Effects of alternating interactions and boundary conditions on quantum entanglement of a three-leg Heisenberg ladder

Abstract

The spin-1/2 three-leg antiferromagnetic Heisenberg spin ladder is studied under open boundary condition (OBC) and cylinder boundary condition (CBC), using the density matrix renormalization group and matrix product state methods. Specifically, we calculate the energy density, entanglement entropy, and concurrence while discussing the effects of the interleg interaction $J_2$ and the alternating coupling parameter $\gamma$ on these quantities. In the case of OBC and $\gamma =0$, we observe that the intraleg concurrence separation occurs between odd and even bonds within the system and that the introduction of $\gamma$ can completely reverse this distribution of concurrence. On another note, the interleg concurrence is always present, regardless of the presence of $\gamma$. We also find that in the special case where $\gamma =0$, the CBC suppresses interleg concurrence; however, the introduction of $\gamma$ can lead to the emergence of interleg concurrence between chains 1 and 3. This behavior becomes more complex due to the competition between CBC and $\gamma$. Additionally, we find that $\gamma$ induces two types of long-distance entanglement (LDE) in the system under OBC: intraleg LDE and interleg one. When the system size is sufficiently large, both types of LDE reach the same strength and stabilize at a constant value. Compared to the two-leg ladder, the three-leg ladder is more effective in producing LDE. However, the generation of LDE is inhibited in the system under CBC, in which spin frustration exists. In addition, the results of energy, entanglement entropy, and concurrence show that there are essential relations between these quantities and the phase transitions of the system. Furthermore, we predict a phase transition point at approximately $\gamma$ $=0.54$ under OBC. This study provides valuable insights into understanding the phase diagram of this class of systems.