We study applications of the AdS/CFT correspondence to strongly coupled condensed matter theories. Specifically, we focus on Lifshitz spacetime, which was proposed as a gravity dual to field theories with Lifshitz scaling symmetry. We first show that higher derivative corrections, such as those arising from string theory, can resolve the apparent tidal singularity of pure Lifshitz spacetime in the deep infrared. We do so by explicitly constructing a toy-model of 4-derivative gravity coupled to Maxwell-dilaton theory to show that the singular horizon can be resolved into a nonsingular AdS2xR2 geometry. Next, we demonstrate that the non-relativistic Lifshitz symmetry leads to an effective tunneling barrier for matter fields propagating in Lifshitz spacetime. In particular, the tunneling barrier causes scalar modes to either grow or decay exponentially near the boundary. We investigate two consequences of this behavior: First, we show that the boundary-to-bulk correlator, or smearing function, is not well-defined in Lifshitz spacetime, due to a divergence at large momenta and small frequencies. Second, we show that the boundary retarded Green;;s function for scalar operators is insensitive to small changes in the near-horizon geometry. This insensitivity manifests itself in an exponentially small spectral function at low energies and large momenta. We show that this exponential behavior of the spectral weight is robust with respect to higher derivative corrections in the bulk, and is therefore a concrete prediction of AdS/CFT for condensed matter systems. We conclude by giving a field theory interpretation of the exponential behavior in terms of a non-perturbative resummation of Feynman diagrams.
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Lifshitz Spacetime as a Window into Condensed Matter Physics.
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