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Abstract
摘要 |
We study a holographic model with vector condensate by coupling the antide Sitter gravity to an Abelian gauge field and a charged vector field in (3+1) dimensional spacetime. In this model there exists a non-minimal coupling of the vector filed to the gauge field. In the probe limit, we find that there is a critical temperature below which the charged vector condenses via a second order phase transition. The DC conductivity becomes infinite and the AC conductivity develops a gap in the condensed phase. We study the effect of a background magnetic field on the system. It is found that the background magnetic field can induce the condensate of the vector field even in the case without chemical potential/charge density. In the case with non-vanishing charge density, the transition temperature raises with the applied magnetic field, and the condensate of the charged vector operator forms a vortex lattice structure in the spatial directions perpendicular to the magnetic field. We then solve the full coupled equations of motion of the system without magnetic field. The vector hairy black hole solutions are dual to a thermal state with the U(1) symmetry as well as the spatial rotational symmetry breaking spontaneously. Depending the mass and charge of the vector field, we find a rich phase structure: zeroth order, first order and second order phase transitions can happen in this model. We also find “retrograde condensation” in which the hairy black hole solution exists only for the temperatures above a critical value with the free energy much larger than the black hole without hair. We construct the phase diagram for this system in terms of the temperature and charge of the vector field. |
