Abstract
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(logℓ) 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].
Highlights
The problem of quantifying the amount of entanglement which may be “stored” in the ground state of a many body quantum system has attracted the interest of the quantum information and theoretical physics communities for a long time
1 + 1-dimensional many body quantum systems have received considerable attention over the past decade. Much work in this area has been inspired by the results of Calabrese and Cardy [1] which used principles of Conformal Field Theory (CFT) to study a particular measure of entanglement, the entanglement entropy (EE) [2]
The paper is organized as follows: In sections 2 and 3 we review basic CFT and QFT results, regarding the short distance behaviour of two-point functions of twist fields and how these two-point functions may be expressed in terms of the form factors (4)
Summary
The problem of quantifying the amount of entanglement which may be “stored” in the ground state of a many body quantum system has attracted the interest of the quantum information and theoretical physics communities for a long time. Formulae (1) are advantageous in that partition functions in replica theories may be computed systematically by various approaches, and disadvantageous because the analytic continuations involved are often very difficult to perform and there is no generic proof of existence and uniqueness It was first noted in [1] that the function Tr(ρAn ) may be expressed as a two-point function of fields with conformal dimension given by n c 24 n− 1 n (2). At short-distances we expect the massive QFT to be described by its corresponding ultraviolet limit (that is, the massless (non-compactified) free Boson CFT) We expect these two-point functions to exhibit power-law behaviours with powers related to the dimension of twist fields. Appendix B provides a discussion and assessment of the error of some of our numerical procedures
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