Holographic S-wave Research Articles

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.

Gauss–Bonnet holographic superconductors in lower dimensions

We disclose the effects of Gauss–Bonnet gravity on the properties of holographic s-wave and p-wave superconductors, with higher-order corrections, in lower-dimensional spacetime. We employ shooting method to solve equations of motion numerically and obtain the effect of different values of mass, nonlinear gauge field and Gauss–Bonnet parameters on the critical temperature and condensation. Based on our results, increasing each of these three parameters leads to lower temperatures and larger values of condensation. This phenomenon is rooted in the fact that conductor/superconductor phase transition faces with difficulty for higher effects of nonlinear and Gauss–Bonnet terms in the presence of a massive field. In addition, we study the electrical conductivity in holographic setup. In four dimension, real and imaginary parts of conductivity in holographic s- and p-wave models behave similarly and follow the same trend as higher dimensions by showing the delta function and divergence behavior at low-frequency regime that Kramers–Kronig relation can connect these two parts of conductivity to each other. We observe the appearance of a gap energy at $$\omega _{\mathrm{g}}\approx 8T_{\mathrm{c}}$$ which shifts toward higher frequencies by diminishing temperature and increasing the effect of nonlinear and Gauss–Bonnet terms. Conductivity in three dimensions is far different from other dimensions. Even the real and imaginary parts in s- and p-wave modes pursue various trends. For example in $$\omega \rightarrow 0$$ limit, imaginary part in holographic s-wave model tends to infinity but in p-wave model approaches to zero. However, the real parts in both models show a delta function behavior. In general, real and imaginary parts of conductivity in all cases that we study tend to a constant value in $$\omega \rightarrow \infty $$ regime.

Excited states of holographic superconductors with hyperscaling violation

We employ the numerical and analytical methods to study the effects of the hyperscaling violation on the ground and excited states of holographic superconductors for d=2,z=2. For both the holographic s-wave and p-wave models with the hyperscaling violation, we observe that the excited state has a lower critical temperature than the corresponding ground state, which is similar to the relativistic case, and the difference of the dimensionless critical chemical potential between the consecutive states decreases as the hyperscaling violation increases. Interestingly, as we amplify the hyperscaling violation in the s-wave model, the critical temperature of the ground state first decreases and then increases, but that of the excited states always decreases. In the p-wave model, regardless of the ground state or the excited states, the critical temperature always decreases with increasing the hyperscaling violation. In addition, we find that the hyperscaling violation affects the conductivity σ which has n peaks for the n-th excited state, and changes the relation in the gap frequency for the excited states in both s-wave and p-wave models.

Holographic p-wave superconductivity from higher derivative theory

We construct a holographic SU(2) p-wave superconductor model with Weyl corrections. The high derivative (HD) terms do not seem to spoil the generation of the p-wave superconducting phase. We mainly study the properties of AC conductivity, which is absent in holographic SU(2) p-wave superconductor with Weyl corrections. The conductivities in superconducting phase exhibit obvious anisotropic behaviors. Along y direction, the conductivity sigma _{yy} is similar to that of holographic s-wave superconductor. The superconducting energy gap exhibits a wide extension. For the conductivity sigma _{xx} along x direction, the behaviors of the real part in the normal state are closely similar to that of sigma _{yy}. However, the anisotropy of the conductivity obviously shows up in the superconducting phase. A Drude-like peak at low frequency emerges in Resigma _{xx} once the system enters into the superconducting phase, regardless of the behaviors in normal state.

Holographic s-wave superconductors with conformal anomaly correction

We build a holographic s-wave conductor/superconductor model and an insulator/superconductor model in the four-dimensional conformal anomaly corrected (CAC) AdS gravity. The effects of CAC parameter $$\alpha $$ are studied using both numerical and analytical methods in the probe approximation. Concretely, when the CAC parameter increases, the critical temperature increases for the conductor/superconductor phase transition, while the critical chemical potential decreases for the insulator/superconductor case, which suggests that the increasing CAC parameter enhances both superconducting phase transitions. Meanwhile, below the critical temperature or beyond the critical chemical potential, the scalar hair begins to condense, and the condensed phases are found to be thermodynamically stable. The critical behaviors obtained from numerics are confirmed by our analytical analysis. In addition, the energy gap in the conductor/superconductor model decreases monotonically with the increasing CAC parameter.

Nonequilibrium dynamical transition process between excited states of holographic superconductors

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 the eigenfunctions is chosen as the initial data, the system will go through a series of the intermediate states corresponding to different overtone numbers, which can be regarded as the dynamical transition process between the excited states of holographic superconductor. Because the intermediate states are metastable, the bulk system eventually evolves to the stationary state that corresponds the ground state of the holographic superconductor. We also provide a global and physical picture of the evolution dynamics of the black hole and the corresponding superconducting phase transition from the funneled landscape view, quantifying the weights of the states and characterizing the transitions and cascades towards the ground state.

An analytic study on the excited states of holographic superconductors

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 state and the difference of the dimensionless critical chemical potential between the consecutive states is around 5. Moreover, we analytically confirm that the holographic superconductor phase transition with the excited states belongs to the second order, which can be used to back up the numerical findings for both s-wave and p-wave superconductors.

Holographic s-wave superconductors with Horndeski correction

Via both numerical and analytical methods, we build the holographic s-wave insulator/superconductor model in the five-dimensional AdS soliton with the Horndeski correction in the probe limit and study the effects of Horndeski parameter k on the superconductor model. For the fixed mass squared of the scalar field (m^2), the critical chemical potential mu _c increases with the larger Horndeski parameter k, which means that the increasing Horndeski correction hinders the superconductor phase transition. Meanwhile, above the critical chemical potential, the obvious pole arises in the low frequency of the imaginal part of conductivity, which signs the appearance of superconducting state. What is more, the energy of quasiparticle excitation decreases with the larger Horndeski correction. Furthermore, the critical exponent of the condensate (charge density) is frac{1}{2} (1), which is independent of the Horndeski correction. In addition, the analytical results agree well with the numerical results. Subsequently, the conductor/superconductor model with Horndeski correction is analytically realized in the four- and five-dimensional AdS black holes. It is observed that the increasing Horndeski correction decreases the critical temperature and thus hinders the superconductor phase transition, which agrees with the numerical result in the previous works.

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.

Holographic Lifshitz superconductors with six derivative correction

We numerically realize the holographic s-wave conductor/superconductor model in the four-dimensional Lifshitz black hole with six derivative (SD) correction, and mainly study the influences of the Lifshitz parameter z as well as the SD parameter γ on the superconductor model. The results show that the increasing z inhibits the superconductor phase transition for large enough γ but enhances the phase transition for small enough γ. Meanwhile, the larger γ enhances the superconductor phase transition for small enough z (such as z=1,65), but inhibits the phase transition for large enough z, for example, z=85. Furthermore, for fixed γ, there exists a pronounced peak increasing with the improving z. However, for fixed z, the pronounced peak and the energy gap become weaker and weaker with the increasing γ. In addition, with the increasing z, the energy gap decreases for small enough γ but increases for large enough γ, which agrees well with the phase diagram of the critical temperature.

Nonequilibrium condensation process of a holographic superconductor in de Rham-Gabadadze-Tolley massive gravity

We study the nonequilibrium condensation process of a holographic s-wave superconductor constructed in the background of de Rham-Gabadadze-Tolley massive gravity theory. When the temperature is lower than a critical value, this model has an asymptotic anti--de Sitter (AdS) bald black hole solution and a black hole solution with nontrivial scalar hair, which can be identified as the normal phase and the superconducting phase in boundary theory, respectively. We consider Gaussian-type perturbation of a scalar field on the background of bald AdS black hole spacetime, and we numerically solve the full nonlinear dynamics of the gravitational system in the bulk. With the full time-dependent solution of the gravitational system, we observe a dynamical process from the perturbed bald AdS black hole to the black hole with scalar hair. According to holographic duality, this process can be regarded as the dynamical phase transition process from normal state to s-wave superconducting state in boundary theory. We also investigate the time evolution of the superconducting condensate operator and clarify how the condensation process from the far-from-equilibrium state proceeds in boundary theory. By fitting the evolution data of the superconducting order parameter at early time, we show that the initial nonequilibrium condensation process can be predicted by linear quasinormal modes of scalar field perturbation on the background of the bald AdS black hole. Finally, we study the time evolution of the event and apparent horizons.

Holographic superconductor with nonlinear Born–Infeld-type electrodynamics

Holographic s-wave superconductors in the framework of nonlinear Born–Infeld-type electrodynamics are investigated in the background of Schwarzschild anti-de Sitter black holes. As particular cases, at some model parameters, we obtain results for Born–Infeld and exponential electrodynamics. We explore the analytical Sturm–Liouville eigenvalue problem in the probe limit where the scalar and electromagnetic fields do not affect the background metric. The critical temperatures of phase transitions and the order parameter are calculated which depend on the model parameters. We show that the critical exponent near the critical temperature is 1/2. Making use of the matching method, we derive analytical expressions for the condensation values and the critical temperature. The conductivity by the analytical method is calculated.

Analytical study of holographic superconductors with exponential nonlinear electrodynamics

Based on the Sturm–Liouville eigenvalue problem, we analytically study several properties of holographic s-wave superconductors with exponential nonlinear (EN) electrodynamics in the background of Schwarzschild anti-de Sitter black holes. We assume the probe limit in which the scalar and gauge fields do not back react on the background metric. We show that for this system, one can still obtain an analytical relation between the critical temperature and the charge density. Interestingly enough, we find that EN electrodynamics decreases the critical temperature, Tc, of the holographic superconductors compared to the linear Maxwell field. This implies that the nonlinear electrodynamics make the condensation harder. The analytical results obtained in this paper are in good agreement with the existing numerical results. We also compute the critical exponent near the critical temperature and find out that it is still 1/2, which seems to be a universal value in mean field theory.

Holographic s-wave and p-wave Josephson junction with backreaction

In this paper, we study the holographic models of s-wave and p-wave Josephoson junction away from probe limit in (3+1)-dimensional spacetime, respectively. With the backreaction of the matter, we obtained the anisotropic black hole solution with the condensation of matter fields. We observe that the critical temperature of Josephoson junction decreases with increasing backreaction. In addition to this, the tunneling current and condenstion of Josephoson junction become smaller as backreaction grows larger, but the relationship between current and phase difference still holds for sine function. Moreover, condenstion of Josephoson junction deceases with increasing width of junction exponentially.

Condensate flow in holographic models in the presence of dark matter

Holographic model of a three-dimensional current carrying superconductor or superfluid with {\it dark matter} sector described by the additional $U(1)$-gauge field coupled to the ordinary Maxwell one, has been studied in the probe limit. We investigated analytically by the Sturm-Liouville variational method, the holographic s-wave and p-wave models in the background of the AdS soliton as well as five-dimensional AdS black hole spacetimes. The two models of p-wave superfluids were considered, the so called $SU(2)$ and the Maxwell-vector. Special attention has been paid to the dependence of the critical chemical potential and critical transition temperature on the velocity of the condensate and {\it dark matter} parameters. The current $J$ in holographic three-dimensional superconductor studied here, shows the linear dependence on $T_c-T$ for both s and p-wave symmetry. This is in a significant contrast with the previously obtained results for two-dimensional superconductors, which reveal the $(T-T_c)^{3/2}$ temperature dependence. The coupling constant $\alpha$, as well as, chemical potential $\mu_D$ and the velocity $S_D$ of the {\it dark matter}, affect the critical chemical potential of the p-wave holographic $SU(2)$ system. On the other hand, $\alpha$, {\it dark matter} velocity $S_D$ and density $\rho_D$ determine the actual value of the transition temperature of the same superconductor/superfluid set up. However, the {\it dark matter} does not affect the value of the current.

Holographic superconductors with the general RF2-type couplings

We explore the effects of the general non-minimally coupled [Formula: see text]-type couplings on the holographic s-wave superconductors numerically in the Schwarzschild-anti-de Sitter (AdS) background. We calculate the condensation and conductivity of the model for the coupling parameters a1 and [Formula: see text]. We obtain that the bigger deviations of the parameter a1 from the minimal case lead to the larger deviations of the gap frequency from the universal value [Formula: see text]. Moreover, the smaller [Formula: see text] and a1 cause to gradually stronger and narrower coherence peak.

Analytical study on holographic superfluid in AdS soliton background

We analytically study the holographic superfluid phase transition in the AdS soliton background by using the variational method for the Sturm-Liouville eigenvalue problem. By investigating the holographic s-wave and p-wave superfluid models in the probe limit, we observe that the spatial component of the gauge field will hinder the phase transition. Moreover, we note that, different from the AdS black hole spacetime, in the AdS soliton background the holographic superfluid phase transition always belongs to the second order and the critical exponent of the system takes the mean-field value in both s-wave and p-wave models. Our analytical results are found to be in good agreement with the numerical findings.

Holographic [formula omitted]-wave condensation and Meissner-like effect in Gauss–Bonnet gravity with various non-linear corrections

In this paper we have studied the onset of holographic s-wave condensate in the (4+1) dimensional planar Gauss–Bonnet-AdS black hole background with several non-linear corrections to the gauge field. In the probe limit, performing explicit analytic computations, with and without magnetic field, we found that these higher order corrections indeed affect various quantities characterizing the holographic superconductors. Also, performing a comparative study of the two non-linear electrodynamics it has been shown that the exponential electrodynamics has stronger effects on the formation of the scalar hair. We observe that our results agree well with those obtained numerically (Zhao et al., 2013).

Generalized superconductors and holographic optics

We study generalized holographic s-wave superconductors in four dimensional R-charged black hole and Lifshitz black hole backgrounds, in the probe limit. We first establish the superconducting nature of the boundary theories, and then study their optical properties. Numerical analysis indicates that a negative Depine-Lakhtakia index may appear at low frequencies in the theory dual to the R-charged black hole, for certain temperature ranges, for specific values of the charge parameter. The corresponding cut-off values for these are numerically established in several cases. Such effects are seen to be absent in the Lifshitz background where this index is always positive.

Holographic type II Goldstone bosons

The Goldstone theorem implies the appearance of an ungapped mode whenever a continuous global symmetry is spontaneously broken. In general it does not say anything about the precise form of the dispersion relation nor does it imply that there is one massless mode for each broken symmetry generator. It is a well-established fact that even for relativistic field theories in the presence of a chemical potential Goldstone modes with quadratic dispersion relation, the type II Goldstone bosons, appear in the spectrum. We develop two holographic models that feature type II Goldstone modes as part of the quasinormal mode spectrum. The models are based on simple generalizations with U(2) symmetry of the well-studied holographic s-wave superfluid. Our results include Goldstone modes without broken generators but with unusual realization of symmetries and a frequency dependent conductivity of striking resemblance to the one of Graphene.