Marina Huerta

Towards a derivation of holographic entanglement entropy

We provide a derivation of holographic entanglement entropy for spherical entangling surfaces. Our construction relies on conformally mapping the boundary CFT to a hyperbolic geometry and observing that the vacuum state is mapped to a thermal state in the latter geometry. Hence the conformal transformation maps the entanglement entropy to the thermodynamic entropy of this thermal state. The AdS/CFT dictionary allows us to calculate this thermodynamic entropy as the horizon entropy of a certain topological black hole. In even dimensions, we also demonstrate that the universal contribution to the entanglement entropy is given by A-type trace anomaly for any CFT, without reference to holography.

On the RG running of the entanglement entropy of a circle

We show, using strong subadditivity and Lorentz covariance, that in three dimensional space-time the entanglement entropy of a circle is a concave function. This implies the decrease of the coefficient of the area term and the increase of the constant term in the entropy between the ultraviolet and infrared fixed points. This is in accordance with recent holographic c-theorems and with conjectures about the renormalization group flow of the partition function of a three sphere (F-theorem). The irreversibility of the renormalization group flow in three dimensions would follow from the argument provided there is an intrinsic definition for the constant term in the entropy at fixed points. We discuss the difficulties in generalizing this result for spheres in higher dimensions.

Remarks on entanglement entropy for gauge fields

Fil: Casini, Horacio German. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina; Comision Nacional de Energia Atomica. Gerencia del Area Investicaciones y Aplicaciones No Nucleares;

Mutual information and the F-theorem

A bstractMutual information is used as a purely geometrical regularization of entanglement entropy applicable to any QFT. A coefficient in the mutual information between concentric circular entangling surfaces gives a precise universal prescription for the monotonous quantity in the c-theorem for d = 3. This is in principle computable using any regularization for the entropy, and in particular is a definition suitable for lattice models. We rederive the proof of the c-theorem for d = 3 in terms of mutual information, and check our arguments with holographic entanglement entropy, a free scalar field, and an extensive mutual information model.

Entanglement entropy for a Maxwell field: Numerical calculation on a two dimensional lattice

We study entanglement entropy (EE) for a Maxwell field in (2 + 1) dimensions. We do numerical calculations in two dimensional lattices. This gives a concrete example of the general results of our recent work [1] on entropy for lattice gauge fields using an algebraic approach. To evaluate the entropies we extend the standard calculation methods for the entropy of Gaussian states in canonical commutation algebras to the more general case of algebras with center and arbitrary numerical commutators. We find that while the entropy depends on the details of the algebra choice, mutual information has a well defined continuum limit as predicted in [1]. We study several universal terms for the entropy of the Maxwell field and compare with the case of a massless scalar field. We find some interesting new phenomena: An “evanescent” logarithmically divergent term in the entropy with topological coefficient which does not have any correspondence with ultraviolet entanglement in the universal quantities, and a non standard way in which strong subadditivity is realized. Based on the results of our calculations we propose a generalization of strong subadditivity for the entropy on some algebras that are not in tensor product.

Local temperatures and local terms in modular Hamiltonians

We show there are analogues to the Unruh temperature that can be defined for any quantum field theory and region of the space. These local temperatures are defined using relative entropy with localized excitations. We show important restrictions arise from relative entropy inequalities and causal propagation between Cauchy surfaces. These suggest a large amount of universality for local temperatures, specially the ones affecting null directions. For regions with any number of intervals in two space-time dimensions the local temperatures might arise from a term in the modular Hamiltonian proportional to the stress tensor. We argue this term might be universal, with a coefficient that is the same for any theory, and check analytically and numerically this is the case for free massive scalar and Dirac fields. In dimensions

Entanglement entropy of a Maxwell field on the sphere

d\ge 3

Positivity, entanglement entropy, and minimal surfaces

the local terms in the modular Hamiltonian producing these local temperatures cannot be formed exclusively from the stress tensor. For a free scalar field we classify the structure of the local terms.

Anisotropic Unruh temperatures

We compute the logarithmic coefficient of the entanglement entropyon a sphere for a Maxwell field in d = 4 dimensions. In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of l = 0 has been removed. This shows the relation c M = 2(c S − c S l=0 log ) between the logarithmic coefficient in the entropy for a Maxwell field c M , the one for a d = 4 massless scalar c S , and the logarithmic coefficientc S l=0 log for a d = 2 scalar with Dirichlet boundary condition at the origin. Using the accepted values for these coefficientsc S = −1/90 and c S l=0 log = 1/6 we get c M = −16/45, which coincides with Dowker’s calculation, but does not match the coefficient− 31 in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers c M , c S and c S l=0 log , verifying the relation, as well as checked they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.

Numerical determination of the entanglement entropy for a Maxwell field in the cylinder

A bstractThe path integral representation for the Renyi entanglement entropies of integer index n implies these information measures define operator correlation functions in QFT. We analyze whether the limit n → 1, corresponding to the entanglement entropy, can also be represented in terms of a path integral with insertions on the region’s boundary, at first order in n − 1. This conjecture has been used in the literature in several occasions, and specially in an attempt to prove the Ryu-Takayanagi holographic entanglement entropy formula. We show it leads to conditional positivity of the entropy correlation matrices, which is equivalent to an infinite series of polynomial inequalities for the entropies in QFT or the areas of minimal surfaces representing the entanglement entropy in the AdS-CFT context. We check these inequalities in several examples. No counterexample is found in the few known exact results for the entanglement entropy in QFT. The inequalities are also remarkable satisfied for several classes of minimal surfaces but we find counterexamples corresponding to more complicated geometries. We develop some analytic tools to test the inequalities, and as a byproduct, we show that positivity for the correlation functions is a local property when supplemented with analyticity. We also review general aspects of positivity for large N theories and Wilson loops in AdS-CFT.