In this thesis we study aspects of transport in strongly coupled quantum systems with broken translational symmetry. Using holographic duality, we also examine the associated dynamical problem in asymptotically Anti-de Sitter, spatially modulated black holes.
More precisely, in chapter 2 we consider the transport of conserved charges in spatially inhomogeneous quantum systems with a discrete lattice symmetry. When the DC conductivities are finite, we derive a set of generalised Einstein relations, relating the diffusion constants of the conserved charges to the DC conductivities and static susceptibilities. We also develop a long-wavelength expansion in order to explicitly construct the heat and charge diffusive modes within hydrodynamics on curved manifolds. In chapter 3 we used analogous techniques to construct the thermoelectric diffusive quasinormal modes in a large class of black hole spacetimes that are holographically dual to strongly coupled field theories in which spatial translations are broken explicitly. These modes satisfy a set of constraints on the black hole horizon, from which we find that their dispersion relations are given by the generalised Einstein relations. In chapter 4 we define a boost incoherent current in spontaneously modulated phases, and we show that in holographic theories, its DC conductivity can be obtained from solving a system of horizon Stokes equations
