The emergence of long-range entanglement and odd-even effect in periodic generalized cluster models

Abstract

We investigate the entanglement properties in a generalized cluster model under periodic boundary condition. By evaluating the entanglement entropy and the quantum conditional mutual information entropy under three or 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. This non-vanishing 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 finite 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 spin systems.

Figures (7)

• Figure 1: Entanglement entropy $S_l$ as a function of the block size $l$$(l\geq m)$ with (a) $N=1001$ and $m=3$, (b) $N=1001$ and $m=4$, (c) $N=1000$ and $m=3$, and (d) $N=1000$ and $m=4$.
• Figure 2: Schematic representation of the two types of partitions used in the calculation of the quantum mutual information entropy. (a) The system is divided into three subsystems, with the resulting entropy denoted by $S_{cmi}^{t}$. (b) The system is divided into four parts, with the corresponding entropy denoted by $S_{cmi}^{q}$.
• Figure 3: Quantum mutual information entropy as a function of system size. Results are shown for the two representative cases: (a) and (b) $m=3, N \in \text{odd}$, where both $S_{cmi}^{t}$ and $S_{cmi}^{q}$ are nonzero, and (c) and (d) other cases, we select a representative example $m=4, N \in \text{odd}$, where the entropy vanish.
• Figure 4: The low energy of the mixed cluster model with quantum fluctuations with (a) $m=3$ and $N=25$ ; (b) $m=4$ and $N = 25$. $E_{0}$ represents the energy of ground state.
• Figure 5: Entanglement entropy $S_l$ as a function of the block size $l$$(l\geq m)$ with (a)-(b) $N=1001$ and $m=3$, and (c)-(d) $N=1001$ and $m=4$.