# Analytic investigation of rotating holographic superconductors

## Abstract

In this paper we have investigated, in the probe limit, *s*-wave holographic superconductors in rotating \(AdS_{3+1}\) spacetime using the matching method as well as the Stürm–Liouville eigenvalue approach. We have calculated the critical temperature using the matching technique in such a setting and our results are in agreement with previously reported results obtained using the Stürm–Liouville approach. We have then obtained the condensation operators using both analytical methods. The results obtained by both these techniques share the same features as found numerically. We observe that the rotation parameter of the black hole affects the critical temperature and the condensation operator in a non-trivial way.

## 1 Introduction

High-\(T_{c}\) superconductors were discovered by Bednorz and Müller [1]. These are prototype of the so called strongly correlated systems in condensed matter physics. Such systems generically fall under strongly coupled field theories. However, due to strong coupling these systems are hard to tame using traditional field theoretic approaches. In the last decade, the AdS/CFT correspondence has emerged as a powerful tool to study such systems. The AdS/CFT correspondence originated from string theory and was discovered by Maldacena in 1997 [2]. The conjecture implies that a gravity theory in a (\(d+1\))-dimensional anti-de Sitter (AdS) spacetime is an exact dual to a conformal field theory (CFT) living on the *d*-dimensional boundary of the bulk spacetime.

The importance of the gauge/gravity correspondence in strongly coupled systems was realized in 2008 when Gubser showed that for an asymptotic AdS black hole, near to its horizon, *U*(1) gauge symmetry spontaneously breaks giving rise to the phenomenon of superconductivity in the vicinity of the black hole horizon [3]. Immediately using this result Hartnoll, Herzog, and Horowitz used the AdS/CFT correspondence to study holographic superconductors which mimics the properties of high-\(T_{c}\) superconductors [4, 5]. Since then various aspects of such holographic superconductors have been explored in various black hole space-time settings [6, 7, 8, 9, 10, 11]. Most of the holographic superconductors considered so far are constructed using non-rotating black hole spacetime. However, even before the discovery of AdS/CFT correspondence, rotating black holes were known to show Meissner like effect [12, 13]. This motivates us to investigate about the role of rotating black holes in holographic superconductor models. Such a study was first initiated in [14] where the study of spontaneous symmetry breaking in \(3+1\)-dimensional rotating, charged AdS black hole was carried out. It was observed that the superconducting order gets destroyed for a particular value of the rotation just below the critical temperature. Recently, a holographic superconductor model in rotating black hole spacetime was studied in [15]. Here, the Stürm–Liouville (SL) eigenvalue approach was used to obtain the critical temperature analytically. However, the condensation operator was studied numerically. Our aim in this paper is to provide an analytical approach to rotating holographic superconductors and obtain the critical temperature as well as the condensation operator. We shall first use the matching technique to obtain the critical temperature for both possible values of the conformal dimension. This technique was introduced in [16] and involves the matching of the solutions to the matter and gauge field equations near the black hole horizon and the AdS boundary. We then compare our results with those obtained from the SL technique in [15]. Using the matching approach, we then obtain the values of the condensation operator. Further, we also apply the SL eigenvalue approach to obtain the condensation values and compare the results with those obtained by us using the matching technique and also with the numerical results in the literature. All the calculations in this paper have been done in the probe limit where we do not consider the backreaction of the matter sector into the metric in our model.

The paper is organized in the following way. In Sect. 2, we set up a simple model for holographic superconductors in an uncharged rotating black hole background in \(AdS_{3+1}\) spacetime. In Sect. 3, we calculate the critical temperature and the condensation operator values using matching method, where we match the solutions to the field equations obtained at the boundaries at some appropriate point between the boundaries. Then in Sect. 4, we go on to calculate the critical temperature as well as the condensation operator values using the SL eigenvalue analysis. In the last Sect. 5 of this paper, we summarized our findings.

## 2 Setting up of the holographic superconductor

*a*being the rotation parameter of the black hole. For convenience, we take unit AdS radius and the cosmological constant \(\Lambda =- \ 3\). The Hawking temperature associated with the above black hole geometry is given by [15]

*r*, we set \(\Psi =\psi (r)\). We also make the ansatz \(A_{\mu }=\delta ^{t}_{\mu }\Phi (r)+\delta ^{\varphi }_{\mu }\Omega (r)\) because of the presence of the \(g_{t\phi }\) term in metric (1). Now varying the Lagrangian density \({\mathcal {L}}\), we get the following equations for the matter field \(\psi (r)\) and the gauge fields \(\Phi (r)\) and \(\Omega (r)\)

*r*.

*z*.

## 3 Critical temperature and condensation values using matching method

In this section, we shall apply the so called matching method [18] to find the critical temperature and the condensation operator values. In this approach one writes down the approximate solutions of the field equations for \(\psi , \Phi \) and \(\Omega \) near the horizon and around the AdS boundary and then match them at some point to determine the unknown coefficients in the solutions. Before we proceed, we note that the finiteness of \(A_{\mu }\) at the horizon gives the boundary conditions \(\Phi (1)=0\) and \(\Omega (1)=0\). We carry out our analysis for \(m^{2}=-2\) which is within the Breitenlohner–Freedman (BF) mass bound [19, 20].

### 3.1 Solution near the event horizon (\(z = 1\))

### 3.2 Solution near the asymptotic AdS region (\(z = 0\))

### 3.3 Matching and phase transition

*T*and \(T_{c}\) with \(T_{c}\) being given by

### 3.4 Condensation operator with \(\Delta = 1\)

*a*, we find

*a*from \([0,a_{min}]\) where the critical temperature decreases and hence superconductivity is not favoured. However, for the rotation parameter \(a > a_{min}\), the crtical temperature increases indicating a situation favouring superconductivity. It is interesting to note that such an observation was made earlier in the literature in the context of Kerr black holes [12, 13]. There it was observed that higher rotation value of the black hole favours superconductivity by expelling magnetic field. We also observe that the value of the condensation operator decreases with the increase in the rotation parameter of the black hole. These observations have been presented in Fig. 1a, b where the matching method results have also been compared with the SL results.

### 3.5 Condensation operator with \(\Delta = 2\)

In the table below, we have presented the results for the critical temperature \(T_{c}\) for different choices of the matching points \(z_{m}\) for the two possible values of the condensation operators. It can be seen that the results agree with the SL results given in [15] (Table 1).

## 4 Stürm–Liouville analysis

Critical temperature at different matching points for \(\Delta = 1, 2\)

Matching point, \(z_{m}\) | \(T_{c}\) from matching method | |
---|---|---|

\(\Delta = 1\) | \(\Delta = 2\) | |

0.1 | 0.1675 \(\eta \sqrt{\rho }\) | 0.1486 \( \eta \sqrt{\rho }\) |

0.3 | 0.1582 \(\eta \sqrt{\rho }\) | 0.1228 \(\eta \sqrt{\rho }\) |

0.5 | 0.1419 \(\eta \sqrt{\rho }\) | 0.1038 \(\eta \sqrt{\rho }\) |

0.7 | 0.1203 \(\eta \sqrt{\rho }\) | 0.0858 \(\eta \sqrt{\rho }\) |

### 4.1 Analysis for \(\Delta =1\)

*a*and Eq. (63) by (\(1 + a^2\)) and adding them up together with the condition (49), we arrive at the following equation that is entirely given in terms of \(\Lambda (z)\)

*z*between 0 and 1, we get the following result

*z*on both sides of the above equation gives

*a*affects the condensate.

### 4.2 Analysis for \(\Delta =2\)

*a*and Eq. (86) by (\(1 + a^2\)) and adding them up together with the boundary condition given in Eq. (49), we arrive at the following equation

*z*between 0 and 1, we arrive at following result

*z*on both sides of the above equation, we get

*T*and \(T_{c}\), we arrive at the following result

## 5 Conclusions

We now summarize our findings. In this paper we have analytically investigated a model for the rotating holographic *s*-wave superconductor in the probe limit. We have calculated the critical temperature and the condensation operator values for the two possible conformal dimensions \(\Delta = 1, 2\) using matching method as well as Stürm–Liouville eigenvalue analysis. From our investigation we notice that if we increase the rotation parameter, *a*, of the black hole, the critical temperature first decreases and thereafter it again starts to rise from the value of \(a \approx 0.5165\). This behaviour of the critical temperature obtained using the matching method as well as the Stürm–Liouville analysis, for \(\Delta = 1, 2\), is shown in Fig. 1a. From the figure, it is evident that there is a value of the rotation at which the critical temperature is a minimum which indicates that the superconductivity is not favoured for this value of the rotation parameter of the black hole. This is similar to the observations made in [14] where it was observed that below \(T_{c}\), the transition temperature at zero rotation, there exists a critical value of the rotation which breaks the superconductivity order. In Fig. 1b, we find that the condensation operator values show a second order phase transition but keep on decreasing with increase in the value of the rotation parameter of the black hole. For higher values of the rotation parameter, value of the condensation operators falls sharply. However, it is interesting to note that for rapidly rotating black holes (\(a \rightarrow 1\)), a small amount of condensate forms with a higher value of the critical temperature than their non-rotating counterpart as can be clearly seen in Fig. 1a, b. This is interesting since such an observation was made long back where it was demonstrated that higher values of spin is favourable for superconductivity [12, 13].

In Fig. 1b, (up) and (down), upper most curve represents values of the condensation operator for non-rotating black hole, that is \(a = 0\), and subsequently lower curves are associated with increasing values of the rotation parameter of the black hole such that the lowest one corresponds to the maximally rotating black hole with the rotation parameter \(a = 1\).

## Notes

### Acknowledgements

SG acknowledges the support by DST SERB under Start Up Research Grant (Young Scientist), File no. YSS/2014/000180. SG also acknowledge the Visiting Associateship at IUCAA, Pune. The authors would also like to thank the referee for useful comments.

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