Entanglement in the Quantum Hall Matrix Model

TL;DR

This paper calculates entanglement entropy in the Quantum Hall Matrix Model, revealing boundary and bulk contributions that shed light on the holographic emergence of spacetime.

Contribution

It introduces a method to compute entanglement entropy in a gauged matrix quantum mechanics with an emergent spatial disk, highlighting boundary and bulk effects.

Findings

Logarithmic entanglement entropy from boundary modes

Bulk area law contribution from off-diagonal matrix elements

Entanglement entropy regularized by finite N

Abstract

Characterizing the entanglement of matrix degrees of freedom is essential for understanding the holographic emergence of spacetime. The Quantum Hall Matrix Model is a gauged $U(N)$ matrix quantum mechanics with two matrices whose ground state is known exactly and describes an emergent spatial disk with incompressible bulk dynamics. We define and compute an entanglement entropy in the ground state associated to a cut through the disk. There are two contributions. A collective field describing the eigenvalues of one of the matrices gives a gauge-invariant chiral boundary mode leading to an expected logarithmic entanglement entropy. Further, the cut through the bulk splits certain 'off-diagonal' matrix elements that must be duplicated and associated to both sides of the cut. Sewing these duplicated modes together in a gauge-invariant way leads to a bulk 'area law' contribution to the entanglement entropy. All of these entropies are regularized by finite $N$.