The geometry and entanglement entropy of surfaces in loop quantum gravity

TL;DR

This paper explores the quantized geometry of surfaces in loop quantum gravity, defining operators for curvature and analyzing entanglement entropy, revealing a proportionality between entropy gradient and mean curvature.

Contribution

It introduces new operators for scalar and mean curvature in loop quantum gravity and links entanglement entropy variations to surface geometry.

Findings

Entropy gradient is proportional to mean curvature on certain states.

Entanglement entropy remains constant under small deformations of minimal surfaces.

Provides insights into the geometric and entanglement structure of surfaces in loop quantum gravity.

Abstract

In loop quantum gravity, the area element of embedded spatial surfaces is given by a well-defined operator. We further characterize the quantized geometry of such surfaces by proposing definitions for operators quantizing scalar curvature and mean curvature. By investigating their properties, we shed light on the nature of the geometry of surfaces in loop quantum gravity. We also investigate the entanglement entropy across surfaces in the case where spin network edges are running within the surface. We observe that, on a certain class of states, the entropy gradient across a surface is proportional to the mean curvature. In particular, the entanglement entropy is constant for small deformations of a minimal surface in this case.