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 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. 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.Comment: 27 pages, 15 figure

Remarks on entanglement entropy for gauge fields

In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center --- a set of operators which commute with all others --- in the gauge invariant operator algebra corresponding to finite region. A unique entropy can be assigned to algebras with center, giving place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.Comment: 26 pages, 7 figure

Mutual information and the F-theorem

Mutual 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.Comment: 80 pages, 16 figure

Anisotropic Unruh temperatures

The relative entropy between very high-energy localized excitations and the vacuum, where both states are reduced to a spatial region, gives place to a precise definition of a local temperature produced by vacuum entanglement across the boundary. This generalizes the Unruh temperature of the Rindler wedge to arbitrary regions. The local temperatures can be read off from the short distance leading have a universal geometric expression that follows by solving a particular eikonal type equation in Euclidean space. This equation generalizes to any dimension the holomorphic property that holds in two dimensions. For regions of arbitrary shapes the local temperatures at a point are direction dependent. We compute their explicit expression for the geometry of a wall or strip.Fil: Arias, Raúl Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; ArgentinaFil: Casini, Horacio German. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; ArgentinaFil: Huerta, Marina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; ArgentinaFil: Pontello, Diego Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentin

Entanglement entropy of a Maxwell field on the sphere

We compute the logarithmic coefficient of the entanglement entropy on asphere for a Maxwell field in d=4 dimensions. In spherical coordinates theproblem decomposes into one dimensional ones along the radial coordinate foreach angular momentum. We show the entanglement entropy of a Maxwell field isequivalent to the one of two identical massless scalars from which the mode ofl=0 has been removed. This shows the relation c^M_{log}=2(c^S_{log}-c^{S_{l=0}}_{log}) between the logarithmic coefficient in theentropy for a Maxwell field c^M_{log}, the one for a d=4 massless scalarc_{log}^S, and the logarithmic coefficient c^{S_{l=0}}_{log} for a d=2scalar with Dirichlet boundary condition at the origin. Using the acceptedvalues for these coefficients c_{log}^S=-1/90 and c^{S_{l=0}}_{log}=1/6we get c^M_{log}=-16/45, which coincides with Dowker´s calculation, but doesnot match the coefficient -rac{31}{45} in the trace anomaly for a Maxwellfield. We have numerically evaluated these three numbers c^M_{log},c^S_{log} and c^{S_{l=0}}_{log}, verifying the relation, as well aschecked they coincide with the corresponding logarithmic term in mutualinformation of two concentric spheres.Fil: Casini, Horacio German. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); ArgentinaFil: Huerta, Marina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentin