# A Wigner molecule at extremely low densities: a numerically exact study

## TL;DR

This study provides a numerically exact analysis of Wigner localization in a low-density, quasi-one-dimensional two-electron system confined on a ring, highlighting the localization tensor as an effective measure.

## Contribution

It introduces the localization tensor as a reliable tool for detecting Wigner localization across different system sizes using exact diagonalization.

## Key findings

- Localization tensor accurately captures the transition to Wigner localization.
- Wigner localization observed at very low densities in a quasi-1D two-electron system.
- Numerically exact results validate the use of the localization tensor.

## Abstract

In this work we investigate Wigner localization at very low densities by means of the exact diagonalization of the Hamiltonian. This yields numerically exact results. In particular, we study a quasi-one-dimensional system of two electrons that are confined to a ring by three-dimensional gaussians placed along the ring perimeter. To characterize the Wigner localization we study several appropriate observables, namely the two-body reduced density matrix, the localization tensor and the particle-hole entropy. We show that the localization tensor is the most promising quantity to study Wigner localization since it accurately captures the transition from the delocalized to the localized state and it can be applied to systems of all sizes.