Fermionic entanglement in the Lipkin model

TL;DR

This paper investigates fermionic entanglement in the Lipkin model's ground state, revealing how different entanglement measures relate to phase transitions and how they can be modeled with mean-field and RPA approaches.

Contribution

It provides a detailed analysis of fermionic entanglement measures in the Lipkin model and compares their behavior with phase transitions, highlighting the need for advanced correlations in modeling.

Findings

One-body entanglement entropy correlates with the mean-field order parameter.

Fermionic concurrence peaks at the phase transition.

Mean-field with symmetry restoration accurately describes some measures, but RPA is needed for concurrence.

Abstract

We examine the fermionic entanglement in the ground state of the fermionic Lipkin model and its relation with bipartite entanglement. It is first shown that the one-body entanglement entropy, which quantifies the minimum distance to a fermionic Gaussian state, behaves similarly to the mean-field order parameter and is essentially proportional to the total bipartite entanglement between the upper and lower modes, a quantity meaningful only in the fermionic realization of the model. We also analyze the entanglement of the reduced state of four single-particle modes (two up-down pairs), showing that its fermionic concurrence is strongly peaked at the phase transition and behaves differently from the corresponding up-down entanglement. We finally show that the first measures and the up-down reduced entanglement can be correctly described through a basic mean-field approach supplemented with symmetry restoration, whereas the concurrence requires at least the inclusion of RPA-type correlations for a proper prediction. Fermionic separability is also discussed.