Holographic P-wave Superconductor Research Articles

Fermi arc in p-wave holographic superconductors

We have investigated the fermionic spectral function in p-wave holographic superconductors. We show that the vector model with minimal coupling reveals a p-wave spectral function with Fermi arc. This should be contrasted with the previous investigation where p-wave arc was demonstrated in the presence of a tensor field. We study the momentum dependent order parameter, the ω-gap in the real part of the conductivity and the fermion spectral function. In addition, we juxtapose the fermionic spectral gap with the order parameter in the holographic set. We demonstrate the impact of coupling constants, temperature and chemical potential on the spectral function.

Holographic p-Wave Superconductor with Excited States in 4D Einstein–Gauss–Bonnet Gravity

We construct a holographic p-wave superconductor with excited states in the 4D Einstein–Gauss–Bonnet gravity using the Maxwell complex vector field model. In the probe limit, we observe that, the higher curvature correction or the higher excited state can hinder the vector condensate to be formed in the full parameter space, which is different from the holographic s-wave superconductor. Regardless of the choice of the vector mass by selecting the value of m2L2 or m2Leff2, we note that the critical chemical potential becomes evenly spaced for the number of nodes and that the difference of the critical chemical potential between the consecutive states depends on the curvature correction. Moreover, we find that the higher curvature correction or the higher excited state will alter the universal relation of the gap frequency, and the pole and delta function of the conductivity for the excited states can be broadened into the peaks with the finite width as the curvature correction increases.

What lies beyond the horizon of a holographic p-wave superconductor

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.

Holographic p-wave superconductors with momentum relaxation

In the probe limit, we construct the p-wave conductor/superconductor phase transition with momentum relaxation originated from the axion fields by both numerical and analytical methods, and then study the effects of relaxation parameter (k/μ) on the superconductor model. Concretely, the increasing k/μ decreases monotonously the critical temperature and thus hinders the phase transition. The stable value of condensate (C00≡〈Jx〉1/(Δ+1)/μ|T≈0.15Tc) and the coefficient (C1) of condensate (〈Jx〉/μ1/(Δ+1)=C11−T/Tc|T≈Tc) always decrease with the larger k/μ. Near the critical point, the system suffers from a second order phase transition. Meanwhile, for fixed value of k/μ, the larger mass of vector field also suppresses the vector condensate to emerge. As for the above conclusions, the analytical results from the improved trial function agree with the numerical ones. The calculation of the ratio of energy gap to the critical temperature shows that the present superconductor indeed involves strong interaction, which increases with the increasing k/μ.

Lifshitz scaling effects on the holographic p-wave superconductors coupled to nonlinear electrodynamics

We employ gauge/gravity duality to study the effects of Lifshitz scaling on the holographic p-wave superconductors in the presence of Born–Infeld nonlinear electrodynamics. By using the shooting method in the probe limit, we calculate the relation between critical temperature T_mathrm{{c}} and rho ^{z/d} numerically for different values of mass, nonlinear parameter b and Lifshitz critical exponent z in various dimensions. We observe that critical temperature decreases by increasing b, z or the mass parameter m which makes conductor/superconductor phase transition harder to form. In addition, we analyze the electrical conductivity and find the behavior of the real and the imaginary parts as a function of frequency, which depend on the model parameters. However, some universal behaviors are seen. For instance at low frequencies, the real part of conductivity shows a delta function behavior, while the imaginary part has a pole, which means that these two parts are connected to each other through the Kramers–Kronig relation. The behavior of the real part of the conductivity in the large frequency regime can be achieved by mathrm{{Re}}[sigma ]=omega ^{D-4}. Furthermore, with increasing the Lifshitz scaling z, the energy gap and the minimum values of the real and imaginary parts become unclear.

Effects of massive gravity on p-wave holographic superconductor

In this paper, we have analytically investigated p-wave holographic superconductor in the framework of massive gravity in the probe limit. We obtained the analytical expressions for the critical temperature, the value of the condensate operator, and the difference of the free energy between the superconductor and normal phases. We studied the behavior of these quantities in the presence of the mass of graviton, which is found to be dependent almost on the sign of the massive gravity couplings. The critical temperature becomes (lower)higher and the condensate value gets (larger)smaller with (decreasing)increasing the massive gravity couplings or increasing the mass of graviton with the sufficiently (negative)positive massive gravity couplings. This fact corresponds to that the superconductor phase is (more)less thermodynamically favored.

P-wave holographic superconductors with massive vector condensate in Born\u2013Infeld electrodynamics

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 model constructed out of Einstein–Yang–Mills theory. We have then computed the conductivity of these holographic superconductor models using a self-consistent approach and have shown that the DC conductivity diverges.

Kibble-Zurek scaling in a holographic p -wave superconductor

We study the Kibble-Zurek mechanism in a 2d holographic p-wave superconductor model with a homogeneous source quench on the critical point. We derive, on general grounds, the scaling of the Kibble-Zurek time, which marks breaking-down of adiabaticity. It is expressed in terms of four critical exponents, including three static and one dynamical exponents. Via explicit calculations within a holographic model, we confirm the scaling of the Kibble-Zurek time and obtain the scaling functions in the quench process. We find the results are formally similar to a homogeneous quench in a higher dimensional holographic s-wave superconductor. The similarity is due to the special type of quench we take. We expect differences in the quench dynamics if the condition of homogeneous source and dominance of critical mode are relaxed.

Conductivity of the holographic p-wave superconductors with higher order corrections

We investigate the holographic p-wave superconductors in the presence of the higher order corrections on the gravity as well as on the gauge field side. On the gravity side, we add the Gauss–Bonnet curvature correction terms, while on the gauge field side we take the nonlinear Lagrangian in the form {mathcal {F}}+b {mathcal {F}}^{2}, where {mathcal {F}} is the Maxwell Lagrangian and b indicates the strength of the nonlinearity. We employ the shooting method for the numerical calculations in order to obtain the ratio of the critical temperature T_{c} over rho ^{1/(d-2)}. We observe that by increasing the values of the mass and the nonlinear parameters the critical temperature decreases and thus the condensation becomes harder to form. In addition, the stronger Gauss–Bonnet parameter alpha hinders the superconducting phase in Gauss–Bonnet gravity. Furthermore, we calculate the electrical conductivity based on the holographic setup. The real and imaginary parts are related by the Kramers–Kronig relation, which indicates a delta function and a pole in low frequency regime, respectively. However, at enough large frequencies the trend of the real part can be interpreted by Re[sigma ]=omega ^{(d-4)}. Moreover, in holographic model the ratio omega _{g}/T_{c} is always much larger than the BCS value 3.5 due to the strong coupling of holographic superconductors. In both kinds of gravity, decreasing the temperature or increasing the effect of nonlinearity shifts the gap frequency toward larger values. Besides, the gap frequency occurred at larger values by enlarging the Gauss–Bonnet parameter. In general, the behavior of the conductivity depends on the choice of the mass, the nonlinear and the Gauss–Bonnet parameters.

P-Wave holographic superconductor in scalar hairy black holes

We study the properties of the p-wave holographic superconductor for the scalar hairy black holes in the probe limit. The black hole solutions in question possess planar topology, which are derived from the Einstein gravity theory minimally coupled to a scalar field with a generic scalar potential. These solutions can be viewed as characterized by two independent parameters, namely, α and k0, where AdS vacuum is manifestly restored when α→∞. Consequently, the p-wave holographic superconductor is investigated by employing the above static planar black hole spacetime as the background metric, where a Maxwell field is introduced to the model by nonminimally coupling it to a complex vector field. The latter is shown to condensate and furnish the superconducting phase when the temperature is below a critical value. By numerical calculations, we examine in detail how the scalar field in the background affects the properties of the superconductivity. It is found that the critical temperature depends crucially on the parameters α and k0, which subsequently affects the condensation process. By employing the Kubo formula, the real, as well as imaginary parts of the conductivity, are calculated and presented as functions of frequency. The results are discussed regarding the poles of the Green function, and the typical values of the BCS theory.

One-dimensional backreacting holographic p-wave superconductors

We analytically and numerically study the properties of one-dimensional holographic p-wave superconductors in the presence of backreaction. We employ the Sturm–Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the hbox {AdS}_{{3}}/hbox {CFT}_{{2}} correspondence and determine the relation between the critical temperature T_{c} and the chemical potential mu for different values of the mass m of a charged spin-1 field rho _{mu } and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find that increasing the backreaction and the mass parameter causes the greater values for {T_{c}}/{mu }. Thus, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent beta is 1 / 2, which is the same as in the mean field theory. Moreover, both methods confirm the existence of a second order phase transition.

Holographic p-wave superconductor in Lifshitz gravity with RF2 correction

In the probe limit, we numerically construct the holographic p-wave superconductor model in the Lifshitz gravity with the high-order derivative correction. Concretely, we study the effects of both Lifshitz scaling and the RF2 correction on the Maxwell-complex-vector model in the five-dimensional Lifshitz black hole and soliton backgrounds, respectively. We find that the increasing Lifshitz parameter u inhibits the conductor/superconductor phase transition and suppresses the frequency dependent conductivity. For the coexistence of both Lifshitz parameter and RF2 correction, the larger Lifshitz parameter u suppresses the effect of RF2 correction on the superconductor model. Moreover, the Lifshitz parameter inhibits the insulator/superconductor phase transition at first, and then enhances the phase transition when the value of Lifshitz parameter is large enough. Meanwhile, the larger parameter u enlarges the energy of quasiparticle excitation. Near the critical point, both systems undergo a second-order phase transition, which is upheld by the comparison of grand potential between normal state and condensate state.

Holographic p-wave superconductor with high-order derivative correction

In the probe limit, we numerically study the holographic p-wave superconductor phase transition in the high-order derivative theory. Concretely, we study the influences of the high-order derivative correction term αRF2 on the Maxwell complex vector model (MCV) in the five-dimensional AdS black hole and soliton backgrounds, respectively. In the black hole background, the improving correction parameter α increases the critical temperature and thus enhances the conductor/superconductor phase transition. Meanwhile, as the RF2 correction becomes stronger, the ratio of the energy gap to the critical temperature decreases from 9.858 to 5.995, which obviously deviates from the universal value. In the soliton background, we find that the correction does not affect the critical chemical potential. However, as the correction parameter α increases, the vector condensate grows faster, which might suggest that the improving α enhances the insulator/superconductor in some sense. The location of the second pole of imaginary part of conductivity increases with α, which implies that the energy of the quasiparticle excitation increases with the improving correction. In addition, the effects of α on the superfluid density agree with the one on the critical value as well as the condensate in both models. Furthermore, the critical exponent of condensate and superfluid density near the critical point is always 1/2 and 1, respectively.

A Maxwell-vector p-wave holographic superconductor in a particular background AdS black hole metric

We study the p-wave holographic superconductor for AdS black holes with planar event horizon topology for a particular Lovelock gravity, in which the action is characterized by a self-interacting scalar field nonminimally coupled to the gravity theory which is labeled by an integer k. As the Lovelock theory of gravity is the most general metric theory of gravity based on the fundamental assumptions of general relativity, it is a desirable theory to describe the higher dimensional spacetime geometry. The present work is devoted to studying the properties of the p-wave holographic superconductor by including a Maxwell field which nonminimally couples to a complex vector field in a higher dimensional background metric. In the probe limit, we find that the critical temperature decreases with the increase of the index k of the background black hole metric, which shows that a larger k makes it harder for the condensation to form. We also observe that the index k affects the conductivity and the gap frequency of the holographic superconductors.

Holographic P -wave superconductors in 1+1 dimensions

We study $(1+1)$-dimensional p-wave holographic superconductors described by three dimensional Einstein-Maxwell gravity coupled to a massive complex vector field in the context of $AdS_3/CFT_2$ correspondence. In the probe limit where the backreation of matter fields is neglected, we show that there occurs a formation of a vector hair around the black hole below a certain critical temperature. In the dual strongly coupled $(1+1)$-dimensional boundary theory, this holographically corresponds to the formation of a charged vector condensate spontaneously breaking both the $U(1)$ and $SO(1,1)$ symmetries. We numerically compute the ac conductivity for the superconducting phase of the boundary field theory and find that the presence of a magnetic moment term in the dual bulk theory effects the conductivity in the boundary field theory.

Holographic Superconductivity in the Presence of Dark Matter: Basic Issues

The holographic approach to study strongly coupled superconductors in the presence of dark matter is reviewed. We discuss the influence of dark matter on the superconducting transition temperature of both s-wave and p-wave holographic superconductors. The upper critical field, coherence length, penetration depth of holographic superconductors as well as the metal-insulator transitions have also been analysed. Issues related to the validity of AdS/CFT correspondence for the description of superconductors studied in the laboratory and possible experiments directed towards the detection of dark matter are discussed. In doing so we shall compare our assumptions and assertions with those generally accepted in the elementary particle experiments aimed at the detection of dark matter particles. The superconductivity is a macroscopic quantum phenomenon 1 appearing in many condensed matter systems and characterized by the expulsion of the magnetic field and zero resistance state at sufficiently low temperature. The superconducting rings show quantization of magnetic flux. This is also true for vortices in the type two superconductors. A number of other quantum effects appear if the superconductor is in contact with normal metal (Andreev reflection) or other superconductors (Josephson effect). All these properties and phenomena are at the heart of numerous applications of superconductors. Zero resistivity state allows the construction and operation of powerful superconducting electromagnets for important scientific and medical applications. The superconducting quantum interference devices (SQUIDs) are the most sensitive magnetometers. SQUIDs allow for the precise measurements of the flux quantum � = h/2e, where e is the electron charge and h the Planck constant. These devices have been proposed and used in the search for the magnetic monopole; the elementary particle postulated 2 by Dirac in 1931. The search 3 for this elusive particle continues 4 , with the hope that it might address an important question of composition of dark matter 5 . According to standard theory of gravity based on Newtonian dynamics the explanation of many astrophysical observations require the existence of large amount of matter that cannot be seen with telescopes, and is thus termed dark matter. The Planck satellite mission reveals that dark matter constitutes 26.8% of the total mass of the Universe, while ordinary matter makes only 4.9% of it. The rest 6 is the mysterious dark energy, 68.3%.

Holographic vector superconductor in Gauss–Bonnet gravity

In the probe limit, we numerically study the holographic p-wave superconductor phase transitions in the higher curvature theory. Concretely, we study the influences of Gauss–Bonnet parameter α on the Maxwell complex vector model (MCV) in the five-dimensional Gauss–Bonnet–AdS black hole and soliton backgrounds, respectively. In the two backgrounds, the improving Gauss–Bonnet parameter α and dimension of the vector operator Δ inhibit the vector condensate. In the black hole, the condensate quickly saturates a stable value at lower temperature. Moreover, both the stable value of condensate and the ratio ωg/Tc increase with α. In the soliton, the location of the second pole of the imaginary part increases with α, which implies that the energy of the quasiparticle excitation increases with the improving higher curvature correction. In addition, the influences of the Gauss–Bonnet correction on the MCV model are similar to the ones on the SU(2) p-wave model, which confirms that the MCV model is a generalization of the SU(2) Yang–Mills model even without the applied magnetic field to some extent.

Holographic p-wave superconductor with disorder

We implement the effects of disorder on a holographic p-wave superconductor by introducing a random chemical potential which defines the local energy of the charge carriers. Since there are various possibilities for the orientation of the vector order parameter, we explore the behavior of the condensate in the parallel and perpendicular directions to the introduced disorder. We clarify the nature of various branches representing competing solutions and construct the disordered phase diagram. We find that moderate disorder enhances superconductivity as determined by the value of the condensate. Though we mostly focus on uncorrelated noise, we also consider a disorder characterized by its spectral properties and study in detail its influence on the spectral properties of the condensate and charge density. We find fairly universal responses of the resulting power spectra characterized by linear functions of the disorder power spectrum.

P-wave holographic superconductors from Born-Infeld black holes

We obtain (2+1) dimensional p-wave holographic superconductors from charged Born-Infeld black holes in the presence of massive charged vector fields in a bulk AdS4 Einstein-Born-Infeld theory through the AdS4-CF T3 correspondence. Below a certain critical transition temperature the charged black hole develops vector hair that corresponds to charged vector condensate in the strongly coupled (2+1) dimensional boundary field theory that breaks both the U(1) symmetry as well as the rotational invariance. The holographic free energy is computed for the boundary field theory which shows that the vector order parameter exhibits a rich phase structure involving zeroth order, first order, second order and retrograde phase transitions for different values of the backreaction and the Born-Infeld parameters. We numerically compute the ac conductivity for the p-wave superconducting phase of the strongly coupled (2+1) dimensional boundary field theory which also depends on the relative values of the parameters in the theory.

Holographic p-wave superconductor models with Weyl corrections

We study the effect of the Weyl corrections on the holographic p-wave dual models in the backgrounds of AdS soliton and AdS black hole via a Maxwell complex vector field model by using the numerical and analytical methods. We find that, in the soliton background, the Weyl corrections do not influence the properties of the holographic p-wave insulator/superconductor phase transition, which is different from that of the Yang–Mills theory. However, in the black hole background, we observe that similarly to the Weyl correction effects in the Yang–Mills theory, the higher Weyl corrections make it easier for the p-wave metal/superconductor phase transition to be triggered, which shows that these two p-wave models with Weyl corrections share some similar features for the condensation of the vector operator.