Entanglement content of quantum particle excitations. Part II. Disconnected regions and logarithmic negativity

Olalla A Castro-Alvaredo,István M Szécsényi,Cecilia De Fazio,Benjamin Doyon

doi:10.1007/jhep11(2019)058

Olalla A Castro-Alvaredo, István M Szécsényi + Show 2 more

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https://doi.org/10.1007/jhep11(2019)058

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Abstract

In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement.

Highlights

In recent years there has been much progress in the understanding of entanglement measures in one-dimensional many body quantum systems
In two recent works [32, 33] we have investigated the entanglement entropy of gapped systems in zero-density excited states, where excitations have aparticle interpretation
In the present work we extend this intuition, with precise results for the free boson theory, to the entanglement entropy of disconnected regions and the logarithmic negativity

Summary

Introduction

In recent years there has been much progress in the understanding of entanglement measures in one-dimensional many body quantum systems (see e.g. the review articles in [1–3]). Much of this work has focussed on one particular measure of entanglement, the entanglement entropy [27] and on a particular state, the ground state. In two recent works [32, 33] we have investigated the entanglement entropy of gapped systems in zero-density excited states, where excitations have a (quasi-)particle interpretation. We found a very intuitive mathematical structure for the entropy differences between the ground state and such excited states. This structure allowed for simple closed formulae in particular limits

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