We study the dynamical stability of holographic superconductors. We first classify perturbations around black hole background solutions into vector and scalar sectors by means of a 2-dimensional rotational symmetry. We prove the stability of the vector sector by explicitly constructing the positive definite Hamiltonian. To reveal a mechanism for the stabilization of a superconducting phase, we construct a quadratic action for the scalar sector. From the action, we see the stability of black holes near a critical point is determined by the equation of motion for a charged scalar field. We show the effective mass of the charged scalar field in hairy black holes is always above the Breitenlohner-Freedman bound near the critical point due to the backreaction of a gauge field. It implies the stability of the superconducting phase. We also argue that the stability continues away from the critical point.
We present an analytic treatment near the phase transition for the critical temperature of (3+1)-dimensional holographic superconductors in Einstein-Gauss-Bonnet gravity with backreaction. We find that the backreaction makes the critical temperature of the superconductor decrease and condensation harder. This is consistent with previous numerical results.
We study (3+1)-dimensional holographic superconductors in Einstein-Gauss-Bonnet gravity both numerically and analytically. It is found that higher curvature corrections make condensation harder. We give an analytic proof of this result, and directly demonstrate an analytic approximation method that explains the qualitative features of superconductors as well as giving quantitatively good numerical results. We also calculate conductivity and $ω_g / T_c $, for $ω_g$ and $T_c$ the gap in the frequency dependent conductivity and the critical temperature respectively. It turns out that the `univers…
Holographic superconductor is an important arena for holography, as it allows concrete calculations to further understand the dictionary between bulk physics and boundary physics. An important quantity of recent interest is the holographic complexity. Conflicting claims had been made in the literature concerning the behavior of holographic complexity during phase transition. We clarify this issue by performing a numerical study on one-dimensional holographic superconductor. Our investigation shows that holographic complexity does not behave in the same way as holographic entanglement entropy. …
We investigate the Stückelberg holographic superconductor in present of the constant external magnetic field. We observe that a critical value of magnetic field exists as the cases in usual holographic superconductor. Furthermore, we find that the applied magnetic field strongly influence the phase transition of this model and have a jump in the condensate at the critical temperature even for $c_{4}=1$.
We develop a holographic superconductor in BTZ black hole background with backreactions. We investigate the influence of the backreaction on the condensation of the scalar hair and the dynamics of perturbation in the background spacetime. When the Breitenlohner-Freedman bound is approached, we argue that only one of two possible operators can reflect the real property of the condensation in the holographic superconductor. This argument is supported by the investigation in dynamics.
In this paper, we have studied the effect of Born-Infeld electrodynamics in holographic $p$-wave superconductors with massive vector condensation. We have analysed this model in the probe limit using a variational method known as the Stürm-Liouville eigenvalue approach. For this $p$-wave holographic superconductor model, we have calculated the critical temperature $T_{c}$ as well as the value of the condensation operator for two different choices of $m^{2}$. We have also pointed out the similarities and dissimilarities between this model for $m^{2} = 0$ and $p$-wave holographic superconductor …
In this paper, we generalize the study of the model of holographic superconductor in the excited states to the framework of massive gravity. By taking into account the effect of massive graviton, we numerically present a family of solutions of the holographic superconductor in excited states, and find that the critical temperatures can be higher due to the effect of massive graviton comparing with the superconductor in Einstein gravity. We also investigate the condensates and conductivities in the ground state and the excited states with studying various parameters that determine the framework…
We employ the variational method for the Sturm-Liouville eigenvalue problem to analytically investigate the properties of the holographic superconductors. We find that the analytic method is still powerful when the backreaction is turned on. Reducing step size in the iterative procedure, we observe that the consistency of results between the analytic and numerical computations can be further improved. The obtained analytic result can be used to back up the numerical computations in the holographic superconductor in the fully backreacted spacetime.
We study $(2+1)$-dimensional holographic superconductors in the presence of non-minimally coupled electromagnetic field to gravity by considering an arbitrary linear combination of $RF^2$-type invariants with three parameters. Our analytical analysis shows that the non-minimal couplings affect the condensate and the critical temperature.
We study a model describing a four-dimensional holographic superconductor whose properties depend non-trivially on a particular spatial direction, namely a Josephson junction. We analyze the parameter dependence of the condensate and compare it to the condensed matter expectations.
Holographic models of superconductors successfully reproduce certain experimental features of high-temperature superconductors, such as a large gap-to-T_c ratio compared to that of conventional superconductors. By deconstructing the extra dimension of these holographic models, similar phenomenology is described by a class of models defined in the natural dimension of the superconducting system. We analyze the sensitivity of certain observables in holographic and deconstructed holographic superconductors to details of the extra-dimensional spacetime. Our results support the notion that certain …
We study the planar anti-de Sitter black hole in the p-wave holographic superconductor model. We identify a critical coupling value which determines the type of phase transition. Beyond the horizon, at specific temperatures flat spacetime emerges. Numerical analysis close to these temperatures demonstrates the appearance of a large number of alternating Kasner epochs.
The variational method for Sturm-Liouville eigenvalue problem was employed to study analytically properties of the holographic superconductor with dark matter sector, in which a coupling between Maxwell field and another U(1)-gauge field was considered. The backreaction of the dark matter sector on gravitational background in question was also examined.
A d-wave holographic superconductor is studied under a constant magnetic field by perturbation method, we obtain both droplet and triangular vortex lattice solution. The results are the same as the s-wave holographic superconductor. The non-Abelian holographic superconductor with $p+ip$-wave background is also studied under magnetic field, unlike the d-wave and s-wave models, we find that the non-Abelian model has only droplet solution.
Based on the Sturm-Liouville eigenvalue problem, we develop a general analytic technique to investigate the excited states of the holographic superconductors. By including more higher order terms in the expansion of the trial function, we observe that the analytic results agree well with the numeric data, which indicates that the Sturm-Liouville method is very powerful to study the holographic superconductors even if we consider the excited states. For both the holographic s-wave and p-wave models, we find that the excited state has a lower critical temperature than the corresponding ground st…
We study a fully backreacted holographic model of a four-dimensional superconductor by including a higher curvature interaction in the bulk action. We study how the critical temperature and the field theory condensate vary in this model and conclude that positive higher curvature couplings make the condensation harder. We also compute the conductivity, finding significant deviations from the conjectured universal frequency gap to critical temperature ratio.
We study the charged black hole of hyperbolic horizon with scalar hair (charged Martinez-Troncoso-Zanelli: CMTZ black hole) as a model of analytic hairy black hole for holographic superconductor. For this purpose, we investigate the second order phase transition between CMTZ and hyperbolic Reissner-Nordström-AdS (HRNAdS) black holes. However, this transition unlikely occur. As an analytic treatment for holographic superconductor, we develop superconductor in the bulk and superfluidity on the boundary using the CMTZ black hole below the critical temperature. The presence of charge destroys the …
We study an analog hydrodynamic model that mimics a 3+1 AdS planar BH spacetime dual to a 2+1-dimensional superconductor. We demonstrate that the AdS$_4$ bulk and its holographic dual could be realized in nature in an analog gravity model based on fluid dynamics. In particular we mimic the metric of an $O_2$ holographic superconductor and calculate the entanglement entropy of a conveniently designed subsystem at the boundary of the analog AdS$_4$ bulk.
We employ the variational method for the Sturm-Liouville eigenvalue problem to analytically study the phase transition between the holographic insulator and superconductor in the Gauss-Bonnet gravity. By investigating the s-wave and p-wave holographic insulator/superconductor models, we find that this analytic method is more effective to obtain the analytic results on the condensation and the critical phenomena in the AdS soliton background in Gauss-Bonnet gravity. Our analytic result can be used to back up the numerical computations in the AdS soliton with Gauss-Bonnet correction.
In this paper, based on the Sturm-Liouville eigenvalue approach, we analytically investigate the properties of holographic superconductors in the background of pure Einstein and Gauss-Bonnet gravity taking into account the backreaction of the spacetime. Higher value of the backreaction parameter results in a harder condensation to form in both cases. The analytical results obtained are found to be in good agreement with the existing numerical results.
A holographic superconductor is constructed in the background of a massive gravity theory. In the normal state without condensation, the conductivity exhibits a Drude peak that approaches a delta function in the massless gravity limit as studied by David Vegh. In the superconducting state, besides the infinite DC conductivity, the AC conductivity has Drude behavior at low frequency followed by a power law-fall. These results are in agreement with that found earlier by Horowitz and Santos, who studied a holographic superconductor with an implicit periodic potential beyond the probe limit. The r…
In this paper, we analytically investigate the noncommutative effects of a charged black hole on holographic superconductors. The effects of charge of the black hole is investigated in our study. Employing the Sturm-Liouville eigenvalue method, the relation between the critical temperature and charge density is analytically investigated. The condensation operator is then computed. It is observed that condensate gets harder to form for large values of charge of the black hole.
These lectures give an introduction to the theory of holographic superconductors. These are superconductors that have a dual gravitational description using gauge/gravity duality. After introducing a suitable gravitational theory, we discuss its properties in various regimes: the probe limit, the effects of backreaction, the zero temperature limit, and the addition of magnetic fields. Using the gauge/gravity dictionary, these properties reproduce many of the standard features of superconductors. Some familiarity with gauge/gravity duality is assumed. A list of open problems is included at the …
We study holographic superconductors in the Schwarzschild-AdS black hole with a global monopole through a charged complex scalar field. We calculate the condensates of the charged operators in the dual conformal field theories (CFTs) and discuss the effects of the global monopole on the condensation formation. Moreover, we compute the electric conductive using the probe approximation and find that the properties of the conductive are quite similar to those in the Schwarzschild-AdS black hole. These results can help us know more about holographic superconductors in the asymptotic AdS black hole…
We study holographic superconductors in five dimensional Einstein-Gauss-Bonnet gravity both numerically and analytically. We find the critical temperature of the superconductor decreases as backreaction is increased, although the effect of the Gauss-Bonnet coupling is more subtle: the critical temperature first decreases then increases as the coupling tends towards the Chern-Simons value in a backreaction dependent fashion. We compute the conductivity of the system, finding the energy gap, and show that the effect of both backreaction and higher curvature is to increase the gap ratio $ω_g/T_c$…
We study general models for holographic superconductors in Einstein-Gauss-Bonnet gravity. We find that different values of Gauss-Bonnet correction term and model parameters can determine the order of phase transitions and critical exponents of second-order phase transitions. Moreover we find that the size and strength of the conductivity coherence peak can be controlled. The ratios $ω_g/Tc$ for various model parameters have also been examined.
It is known that time-dependent perturbations can enhance superconductivity and increase the critical temperature. If this phenomenon happens to high-T_c superconductors, one could obtain room-temperature superconductors, but this is still an open issue experimentally. Meanwhile, we would like to understand this phenomenon from gravity dual and see if the enhancement is possible for holographic superconductors. Previous work (arXiv:1104.4098 [hep-th]) has studied this issue by adding a "time-dependent chemical potential," but their analysis is questionable as a true dynamic equilibrium. In par…
We study the effects of the Born-Infeld electrodynamics on the holographic superconductors in the background of a Schwarzschild AdS black hole spacetime. We find that the presence of Born-Infeld scale parameter decreases the critical temperature and the ratio of the gap frequency in conductivity to the critical temperature for the condensates. Our results means that it is harder for the scalar condensation to form in the Born-Infeld electrodynamics.
In this paper, we investigate holographic superconductors dual to asymptotically anti-de Sitter black holes in an Einstein-Maxwell-scalar model with a non-minimal coupling between the scalar and Maxwell fields. In the probe limit, it shows that the scalar condensate occurs below the critical temperature $T_{c}$, and decreases with the increase of the coupling constant $α$. On the other hand, the the critical temperature $T_{c}$ increases as the coupling constant $α$ grows. We also calculate the optical conductivity of the holographic superconductor, and observe that a gap forms below $T_{c}$. …
We construct a holographic p-wave superconductor model in the background of quintessence AdS black hole with an SU(2) Yang-Mills gauge field and then probe the effects of quintessence on the holographic p-wave superconductor. We investigate the relation between the critical temperature and the state parameter of quintessence, and present the numerical results for electric conductivity. It is shown that the condensation of the vector field becomes harder as the absolute value of the state parameter increases. Unlike the scalar condensate in the s-wave model, the condensation of the vector field…
In this paper a holographic model of $s$-wave superconductor with anisotropic Lifshitz scaling has been considered. In the presence of an external magnetic field our holographic model exhibits both vortex and droplet solutions. Based on analytic methods we have shown that the anisotropy has no effect on the vortex and droplet solutions whereas it may affect the condensation. Our vortex solution closely resembles the Ginzburg-Landau theory and a relation between the upper critical magnetic field and superconducting coherence length has been speculated from this comparison. Using Sturm-Liouville…
We investigate the properties of the backreacting holographic superconductors from the coupling of a scalar field to the Einstein tensor in the background of a d-dimensional AdS black hole. Imposing the Dirichlet boundary condition of the trial function without the Neumann boundary conditions, we improve the analytical Sturm-Liouville method with an iterative procedure to explore the pure effect of the Einstein tensor on the holographic superconductors and find that the Einstein tensor hinders the condensate of the scalar field but does not affect the critical phenomena. Our analytical finding…
We study the holographic superconductors in f(R) gravity, and show how the critical temperature and the condensate of the dual operator depend on the modifications to Einstein gravity. We first review the black holes with constant curvature, which are clear in f(R) gravity. The dependencies are represented with the modified Newtonian constant and the effective cosmological constant trivially. Then we consider a planar black hole solution for a specific f(R) without imposing the constant curvature, which is asymptotic to AdS spacetime. The corrections to the thermal dynamics of the black holes,…
We do more discussions of holographic s-wave superconductors in the background of AdS3 spacetime. We analytically compute the holographic superconductor model with the motion equation has two different characteristic root on the framework of Maxwell electrodynamics field. We study superconductor models on the framework of Born-Infeld electrodynamics field and on the framework of the Stückelberg form in the background of AdS3 spacetime.
We investigate the behaviors of entanglement entropy in the holographical insulator/superconductor phase transition. We calculate the holographic entanglement entropy for two kinds of geometry configurations in a completely back-reacted gravitational background describing the insulator/superconductor phase transition. The non-monotonic behavior of the entanglement entropy is found in this system. In the belt geometry case, there exist four phases characterized by the chemical potential and belt width.
We apply the soft wall holographic model from hadron physics to a description of the high-$T_c$ superconductivity. In comparison with the existing bottom-up holographic superconductors, the proposed approach is more phenomenological. On the other hand, it is much simpler and has more freedom for fitting the conductivity properties of the real high-$T_c$ materials. We demonstrate some examples of emerging models and discuss a possible origin of the approach.
We study a general class of holographic superconductor models via the Stückelberg mechanism in the non-minimal derivative coupling theory in which the charged scalar field is kinetically coupling to Einstein's tensor. We explore the effects of the coupling parameter on the critical temperature, the order of phase transitions and the critical exponents near the second-order phase transition point. Moreover, we compute the electric conductive using the probe approximation and check the ratios $ω_g/T_c$ for the different coupling parameters.
A quick review on the analytical aspects of holographic superconductors (HSC) with Weyl corrections has been presented. Mainly we focus on matching method and variations approaches. Different types of such HSC have been investigated, s-wave, p-wave and Stúckelberg ones. We also review the fundamental construction of a p-wave type , in which the non-Abelian gauge field is coupled to the Weyl tensor. The results are compared from numerics to analytical results.
The phase transition of Reissner-Nordstrom AdS_4 interacting with a massive charged scalar field has been further revisited. We found exactly one stable and one unstable quasinormal mode regions for the scalar field. The two of them are separated by the first marginally stable solution.
In this work, we formulate a Non-Commutative (NC) extension of AdS-CFT correspondence that is manifested in the modification of behavior of a holographic superconductor. The noncommutativity is introduced in the model through the NC corrected AdS charged black hole metric developed by Nicolini, Smailagic and Spallucci. First of all we discuss thermodynamic properties of this black hole in Euclidean formalism. In particular, we compute trace of the boundary energy-momentum tensor which, as expected, is non-zero due to the NC scale introduced in the model. Our findings indicate that the non-comm…
We investigate systematically the effect of the nonlinear correction to the usual Maxwell electrodynamics on the holographic dual models in the backgrounds of AdS black hole and AdS soliton. Considering three types of typical nonlinear electrodynamics, we observe that in the black hole background the higher nonlinear electrodynamics correction makes the condensation harder to form and changes the expected relation in the gap frequency, which is similar to that caused by the curvature correction. However, in strong contrast to the influence of the curvature correction, we find that in the AdS s…
We study holographic superconductors in Einstein-Gauss-Bonnet gravity. We consider two particular backgrounds: a $d$-dimensional Gauss-Bonnet-AdS black hole and a Gauss-Bonnet-AdS soliton. We discuss in detail the effects that the mass of the scalar field, the Gauss-Bonnet coupling and the dimensionality of the AdS space have on the condensation formation and conductivity. We also study the ratio $ω_g/T_c $ for various masses of the scalar field and Gauss-Bonnet couplings.
Following a methodology similar to \cite{Alishahiha:2015rta}, we derive a holographic complexity for two dimensional holographic superconductors (gauge/string superconductors) with backreactions. Applying a perturbation method proposed by Kanno in Ref. \cite{kanno}, we study behaviors of the complexity for a dual quantum system near critical points. We show that when a system moves from the normal phase ($T>T_c$) to the superconductor phase ($T<T_c$), the holographic complexity will be divergent.
In this note we remark that the butterfly effect can be used to diagnose the phase transition of superconductivity in a holographic framework. Specifically, we compute the butterfly velocity in a charged black hole background as well as anisotropic backgrounds with Q-lattice structure. In both cases we find its derivative to the temperature is discontinuous at critical points. We also propose that the butterfly velocity can signalize the occurrence of thermal phase transition in general holographic models.
We study the effect of the quadratic field strength correction to the usual Maxwell field on the holographic dual models in the backgrounds of AdS black hole and AdS soliton. We find that in the black hole background, the higher correction to the Maxwell field makes the condensation harder to form and changes the expected relation in the gap frequency. This effect is similar to that caused by the curvature correction. However, in the soliton background we find that different from the curvature effect, the correction to the Maxwell field does not influence the holographic superconductor and ins…
In this paper, we investigate the effects of nonlinear exponential electrodynamics as well as backreaction on the properties of one-dimensional $s$-wave holographic superconductors. We continue our study both analytically and numerically. In analytical study, we employ the Sturm-Liouville method while in numerical approach we perform the shooting method. We obtain a relation between the critical temperature and chemical potential analytically. Our results show a good agreement between analytical and numerical methods. We observe that the increase in the strength of both nonlinearity and backre…
With the probe limit, we investigate the behavior of the electric permittivity and effective magnetic permeability and related optical properties in the s-wave holographic superconductors. In particular, our result shows that unlike the strong coupled systems which admit a gravity dual of charged black holes in the bulk, the electric permittivity and effective magnetic permeability are unable to conspire to bring about the negative Depine-Lakhtakia index at low frequencies, which implies that the negative phase velocity does not appear in the holographic superconductors under such a situation.
We explore the effects of the general non-minimally coupled $RF^2$-type couplings on the holographic s-wave superconductors numerically in the Schwarzschild-AdS background. We calculate the condensation and conductivity of the model for the coupling parameters $a_1$ and $β$. We obtain that the bigger deviations of the parameter $a_1$ from the minimal case lead to the larger deviations of the gap frequency from the universal value $ω_g/T_c \approx 8$. Moreover the smaller $β$ and $a_1$ cause to gradually stronger and narrower coherence peak.
In this letter we construct the Stückelberg holographic superconductor with Weyl corrections. Under such corrections, the Weyl coupling parameter $γ$ plays an important role in the order of phase transitions and the critical exponents of second order phase transitions. So do the model parameters $c_α$, $α$ and $c_{4}$. Moreover, we show that the Weyl coupling parameter $γ$ and the model parameters $c_α$, $α$, $c_{4}$ which together control the size and strength of the conductivity coherence peak and the ratio of gap frequency over critical temperature $ω_{g}/T_{c}$.
We investigate an anisotropic model of superconductors in the Einstein-Maxwell-dilaton theory with a charged scalar field. It is found that the critical temperature decreases as the anisotropy becomes large. We then estimate the energy gap of the superconductor, and find that the ratio of the energy gap to the critical temperature increases as the anisotropy increases and so it is larger than that in the isotropic case. We also find that peudogap appears due to the anisotropy.
In this paper we have analytically investigated holographic superconductors in four dimensional Einstein-Gauss-Bonnet gravity background. Recently the novel four dimensional Einstein-Gauss-Bonnet gravity has been formulated by rescaling the Gauss-Bonnet coupling constant $α\rightarrow \fracα{d-4}$ and taking the limit $d\rightarrow 4$, and this predicts several interesting new features. To know the effect of the curvature correction on the $(2+1)$-dimensional superconductors, we have consider the $3+1$-dimensional Einstein-Gauss-Bonnet gravity. Using the Sturm-Liouville eigenvalue approach, th…
The purpose of these lecture notes is to give a quick and introductory overview of holographic superconductors. Besides the actual description of the standard holographic superconductor, attention is paid to the motivations and the relation with the previous, non-holographic context.
We employ the matching method to analytically investigate the holographic superconductors with Lifshitz scaling in an external magnetic field. We discuss systematically the restricted conditions for the matching method and find that this analytic method is not always powerful to explore the effect of external magnetic field on the holographic superconductors unless the matching point is chosen in an appropriate range and the dynamical exponent $z$ satisfies the relation $z=d-1$ or $z=d-2$. From the analytic treatment, we observe that Lifshitz scaling can hinder the condensation to be formed, w…
We consider holographic superconductors related to the Schwarzschild black hole in the low energy limit of Hořava-Lifshitz spacetime. The non-relativistic electromagnetic and scalar fields are introduced to construct a holographic superconductor model in Hořava-Lifshitz gravity and the results show that the $α_2$ term plays an important role, modifying the conductivity curve line by means of an attenuation the conductivity.
Using a simple analytic approach, we study the universal properties of second-order phase transition in holographic superconductor models. We explore a general model in arbitrary dimensions in which the condensation occurs via the Stückelberg spontaneous symmetry breaking mechanism. All the possible second-order phase transitions and their universal characteristics can be identified analytically. The relationship between the critical temperature and charge density is generic, and the critical exponents can be greater than the typical mean field value 1/2. In addition, the related numerical fac…
