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Universal properties of the entanglement entropy in quantum integrable models

Emanuele Levi

Published Jan 1, 2013• 0 citations
OpenAlex

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].

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This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the "replica trick" and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.

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In two recent papers we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised $\mathrm{T}\bar{\mathrm{T}}$-deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to $\mathrm{T}\bar{\mathrm{T}}$-perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for $\mathrm{T}\bar{\mathrm{T}}$-perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of a recent publication where the same problem has been addressed.

arXiv:2306.11064•Jun 19, 2023•0 citations
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].
In this thesis I present the results I have been developing during my PhD studies at City University London. The original results are based on D Bianchini et al, D Bianchini, O Castro-Alvaredo and B Doyon, D Bianchini and F Ravanini, D Bianchini et al and D Bianchini and O Castro-Alvaredo. In all but one publications, we compute the entanglement of various systems. Using the celebrated “replica trick” we compute the entanglement entropy of non unitary systems using integrable tools in continuum and discrete models. In particular, in the first article we generalise the method described in the seventh article in order to take into account non unitary conformal systems. In the second article we use a form factor expansion to probe a non unitary system outside the critical point. In the fourth article we derive the explicit expressions of one dimensional quantum Hamiltonians which provide a lattice realisation of off critical non unitary minimal models. Using a Corner Transfer Matrix approach we compute the scaling of the entanglement of such spin chains. In the fifth article we study the scaling of various twist field correlation functions in order to compute the entanglement entropy and the logarithmic negativity in free boson massive theories.
This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the "replica trick" and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.
We carry out a numerical study of the bipartite entanglement entropy in the gapped regime of two paradigmatic quantum spin chain models: the Ising chain in an external magnetic field and the antiferromagnetic XXZ model. The universal scaling limit of these models is described by the massive Ising field theory and the SU(2)SU(2)SU(2)-Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behavior of the entropy. We numerically confirm that in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to a modified Bessel function of the first kind, K0(2mr){K}_{0}(2mr)K0​(2mr), where mmm is a mass scale (the inverse correlation length) and rrr the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted by J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon [J. Stat. Phys. 130, 129 (2008)] and B. Doyon [Phys. Rev. Lett. 102, 031602 (2009)] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behavior following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length.
Entanglement entropy from form factors in TT‾ \textrm{T}\overline{\textrm{T}} TT-deformed integrable quantum field theories
Abstract In two recent papers [1, 2] we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ -deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ -perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ -perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of the recent publication [3] where the same problem has been addressed.
Recently, Cardy, Castro Alvaredo, and the author obtained the first exponential correction to saturation of the bipartite entanglement entropy at large region lengths in massive two-dimensional integrable quantum field theory. It depends only on the particle content of the model, and not on the way particles scatter. Based on general analyticity arguments for form factors, we propose that this result is universal, and holds for any massive two-dimensional model (also out of integrability). We suggest a link of this result with counting pair creations far in the past.
Entanglement Entropy from Form Factors in TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-Deformed Integrable Quantum Field Theories
In two recent papers we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of a recent publication where the same problem has been addressed.