Phase structure of the one-dimensional $\mathbb{Z}_2$ lattice gauge theory with second nearest-neighbor interactions

We investigate the ground-state phase diagram of a one-dimensional $\mathbb{Z}_2$ lattice gauge theory (LGT) model with hard-core bosons at half-filling, extending previous studies by including second nearest-neighbor (2NN) interactions. Using matrix product state techniques within the density matrix renormalization group, we compute charge gap, static structure factor, and pair-pair correlation functions for various interaction strengths and field parameters. We analyze two representative neatest-neighbor interaction strengths ($V_1$) that correspond to the Luttinger liquid (LL) and Mott insulator (MI) phases in the absence of the 2NN interactions. We introduce the 2NN coupling $V_2$ and investigate its impact on the system. Our results reveal very rich behavior. As the 2NN repulsion increases, in the case of small $V_1$, we observe a direct transition from the LL phase to a charge-ordered insulator (COI) phase, whereas for large $V_1$, we observe a transition from the MI phase (previously found with only $V_1$ included), going through an intermediate LL region, and finally reaching the COI regime. Additionally, the inclusion of 2NN interactions enhances charge order and suppresses pair coherence, evidenced by sharp peaks in the structure factor and rapid decay in pair-pair correlators. Our work extends the well-studied phase structure of 1D $\mathbb{Z}_2$ LGT models and demonstrates the interplay between gauge fields, confinement, and extended interactions.