The quantum sine-Gordon model is the simplest massive interacting integrable quantum field theory whose two-particle scattering matrix is generally non-diagonal. As such, it is a model that has been extensively studied, especially in the context of the bootstrap program. In this paper we compute low particle-number form factors of a special local field known as the branch point twist field, whose correlation functions are building blocks for measures of entanglement. We consider the attractive regime where the theory possesses a particle spectrum consisting of a soliton, an antisoliton (of opposite U(1) charges) and several (neutral) breathers. In the breather sector we exploit the fusion procedure to compute form factors of heavier breathers from those of lighter ones. We apply our results to the study of the entanglement dynamics after a small mass quench and for short times. We show that in the presence of two or more breathers the von Neumann and Rényi entropies display undamped oscillations in time, whose frequencies are proportional to the even breather masses and whose amplitudes are proportional to the breather's one-particle form factor.
The quantum sine-Gordon model is the simplest massive interacting integrable quantum field theory whose two-particle scattering matrix is generally non-diagonal. As such, it is a model that has been extensively studied, especially in the context of the bootstrap program. In this paper we compute low particle-number form factors of a special local field known as the branch point twist field, whose correlation functions are building blocks for measures of entanglement. We consider the attractive regime where the theory possesses a particle spectrum consisting of a soliton, an antisoliton (of opposite U(1) charges) and several (neutral) breathers. In the breather sector we exploit the fusion procedure to compute form factors of heavier breathers from those of lighter ones. We apply our results to the study of the entanglement dynamics after a small mass quench and for short times. We show that in the presence of two or more breathers the von Neumann and Rényi entropies display undamped oscillations in time, whose frequencies are proportional to the even breather masses and whose amplitudes are proportional to the breather's one-particle form factor.
In this paper we continue the program initiated in Part I, that is the study of entanglement measures in the sine-Gordon model. In both parts, we have focussed on one specific technique, that is the well-known connection between branch point twist field correlators and measures of entanglement in 1+1D integrable quantum field theory. Our papers apply this technique for the first time to a non-diagonal theory with an involved particle spectrum, the sine-Gordon model. In this Part II we focus on a different entanglement measure, the symmetry resolved entanglement, and develop its associated twist field description, exploiting the underlying U(1) symmetry of the theory. In this context, conventional branch point twist fields are no longer the fields required, but instead we must work with one of their composite generalisations, which can be understood as the field resulting from the fusion of a standard branch point twist field and the sine-Gordon exponential field associated with U(1) symmetry. The resulting composite twist field has correlators which as usual admit a form factor expansion. In this paper we write the associated form factor equations and solve them for various examples in the breather sector by using the method of angular quantisation. We show that, in the attractive regime, this is the sector which provides the leading contribution to the symmetry resolved entropies, both R\'enyi and von Neumann. We compute the latter in the limit of a large region size and show that they satisfy the property of equipartition, that is the leading contribution to the symmetry resolved entanglement is independent of the symmetry sector.
A bstract Quantum sinh-Gordon model in 1 + 1 dimensions is one of the simplest and best-studied massive integrable relativistic quantum field theories. We consider this theory on a multi-sheeted Riemann surfaces with a flat metric, which can be seen as a pile of planes connected to each other along cut lines. The cut lines end at branch points, which are represented by a twist operator Tn T n . Operators of such kind are interesting in the framework of the problem of computing von Neumann and Renyi entanglement entropies in the original model on the plane. The composite branch-point twist operators (CTO) are a natural generalization of the twist operators, obtained by placing a local operator to a branch point by means of a certain limiting procedure. Correlation function in quantum field theory can be, in principle, found by means of the spectral decomposition. It allows one to express them in terms of form factors of local operators, i.e. their matrix elements in the basis of stationary states. In integrable models complete sets of exact form factors of all operators can be found exactly as solutions of a system of bootstrap equations. Nevertheless, identification of these solution to the operators in terms of the basic fields remains problematic. In this work, we develop a technique of computing form factors of a class of CTO determined in terms of the basic field in the semiclassical approximation.
A bstract We extend the branch point twist field approach for the calculation of entanglement entropies to time-dependent problems in 1+1-dimensional massive quantum field theories. We focus on the simplest example: a mass quench in the Ising field theory from initial mass m 0 to final mass m . The main analytical results are obtained from a perturbative expansion of the twist field one-point function in the post-quench quasi-particle basis. The expected linear growth of the Rényi entropies at large times mt ≫ 1 emerges from a perturbative calculation at second order. We also show that the Rényi and von Neumann entropies, in infinite volume, contain subleading oscillatory contributions of frequency 2 m and amplitude proportional to ( mt ) − 3 / 2 . The oscillatory terms are correctly predicted by an alternative perturbation series, in the pre-quench quasi-particle basis, which we also discuss. A comparison to lattice numerical calculations carried out on an Ising chain in the scaling limit shows very good agreement with the quantum field theory predictions. We also find evidence of clustering of twist field correlators which implies that the entanglement entropies are proportional to the number of subsystem boundary points.
Entanglement Entropy from Form Factors in 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-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-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-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.
Abstract We study the dynamics of the sine-Gordon model after a quantum quench into the attractive regime, where the spectrum consists of solitons, antisolitons and breathers. In particular, we analyse the time-dependent expectation value of the vertex operator, exp ( i β Φ / 2 ) , starting from an initial state in the ‘squeezed state form’ corresponding to integrable boundary conditions. Using an expansion in terms of exact form factors, we compute analytically the leading contributions to this expectation value at late times. We show that form factors containing breathers only contribute to the late-time dynamics if the initial state exhibits zero-momentum breather states. The leading terms at late times exponentially decay, and we compute the different decay rates. In addition, the late-time contributions from the zero-momentum breathers display oscillatory behaviour, with the oscillation frequency given by the breather mass renormalised by interaction effects. Using our result, we compute the low-energy contributions to the power spectrum of the vertex operator. The oscillatory terms in the expectation value are shown to produce smooth peaks in the power spectrum located near the values of the bare breather masses.
Entanglement entropy from form factors in 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 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 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 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.