We develop a renormalization method for holographic entanglement entropy based on area renormalization of entangling surfaces. The renormalized entanglement entropy is derived for entangling surfaces in asymptotically locally anti-de Sitter spacetimes in general dimensions and for entangling surfaces in four dimensional holographic renormalization group flows. The renormalized entanglement entropy for disk regions in AdS 4 spacetimes agrees precisely with the holographically renormalized action for AdS 4 with spherical slicing and hence with the F quantity, in accordance with the Casini-HuertaMyers map. We present a generic class of holographic RG flows associated with deformations by operators of dimension 3/2 < ∆ < 5/2 for which the F quantity increases along the RG flow, hence violating the strong version of the F theorem. We conclude by explaining how the renormalized entanglement entropy can be derived directly from the renormalized partition function using the replica trick i.e. our renormalization method for the entanglement entropy is inherited directly from that of the partition function. We show explicitly how the entanglement entropy counterterms can be derived from the standard holographic renormalization counterterms for asymptotically locally anti-de Sitter spacetimes.

We study models of translational symmetry breaking in which inhomogeneous matter field profiles can be engineered in such a way that black-brane metrics remain isotropic and homogeneous. We explore novel Lagrangians involving square root terms and show how these are related to massive gravity models and to tensionless limits of branes. Analytic expressions for the DC conductivity and for the low frequency scaling of the optical conductivity are derived in phenomenological models, and the optical conductivity is studied in detail numerically. The square root Lagrangians are associated with linear growth in the DC resistivity with temperature and also lead to minima in the optical conductivity at finite frequency, suggesting that our models may capture many features of heavy fermion systems.

The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension 3/2 < < 5/2. Therefore, the strongest version of the F theorem is in general violated.

Abstract:We explore the behaviour of renormalized entanglement entropy in a variety of holographic models: non-conformal branes; the Witten model for QCD; UV conformal RG flows driven by explicit and spontaneous symmetry breaking and Schrödinger geometries. Focussing on slab entangling regions, we find that the renormalized entanglement entropy captures features of the previously defined entropic c-function but also captures deep IR behaviour that is not seen by the c-function. In particular, in theories with symmetry breaking, the renormalized entanglement entropy saturates for large entangling regions to values that are controlled by the symmetry breaking parameters.

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