Holographic Entanglement Entropy

Abstract

We review the developments in the past decade on holographic entanglement entropy, a subject that has garnered much attention owing to its potential to teach us about the emergence of spacetime in holography. We provide an introduction to the concept of entanglement entropy in quantum field theories, review the holographic proposals for computing the same, providing some justification for where these proposals arise from in the first two parts. The final part addresses recent developments linking entanglement and geometry. We provide an overview of the various arguments and technical developments that teach us how to use field theory entanglement to detect geometry. Our discussion is by design eclectic; we have chosen to focus on developments that appear to us most promising for further insights into the holographic map. This is a preliminary draft of a few chapters of a book which will appear sometime in the near future, to be published by Springer, as part of their Lecture Notes in Physics series. The book in addition contains a discussion of application of holographic ideas to computation of entanglement entropy in strongly coupled field theories, and discussion of tensor networks and holography, which we have chosen to exclude from the current manuscript. i AcknowledgmentsWe have been extremely fortunate to benefit from the wisdom and deep physical intuition of our wonderful collaborators Veronika Hubeny and Shinsei Ryu who played a pivotal role in helping us develop the basic picture relating quantum entanglement and holography. The importance of their role in shaping the story of holography entanglement entropy cannot be overstated.We have also enjoyed many excellent collaborations in our explorations over the past decade on this subject: thanks to

Entanglement, replicas, and Thetas

“…A strong form of the thermal entropy relation, as explained in [6,8], says that the n th replica partition function Z n (z 12 , τ ) should reduce in the limit z 12 ∼ 0 to Z 1 (τ ) n (up to the prefactor that encodes the singularity induced by the collision of the two twist operators), while in the limit z 12 ∼ 1 it should go over to Z 1 (nτ ). What was shown in [12] is that the former prescription only agrees with this prediction at small intervals z 12 ∼ 0, while 1 Physically the length of the entangling interval is real, and the modular transformation S transforms z12 from real to imaginary values. Thus the modular transformation of entanglement entropy is a "temporal" version of it.…”

“…Thus it is natural to ask if the result is modular covariant. 1 In this context the following puzzle was raised in [12]. Firstly it was observed that, although the replica partition function for any fixed spin structure is not modular covariant with respect to the full modular group of the torus, one can obtain a modular covariant answer by summing over all four torus spin-structures for the fermions.…”

“…The study of entanglement measures in quantum field theories has provided a rich set of results that illuminate these theories as well as their holographic duals (for a comprehensive review, see [1]). In this work we focus on some of the simplest known quantum field theories, namely free conformal field theories in 1 + 1 dimensions, and address some puzzles that arises when one computes Rényi and entanglement entropies for systems of finite size and at finite temperature.…”

Entanglement in Lifshitz-type quantum field theories

“…Of course the correlators only depend on (m − n) due to the translation invariance of the system. 4 Note that according to the above expressions, in the case of periodic boundary condition the massless limit, i.e., m → 0, is not well-defined due to the existence of a zero mode. This makes us consider a non-zero mass in order to regularize the corresponding divergences.…”

“…Studying the physics of non local correlations due to the quantum entanglement in quantum many-body systems, quantum field theories and especially holographic field theories has gained increasing attention during the last decade [1][2][3][4]. Furthermore, in order to quantify entanglement and have a deeper understanding of it, different measures has been studied so far such as entanglement entropy (EE), mutual information and logarithmic negativity [5][6][7][8].…”