Emergent Area Laws from Entangled Matrices

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.