Quantum field theory text on entanglement entropy
As far as I am aware, there are not really any proper books on the subject. However, there are quite a few review papers that are worth consulting. Just a note: despite you asking for resources about QFT, entanglement entropy is best understood in CFT's and much of the language of these following is written with this in mind. You should know some CFT (at the level of David Tong's notes) in order to really grasp the material.
Firstly, the standard text is Entanglement Entropy and Quantum Field Theory by Cardy and Calabrese. This is a good technical overview but is not amazing at giving you the intuition you might want, it's quite terse. It's worth also reading their other paper, Entanglement Entropy and Conformal Field Theory along side this one as it's often a bit more useful and introduces certain topics in a better way (for example twist fields). Both of these cover the replica trick, entanglement entropy in free QFT's (CFT's), the corner transfer matrix etc although it doesn't go into many of the details of higher dimensions than D = 2.
Another great review is Entanglement entropy in free quantum field theory which written in a more pedagogical style and is very well referenced. It is in my opinion less complete than the Cardy & Calabrese papers but largely covers the same stuff, however it does talk about universal properties of higher dimensional QFT's and has a few different methods in.
To learn about geometric entanglement entropy (not talked about in great detail in the above reviews), the original paper On Geometric Entropy by Callan and Wilczek is very readable, although it is missing some background which can be useful.
Another great resource that I have found useful is the introduction in the PhD Thesis called Universal Properties of the Entanglement Entropy in Quantum Integrable Models by Levi Emanuele. This thesis makes excellent reading alongside the Calabrese/Cardy papers as it goes into alot more detail in many places, although basically covers the same stuff (at least, the general methods). The tone is a little more mathematical (which I like) and I personally like that the author has motivated things in a way that none of the other authors seem to have done.
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Qmechanic
Updated on February 22, 2020Comments

Qmechanic over 3 years
I am looking for a quantum field theory book in which entanglement entropy for quantum fields is explained but I can not find such a book. Is there such a book?

M111 almost 6 yearsThanks for the references. I also found [this]("esc.fnwi.uva.nl/thesis/centraal/files/f319625911.pdf") thesis by Vincent Min pretty useful, since it had some of the explicit calculations that was missing in others. The book "Holographic Entanglement Entropy" by Mukund Rangamani and Tadashi Takayanagi is quite informative