Abstract
We discuss the possible phases dual to the AdS hairy black holes in Horndeski theory. In the probe limit breaking the translational invariance, we study the conductivity and we find a non-trivial structure indicating a collective excitation of the charge carriers as a competing effect of momentum dissipation and the coupling of the scalar field to Einstein tensor. Going beyond the probe limit, we investigate the spontaneous breaking of translational invariance near the critical temperature and discuss the stability of the theory. We consider the backreaction of the charged scalar field to the metric and we construct numerically the hairy black hole solution. To determine the dual phases of a hairy black hole, we compute the conductivity. When the wave number of the scalar field is zero, the DC conductivity is divergent due to the conservation of translational invariance. For nonzero wave parameter with finite DC conductivity, we find two phases in the dual theory. For low temperatures and for positive couplings, as the temperature is lower, the DC conductivity increases therefore the dual theory is in metal phase, while if the coupling is negative we have the opposite behavior and it is dual to an insulating phase. We argue that this behavior of the coupling of the scalar field to Einstein tensor can be attributed to its role as an impurity parameter in the dual theory.
Highlights
One of the first applications of gauge/gravity duality [1,2,3] was the building of a holographic superconductor and the study of its properties
In the probe limit breaking the translational invariance, we study the conductivity and we find a nontrivial structure indicating a collective excitation of the charge carriers as a competing effect of momentum dissipation and the coupling of the scalar field to Einstein tensor
We argue that this behavior of the coupling of the scalar field to Einstein tensor can be attributed to its role as an impurity parameter in the dual theory
Summary
One of the first applications of gauge/gravity duality [1,2,3] was the building of a holographic superconductor and the study of its properties. The basic mechanism of generating a holographic superconductor is the formation in the gravity sector of a hairy black hole below a critical temperature and the generation of a condensate of the charged scalar field through its coupling to a Maxwell field of the gravity sector [4]. In this work we spontaneously break the translational invariance in an Einstein-Maxwell-scalar gravity theory in which the charged scalar field is coupled kinetically to Einstein tensor and we study the possible phases generated on the dual boundary theory. In the probe limit we calculated the conductivity and we found that depending on the parameter of the translational symmetry breaking and on the coupling of the scalar field to curvature we have on the dual theory a coexistence of phases on the boundary theory.
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