We construct an ionic lattice background in the framework of Einstein-Maxwell-dilaton theory in four dimensional space time. The optical conductivity of the dual field theory on the boundary is investigated. Due to the lattice effects, we find the imaginary part of the conductivity is manifestly suppressed in the zero frequency limit, while the DC conductivity approaches a finite value such that the previous delta function reflecting the translation symmetry is absent. Such a behavior can be exactly fit by the Drude law at low frequency. Moreover, we find that the modulus of the optical conductivity exhibits a power-law behavior at intermediate frequency regime. Our results provides further support for the universality of such power-law behavior recently disclosed in Einstein-Maxwell theory

We construct a gravity dual for charge density waves (CDW) in which the translational symmetry along one spatial direction is spontaneously broken. Our linear perturbation calculation on the gravity side produces the frequency dependence of the optical conductivity, which exhibits the two familiar features of CDW, namely the pinned collective mode and gapped single-particle excitation. These two features indicate that our gravity dual also provides a new mechanism to implement the metal to insulator phase transition by CDW, which is further confirmed by the fact that d.c. conductivity decreases with the decreased temperature below the critical temperature.

We construct the simplest gravitational dual model of a superconductor on Qlattices. We analyze the condition for the existence of a critical temperature at which the charged scalar field will condense. In contrast to the holographic superconductor on ionic lattices, the presence of Q-lattices will suppress the condensate of the scalar field and lower the critical temperature. In particular, when the Q-lattice background is dual to a deep insulating phase, the condensation would never occur for some small charges. Furthermore, we numerically compute the optical conductivity in the superconducting regime. It turns out that the presence of Q-lattice does not remove the pole in the imaginary part of the conductivity, ensuring the appearance of a delta function in the real part. We also evaluate the gap which in general depends on the charge of the scalar field as well as the Q-lattice parameters. Nevertheless, when the charge of the scalar field is relatively large and approaches the probe limit, the gap becomes universal with ω g 9T c which is consistent with the result for conventional holographic superconductors.