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.
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.
We reply to the comments by P.Midodashvili about our previous paper [1]. We argue that, contrary to the conclusions in Refs. [2,3], the Generalized Uncertainty Principle proposed by Ng and van Dam in Ref. [4] is compatible with the Holographic Principle in spacetimes with extra dimensions only for a very special (and somehow unrealistic) choice of the relation between the size and mass of the clock.
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.
The holographic superconductors, as one of the most important application of gauge/gravity duality, promote the study of strongly coupled superconductors via classical general relativity living in one higher dimension. One of the interesting properties in holographic superconductor is the appearance of first and second order phase transitions. Recently, another active studies in holographic framework is the holographic entanglement entropy and complexity evaluated from gravity side. In this note, we study the properties of the holographic entanglement entropy and complexity crossing both first…
In this Letter, we consider a Non-Commutative (NC) extension of AdS-CFT correspondence and its effects on holographic superconductors. NC corrections are incorporated via the NC generalization of Schwarzschild black hole metric in AdS with the probe limit. We study NC effects on the relations connecting the charge density and the critical temperature of the Holographic Superconductors. Furthermore, condensation operator of the superconductor has been analyzed. Our results suggest that generically, NC effects increase the critical temperature of the holographic superconductor.
We use the matching method to investigate the first-order phase transition in holographic superconductor and superfluid. We first use the simple holographic superconductor model to describe the matching method and mention how to see the first-order phase transition. Next, we study the holographic superconductor with Stückelberg term and see that the analytic results indicate the existence of first-order phase transition. Finally, we study the holographic superfluid and show that the first-order phase transition can be found for some values of parameters. We determine the critical value analyti…
We study general models of holographic superconductors away from the probe limit. We find that the backreaction of the spacetime can bring richer physics in the phase transition. Moreover we observe that the ratio $ω_g/T_c$ changes with the strength of the backreaction and is not a universal constant.
We consider fermion correlators in non-abelian holographic superconductors. The spectral function of the fermions exhibits several interesting features such as support in displaced Dirac cones and an asymmetric distribution of normal modes. These features are compared to similar ones observed in angle resolved photoemission experiments on high T_c superconductors. Along the way we elucidate some properties of p-wave superconductors in AdS_4 and discuss the construction of SO(4) superconductors.
We study the entanglement entropy of general holographic dual models both in AdS soliton and AdS black hole backgrounds with full backreaction. We find that the entanglement entropy is a good probe to explore the properties of the holographic superconductors and provides richer physics in the phase transition. We obtain the effects of the scalar mass, model parameter and backreaction on the entropy, and argue that the jump of the entanglement entropy may be a quite general feature for the first order phase transition. In strong contrast to the insulator/superconductor system, we note that the …
Using a five dimensional AdS soliton in an Einstein-Yang-Mills theory with SU(2) gauge group we study p-wave holographic insulator/superconductor phase transition. To explore the phase structure of the model we consider the system in the probe limit as well as fully back reacted solutions. We will also study zero temperature limit of the p-wave holographic superconductor in four dimensions.
We studied entanglement entropy for a time dependent two dimensional holographic superconductor. We showed that the conserved charge of the system plays the role of the critical parameter to have condensation.
We study a truncation of four-dimensional maximal gauged supergravity that provides a realization of the minimal model of a holographic superconductor. We find various flow solutions in this truncation at zero and finite temperature with a non-trivial profile for the charged scalar. Below a critical temperature we find holographic superconductor solutions that represent the thermodynamically preferred phase. Depending on the choice of boundary conditions, the superconducting phase transition is either first or second order. For vanishing temperature we find a flow with a condensing charged sca…
In the last years it has been shown that some properties of strongly coupled superconductors can be potentially described by classical general relativity living in one higher dimension, which is known as holographic superconductors. This paper gives a quick and introductory overview of some holographic superconductor models with s-wave, p-wave and d-wave orders in the literature from point of view of bottom-up, and summarizes some basic properties of these holographic models in various regimes. The competition and coexistence of these superconductivity orders are also studied in these supercon…
We study "holographic superconductors" in various spacetime dimensions. We compute most of the static critical exponents in the linear perturbations and show that they take the standard mean-field values. We also consider the dynamic universality class for these models and show that they belong to model A with dynamic critical exponent z=2.
We find, by an appropriate extension of the standard holographic superconductor setup, static bulk solutions which describe holographic duals to non-Abelian vortices. In the core of these vortices a scalar field condenses, breaking a non-Abelian global symmetry which leads to additional zero modes called orientational moduli. These moduli appear in the bulk as Goldstone bosons associated to the condensation of a neutral scalar field.
We construct a type-II holographic superconductor from an anisotropic rotating background metric. We study the effects of the magnetic field on the vortex structure, including continuous deformations both from triangular to square lattices or vice versa. Our holographic model reproduces known experimental vortex lattice deformations and the increment of the vortex population by increasing the external magnetic field.
We study the one dimensional s-wave holographic superconductor by turning on the vector potential $A_x$ in the bulk, which behaves as $A_x=A_x^{(0)} \ln z+ A_x^{(1)}$ on the boundary. By solving the model with fixed $A_x^{(0)}$, we find that if we identify the $A_x^{(0)}$ with the supercurrent $j_x$ of the holographic superconductor, the results agree with the Gindzburg- Landau theory. For example, $A_x^{(0)}$ will break the superconductivity, and the critical value of $A_x^{(0)}$ is proportional to $(T_c-T)^{3/2}$.
We extend earlier treatments of holographic superconductors by studying cases where operators of different dimension condense in both 2+1 and 3+1 superconductors. We also compute a correlation length. We find surprising regularities in quantities such as $ω_g/T_c$ where $ω_g$ is the gap in the frequency dependent conductivity. In special cases, new bound states arise corresponding to vector normal modes of the dual near-extremal black holes.
We consider holographic superconductors in a rotating black string spacetime. In view of the mandatory introduction of the $A_φ$ component of the vector potential we are left with three equations to be solved. Their solutions show that the effect of the rotating parameter $a$ influences the critical temperature $T_c$ and the conductivity $σ$ in a simple but not trivial way.
We show that D=5 electrically charged AdS black branes can have instabilities associated with spatially modulated p-wave superconductors with helical structure. We show that the instabilities are present within N=4 $SU(2)\times U(1)$ gauged supergravity and hence within D=10 and D=11 supergravity.
We study the holographic phase transition of superconductor dual to a Lifshitz black brane probed by an anisotropic scalar field in the probe limit in Hourava-Lifshitz gravity. With the use of numerical and analytical method, we investigate how the critical temperature of the condensation is affected by the Lifshitz exponent $z$, $α-$correction term in the action as well as the dimensions of the gravity. We also numerically explore the condensation of the dual operator and optical conductivity of the holographic system. Various interesting properties of the holographic condensation affected by…
We study the gravity duals of striped holographic superconductors in the AdS black hole and AdS soliton backgrounds. We show the dependences of the condensation and the critical temperature/critical chemical potential on the inhomogeneity in these two different spacetimes. By exploring the dynamics of the normal phase through the scalar field perturbation, we argue that the pair susceptibility and the conductivity can be possible phenomenological indications to disclose the property of inhomogeneity.
In four lectures, delivered at the TASI 2010 summer school, I cover selected topics in the application of the gauge-string duality to nuclear and condensed matter physics. On the nuclear side, I focus on multiplicity estimates from trapped surfaces in AdS_5, and on the consequences of conformal symmetry for relativistic hydrodynamics. On the condensed matter side, I explain the fermion response to the zero-temperature limit of p-wave holographic superconductors.
We study a general flat space/boson star transition model in quasi-local ensemble through approaches familiar from holographic superconductor theories. We manage to find a parameter $ψ_{2}$, which is proved to be useful in disclosing properties of phase transitions. In this work, we explore effects of the scalar mass, scalar charge and St$\ddot{u}$ckelberg mechanism on the critical phase transition points and the order of transitions mainly from behaviors of the parameter $ψ_{2}$. We mention that properties of transitions in quasi-local gravity are strikingly similar to those in holographic su…
We introduce and study a simple unbalanced holographic superconductor model with two scalar order parameters. The attention is focused on the possibility of coexisting orderings corresponding to the concomitant condensation of two scalar operators. Through a probe analysis we show that an attractive or repulsive direct interaction between the two bulk scalars leads respectively to competition and enhancement of the associated condensates. The system at hand is a toy model for studying generic multiple ordering in a strongly coupled context and some comments are given about its applicability to…
In this paper, we have analytically computed the conductivity of holographic superconductors in the framework of Born-Infeld electrodynamics taking into account the backreaction of the matter fields on the bulk spacetime metric. The effect of the Born-Infeld electrodynamics is incorporated in the metric. The band gap energy is found to be corrected by the backreaction and Born-Infeld parameters. The conductivity expression is then compared with that obtained from a self consistent approach.
We study various physical quantities associated with holographic s-wave superconductors as functions of the scaling dimensions of the dual condensates. A bulk scalar field with negative mass squared $m^2$, satisfying the Breitenlohner-Freedman stability bound and the unitarity bound, and allowed to vary in $0.5$ unit intervals, were considered. We observe that all the physical quantities investigated are sensitive to the scaling dimensions of the dual condensates. For all the $m^2$, the characteristic lengths diverge at the critical temperature in agreement with the Ginzburg-Landau theory. The…
We study the scalar condensation of a general holographic superconductor model in AdS black hole background away from the probe limit. We find the model parameters together with the scalar mass and backreaction can determine the order of phase transitions completely. In addition, we observe two types of discontinuities of the scalar operator in the case of first order phase transitions. We analyze in detail the effects of the scalar mass and backreaction on the formation of discontinuities and arrive at an approximate relation between the threshold model parameters. Furthermore, we obtain supe…
Frustration is an important phenomenon in condensed matter physics because it can introduce a new order parameter such as chirality. Towards understanding a mechanism of the frustration in strongly correlated systems, we study a holographic superconductor model with three scalar fields and an interband Josephson coupling, which is important for the frustration. We analyze free energy of solutions of the model to determine ground states. We find chiral ground states, which have nonzero chirality.
In this letter we study aspects of the holographic superconductors analytically in the presence of a constant external magnetic field. We show that the critical temperature and critical magnetic field can be calculated at nonzero temperature. We detect the Meissner effect in such superconductors. A universal relation between black hole mass $ M$ and critical magnetic field $H_c$ is proposed as $\frac{H_c}{M^{2/3}}\leq 0.687365$. We discuss some aspects of phase transition in terms of black hole entropy and the Bekenstein's entropy to energy upper bound.
In this article we have analytically deduced the frequency dependent expression of conductivity and the band gap energy in $AdS_{4}$ Schwarzschild background for p-wave holographic superconductors considering Einstein-Yang-Mills theory. We also used the self consistent approach to obtain the expressions of conductivity for different frequency ranges at low temperature. We then compared the imaginary part of conductivity at low frequency region. The band gap energy obtained from these two methods seem to agree very well.
We construct a toy model for holographic superconductor with non linear Maxwell field in the frame of modified gravity. By probe the bulk background by non linear Maxwell fields we show that superconductivity happens under a specific critical temperature. The effect of the non linear Maxwell field and non linear curvature corrections have been studied by analytical matching methods. We conclude that the non linearity in Maxwell field and curvature coupling make condensation harder.
We use the variational method for the Sturm-Liouville eigenvalue problem to analytically calculate some properties of holographic superconductors with Gauss-Bonnet gravity in probe limit. By studying the holographic p-wave and s-wave superconductors in (3+1)-dimensional boundary field theories, it is found that near the critical temperature, the critical exponent of the condensation is 1/2 which is the universal value in mean-field theory. We also find that when Gauss-Bonnet coefficients grow bigger the operators on the boundary field theory will be harder to condense. These are in good agreem…
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.
We study the dynamics of the holographic $s$-wave superconductors described by the Einstein-Maxwell-complex scalar field theory with a negative cosmological constant. If the eigenfunction of the linearized equation of motion of the scalar field in the planar RNAdS black hole background is chosen as the initial data, the bulk system will evolve to the intermediate state that corresponds to the excited state superconductor on the boundary. The process can be regarded as the non-equilibrium condensation process of the excited state of holographic superconductor. When the linear superposition of t…
In this work we discuss the zero temperature limit of a "p-wave" holographic superconductor. The bulk description consists of a non-Abelian SU(2) gauge fields minimally coupled to gravity. We numerically construct the zero temperature solution which is the gravity dual of the superconducting ground state of the "p-wave" holographic superconductors. The solution is a smooth soliton with zero horizon size and shows an emergent conformal symmetry in the IR. We found the expected superconducting behavior. Using the near horizon analysis we show that the system has a "hard gap" for the relevant gau…
We investigate a holographic superconductor that admits an analytic treatment near the phase transition. In the dual 3+1 dimensional field theory, the phase transition occurs when a scalar operator of scaling dimension two gets a vacuum expectation value. We calculate current-current correlation functions along with the speed of second sound near the critical temperature. We also make some remarks about critical exponents. An analytic treatment is possible because an underlying Heun equation describing the zero mode of the phase transition has a polynomial solution. Amusingly, the treatment he…
We consider a holographic superconductor with homogeneous impurities added. We start with the holographic Abelian-Higgs model for s-wave superconductivity, and turn on a coupling between the gauge field and a new massive gauge field that is introduced for impurities, whose effect is examined in the probe limit. We find that the condensation of the massive gauge field is induced in the superconducting phase. When the coupling is sufficiently large, the mass gap in the optical conductivity disappears. A resonance peak is found in the conductivity for the massive vector field.
Using a noncommutative inspired anti-de Sitter-Einstein-Born-Infeld black hole with both non-commutative mass and charge distributions, we analyse a holographic superconductor in the context of the AdS/CFT correspondence. We show that the coupling of the Born-Infeld parameter and the noncommutative perturbations provides an enhancing mechanism for higher values of the expectation value of the condensate and that noncommutativity counter-balances nonlinearity of the electrodynamics to keep the ratio $ω_g/T_c \approx 8$.
We studied holographic insulator/superconductor phase transition in the framework of Born-Infeld electrodynamics both numerically and analytically. First we numerically study the effects of the Born-Infeld electrodynamics on the phase transition, find that the critical chemical potential is not changed by the Born-Infeld parameter. Then we employ the variational method for the Sturm-Liouville eigenvalue problem to analytically study the phase transition. The analytical results obtained are found to be consistent with the numerical results.
We develop a general holographic superconductor away from the probe limit by considering the corrections both in the gravity and in the gauge matter fields. We find the consistent effects of the high curvature correction in the gravity, the nonlinear correction in the gauge matter field and the backreaction on the dynamics of bulk AdS background and boundary CFT. They all have the effect to protect the stability of the bulk spacetime and hinder the formation of the scalar hair condensation on the boundary.
We consider a holographic model with a scalar field, a tensor field and a direct coupling between them as a superconductor with an s-wave and a d-wave. We find a rich phase structure in the model. The model exhibits a phase of coexistence of the s-wave and the d-wave, or a phase of an order competition. Furthermore, it has a triple point.
We investigate the vortex lattice solution in a (2+1)-dimensional holographic model of superconductors constructed from a charged scalar condensate. The solution is obtained perturbatively near the second-order phase transition and is a holographic realization of the Abrikosov lattice. Below a critical value of magnetic field, the solution has a lower free energy than the normal state. Both the free energy density and the superconducting current are expressed by nonlocal functions, but they reduce to the expressions in the Ginzburg-Landau (GL) theory at long wavelength. As a result, a triangul…
It is known that a classical SU(2) Einstein-Yang-Mills theory in 3+1 dimensional anti-de Sitter spacetime can provide a holographic dual to a 2+1 dimensional time reversal symmetry breaking superconductor with a pseudogap. We study the properties of this holographic superconductor in the presence of an applied constant external magnetic field, neglecting backreaction on the geometry. The superconductor is immersed into a constant external magnetic field by adding a radially (the extra dimension) dependent magnetic field to the black hole. As for real superconductors, there is a critical magnet…
We construct a holographic p-wave superconductor model in the framework of quasi-topological gravity in the probe limit. The relation between the critical temperature and the coupling parameters of higher curvature terms is investigated. The numerical results for conductivity are presented as well. It turns out that our data fits the Drude model very well in the low frequency limit, and the values of DC conductivity as well as the relaxation time are obtained numerically.
We explore a class of holographic superconductors built using non-abelian condensates on probe branes in conformal and non-conformal backgrounds. These are shown to exhibit behaviour of the specific heat which resembles that of heavy fermion compounds in the superconducting phase. Instead of showing BCS-like exponential behaviour, the specific heat is polynomial in the temperature. It exhibits a jump at the critical temperature, in agreement with real-world superconductors. We also analyse the behaviour of the energy gap and the AC and DC conductivities, and find that the systems can be either…
A gravity dual of a superconductor at finite temperature has been recently proposed. We present the vortex configuration of this model and study its properties. In particular, we calculate the free energy as a function of an external magnetic field, the magnetization and the superconducting density. We also find the two critical magnetic fields that define the region in which the vortex configurations are energetically favorable.
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 affect on 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 eigen value 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 tr…
We investigate holographic superconductors in asympototically geometries with hyperscaling violation. The mass of the scalar field decouples from the UV dimension of the dual scalar operator and can be chosen as negative as we want, without disturbing the Breitenlohner-Freedman bound. We first numerically find that the scalar condenses below a critical temperature and a gap opens in the real part of the conductivity, indicating the onset of superconductivity. We further analytically explore the effects of the hyperscaling violation on the superconducting transition temperature. We find that th…
In this paper, we first study the Lifshitz-dilaton holographic superconductors with nonlinear Born-Infeld (BI) gauge field and obtain the critical temperature of the system for different values of Lifshitz dynamical exponent, $z$, and nonlinear parameter $b$. We find that for fixed value of $b$, the critical temperature decreases with increasing $z$. This indicates that the increase of anisotropy between space and time prevents the phase transition. Also, for fixed value of $z$, the critical temperature decrease with increasing $b$. Then, we investigate the optical properties of ($2+1$) and ($…
We consider fermions in a zero-temperature superconducting anti-de Sitter domain wall solution and find continuous bands of normal modes. These bands can be either partially filled or totally empty and gapped. We present a semi-classical argument which approximately captures the main features of the normal mode spectrum.
In a recent paper, Kiritsis and Li presented a holographic model to study the competition between different orders at finite doping in holographic superconductors. In the present work, we introduce fermions into such model and study the fermionic spectral functions in the normal phase at zero and finite temperatures. Combining analytic and numerical methods, we found that there is a crossover from a strange metal with short lived excitations at small doping, into a Fermi liquid with well defined quasiparticles at large doping. The critical doping at which excitations becomes long lived increas…
We consider the gravity dual of strongly coupled system at a Lifshitz-fixed point and finite temperature, which was constructed in a recent work arXiv:0909.0263. We construct an Abelian Higgs model in that background and calculate condensation and conductivity using holographic techniques. We find that condensation happens and DC conductivity blows up when temperature turns below a critical value.
By employing the gauge/gravity duality, we disclose the effects of the non-minimal coupling between the scalar and Maxwell fields on the holographic superconductors in the probe limit. As the background spacetime, we consider a five dimensional Gauss-Bonnet (GB) black hole with a flat horizon. We find out that the critical temperature decreases for larger values of GB coupling constant or smaller values of non-minimal coupling constant, which means that the condensation is harder to form. In addition, we study the electrical conductivity in the holographic setup. We show that at low frequency …
We study competition between s-wave order and d-wave order through two holographic superconductor models. We find that once the coexisting phase appears, it is always thermodynamically favored, and that the coexistence phase is narrow and one condensate tends to kill the other. The phase diagram is constructed for each model in terms of temperature and the ratio of charges of two orders. We further compare the behaviors of some thermodynamic quantities, and discuss the different aspects and identical ones between two models.
We describe the simplest holographic model for an s-wave unbalanced superconductor in 2+1 dimensions. We study its phase diagram and linear response features with particular attention to the possibility of spatially modulated phases (LOFF) and mixed spin-electric properties. The normal phase of the model at hand allows us to analyze a strong-coupling generalization of Mott two-current model for spintronic systems; the superconducting phase features an interesting DC spin-superconductivity without spin-symmetry breaking .
We construct and analyze holographic superconductors with generalized higher derivative couplings, in single R-charged black hole backgrounds in four and five dimensions. These systems, which we call very general holographic superconductors, have multiple tuning parameters and are shown to exhibit a rich phase structure. We establish the phase diagram numerically as well as by computing the free energy, and then validated the results by calculating the entanglement entropy for these systems. The entanglement entropy is shown to be a perfect indicator of the phase diagram. The differences in th…
We obtain the thermodynamic geometry of a (2+1) dimensional strongly coupled quantum field theory at a finite temperature in a holographic set up, through the gauge/gravity correspondence. The bulk dual gravitational theory is described by a (3+1) dimensional charged AdS black hole in the presence of a massive charged scalar field. The holographic free energy of the (2+1) dimensional strongly coupled boundary field theory is computed analytically through the bulk boundary correspondence. The thermodynamic metric and the corresponding scalar curvature is then obtained from the holographic free …
The purpose of this report is to provide a framework for defining phase transition processes in two dimensional holographic superconductors, and to illustrate how they are useful to be described by holographic entanglement entropy. We study holographic entanglement entropy in a two dimensional fully backrected model for holographic superconductors. We prove that phase transition could be observe using a discontinuty in the first order of entropy.
We construct numerically static vortex solutions in a holographic model of two-band superconductor with an interband Josephson coupling in both the superfluid and superconductor regime. We investigate the effects of the interband coupling on the order parameter of each superconducting band in the vortex solution, and we find that it is different for each of the two bands. We compute also the free energy, critical magnetic field, magnetic penetration length and coherence lengths for the two bands, and we study their dependence on the interband coupling and temperature. Interestingly, we find th…
