Emergence of long-range entanglement and odd-even effect in periodic generalized quantum cluster models

TL;DR

This paper explores how long-range entanglement emerges in a generalized quantum cluster model with periodic boundary conditions, highlighting the odd-even effects related to system size and interaction range, and demonstrating robustness against quantum fluctuations.

Contribution

It reveals the conditions under which long-range entanglement appears in the model, specifically when both system size and interaction range are odd, and shows this entanglement persists under large transverse fields.

Findings

Long-range entanglement occurs when both N and m are odd.

Four-part quantum conditional mutual information signals long-range entanglement.

Entanglement persists even with large transverse fields.

Abstract

We investigate the entanglement properties in a generalized quantum cluster model under periodic boundary condition. By evaluating the quantum conditional mutual information entropy under four subsystem partitions, we identify clear signatures of long-range entanglement. Specifically, when both the system size $N$ and the interaction range $m$ are odd, the system exhibits nonzero four-part quantum conditional mutual information entropies in infinitesimal but finite field. This nonvanishing four-part quantum conditional mutual information entropy directly signals the presence of long-range entanglement. In contrast, all other combination of $N$ and $m$ yield vanishing four-part quantum conditional mutual information entropy. Remarkably, in the case of $N, m \in \text{odd}$, these long-range entangled features persist even in the presence of a large transverse field, demonstrating their robustness against quantum fluctuations. These results demonstrate how the interplay between system size and interaction range governs the emergence of long-range entanglement in one-dimensional generalized quantum cluster model.