The von Neumann entanglement entropy for Wigner-crystal states in one dimensional N-particle systems

TL;DR

This paper analytically investigates the von Neumann entanglement entropy in one-dimensional N-particle systems with inverse power law interactions, revealing how it depends on particle number and interaction type in the strong coupling limit.

Contribution

It derives the Schmidt decomposition, natural orbitals, and asymptotic entanglement entropy expressions for strongly interacting 1D particles, providing new analytical insights.

Findings

Analytic form of the Schmidt decomposition in the strong interaction limit

Closed-form asymptotic natural orbitals and occupancies

Explicit dependence of von Neumann entropy on particle number and interaction type

Abstract

We study one-dimensional systems of $N$ particles in a one-dimensional harmonic trap with an inverse power law interaction $\sim|x|^{-d}$. Within the framework of the harmonic approximation we derive, in the strong interaction limit, the Schmidt decomposition of the one-particle reduced density matrix and investigate the nature of the degeneracy appearing in its spectrum. Furthermore, the ground-state asymptotic occupancies and their natural orbitals are derived in closed analytic form, which enables their easy determination for a wide range of values of $N$. A closed form asymptotic expression for the von Neumann entanglement entropy is also provided and its dependence on $N$ is discussed for the systems with $d=1$ (charged particles) and with $d=3$ (dipolar particles).