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Role of spatial higher order derivatives in momentum space entanglement

S. Santhosh Kumar (IISER-TVM), S. Shankaranarayanan (IISER-TVM)
(Submitted on 28 Feb 2017)

We study the momentum space entanglement between different energy modes of interacting scalar fields propagating in general (D + 1)-dimensional flat space-time. As opposed to some of the recent works [1], we use Lorentz invariant normalized ground state to obtain the momentum space entanglement entropy. We show that the Lorenz invariant definition removes the spurious power-law behaviour obtained in the earlier works [1]. More specifically, we show that the cubic interacting scalar field in (1 + 1) dimensions leads to logarithmic divergence of the entanglement entropy and consistent with the results from real space entanglement calculations. We study the effects of the introduction of the Lorentz violating higher derivative terms in the presence of non-linear self inter- acting scalar field potential and show that the divergence structure of the entanglement entropy is improved in the presence of spatial higher derivative terms.

https://arxiv.org/pdf/1702.08655.pdf

The rest of the paper is organized as follows: Sec. (II) discusses the approach used to evaluate entanglement entropy in momentum space and gives explicit formula for calculating the entanglement entropy of a scalar ?eld in any dimension. In Sec. (III), we discuss the model action in any (D+1)- dimensional space-time using the perturbative expansion. In Sec. (IV), we discuss the results for two speci?c cases and generalize to any space dimensions. We show that the entropy relation derived here is di?erent from the ones obtained in Ref. [1] by an extra power factor which has quite compelling implications in the renormalization. It is shown that the divergence in the entropy is tunable by changing the dimension of the space-time and the power of the self interaction. Sec. (V), concludes with the discussion about our results and its possible connection to the renormalization of entanglement entropy. In this work, we set c = ~ = 1.

[20] A. Tagliacozzol, Private communication

Real space entanglement entropy is symmetric, however, the momentum space entanglement de?ned in Ref. [1] is not and hence can not be considered as an universal quantity [15]. Our analysis in Sec. (IV-A), in the light of Ref. [16], shows that at least for the (1+1)-dimensional ?eld theory, the momentum space entanglement entropy can indeed be considered as an universal quantity. More speci?cally, since the UV and IR are related by a simple rescaling of the variables in (1+1)-dimensions [16], the momentum space entanglement entropy evaluated by integrating over the UV modes or IR modes is identical. Our aim is to extend the analysis for higher dimensions. This is currently under investigatio