On Holographic Entanglement Density

TL;DR

This paper investigates how holographic entanglement entropy varies with deformations in conformal field theories, revealing insights into states of matter and conditions for area theorem validity or violation.

Contribution

It introduces the concept of entanglement density and analyzes its behavior under various deformations, providing new criteria for characterizing matter states and understanding area theorem violations.

Findings

ED approaches zero from below in Lorentz-invariant RG flows.

ED can approach thermal entropy density from above when Lorentz symmetry is broken.

Small-L behavior of ED is determined by operator dimension or first law of EE.

Abstract

We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions $d$, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature $T$, a chemical potential $\mu$, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the "entanglement density" (ED) as the change in EE due to the deformation, divided by the sub-region's volume. Using the deformed CFTs above, we show how the ED's dependence on the strip width or sphere radius, $L$, is useful for characterizing states of matter. For example, the ED's small-$L$ behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the "area theorem" states that the coefficient of the EE's area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as $L \to \infty$. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.