Reissner-Nordstrom-AdS black brane which was introduced in Ref. dyonic . This model gives a starting point for exploration of more complicated magnetic phenomena and quantum phase transition. It was considered that the magnetic moment could be realized by a real antisymmetric tensor field which is coupled to the background gauge field strength in the bulk. It was found that the spontaneous magnetization happens in the absence of external magnetic field, and it can be realized as the paramagnetic-ferromagnetic phase transition. This model was extended by introducing two antisymmetric tensor fields which correspond with two magnetic sublattices in the materials p.Acai6 . In the framework of usual Maxwell electrodynamics, holographic paramagnetism-ferromagnetism phase transition have been investigated p.Acai6 ; Coexistence.Cai ; Yokoi ; Cai3 ; Cai4 ; Insulator.Cai ; Understanding.Cai ; Lifshitz5 . However, it is interesting to investigate the effects of nonlinear electrodynamics on the properties of the holographic paramagnetic-ferromagnetic phase transition. Considering three types of nonlinear electrodynamics, namely, Born-Infeld, logarithmic and exponential nonlinear electrodynamics, and using the numerical methods, it has been observed that in the Schwarzschild AdS black hole background, the higher nonlinear electrodynamics corrections make the magnetic moment harder to form in the absence of external magnetic field Zhang2 ; Wu1 . Although, the properties of holographic superconductor with conformally invariant Power-Maxwell electrodynamics have been studied in PM1 ; PM2 ; Shey1 ; Shey2 ; Shey3 , the properties of holographic paramagnetic-ferromagnetic phase transition coupled to the Power-Maxwell field have not been explored yet. For applications of gauge/gravity duality like holographic superconductor model, the gravitational model could not be studied as well as others which satisfy the behavior of boundary theory and the condition of string theory. Since in this viewpoint, the Power-Maxwell nonlinear electrodynamics can be representation the fields with higher order terms, so it can be useful for our investigation. In this paper, we are going to extend the study on the holographic paramagnetic-ferromagnetic phase transition by taking into account the nonlinear Power-Maxwell electrodynamics. In particular, we shall investigate how the Power-Maxwell electrodynamics influence the critical temperature and magnetic moment. Interestingly, we find that the effect of sublinear Power-Maxwell field can lead to the easier formation of the magnetic moment at higher critical temperature. We shall focus on 4D and 5D holographic paramagnetic-ferromagnetic phase transition in probe limit by neglecting the back reaction of both gauge and the $2-$form fields on the background geometry. We employ the numerical shooting method to investigate the features of our holographic model.

This paper is organized as follows. In section II, we introduce the action and basic field equations in the presence of Power-Maxwell electrodynamics. In section III, we employ the shooting method for our numerical calculation and obtain the critical temperature and magnetic moment. In that section, we also study the magnetic susceptibility density. In the last section, we summarize our results.

II Holographic Set-up

We consider a holographic ferromagnetism model in Einstein gravity in a $d$-dimensional AdS spacetime which is given by the action,

[TABLE]

where $\kappa^{2}=8\pi G$ with $G$ is Newtonian gravitational constant, $g$ is the determinant of metric, $R$ is Ricci scalar and $\Lambda=-{(d-1)(d-2)}/{2l^{2}}$ is the cosmological constant of $d-$dimensional AdS spacetime with radius $l$. $L_{1}(F)=(-F)^{\alpha}/4$, where $F=F_{\mu\nu}F^{\mu\nu}$ in which $F_{\mu\nu}=\nabla_{[\mu}A_{\mu]}$ and $A_{\mu}$ is the gauge potential of U(1) gauge field and $\alpha$ is the power parameter of the Power-Maxwell field. In the case where $\alpha$ tend to zero the Power-Maxwell Lagrangian will reduce to the Maxwell case ($L_{1}\rightarrow-F_{\mu\nu}F^{\mu\nu}/4$) and the Einstein-Maxwell theory is recovered. Besides, the Power-Maxwell action is invariant under conformal transformation $g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$ and $A_{\mu}\to A_{\mu}$ in $d$-dimension is given by

[TABLE]

where $F$ is the Maxwell invariant. The associated energy-momentum tensor of the above action is given by

[TABLE]

One can easily check that the above energy-momentum tensor is traceless for $\alpha={d}/{4}$. In this paper, for generality, we consider not only the conformal case, but also the arbitrary value of $\alpha$. This allows us to consider more solutions from different aspects shamsip30 ; dehyadegari . The term $L_{2}$ in action (LABEL:Act) is defined as Cai3

[TABLE]

where $\lambda$ and $J$ are two constants with $J<0$ for producing the spontaneous magnetization and $\lambda^{2}$ characterizes the back reaction of the two polarization field $M_{\mu\nu}$ and the Maxwell field strength on the background geometry. In addition, $m$ is the mass of $2$-form field $M_{\mu\nu}$ being greater than zero Cai3 and $dM$ is the exterior differential $2$-form field $M_{\mu\nu}$. The nonlinear potential of $2$-form field $M_{\mu\nu}$, $V(M_{\mu\nu})$, describes the self interaction of polarization tensor which should be expanded as the even power of $M_{\mu\nu}$. In this model, we take the following form for the potential

[TABLE]

where $*$ is the Hodge star operator. We choose this form just for simplicity. This potential shows a global minimum at some nonzero value of $\rho$ Cai3 .

Varying the action (LABEL:Act) with respect to $M_{\mu\nu}$ and $A_{\nu}$, the field equation read, respectively,

[TABLE]

In the probe limit, we can neglect the back reaction of the 2-form field. As the background geometry, we consider the $d-$dimensional Schwarzschild AdS black hole which its metric reads

[TABLE]

with

[TABLE]

where $r_{+}$ is the event horizon radius of the black hole. The Hawking temperature of black hole on the horizon which will be interpreted as the temperature of CFT, is given by q.pan

[TABLE]

In order to explore the effects of the power parameter $\alpha$ on the holographic ferromagnetic phase transition, we take the self-consistent ansatz with matter fields as follows,

[TABLE]

[TABLE]

where $B$ is a constant magnetic field which is considered as an external magnetic field of dual boundary field theory. Inserting this ansatz into Eqs. (5) and (6), we arrive at

[TABLE]

where the prime denotes the derivative with respect to $r$. Obviously, the above equations reduce to the corresponding equations in Ref. Cai3 when $\alpha\rightarrow 1$ and $d=4$. We should specify boundary conditions for the fields to solve Eq. (12) numerically. At the horizon, we need to impose a regular boundary condition. Therefore, in additional to $f(r_{+})=0$, because the norm of the gauge field, namely $g_{\mu\nu}A^{\mu}A^{\nu}$, should be finite at the horizon, we require $\phi(r_{+})=0$ and $\rho(r_{+})=\frac{(d-1)r_{+}}{m^{2}}\rho^{\prime}+\frac{B}{m^{2}}$. The behaviors of model functions governed by the field equations (12) near the boundary ($r\rightarrow\infty$) are given by

[TABLE]

where $\mu$ and $\sigma$ are respectively interpreted as the chemical potential and charge density of dual field theory, and

[TABLE]

According to AdS/CFT correspondence, $\rho_{+}$ and $\rho_{-}$ are two constants correspond to the source and vacuum expectation value of dual operator when $B=0$. Therefore, condensation happens spontaneously below a critical temperature when we set $\rho_{+}=0$. By considering $B\neq 0$, the asymptotic behavior is governed by external magnetic field $B$. It is important to note that the boundary condition for the gauge field $\phi$ depends on the power parameter $\alpha$ of the Power-Maxwell field unlike other nonlinear electrodynamics such as Born-Infeld-like electrodynamics Zhao ; Jing . Using boundary condition Eq. (13) and the fact that $\phi$ should be finite as $r\rightarrow\infty$, we require that $\frac{d-2}{2\alpha-1}-1>0$, which restricts the values of $\alpha$ to be $\alpha<\frac{d-1}{2}$. On the other hand since $\frac{d-2}{2\alpha-1}>1$, it must be a positive real number, it leads to the range of the parameter $\alpha$ to be $1/2<\alpha<\frac{d-1}{2}$.

In order to see the main properties of the model, we consider the probe limit for simplicity. It is worth noting that we have two kinds of probe limit. In the former case, one may take the model parameter $\lambda\rightarrow 0$ as in Ref. dyonic by neglecting the back-reaction of the massive $2$-form field on the background geometry and the Maxwell field. In that case, the effect of the Maxwell field on the background geometry has been considered. In this probe limit, the influence of external field on the materials is considered, but they neglect the back reaction of the electromagnetic response on the external field and structures of materials. In the latter case one may neglect all back reaction of matter fields including the Maxwell field on the background geometry. In our model, we follow the latter type of the probe limit in which the interaction between the electrodynamic response and external field is taken into account and the $\lambda$ parameter can take any small value. The charge density of the system is given by Cai4

[TABLE]

We see that the properties of this black hole solution depends on the value of $1-\frac{\lambda^{2}}{4m^{2}}$ when $B\to 0$ and $\alpha\to 1$ in above equation. To investigate the physical properties when $1-\frac{\lambda^{2}}{4m^{2}}\neq 0$, one can compute the partial derivative of the charge density with respect to chemical potential and it is easy to check that when $1-\frac{\lambda^{2}}{4m^{2}}>0$ the value of this partial derivative become positive Cai4 . It gives a chemical stable dual boundary system. For $1-\frac{\lambda^{2}}{4m^{2}}<0$, this parameter is negative which leads the dual boundary system to be chemical instability. So in this probe limit, the parameters have to satisfy the condition $\lambda^{2}<4m^{2}$. As an example, we choose $\lambda=1/2$ which is small enough for this parameter.

In the following sections, we will study the holographic ferromagnetic-paramagnetic phase transition numerically.