# Entanglement of Purification in Many Body Systems and Symmetry Breaking

## TL;DR

This paper investigates the behavior of entanglement of purification in lattice models, revealing non-monotonicity, symmetry breaking, and the interplay of classical and quantum correlations in many-body systems.

## Contribution

It provides numerical analysis of EoP in free scalar and Ising models, showing non-monotonic behavior, symmetry breaking, and phase-dependent effects.

## Key findings

- EoP is non-monotonic at small system sizes
- EoP becomes monotonic with plateau at large sizes
- Reflection symmetry can be broken in the ferromagnetic phase

## Abstract

We study the entanglement of purification (EoP), a measure of total correlation between two subsystems $A$ and $B$, for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EoP becomes a non-monotonic function of the distance between $A$ and $B$ when the total number of lattice sites is small. When it is large, the EoP becomes monotonic and shows a plateau-like behavior. Moreover, we show that the original reflection symmetry which exchanges $A$ and $B$ can get broken in optimally purified systems. In the Ising model, we find this symmetry breaking in the ferromagnetic phase. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.