Analytical and numerical study of backreacting one-dimensional holographic superconductors in the presence of Born–Infeld electrodynamics
The European Physical Journal C
August 2018, 78:654 | Cite as
Analytical and numerical study of backreacting one-dimensional holographic superconductors in the presence of Born–Infeld electrodynamics
AuthorsAuthors and affiliations
Mahya MohammadiAhmad SheykhiMahdi Kord Zangeneh
Open Access
Regular Article - Theoretical Physics
First Online: 17 August 2018
Received: 07 May 2018
Accepted: 04 August 2018
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Abstract
We analytically as well as numerically study the effects of Born–Infeld nonlinear electrodynamics on the properties of \((1+1)\)-dimensional s-wave holographic superconductors. We relax the probe limit and further assume the scalar and gauge fields to affect the background spacetime. We thus explore the effects of backreaction on the condensation of the scalar hair. For the analytical method, we employ the Sturm–Liouville eigenvalue problem, and for the numerical method, we employ the shooting method. We show that these methods are powerful enough to analyze the critical temperature and phase transition of the one-dimensional holographic superconductor. We find that increasing the backreaction as well as the nonlinearity makes the condensation harder to form. In addition, this one-dimensional holographic superconductor faces a second order phase transition and the critical exponent has the mean field value \(\beta ={1}/{2}\).
1 Introduction
The best-known theory for describing the mechanism behind superconductivity from a microscopic perspective is the BCS theory proposed by Bardeen, Cooper and Schrieffer. According to BCS theory, the condensation of Cooper pairs into a boson-like state, at low temperature, is responsible for an infinite conductivity in the solid state system [1]. However, when the temperature increases, the Cooper pair decouples and thus the BCS theory is unable to explain the mechanism of superconductivity for high temperature superconductors [1]. The correspondence between gravity in an anti-de Sitter (AdS) spacetime and Conformal Field Theory (CFT) living on the boundary of the spacetime provides a powerful tool for calculating correlation functions in a strongly interacting field theory using a dual classical gravity description [2]. According to the AdS/CFT duality proposal an n-dimensional conformal field theory on the boundary is equivalent to gravity theory in \((n+1)\)-dimensional AdS bulk [2, 3, 4, 5, 6, 7]. The dictionary of AdS/CFT duality implies that each quantity in the bulk has a dual on the boundary. For example, the energy-momentum tensor \(T_{\mu \nu }\) on the boundary corresponds to the bulk metric \(g_{\mu \nu }\) [3, 4]. Based on this duality, Hartnoll et al. proposed a model for a holographic superconductor in 2008 [5]. Their motivation was to shed light on the problem of high temperature superconductors. According to the theory of holographic superconductors, we need a hairy black hole on the gravity side to describe a superconductor on its boundary. During the past decade, the investigation of the holographic superconductor has got a lot of attention (see e.g. [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]).
On the other hand, BTZ (Bandos–Teitelboim–Zanelli) black holes, the well-known solutions of general relativity in \((2+1)\)-dimensional spacetime, provide a simplified model to investigate some conceptual issues in black hole thermodynamics, quantum gravity, string theory, gauge theory and the AdS/CFT correspondence [38, 39, 40, 41, 42]. It has been shown that the quasinormal modes in this spacetime coincide with the poles of the correlation function in the dual CFT. This gives quantitative evidence for AdS/CFT [43]. In addition, BTZ black holes play a crucial role for improving our perception of gravitational interaction in low-dimensional spacetimes [44]. These kinds of solutions have been studied from different point of views [45, 46, 47, 48].
Holographic superconductors dual to asymptotic BTZ black holes have been explored widely (see e.g. [19, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]). In order to construct the \((1+1)\)-dimensional holographic superconductors one should employ the AdS\(_{3}\)/CFT\(_{2}\) correspondence. In [49], the \((1+1)\)-dimensional holographic superconductors were explored in the probe limit and its distinctive features in both normal and superconducting phases were investigated. Employing the variational method of the Sturm–Liouville eigenvalue problem, the one-dimensional holographic superconductors have been analytically studied in [50, 51, 52]. It is also interesting to study the \((1+1)\)-dimensional holographic superconductor away from the probe limit by considering the backreaction. In [53], the effects of the backreaction have been studied for s-wave linearly charged one-dimensional holographic superconductors.
Holographic superconductors have also been studied extensively in the presence of n (...truncated)