Critical and noncritical long range entanglement in the Klein-Gordon field

TL;DR

This paper analyzes the entanglement properties of the vacuum state in a 1D Klein-Gordon field, revealing how entanglement varies with system parameters and distinguishing critical from noncritical regimes.

Contribution

It provides explicit calculations of long-range entanglement in the Klein-Gordon field, highlighting differences between critical and noncritical systems without requiring renormalization.

Findings

Entanglement is finite in the continuum limit without renormalization.

Quantum correlations decay exponentially beyond the segment size in the critical case.

Entanglement diverges as a power law when segments are close, and depends on segment size and separation in noncritical regimes.

Abstract

We investigate the entanglement between two separated segments in the vacuum state of a free 1D Klein-Gordon field, where explicit computations are performed in the continuum limit of the linear harmonic chain. We show that the entanglement, which we measure by the logarithmic negativity, is finite with no further need for renormalization. We find that the quantum correlations decay much faster than the classical correlations as in the critical limit long range entanglement decays exponentially for separations larger than the size of the segments. As the segments become closer to each other the entanglement diverges as a power law. The noncritical regime manifests richer behavior, as the entanglement depends both on the size of the segments and on their separation. In correspondence with the von Neumann entropy long-range entanglement also distinguishes critical from noncritical systems.