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arXiv:1309.7380

Entanglement entropy of a scalar field across a spherical boundary in the Einstein universe

Katja Ried

Published Sep 27, 2013• 1 citation
arXivHTMLPDF

Abstract

A scalar field in the ground state, when partially hidden from observation by a spherical boundary, acquires entanglement entropy $S$ proportional to the area of the surface. This area law is well established in flat space, where it follows almost directly from dimensional arguments. We study its validity in an Einstein universe, whose curvature provides an additional physical parameter on which the entropy could, in principle, depend. The surprisingly simple result is that the entanglement entropy still scales linearly with the area. This is supported by other observations to the effect that the entanglement entropy arises mostly from degrees of freedom near the boundary, making it insensitive to the large-scale geometry of the background space.

DOI
10.48550/arxiv.1309.7380

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A scalar field in the ground state, when partially hidden from observation by a spherical boundary, acquires entanglement entropy SSS proportional to the area of the surface. This area law is well established in flat space, where it follows almost directly from dimensional arguments. We study its validity in an Einstein universe, whose curvature provides an additional physical parameter on which the entropy could, in principle, depend. The surprisingly simple result is that the entanglement entropy still scales linearly with the area. This is supported by other observations to the effect that the entanglement entropy arises mostly from degrees of freedom near the boundary, making it insensitive to the large-scale geometry of the background space.
We revisit the problem of finding the entanglement entropy of a scalar field on a lattice by tracing over its degrees of freedom inside a sphere. It is known that this entropy satisfies the area law---entropy proportional to the area of the sphere---when the field is assumed to be in its ground state. We show that the area law continues to hold when the scalar field degrees of freedom are in generic coherent states and a class of squeezed states. However, when excited states are considered, the entropy scales as a lower power of the area. This suggests that, for large horizons, the ground state entropy dominates, whereas entropy due to excited states gives power-law corrections. We discuss possible implications of this result to black hole entropy.
A bstract We study the entanglement entropy within a spherical region for a free scalar field in a squeezed state in 3 + 1 dimensions. We show that, even for small squeezing, a volume term appears, whose coefficient is essentially independent of the field mass. This is in line with Page’s argument that the entanglement entropy in an arbitrary quantum state is proportional to the number of degrees of freedom of the smaller subsystem. It follows that squeezed states can be considered as arbitrary quantum states, in contrast to the ground or coherent states that give rise to entanglement entropy dominated by a term proportional to the area of the entangling surface.
Subdominant contributions to the entanglement entropy of quantum fields include logarithmic corrections to the area law characterized by universal coefficients that are independent of the ultraviolet regulator and capture detailed information on the geometry around the entangling surface. We determine two universal coefficients of the entanglement entropy for a massive scalar field in a static closed universe $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{3}$ perturbatively and verify the results numerically. The first coefficient describes a well-known generic correction to the area law independent of the geometry of the entangling surface and background. The second coefficient describes a curvature-dependent universal term with a nontrivial dependence on the intrinsic and extrinsic geometries of the entangling surface and curvature of the background. The numerical calculations confirm the analytical results to a high accuracy. The first and second universal coefficients are determined numerically with a relative error with respect to the analytical values of the orders ${10}^{\ensuremath{-}4}$ and ${10}^{\ensuremath{-}2}$, respectively.
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We investigate the dynamics of the ground state entanglement entropy for a discretized scalar field propagating within the Oppenheimer-Snyder collapse metric. Starting from a well-controlled initial configuration, we follow the system as it evolves toward the formation of a horizon and, eventually, a singularity. Our approach employs an Ermakov-like equation to determine the time-dependent ground state of the field and calculates the resulting entanglement entropy by tracing out the degrees of freedom inside a spherical region within the matter sphere. We find that the entanglement entropy exhibits nontrivial scaling and time dependence during collapse. Close to the horizon, the entropy can deviate from the simple area law, reflecting the rapid changes in geometry and field configuration. Although the model is idealized, these results provide insights into the generation and scaling of entanglement in the presence of realistic, dynamically evolving gravitational fields.
A bstract We study the area and volume laws for entanglement of free quantum scalar fields. In addition to the entropy, we use the notion of the capacity of entanglement, which measures entropy fluctuations. We consider flat spacetimes as well as the curved ones relevant for cosmology. Moreover, we put special emphasis on quench phenomena and different geometries of the entangling surfaces. First, we show that, in the Minkowski spacetime, the capacity of entanglement, like entropy, exhibits the area law for two kinds of geometries of the entangling surfaces: the sphere and strip. Moreover, we show that the ratio of both quantities takes the same values for both surfaces. Next, we turn our attention to quenches. Namely, we analyse the dynamics of capacity; in particular, contribution of the volume and surface terms. Moreover, we compare these results with theoretical predictions resulting from the quasiparticles model. In the second part, we consider the above issues for the FLRW spaces; especially, for de Sitter space as well as a metric modeling the transition to radiation-dominated era. Finally, we analyse the abrupt quenches in de Sitter space.