In this review the entanglement and Renyi entropies in quantum field theory are described from different points of view, including the perturbative approach and holographic dualities. The applications of these results to constraining renormalization group flows are presented effectively and illustrated with a variety of examples.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.
Abstract In this paper we continue the study of renormalized entanglement entropy introduced in [1]. In particular, we investigate its behavior near an IR fixed point using holographic duality. We develop techniques which, for any static holographic geometry, enable us to extract the large radius expansion of the entanglement entropy for a spherical region. We show that for both a sphere and a strip, the approach of the renormalized entanglement entropy to the IR fixed point value contains a contribution that depends on the whole RG trajectory. Such a contribution is dominant, when the leading irrelevant operator is sufficiently irrelevant. For a spherical region such terms can be anticipated from a geometric expansion, while for a strip whether these terms have geometric origins remains to be seen.
This work is meant as a review summary of a series of recent results concerning the derivation of a holographic entanglement entropy formula for generic open spin network states in the group field theory (GFT) approach to quantum gravity. The statistical group-field computation of the Rényi entropy for a bipartite network state for a simple interacting GFT is reviewed, within a recently proposed dictionary between group field theories and random tensor networks, and with an emphasis on the problem of a consistent characterisation of the entanglement entropy in the GFT second quantisation formalism.
An entanglement Renyi entropy for a spatial partition of a system is studied in conformal theories which admit a dual description in terms of an anti-de Sitter gravity. The divergent part of the Renyi entropy is computed in 4D conformal N=4 super Yang-Mills theory at a weak coupling. This result is used to suggest a holographic formula which reproduces the Renyi entropy at least in the leading approximation. The holographic Renyi entropy is an invariant functional set on a codimension 2 minimal hypersurface in the bulk geometry. The bulk space does not depend on order n of the Renyi entropy. The holographic Renyi entropy is a sum of local and non-local functionals multiplied by polynomials of 1/n.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
A bstract Open quantum systems are defined as ordinary unitary quantum theories coupled to a set of external degrees of freedom, which are introduced to take on the rôle of an unobserved environment. Here we study examples of open quantum field theories, with the aid of the so-called Feynman-Vernon Influence Functional (“IF”), including field theories that arise in holographic duality. We interpret the system in the presence of an IF as an open effective field theory, able to capture the effect of the unobserved environment. Our main focus is on computing Rényi and entanglement entropies in such systems, whose description from the IF, or “open EFT”, point of view we develop in this paper. The issue of computing the entanglement-Rényi entropies in open quantum systems is surprisingly rich, and we point out how different prescriptions for the IF may be appropriate depending on the application of choice. A striking application of our methods concerns the fine-grained entropy of subsystems when including gravity in the setup, for example when considering the Hawking radiation emitted by black holes. In this case we show that one prescription for the IF leads to answers consistent with unitary evolution, while the other merely reproduces standard EFT results, well known to be inconsistent with unitary global evolution. We establish these results for asymptotically AdS gravity in arbitrary dimensions, and illustrate them with explicit analytical expressions for the IF in the case of matter-coupled JT gravity in two dimensions.