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arXiv:1907.11735 · Journal of High Energy Physics

Entanglement dynamics after a quench in Ising field theory: a branch point twist field approach

Olalla A. Castro-Alvaredo, Máté Lencsés, István M. Szécsényi, Jacopo Viti

Published Dec 1, 2019• 39 citations
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Abstract

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.

DOI
10.1007/jhep12(2019)079

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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.
Abstract In this paper we investigate the properties of the symmetry resolved entanglement entropy after a mass quench in the Ising field theory. Since the theory is free and the post-quench state known explicitly, the one-point function of the relevant (composite) branch point twist field can be computed using form factor techniques, similar to previous work on the branch point twist field and the magnetisation, respectively. We find that the symmetry resolved entropy grows linearly in time at the same rate as the total entropy, and that there are sub-leading oscillatory corrections. This result provides the first explicit computation of the out-of-equilibrium dynamics of the symmetry resolved entropy employing twist fields in quantum field theory and is consistent with existing results based on the quasiparticle picture.
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 I propose a branch point twist field approach to computing the temporal entropy, that is, an entanglement measure across different time regions, as opposed to the usual spacial measures. Considering only the ground state of a gapped theory, I discuss how the shift to time-dependence manifests in form factor calculations. I show that the generalization of the spacial measures to temporal ones reproduces expected features of the temporal entanglement: the entropy is complex, oscillatory and reminiscent of the evolution of entanglement following a global quench. Considering the temporal von Neumann entropy, I argue that spacial and temporal entropies are two sides of the same coin. They both encapsulate universal information about the theory, in particular its mass spectrum. Also in both cases, a quasiparticle picture can be employed to interpret results. This picture connects features of the temporal entropy with those of the spacial entropy following a quantum quench.
Well-known measures of entanglement in one-dimensional many body quantum systems, such as the entanglement entropy and the logarithmic negativity, may be expressed in terms of the correlation functions of local fields known as branch point twist fields in a replica quantum field theory. In this “replica” approach the computation of measures of entanglement generally involves a mathematically non-trivial analytic continuation in the number of replicas. In this paper we consider two-point functions of twist fields and their analytic continuation in the 1+1 dimensional massive (non-compactified) free Boson theory. This is one of the few theories for which all matrix elements of twist fields are known so that we may hope to compute correlation functions very precisely. We study two particular two-point functions which are related to the logarithmic negativity of semi-infinite disjoint intervals and to the entanglement entropy of one interval. We show that our prescription for the analytic continuation yields results which are in full agreement with conformal field theory predictions in the short-distance limit. We provide numerical estimates of universal quantities and their ratios, both in the massless (twist field structure constants) and the massive (expectation values of twist fields) theory. We find that particular ratios are given by divergent form factor expansions. We propose such divergences stem from the presence of logarithmic factors in addition to the expected power-law behaviour of two-point functions at short-distances. Surprisingly, at criticality these corrections give rise to a log⁡(log⁡ℓ) correction to the entanglement entropy of one interval of length ℓ. This hitherto overlooked result is in agreement with results by Calabrese, Cardy and Tonni and has been independently derived by Blondeau-Fournier and Doyon [25].
A bstract We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.
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.