Entanglement renormalization and integral geometry

We revisit the applications of integral geometry in AdS$_3$ and argue that the metric of the kinematic space can be realized as the entanglement contour, which is defined as the additive entanglement density. From the renormalization of the entanglement contour, we can holographically understand the operations of disentangler and isometry in multi-scale entanglement renormalization ansatz. Furthermore, a renormalization group equation of the long-distance entanglement contour is then derived. We then generalize this integral geometric construction to higher dimensions and in particular demonstrate how it works in bulk space of homogeneity and isotropy.Comment: 40 pages, 7 figures. v2: discussions on the general measure added, typos fixed; v3: sections reorganized, various points clarified, to appear in JHE

Higher-Derivative Gravity with Non-minimally Coupled Maxwell Field

We construct higher-derivative gravities with a non-minimally coupled Maxwell field. The Lagrangian consists of polynomial invariants built from the Riemann tensor and the Maxwell field strength in such a way that the equations of motion are second order for both the metric and the Maxwell potential. We also generalize the construction to involve a generic non-minimally coupled $p$-form field strength. We then focus on one low-lying example in four dimensions and construct the exact magnetically-charged black holes. We also construct exact electrically-charged $z=2$ Lifshitz black holes. We obtain approximate dyonic black holes for the small coupling constant or small charges. We find that the thermodynamics based on the Wald formalism disagrees with that derived from the Euclidean action procedure, suggesting this may be a general situation in higher-derivative gravities with non-minimally coupled form fields. As an application in the AdS/CFT correspondence, we study the entropy/viscosity ratio for the AdS or Lifshitz planar black holes, and find that the exact ratio can be obtained without having to know the details of the solutions, even for this higher-derivative theory.Comment: Latex, 23 page

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

This paper introduces a new discrete fracture model accounting for non-isothermal compositional multiphase Darcy flows and complex networks of fractures with intersecting, immersed and non immersed fractures. The so called hybrid-dimensional model using a 2D model in the fractures coupled with a 3D model in the matrix is first derived rigorously starting from the equi-dimensional matrix fracture model. Then, it is dis-cretized using a fully implicit time integration combined with the Vertex Approximate Gradient (VAG) finite volume scheme which is adapted to polyhedral meshes and anisotropic heterogeneous media. The fully coupled systems are assembled and solved in parallel using the Single Program Multiple Data (SPMD) paradigm with one layer of ghost cells. This strategy allows for a local assembly of the discrete systems. An efficient preconditioner is implemented to solve the linear systems at each time step and each Newton type iteration of the simulation. The numerical efficiency of our approach is assessed on different meshes, fracture networks, and physical settings in terms of parallel scalability, nonlinear convergence and linear convergence

Non-Abelian (Hyperscaling Violating) Lifshitz Black Holes in General Dimensions

We consider Einstein gravities coupled to a cosmological constant and multiple $SU(2)$ Yang-Mills fields in general dimensions and find that the theories admit colored Lifshitz solutions with dynamic exponents $z>1$. We also introduce a Maxwell field and construct exact electric charged black holes that asymptote to the $z=D-1$ colored Lifshitz spacetimes and analyse their thermodynamical first law. Furthermore, we introduce a dilaton to the system and construct Lifshitz spacetimes with hyperscaling violations. After turning on the Maxwell field, we obtain a class of hyperscaling violating Lifshitz black holes when $\theta=\frac{2}{D-2}[z-(D-1)]$.Comment: Latex, 12 pages, a published version in PL