Entanglement of Purification in Many Body Systems and Symmetry Breaking

TL;DR

The paper investigates entanglement of purification $E_P$ in many-body quantum systems, using a Gaussian purification Ansatz for a lattice free scalar field and numerical purification techniques for spin chains, including the transverse-field Ising model. It reveals that $E_P$ can be non-monotonic or plateau-like with subsystem separation at small system sizes and can exhibit $Z_2$ symmetry breaking in the optimal purification, signaling a delicate balance between classical and quantum correlations. In the conformal (massless) regime, $E_P$ and mutual information share leading logarithmic scaling with system size, while their subleading dependence on the ratio $d/w$ follows distinct but related forms; the study also connects these phenomena to phase transitions in spin systems and to purified-state structures such as Werner states. Overall, the work highlights how purification-based correlation measures uncover nuanced structure in many-body systems beyond what standard entanglement or mutual information alone reveal, with potential links to holographic interpretations and quantum phase behavior.

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.

Figures (18)

• Figure 1: An example of the setup for our lattice model with $N=20$ and $|A|=|B|=4$. The distance between $A$ and $B$ is $d=1$. There is an $Z_2$ reflection symmetry.
• Figure 2: Half of MI (left) and LN (right) for $|A|=|B|=4$ and $N=60$ as a function of $d$, shown for mass $m=10^{-1},10^{-2},10^{-3},10^{-4}$ (bottom to top).
• Figure 3: Entanglement of purification $E_P$ at $N=60$ for $w=|A|=|B|=1,2,3,4$ (bottom to top) for mass $m=10^{-1}$ (left) and $m=10^{-4}$ (right), no $Z_2$ symmetry being assumed.
• Figure 4: Coupling matrix $K$ (defined in \ref{['EQ_KMATRIX']}) for minimal entanglement of purification between physical sites $AB$ and auxiliary sites $\tilde{A}\tilde{B}$ for mass $m=10^{-4}$, block width $w=|A|=|B|=4$, $N=60$ and various block distances $d$.
• Figure 5: $Z_2$ symmetry breaking at masses $m=10^{-1}$ (left) and $m=10^{-4}$ (right) in terms of asymmetry parameter $\mathcal{A}$ for block width $w=4$ and total size $N=60$.