Analytical calculation on critical magnetic field in holographic superconductors with backreaction

# Analytical calculation on critical magnetic field in holographic superconductors with backreaction

Xian-Hui Ge*)*)*)E-mail: gexh@shu.edu.cn and Hong-Qiang Leng†)†)†)E-mail: lenghq88@shu.edu.cn

## 1 Introduction

The gauge/gravity duality [1, 2, 3] as the most fruitful idea stemming from string theory, has been proved to be a powerful tool for studying the strongly coupled systems in field theory. By using a dual classical gravity description, we can effectively calculate correlation functions in a strongly interacting field theory. Recently, a superconducting phase was established with the help of black hole physics in higher dimensional spacetime[4, 5, 6, 7].

Counting on the numerical calculations, the critical temperature was calculated with and without the backreaction for various conditions [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. The behavior of holographic superconductors in the presence of an external magnetic field has been widely studied in the probe limit[25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. The analytical calculation is useful for gaining insight into the strong interacting system. If the problem can be solved analytically, however vaguely, one can usually gain some insight. As an analytical approach for deriving the upper critical magnetic field, an expression was found in the probe limit by extending the matching method first proposed in [9] to the magnetic case [32], which is shown to be consistent with the Ginzburg-Landau theory.

Most of previous studies on the holographic superconductors focus on the probe limit neglecting the backreaction of matter field on the spacetime. The probe limit corresponds to the case the electric charge or the Newton constant approaches zero. The backreaction of the spacetime becomes important in the case away from the probe limit. At a lower temperature, the black hole becomes hairy and the phase diagram might be modified. Recently, an analytical calculation on the critical temperature of the Gauss-Bonnet holographic superconductors with backreaction was presented in [21] and confirmed the numerical results that backreaction makes condensation harder[19, 20, 13, 14] .

Considering the above facts, it would be of great interest to explore the behavior of the upper critical magnetic field for holographic superconductors in the presence of the backreaction. In this work, we will first consider the effect of the spacetime backreaction to -wave holographic superconductors without the magnetic field. Different from the probe limit case, the backreaction of spacetime actually leads to a charged black hole solution in AdS space at the leading order. We will compute the critical temperature analytically by using this charged black hole metric through the matching method. Secondly we will study the properties of holographic superconductors in the presence of external magnetic field. When we turn on the external magnetic field, the resulting background geometry becomes the dyonic black hole solution in AdS space to the zeroth order. The analytical investigation on the effect of the spacetime backreaction to the upper critical magnetic field has not been carried out so far. Therefore, the contents in this paper will be greatly different from the probe limit case as we consider the spacetime backreaction. Note that in both cases, the small backreaction approximation shall be used to obtain an analytical result. In order to compare the analytic study with numerical results, we will also carry on numerical computation.

The organization of the paper is as follows: We first consider the effects of backreaction on -dimensional -wave holographic superconductors in section 2. Without the magnetic field, the critical temperature with backreaction will be derived first. Then we continue the calculation to the strong external magnetic field case and find an analytical expression for the backreaction on the upper critical magnetic field. From the Einstein equation, we know that the presence of charge and magnetism in -dimensional spacetime yields a dyonic black hole solution. The critical temperature may influenced by the backreaction of the magnetism. We will compare the analytic and numerical results. We present the conclusion will be presented in the last section.

## 2 (2+1)-dimensional s-wave holographic superconductors

In this section, we first investigate the backreaction of electric field on superconductivity and derive the phase transition temperature in this case. After that, we turn to the backreaction of the external magnetic field and calculate the critical magnetic field.

### 2.1 Critical temperature with backreaction in the absence of magnetic fields

We begin with a charged, complex scalar field into the 4-dimensional Einstein-Maxwell action with a negative cosmological constant

 S=116πG4∫d4x√−g{R−2\mathchar28931\relax−14FμνFμν−|∂μψ−iqAμψ|2−m2|ψ|2}, (2.1)

where is the 4-dimensional Newton constant, the cosmological constant and . The hairy black hole solution is assumed to take the following metric ansatz

 ds2=−f(r)e−χ(r)dt2+dr2f(r)+r2l2(dx2+dy2), (2.2)

together with

 Aμ=(ϕ(r),0,0,0),   ψ=ψ(r). (2.3)

The Hawking temperature, which will be interpreted as the temperature of the holographic superconductors, is given by

 T=14πf′(r)e−χ(r)/2∣∣∣r=r+, (2.4)

where a prime denotes derivative with respect to and is the black hole horizon defined by . can be taken to be real by using the transformation. The gauge and scalar equations become

 ϕ′′+(χ′2+2r)ϕ′−2q2ψ2fϕ=0, (2.5) ψ′′+(f′f−χ′2+2r)ψ′+(q2ϕ2eχf2−m2f)ψ=0. (2.6)

The and components of the background Einstein equations yield

 f′+fr−3rl2+κ2r[eχ2ϕ′2+m2ψ2+f(ψ′2+q2ϕ2ψ2eχf2)]=0, (2.7) χ′+2κ2r(ψ′2+q2ϕ2ψ2eχf2)=0. (2.8)

When the Hawking temperature is above a critical temperature, the solution is the well-known AdS-Reissner-Nordstrm (RNAdS) black holes

 f=r2l2−1r(r3+l2+κ2ρ22r+)+κ2ρ22r2,   χ=ψ=0,   ϕ=ρ(1r+−1r), (2.9)

where . At the critical temperature , the coupling of the scalar to gauge field induces an effective negative mass term for the scalar field,the RNAdS solution thus becomes unstable against perturbation of the scalar field. At the asymptotic AdS boundary (), the scalar and the Maxwell fields behave as

 ψ=r\mathchar28929\relax−+r\mathchar28929\relax+,     ϕ=μ−ρr+... (2.10)

where and are interpreted as the chemical potential and charge density of the dual field theory on the boundary. According to the gauge/gravity duality, represents the expectation value of the operator dual to the charged scalar field . The exponent is determined by the mass as . Note that for both of the falloffs are normalizable and we choose the boundary condition that either or is vanishing. We will impose that is fixed and take as in [7]. Moreover, we will consider the values of which must satisfy the Breitenlohner-Freedman (BF) bound with [37] for the dimensionality of the spacetime in the following analysis.

After introducing the new coordinate , the equations of motion become

 −f′+fz−3r2+l2z3+κ2r2+z3[z4eχ2r2+ϕ′2+m2ψ2+f(z4r2+ψ′2+q2ϕ2ψ2eχf2)]=0, (2.11) −χ′+2κ2r2+z3[z4r2+ψ′2+q2ϕ2ψ2eχf2)]=0, (2.12) ϕ′′+12χ′ϕ′−2q2ψ2r2+z4fϕ=0, (2.13) ψ′′−(χ′−f′f)ψ′+r2+z4(q2ϕ2eχf2−m2f)ψ=0, (2.14)

where the prime denotes a derivative with respect to . One may find that the transformation and does not change the form of the Maxwell and the scalar equations, but the gravitational coupling of the Einstein equation changes . The probe limit studied in [7] corresponds to the limit in which the matter sources drop out of the Einstein equations. The hairy black hole solution requires to go beyond the probe limit. In [7], it was suggested to take finite by setting . Recently, the author in [21] proposed to keep finite with setting instead. We will take the latter choice.

In the neighborhood of the critical temperature ,we can choose the order parameter as an expansion parameter because it is small valued

 ϵ≡. (2.15)

We find that given the structure of our equations of motion, only the even orders of in the gauge field and gravitational field, and odd orders of in the scalar field appear here. That is to say, we can expand the scalar field , the gauge field as a series in as

 ϕ=ϕ0+ϵ2ϕ2+ϵ4ϕ4+... (2.16) ψ=ϵψ1+ϵ3ψ3+ϵ5ψ5+... (2.17)

Let us expand the background geometry line elements and around the AdS-Reissner-Nordstrm solution

 f=f0+ϵ2f2+ϵ4f4+... (2.18) χ=ϵ2χ2+ϵ4χ4+... (2.19)

The chemical potential should also expanded as

 μ=μ0+ϵ2δμ2, (2.20)

where is positive. Therefore, near the phase transition, the order parameter as a function of the chemical potential, has the form

 ϵ=(μ−μ0δμ2)1/2. (2.21)

It is clear that that when approaches , the order parameter approaches zero. The phase transition occurs at the critical value . Note that the critical exponent is the universal result from the Ginzburg-Landau mean field theory. The equation of motion for is solved at zeroth order by and this gives a relation . So, to zeroth order the equation for is solved as

 f0(z)=r2+z2l2(1−z)(1+z+z2−κ2l2μ202r2+z3). (2.22)

Now the horizon locates at . We will see that the critical temperature with spacetime backreaction can be determined by solving the equation of motion for to the first order.

At first order, we need solve the equation for by the matching method. The boundary condition and regularity at the horizon requires

 ψ′(1)=r2+m2f′0(1)ψ1(1). (2.23)

In the asymptotic AdS region, behaves like

 ψ1=C+z\mathchar28929\relax+, (2.24)

Now let us expand in a Taylor series near the horizon

 ψ1=ψ1(1)−ψ′1(1)(1−z)+12ψ′′1(1)(1−z)2+... (2.25)

From (2.1), we obtain the second derivative of at the horizon

 ψ′′1(1)=−12(4+f′′0(1)f′0(1)−m2r2+f′0(1))ψ′1(1)−r2+ϕ′0(1)22f′20(1)ψ1(1). (2.26)

Using (2.1) and (2.1), we find the approximate solution near the horizon

 ψ1(z)=ψ1(1)−r2+m2f′0(1)ψ1(1)(1−z)+[−r2+m24f′0(1)(4+f′′0(1)f′0(1)−r2+m2f′0(1))−r2+4ϕ′1(1)2f′0(1)2]ψ1(1)(1−z)2+... (2.27)

In order to determine and , we match the solution (2.1) and (2.1) smoothly at . We find that

 z\mathchar28929\relax+mC+ = ψ1(1)−r2+m2f′0(1)ψ1(1)(1−zm)+[−r2+m24f′0(1)(4+f′′0(1)f′0(1)−r2+m2f′0(1)) (2.28) −r2+4ϕ′1(1)2f′0(1)2]ψ1(1)(1−zm)2, \mathchar28929\relax+z\mathchar28929\relax+−1mC+ = r2+m2f′0(1)ψ1(1)−2[−r2+m24f′0(1)(4+f′′0(1)f′0(1)−r2+m2f′0(1)) (2.29) −r2+4ϕ′1(1)2f′0(1)2]ψ1(1)(1−zm).

Solving the above equation, we obtain the expression for in terms of

 C+=2zm2zm+(1−zm)\mathchar28929\relax+z−\mathchar28929\relax+m(1−1−zm2r2+m2f′0(1))ψ1(1). (2.30)

Substituting the above equation back into (2.29), we get a non-trivial relation provided ,

 2\mathchar28929\relax+2zm+(1−zm)\mathchar28929\relax+−[(1−zm)\mathchar28929\relax+2zm+(1−zm)\mathchar28929\relax++(3−2zm)]r2+m2f′0(1) (2.31) − (1−zm)r2+m22f′′0(1)f′0(1)2+1−zm2r4+m4f′0(1)2−(1−zm)r2+2ϕ′0(1)2f′0(1)2=0.

Note that , and . Plugging these relations back into (2.31), we obtain an equation for

 κ4l42r4+\mathchar28929\relax+2zm+(1−zm)\mathchar28929\relax+μ40−l4(1−zm)2r2+{1+2κ2r2+l4(1−zm)[m2l4(1−zm)\mathchar28929\relax+2r2+(2zm+(1−zm)\mathchar28929\relax+) +6\mathchar28929\relax+l2r2+(2zm+(1−zm)\mathchar28929\relax+)−m2l4r2+(32zm−2)]}μ20 +3m2l2(2−zm)+m4l42(1−zm)−18+3m2l2(1−zm)zm(\mathchar28929\relax+−2)−\mathchar28929\relax+\mathchar28929\relax+=0. (2.32)

The main idea of [21] is to work in the small backreaction approximation together with the matching method so that all the functions can be expanded by and the term in the above equation can be neglected. In this sense, is solved as

 μ0=√21−zmr+l2[3m2l2(2−zm)+m4l42(1−zm) −18+3m2l2(1−zm)zm(\mathchar28929\relax+−2)−\mathchar28929\relax+\mathchar28929\relax+]1/2{1−2κ2l2(1−zm)[m2l2(1−zm)\mathchar28929\relax+4(2zm+(1−zm)\mathchar28929\relax+) +3\mathchar28929\relax+2zm+(1−zm)\mathchar28929\relax+−3m2l2zm4+m2l2]}. (2.33)

Without the term, the expression for can be reduced to the result of the probe limit case. The term in the above equation is positive, which means that increase. By further using the relation , we find an expression for :

 r+=ρ1/2l(1−zm2)1/4[3m2l2(2−zm)+m4l42(1−zm) −18+3m2l2(1−zm)zm(\mathchar28929\relax+−2)−\mathchar28929\relax+\mathchar28929\relax+]−1/4{1+2κ2l2(1−zm)[m2l28(2zm+(1−zm)\mathchar28929\relax+) +3\mathchar28929\relax+2(2zm+(1−zm)\mathchar28929\relax+)−(3zm8−12)m2l2]}. (2.34)

The Hawking temperature is given by

 T=r+4πl2(3−κ2l2μ202r2+). (2.35)

When , the above Hawking temperature reaches the critical point where the order parameter approaches zero. From (2.33) and (2.35), we obtain the critical temperature

 Tc=r+4πl2{3−κ2l2(1−zm)[3m2l2(2−zm)+m4l42(1−zm)−18+3m2l2(1−zm)zm(\mathchar28929\relax+−2)−\mathchar28929\relax+\mathchar28929\relax+]} (2.36)

Together with (2.34), we write the critical temperature in a form as

 Tc=T1(1−2κ2l2T2) (2.37)

where

 −18+3m2l2(1−zm)zm(\mathchar28929\relax+−2)−\mathchar28929\relax+\mathchar28929\relax+]−14, (2.38) T2=36\mathchar28929\relax++m2l28(1−zm)[2zm+(1−zm)\mathchar28929\relax+]+m2l2\mathchar28929\relax+2[2zm+(1−zm)\mathchar28929\relax+] +(12−7zm)m2l28(1−zm)+m4l412. (2.39)

It is easy to check that when , and thus , we have , the exact result obtained in [9] for -dimensional superconductors and . This result is also in good agreement with the numerical result by choosing a proper matching point [7]. We also find that the corrections due to the backreaction is positive for arbitrary in the region . Therefore, we confirm the result found in [21, 19, 20] that the backreaction makes condensation harder. The reason for the decreasing of the critical temperature can be understood from the relation [9]. The value of increases due to the gravitational backreaction and thus decreases.

### 2.2 The upper critical magnetic field with backreaction

In this section, we will explore the effects of the backreaction on the external critical magnetic field. In the neighborhood of the upper critical magnetic field , the scalar field is small and can be regarded as a perturbation. The scalar field becomes a function of the bulk coordinate and the boundary coordinates simultaneously because of the presence of the magnetic field. According to the AdS/CFT correspondence, if the scalar field , the vacuum expectation values at the asymptotic AdS boundary (i.e. )[25, 30]. We can simply write by dropping the overall factor . So, to the leading order, it is consistent to set the ansatz

 At=ϕ0(z),   Ax=0,   Ay=Bc2x. (2.40)

Note that the applied external magnetic is constant and homogenous. Considering the fact that an external magnetic field is included, one may wonder whether such a constant external magnetic field could backreact on the bulk gravity or not. We may need to consider the effects of the spatial component of the gauge field in the superconducting phase and assume the gauge field behaves as

 A=ϕ0(z)dt+b(z)dx. (2.41)

In other words, we need solve the bulk gravity equation for and the resulted metric is anisotropic. following this line, we may obtain a kind of dyonic black hole solution, which includes charge and magnetism. However, we notice that several authors have already discussed such conditions in [36]. In these works, is interpreted as the vector hair of the black hole. At the AdS boundary, behaves as

 b(z)=σ−ξz+... (2.42)

According the AdS/CFT correspondence, is the dual current density and is the dual current source of the holographic superfluid. Of course, it is not proper to regard as a homogenous applied magnetic field. Actually, when we discuss the vortex structure of the holographic superconductors, in general we should consider

 ψ1=ψ1(x,y,z),   At=ϕ(x,y,z),   Ax=Ax(x,y,z),   Ay=Ay(x,y,z). (2.43)

or simply in the polar coordinate as well as the boundary condition that and regular.

In the presence of external magnetic field, not only the matter fields but also the spacetime metric should depend on the coordinates . The background static metric may have the form

 ds2=g00(z,x,y)dt2+gzz(z,x,y)dz2+gxx(z,x,y)dx2+gyy(z,x,y)dy2. (2.44)

In this case, we need solve the Einstein equations

 Rμν−12gμνR−3l2gμν=κ2Tμν, (2.45)

where

 Tμν=FμλFλν−14gμνFλρFλρ−gμν(|Dψ|2+m2|ψ|2)+[Dμψ(Dνψ)∗+Dνψ(Dμψ)∗], (2.46)

together with the Klein-Gordon equation

 1√−gDμ(√−ggμνDνψ)=m2ψ, (2.47)

and the Maxwell equation

 1√−g∂λ(√−ggλμgρσFμσ)=gρσJσ, (2.48)

where we have defined and . In this case, we have three coupled nonlinear partial differential equations involving the metric components, scalar field , the scalar potential and vector potential in which analytic study becomes very difficult to do. Note that we can expand the background geometry in series of

 gμν=g(0)μν+ϵ2g(2)μν+ϵ4g(4)μν+... (2.49)

To solve these equations analytically we will follow the logic as shown in table I. In the absence of the external magnetic field, the backreaction of the electric field to the background geometry leads to the RNAdS black hole solution at the zeroth order. At the linear order, the metric receives no corrections from matter fields and we need only solve the equation of motion for at this moment. As we have done from (2.1) to (2.37), all the equations depends only on the radial coordinate . We obtained the critical temperature with backreaction. When we turn on the external magnetic field, the background spacetime changes because of the presence of . We can still expand , and in series of . At the leading order, the matter field and result in a dyonic black hole solution in AdS space. By solving () at next to leading order, we should obtain the expression for the upper critical magnetic field. The above arguments are actually the logic of the calculation of the whole paper.

After justify the usage of (2.40), we can then solve the equations of motion order by order. The black hole carries both electric and magnetic charge and the bulk Maxwell field yields

 A=Bc2xdy+ϕdt (2.50)

At the zeroth order , we solve the Einstein equation and the line elements of the dyonic black hole metric are given by [38]

 ds2=−f0dt2+r2+z2l2(dx2+dy2)+r2+z4f0dz2, (2.51) ϕ0=μ0−ρr+z,   A(0)y=Bc2x (2.52)

where . The Hawking temperature at the event horizon is evaluated as

 T=r+4πl2(3−κ2l2μ202r2+−κ2l2B2c22r4+). (2.53)

To the linear order, the equation of motion for has its new form

 f0ψ′′1+f′0ψ′1+r2+z4(ϕ20f0−m2)ψ1=−l2z2[∂2x+(∂y−iBc2x)2]ψ1, (2.54)

where the prime denotes derivative with respect to . We use separation of variables

 ψ1=eikyyXn(x)Rn(z), (2.55)

and obtain the equation of a two dimensional harmonic oscillator and a equation for

 −X′′n(x)+(ky−Bc2x)2Xn(x)=λnBc2Xn(x), (2.56) f0R′′n+f′0R′n+r2+z4(ϕ20f0−m2)Rn=λnBc2l2z2Rn,, (2.57)

where is the eigenvalue of the harmonic oscillator equation, denotes the Landau energy level and the prime in (2.56) and (2.57) denote derivative with respect to and , respectively. The equation (2.56)is solved by the Hermite polynomials

 Xn(x)=e−(Bc2x−ky)22Bc2Hn(x). (2.58)

Let us choose the lowest mode in what follows, which is the first to condensate and is the most stable solution after condensation. Actually, the Arikosov vortex lattice is given by a superposition of the lowest energy solutions

 ψ1=R0(z)∑jcjeikjyX0(x), (2.59)

where are coefficients that determine the structure of the vortex lattice.

Now we are going to solve (2.57) by exploring the matching method and find the correction to the upper critical magnetic field away from the probe limit. Again regularity at the horizon requires

 R′0(1)=m2r2+f′0(1)R0(1)+Bc2l2f′0(1)R0(1). (2.60)

The behavior of at the asymptotic AdS boundary is given by

 R0(z)=C+z\mathchar28929\relax+. (2.61)

The scalar potential satisfies the boundary condition at the asymptotic AdS region