On the renormalization of entanglement entropy

Jin-Yi Pang,Jiunn-Wei Chen

doi:10.1007/s43673-021-00032-1

Abstract

The renormalization of entanglement entropy of quantum field theories is investigated in the simplest setting with a λϕ4 scalar field theory. The 3+1 dimensional spacetime is separated into two regions by an infinitely flat 2-dimensional interface. The entanglement entropy of the system across the interface has an elegant geometrical interpretation using the replica trick, which requires putting the field theory on a curved spacetime background. We demonstrate that the theory, and hence the entanglement entropy, is renormalizable at order λ once all the relevant operators up to dimension 4 are included in the action. This exercise has a one-to-one correspondence to entanglement entropy interpretation of the black hole entropy which suggests that our treatment is sensible. Our study suggests that entanglement entropy is renormalizable and is a physical quantity.

Highlights

An interesting attempt to understand the BekensteinHawking formula of black hole entropy SBH = A/4G, with A the black hole horizon area and G the gravitational constant, is to relate it to the entanglement entropy (SE) across the black hole horizon [1,2,3,4,5]
5 Conclusion and discussion Our study suggests that entanglement entropy is renormalizable and is a physical quantity
We have demonstrated the renormalizability of the entanglement entropy of the λφ4 at order λ when the 3+1 dimensional theory is separated into two regions by an infinitely flat 2-dimensional interface

Summary

Introduction

Susskind and Uglum suggested that the divergence just renormalizes the bare gravitational constant G to the renormalized one, GR, such that SBH = SE = A/4GR [6, 7] This suggestion was checked by explicit computations in the massive black hole limit for free fields while treating gravity classical [7]. We do the same exercise to the black hole case by replacing the curve spacetime background by gravity and find that there is a one-to-one correspondence between the non-gravitational theory and the black hole case mathematically While this result suggests that our formulation is sensible from the point of view of general relativity, the deficit angle in a condensed matter system in flat space remains an illusive concept worth further exploration

Review of the replica trick

Renormalized entanglement entropy in the λφ4 theory

Conclusion and discussion