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

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 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.