Effect of magnetic field on holographic insulator/superconductor phase transition in higher dimensional Gauss-Bonnet gravity

In this paper, we have investigated the effect of magnetic field numerically as well as analytically for holographic insulator/superconductor phase transition in higher dimensional Gauss-Bonnet gravity. First we have analysed the critical phenomena with magnetic field using two different numerical methods, namely, quasinormal modes method and the shooting method. Then we have carried out our calculation analytically using the St$\ddot{u}$rm-Liouville eigenvalue method. The methods show that marginally stable modes emerge at critical values of the chemical potential and the magnetic field satisfying the relation $\Lambda^2\equiv\mu^2-B$. We observe that the value of the chemical potential and hence the value of $\Lambda$ increases with higher values of the Gauss-Bonnet parameter and dimension of spacetime for a fixed mass of the scalar field. This clearly indicates that the phase transition from insulator to superconductor becomes difficult in the presence of the magnetic field for higher values of the Gauss-Bonnet parameter and dimension of spacetime. Our analytic results are in very good agreement with our numerical results.


Introduction
to explain phase transition in strongly coupled system. The important result that one gets by using the gauge/gravity correspondence is the formation of a condensate below a certain temperature called the critical temperature. The idea is to construct a gravity theory in one higher dimension and study its properties. The duality is then applied to extract the properties of the boundary theory. There has been a lot of work on the holographic metal/superconductor transition. However to describe an insulator/superconductor phase transition one has to consider a holographic model in the bulk AdS soliton background [28]- [31]. Further in [32], the response of magnetic field on this phase transition has been studied in Einstein gravity background. However, we note that the effect of magnetic field in Gauss-Bonnet (GB) gravity in arbitrary spacetime dimensions would be important to look at. The reason for this is that GB gravity is a higher curvature gravity theory in higher spacetime dimensions and the Mermin-Wagner theorem claims that the phase transition is affected by higher curvature corrections [7]. Study incorporating the GB gravity background effects on holographic insulator/superconductor phase transition without magnetic field in higher dimensions has been done in [33]. The main purpose of this investigation is to see how the phase transition gets affected in the presence of magnetic field in GB gravity background in higher spacetime dimensions. The effect of the magnetic field in this phase transition is different from that in metal/superconductor phase transition. For metal/superconductor phase transition, one gets critical magnetic field B c whereas in this case one gets a relation between a constant magnetic field B and chemical potential µ. In this paper, we have investigated the effect of magnetic field in presence of GB gravity for holographic insulator/superconductor phase transition in higher dimensional spacetime. We have carried out the investigation both numerically as well as analytically. To see the phase transition from insulator to superconductors, we consider the GB AdS d soliton background. We also consider the symmetric gauge to see effect of magnetic field on this phase transition in GB gravity background. First we employ two numerical approaches, namely, quasinormal mode method and the shooting method to study the critical phenomena. Both these approaches are based on the idea of marginally stable modes [4], [32]. In both these techniques, one finds that marginally stable modes emerge at some critical value of the chemical potential and the magnetic field. The emergence of such marginal stable modes indicate that the AdS d soliton background becomes unstable and a condensate of charged scalar field forms. Then we analytically investigate the same phenomena using Stürm-Liouville (SL) eigenvalue method and the analytical results agree with numerical results. It is observed that the square of the critical chemical potential and the magnetic field satisfies a linear relation Λ 2 ≡ µ 2 − B. The value of Λ 2 increases with higher values of GB parametersα and dimension of spacetime d. This shows that phase transition becomes difficult in the presence of magnetic field for higher values ofα and d. The paper is organized as follows. In section 2, we discuss the basic set up of holographic insulator/superconductor phase transition in the presence of magnetic field. We investigate numerically critical phenomena using quasinormal mode in section 3. In section 4, we do same analysis using the shooting method. In section 5, we analytically investigate the critical phenomena in presence of magnetic field using the SL eigenvalue method. Finally, we conclude in section 6.

Set up in the Gauss-Bonnet AdS d soliton background
In this section, we construct the model of the holographic insulator to superconductor phase transition in the background of the Gauss-Bonnet AdS d soliton background. The metric for Gauss-Bonnet AdS d soliton reads [34] where dx i dx i , (i = 1, .., d − 5), represents the line element of (d − 5)-dimensional hypersurface with no curvature, r 0 is the tip of the soliton, α is related to the GB coupling constant α as α = (d − 3)(d − 4)α and L is the AdS radius. Without any horizon, this space time only have a conical singularity at r = r 0 . Imposing a period β = 4πL 2 (d−1)r 0 for the coordinate χ one can remove this singularity. This line element is the solution of following action where R is the Ricci scalar, R GB = R 2 − 4R µν R µν + R αβγδ R αβγδ is the Gauss-Bonnet term and Λ = −(d − 1)(d − 2)/(2L 2 ) is the cosmological constant. The asymptotic behaviour of f (r) reads with the effective asymptotic AdS scale defined by It should be noted that L 2 ef f = L 2 and L 2 ef f = L 2 2 for α → 0 and α → L 2 4 respectively. The Schwarzschild AdS soliton is recovered by taking the limit α → 0 in eq.(2). The matter Lagrangian for a holographic model of insulator/superconductor phase transition reads where F µν = ∂ µ A ν −∂ ν A µ is the field strength tensor, D µ ψ = ∂ µ ψ −iqA µ ψ is the covariant derivative, A µ and ψ represent the gauge and the scalar fields. The equations of motion of matter fields and gauge fields are To solve these equations we need boundary conditions for these fields. From AdS/CFT correspondence, we know that the asymptotic behaviour of the fields are related to operators in the boundary theory in the following way [4] with where ∆ ± are the conformal dimensions, µ and ρ are interpreted as the chemical potential and charge density in the boundary field theory . In this work we consider ψ (−) = 0, so ψ (+) is related to the condensation operators in the boundary field theory.
To study the effect of the magnetic field in insulator/superconductor phase transition, we take the following ansatz which satisfies the gauge field equation (8) and the boundary condition (10). Introducing z = r 0 r and considering an ansatz of the form ψ = F (t, z)R(x, y)H(χ), we obtain from eq.(7) Since ψ is axis symmetric, the last term in the left hand side of eq.(14) is zero. Hence eq.(14) becomes Schrödinger like equation with two-dimensional harmonic potential, having eigenvalue k 2 = qB(n x + n y + 1) where n x , n y ∈ Z + . We expect that the lowest mode l = 0, n x = 0, n y = 0 will be the first most stable solution after condensation. Setting L = 1 and r 0 = 1, we obtain the equation of motion of F (t, r) to be The solution of eq.(14) reads [32] R(x, y) = e − qB which clearly shows the superconducting condensate will be localized to a finite circular region for any finite magnetic field. The region grows for smaller value of magnetic field and it occupies the whole xy−plane when B → 0.

Critical behaviour via quasinormal modes
In this section we study the critical behaviour via quasinormal modes in GB AdS d soliton background. The analysis of quasinormal modes of the perturbation in a fixed background provides a nice way of getting information about the stability of background spacetime. It turns out that the temporal part of the quasinomal modes behave like e −iωt . Hence if the imaginary part of ω is negative, the mode decays in time. This means that the perturbation fades away thereby signalling the stability of the spacetime background. The reverse situation occurs when the imaginary part of ω is positive. The situation when ω = 0 is the critical case and the mode of the perturbation is said to be marginally stable.
The existence of this mode is also expected to be a sign of instability [4]. In [31], [32], this method has been used to study critical behaviour in Einstein gravity. In this paper, we employ this method in the set up of GB gravity to study the effect of magnetic field on holographic insulator/superconductor phase transition in arbitrary spacetime dimensions. In our analysis, we shall consider effects only up to first order in the GB parameterα. Hence we expand the metric component f (z) upto first order of GB parameter (α) In order to study the phase transition in this background, we further define Substituting this in eq(16), we get Multiplying throughout by z 4 f (z)/(z − 1) in the above equation, we obtain where the coefficients are given by This coefficients S(z), T (z), V (z) are all polynomials and can be written as The coefficients s i , t i , v i can be calculated by comparing with eq. (22). For d = 5, the values of s i , t i , v i are given in Table 1. One can compute these values for other spacetime dimensions as well.  (21) is a second order differential equation with a regular singular point z = 1. Hence one can write down a power series solution of this equation near the tip as Substituting eq.(s)(23, 24) into eq.(21), we find

This in turn implies
In the above equation, we have set the upper limit of i to be ∞ using the fact that s j , t j is zero when j > (2d − 1) and v j is zero if j > 3. Introducing a new variable n = i + j, the above equation takes the form ∞ n=0 n j=0 a j j(j − 1)s n−j + jt n−j + v n−j (z − 1) n = 0 We now separate out the n th -term from above equation to get a recursion relation which reads We now set a 0 = 1 for simplicity and use the boundary condition of the scalar field ψ at z = 0, which reads The solution of this algebraic equation with the a n 's given by eq. (28) gives the value of Λ 2 = µ 2 − B which determines the stability of the system. The smallest value of Λ is the most marginal stable mode.
In the subsequent numerical calculations, we restrict q = 1 and N = 600. From numerical results it is observed that if a marginally stable mode arises, the square of the chemical potential µ 2 and magnetic field B satisfy a linear relation, whose slope is unity and intercept with µ 2 axis gives the square of the critical chemical potential in the absence of the magnetic field. As the magnetic field increases, the critical chemical potential becomes higher. So in the presence of the magnetic field, transition from insulator to superconductor will be more difficult. We have shown the first three lowest critical Λ n 's (n denotes the "overtone number") in Table 2 for Einstein gravity (α = 0) which exactly match with the previous findings [32]. In Table 3 This implies that the critical chemical potential increases for a fixed magnetic field. This in turn shows that the phase transition becomes difficult in presence of the magnetic field in GB gravity background in higher spacetime dimensions.

Critical behaviour via shooting method
An alternative way to numerically study the critical behaviour of the phase transition is the so called shooting method [32]. Here we describe how to use the shooting method to study the critical behaviour in GB AdS d soliton background. Using the shooting method we plot the profile of the scalar field and compare it with the results of quasinormal modes approach. We consider the static case in which F is independent of t. Hence setting ω = 0 in eq.(20), we obtain To study the behaviour of the solution near the tip of the soliton z = 1, we substitute in eq. (20). This gives where k ≡ q 2 µ 2 − qB − m 2 . The solution of this equation near z = 1 reads where α and β are two constants. Since the field is finite at the tip, we have to impose the condition β = 0. Near the boundary W (z) behaves as In the following calculations, we will set ψ (−) = 0 in order to turn off the effect of the source on the boundary field theory. Setting q = 1 for simplicity, we note that at the critical point of the phase transition, W (z) is very close to zero. Therefore, we impose the following conditions at the tip z = 1 The second condition follows from eq. (31). For a given d, m 2 andα, we now solve eq.(30) by the shooting method. We start with the above initial value of W (z) at the tip z = 1 and then numerically solve eq. (30), such that the condition ψ (−) = 0 is satisfied at the boundary. This fixes the values of Λ. This Λ implies that particular combinations of chemical potential µ and magnetic field B satisfy the matter field equation for ψ. The values of Λ 2 ≡ µ 2 − B are shown in Table 2 and Table 3 for Einstein and GB gravity  Figure 1. From Table 3 we find that Λ 0 increases with higher spacetime dimension, GB parameter and mass of the scalar field. This implies that condensation in GB gravity background is harder than in Einstein background, and it becomes more difficult as the number of spacetime dimensions increases and the mass of the scalar field increases.

Critical behaviour via the Stürm-Liouville method
In this section we study analytically the critical behaviour using Stürm-Liouville method. From the last two sections, we observed that when the combination of chemical potential µ and magnetic field B, which is Λ 2 ≡ µ 2 − B, exceeds a critical value Λ 0 for given mass, dimension and GB parameter, the condensations of the operators will happen. This can be viewed as a superconductor phase. For Λ < Λ 0 ,the scalar field is zero and this can be interpreted as the insulator phase. Therefore, the critical parameters satisfying Λ 2 0 = µ 2 − B, are the turning points of the holographic insulator/superconductor phase transition. Here we are trying to find an approximate function to relate the parameters q, µ, B, m 2 , d, α near the critical phase transition point. Starting from eq(30), we introduce a trial function Γ(z) into W (z) near z = 0 as