Emergent Area Laws from Entangled Matrices

TL;DR

This paper demonstrates that in matrix-based wavefunctions near a fuzzy sphere geometry, the Gauss law entanglement entropy scales with the boundary area, resembling gravitational entropy, and is linked to an emergent Maxwell coupling.

Contribution

It introduces the concept of Gauss law entanglement in matrix models and shows its proportionality to geometric area after coarse-graining, revealing emergent gravitational-like behavior.

Findings

Gauss law entanglement entropy scales with boundary area.

Entanglement entropy is proportional to the logarithm of a dominant representation's dimension.

The proportionality constant relates to an emergent Maxwell coupling, akin to gravitational entropy.

Abstract

We consider a wavefunction of large $N$ matrices supported close to an emergent classical fuzzy sphere geometry. The $SU(N)$ Gauss law of the theory enforces correlations between the matrix degrees of freedom associated to a geometric subregion and their complement. We call this `Gauss law entanglement'. We show that the subregion degrees of freedom transform under a single dominant, low rank representation of $SU(N)$. The corresponding Gauss law entanglement entropy is given by the logarithm of the dimension of this dominant representation. It is found that, after coarse-graining in momentum space, the $SU(N)$ Gauss law entanglement entropy is proportional to the geometric area bounding the subregion. The constant of proportionality goes like the inverse of an emergent Maxwell coupling constant, reminiscent of gravitational entropy.