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arXiv:1502.03275 · Nuclear Physics B

Entanglement entropy of non-unitary integrable quantum field theory

Davide Bianchini, Olalla A. Castro-Alvaredo, Benjamin Doyon

Published May 14, 2015• 45 citations
arXivHTMLPDF

Abstract

In this paper we study the simplest massive 1+1 dimensional integrable quantum field theory which can be described as a perturbation of a non-unitary minimal conformal field theory: the Lee-Yang model. We are particularly interested in the features of the bi-partite entanglement entropy for this model and on building blocks thereof, namely twist field form factors. Non-unitarity selects out a new type of twist field as the operator whose two-point function (appropriately normalized) yields the entanglement entropy. We compute this two-point function both from a form factor expansion and by means of perturbed conformal field theory. We find good agreement with CFT predictions put forward in a recent work involving the present authors. In particular, our results are consistent with a scaling of the entanglement entropy given by $\frac{c_{\text{eff}}}{3}\log \ell$ where $c_{\text{eff}}$ is the effective central charge of the theory (a positive number related to the central charge) and $\ell$ is the size of the region. Furthermore the form factor expansion of twist fields allows us to explore the large region limit of the entanglement entropy and find the next-to-leading order correction to saturation. We find that this correction is very different from its counterpart in unitary models. Whereas in the latter case, it had a form depending only on few parameters of the model (the particle spectrum), it appears to be much more model-dependent for non-unitary models.

DOI
10.1016/j.nuclphysb.2015.05.013

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In this paper we study the simplest massive 1+1 dimensional integrable quantum field theory which can be described as a perturbation of a non-unitary minimal conformal field theory: the Lee-Yang model. We are particularly interested in the features of the bi-partite entanglement entropy for this model and on building blocks thereof, namely twist field form factors. Non-unitarity selects out a new type of twist field as the operator whose two-point function (appropriately normalized) yields the entanglement entropy. We compute this two-point function both from a form factor expansion and by means of perturbed conformal field theory. We find good agreement with CFT predictions put forward in a recent work involving the present authors. In particular, our results are consistent with a scaling of the entanglement entropy given by ceff3log⁡ℓ\frac{c_{\text{eff}}}{3}\log \ell3ceff​​logℓ where ceffc_{\text{eff}}ceff​ is the effective central charge of the theory (a positive number related to the central charge) and ℓ\ellℓ is the size of the region. Furthermore the form factor expansion of twist fields allows us to explore the large region limit of the entanglement entropy and find the next-to-leading order correction to saturation. We find that this correction is very different from its counterpart in unitary models. Whereas in the latter case, it had a form depending only on few parameters of the model (the particle spectrum), it appears to be much more model-dependent for non-unitary models.
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.
Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.
In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.
In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.
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.
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.