Geometric entanglement in the Laughlin wave function

TL;DR

This paper numerically investigates the geometric entanglement in the Laughlin wave function, revealing a linear relationship with the number of electrons and providing insights into its topological properties.

Contribution

It introduces an iterative algorithm to construct the Slater determinant with the largest overlap and analyzes the geometric entanglement's behavior in Laughlin states.

Findings

Geometric entanglement is approximately linear in electron number.

Linear behavior persists down to two electrons.

Constant term differs from topological entropy expectations.

Abstract

We study numerically the geometric entanglement in the Laughlin wave function, which is of great importance in condensed matter physics. The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. It is found that the geometric entanglement is a linear function of the number of electrons to a good extent. This is especially the case for the lowest Laughlin wave function, namely the one with filling factor of $1/3$. Surprisingly, the linear behavior extends well down to the smallest possible value of the electron number, namely, $ N= 2$. The constant term does not agree with the expected topological entropy.