Entanglement Entropy and Phase Space Density: Lowest Landau Levels and 1/2 BPS states

TL;DR

This paper derives an analytical expression for the entanglement entropy of non-relativistic fermions in Lowest Landau Level states, revealing a perimeter law with corner corrections, relevant to quantum Hall systems and half-BPS states in gauge theories.

Contribution

It provides a novel analytical framework connecting entanglement entropy in LLL states to phase space density, including corner contributions, applicable to quantum Hall and gauge theory contexts.

Findings

Entanglement entropy follows a perimeter law with shape-independent coefficient.

Corner contributions to entanglement entropy are analytically derived.

Results agree with existing calculations for specific subregions.

Abstract

We consider the entanglement entropy of an arbitrary subregion in a system of $N$ non-relativistic fermions in $2+1$ dimensions in Lowest Landau Level (LLL) states. Using the connection of these states to those of an auxiliary $1+1$ dimensional fermionic system, we derive an expression for the leading large-$N$ contribution in terms of the expectation value of the phase space density operator in $1+1$ dimensions. For appropriate subregions the latter can replaced by its semiclassical Thomas-Fermi value, yielding expressions in terms of explicit integrals which can be evaluated analytically. We show that the leading term in the entanglement entropy is a perimeter law with a shape independent coefficient. Furthermore, we obtain analytic expressions for additional contributions from sharp corners on the entangling curve. Both the perimeter and the corner pieces are in good agreement with existing calculations for special subregions. Our results are relevant to the integer quantum Hall effect problem, and to the half-BPS sector of $\mathcal N=4$ Yang Mills theory on $S^3$. In this latter context, the entanglement we consider is an entanglement in target space. We comment on possible implications to gauge-gravity duality.