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arXiv:1011.4706 · Journal of Statistical Mechanics Theory and Experiment

Permutation operators, entanglement entropy, and theXXZspin chain in the limit \Delta \to-1^+

Olalla A. Castro-Alvaredo, Benjamin Doyon

Published Feb 1, 2011• 38 citations
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Abstract

In this paper we develop a new approach to the investigation of the bi-partite entanglement entropy in integrable quantum spin chains. Our method employs the well-known replica trick, thus taking a replica version of the spin chain model as starting point. At each site i of this new model we construct an operator which acts as a cyclic permutation among the n replicas of the model. Infinite products of give rise to local operators, precursors of branch-point twist fields of quantum field theory. The entanglement entropy is then expressed in terms of correlation functions of such operators. Employing this approach we investigate the von Neumann and Rényi entropies of a particularly interesting quantum state occurring as a limit (in a compact convergence topology) of the antiferromagnetic XXZ quantum spin chain. We find that, for large sizes, the entropy scales logarithmically, but not conformally.

DOI
10.1088/1742-5468/2011/02/p02001

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Permutation operators, entanglement entropy, and the XXZ spin chain in the limit Delta ->-1(+)

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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].

•Jan 1, 2013•0 citations
In this paper we develop a new approach to the investigation of the bi-partite entanglement entropy in integrable quantum spin chains. Our method employs the well-known replica trick, thus taking a replica version of the spin chain model as starting point. At each site i of this new model we construct an operator which acts as a cyclic permutation among the n replicas of the model. Infinite products of give rise to local operators, precursors of branch-point twist fields of quantum field theory. The entanglement entropy is then expressed in terms of correlation functions of such operators. Employing this approach we investigate the von Neumann and Rényi entropies of a particularly interesting quantum state occurring as a limit (in a compact convergence topology) of the antiferromagnetic XXZ quantum spin chain. We find that, for large sizes, the entropy scales logarithmically, but not conformally.
In this paper we develop a new approach to the investigation of the bi-partite entanglement entropy in integrable quantum spin chains. Our method employs the well-known replica trick, thus taking a replica version of the spin chain model as starting point. At each site i of this new model we construct an operator T-i which acts as a cyclic permutation among the n replicas of the model. Infinite products of T-i give rise to local operators, precursors of branch-point twist fields of quantum field theory. The entanglement entropy is then expressed in terms of correlation functions of such operators. Employing this approach we investigate the von Neumann and Renyi entropies of a particularly interesting quantum state occurring as a limit (in a compact convergence topology) of the antiferromagnetic XXZ quantum spin chain. We find that, for large sizes, the entropy scales logarithmically, but not conformally.
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.
A large class of strongly correlated quantum systems can be described in certain large-N limits by quadratic in field actions along with self-consistency equations that determine the two-point functions. We use the replica approach and the notion of shifted Matsubara frequency to compute von Neumann and R\'enyi entanglement entropies for generic bipartitioning of such systems. We argue that the von Neumann entropy can be computed from equilibrium spectral functions without partitioning, while the R\'enyi entropy requires recalculating the spectrum in the interacting case. We demonstrate the flexibility of the method by applying it to examples of a two-site problem in presence of decoherence and coupled Sachdev-Ye-Kitaev models.
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.
Abstract All standard measures of bipartite entanglement in one-dimensional quantum field theories can be expressed in terms of correlators of branch point twist fields, here denoted by and . These are symmetry fields associated to cyclic permutation symmetry in a replica theory and having the smallest conformal dimension at the critical point. Recently, other twist fields (composite twist fields), typically of higher dimension, have been shown to play a role in the study of a new measure of entanglement known as the symmetry resolved entanglement entropy. In this paper we give an exact expression for the two-point function of a composite twist field that arises in the Ising field theory. In doing so we extend the techniques originally developed for the standard branch point twist field in free theories as well as an existing computation due to Horváth and Calabrese of the same two-point function which focused on the leading large-distance contribution. We study the ground state two-point function of the composite twist field and its conjugate . At criticality, this field can be defined as the leading field in the operator product expansion of and the disorder field µ. We find a general formula for log 〈 T μ ( 0 ) T μ † ( r ) 〉 and for (the derivative of) its analytic continuation to positive real replica numbers greater than 1. We check our formula for consistency by showing that at short distances it exactly reproduces the expected conformal dimension.
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].