| Gauge/gravity duality creates a new links between quantum theory and gravity.It claims that the dimensions strongly coupled quantum many-body system has a classical gravitational dual,which is the most successful implenmentation of the holographic principle.By studying the weakly coupled dual gravitational system in the higher dimensions,gauge/gravity duality provides a new way to handle the strongly interaction physics.Gauge/gravity duality unify seemingly unrelated fields of physics,and the successful application to high-energy physics,condensed matter systems and hydrodynamics shows its strong vitality.In its application to the condensed matter physics,the holographic superconductor can be described by the condensation process of the charged scalar field,which is caused by the spontaneous breaking of the symmetry under the perturbation instability near the black hole horizon.In this paper,we study the zero temperature holographic superconductor model with two competing order parameters in the gravitational soliton background in the probe limit and fully gravitational backreaction cases,respectively.In the probe limit the gravitational backreaction to the matter fields is ignored,we couple the two complex scalar fields with the same gauge field on the soliton background.When the chemical potential of the gravitational system increases to a certain critical value,the soliton geometry becomes unstable and carries scalar hairs.This process holographically describes the superconductivity phenomenon with scalar order parameters in zero temperature.In addition to the known normal phase and two one component superconduct phases,the two ordered parameters coexisting superconducting phases in the soliton geometric background is found for the first time.Using the pseudo-spectral method and the NewtonRaphson iterative method,we construct the complete phase diagram of the system in the plane,and we find that the numerical results in the probe limit are in good agreement with the qualitative analysis.Furthermore,we calculate the corresponding electrical conductivity and sound speed through the linear response theory.The divergence behavior of the conductivity at the indicates that the symmetry spontaneous breaking phase always represents the superconducting phase.In addition,the resulting sound speed demonstrates the non-smoothness at the critical points as the order parameter condensate,which indicates that the phase transitions can also be identified by the behavior of sound speed.Moreover,as expected from the boundary conformal field theory,the sound speed saturates to at the large chemical potential limit for our model.Considering the backreaction of the gravitational field to the matter fields,we give a proper metric ansatz of the hair soliton background and then obtain the equations of motion for the gravitational field and the matter fields from the action.By calculating the renormalized on-shell action of gravitational system,we obtain the corresponding free energy density in the grand canonical ensemble.We use the numerical methods to obtain the solutions of the system and compare their free energy to pick out the most stable phase.It is found that the phase transition and phase structure of the coexisting phase change with the coupling between the gravitational field and the matter fields.When the gravitational backreaction is weak,the phase transition and the phases sequence along with the chemical potential is similar to the probe limit,that is,the coexistence phase of the two order parameters is connected by one band phases through a second order phase transformation.However,when the influence of gravitational background on the matter fields is strong,we find a series of new phase transition sequences and phase transition types,especially in some parameters values we find the existence of zero order phase transition process.Using the similar procedure,we figure out the complete phase diagram in the plane with different coupling between gravitational field and the matter fields. |
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