Universal properties of the entanglement entropy in quantum integrable models

Levi, Emanuele (2013). Universal properties of the entanglement entropy in quantum integrable models. (Unpublished Doctoral thesis, City University London)

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Abstract

This thesis is a review of the works and ideas I have been developing in my doctoral studies, and it is mainly based on Castro-Alvaredo & Levi [2011]; Castro-Alvaredo et al. [2011]; Levi [2012]; Levi et al. [2013]. The specific aims of these works were to explore the methods developed in Calabrese & Cardy [2004]; Cardy et al. [2008] with the purpose of quantifying entanglement in a quantum field theory, and have a deeper understanding of their predicting power on lattice systems.

The first chapter is meant to be a review of quantum entanglement in many-body physics, and the methods we use to establish the link to QFT. In the second chapter, after a small introduction on conformal field theory, we collect the results of Calabrese & Cardy [2004], focusing in particular on the replica trick and the twist field.

The third chapter is devoted to adapting these tools to massive QFT, as performed in Cardy et al. [2008]. In particular we focus on the form factor program for the twist field, by means of which we are able to outline the behavior of entanglement entropy in massive theories in a non perturbative way. We expand on the results found in Castro-Alvaredo & Levi [2011], where higher particle form factors were studied for the roaming trajectory model, and the SU(3)2-homogenous sine-Gordon model. We then carry out a numerical study of the Δ-function of the twist field for these two models.

In the fourth chapter we focus on the connection between the Δ-function of the twist field and Zamolodchikov c-function, as performed in Castro-Alvaredo et al. [2011]. In addressing this issue we perform a thorough study of the two point function of the twist field and the trace of the stress-energy tensor. This allows us to introduce a class of composite twist fields, which were the main topic of Levi [2012]. In the fifth and last chapter we group the most common methods used to study the entanglement entropy of quantum spin chains. We start with the XY chain analysis, which is performed with a combination of analytical and numerical methods based on free fermion techniques. We then perform a numerical study of the XXZ chain by means of the density matrix renormalization group approach. Eventually we present the results obtained for these two models in Levi et al. [2013].

Item Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Divisions: City University London PhD theses
School of Engineering & Mathematical Sciences > Department of Mathematical Science
URI: http://openaccess.city.ac.uk/id/eprint/3041

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