# Structure of Quantum Entanglement at a Finite Temperature Critical Point

## TL;DR

This paper investigates the nature of quantum entanglement near finite temperature critical points, revealing that certain long-range correlations are classical and introducing a quantum correlation length that remains finite at the transition.

## Contribution

It introduces a scheme using tripartite entanglement negativity to characterize quantum entanglement at finite temperature critical points and demonstrates its classical nature in a mean-field Ising model.

## Key findings

- Tripartite negativity shows no singularity across the transition.
- Long-distance critical fluctuations are purely classical.
- Mixed bosonic states with certain symmetries have zero entanglement.

## Abstract

We propose a scheme to characterize long-range quantum entanglement close to a finite temperature critical point using tripartite entanglement negativity. As an application, we study a model with mean-field Ising critical exponents and find that the tripartite negativity does not exhibit any singularity in the thermodynamic limit across the transition. This indicates that the long-distance critical fluctuations are completely classical, allowing one to define a `quantum correlation length' that remains finite at the transition despite a divergent physical correlation length. Motivated by our model, we also study mixed state entanglement in tight-binding models of bosons with $U(1)$ and time-reversal symmetry. By employing Glauber-Sudarshan `P-representation', we find a surprising result that such states have zero entanglement.