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

TL;DR

The paper investigates the ground-state phase diagram of a 1D $Z_2$ lattice gauge theory with hard-core bosons at half-filling, extended by a second-nearest-neighbor interaction $V_2$. It employs density-matrix renormalization group with matrix product states to compute the charge gap $\Delta$, static structure factor $S(k)$, gauge-invariant pair correlators, and the entanglement entropy $S_{vN}$ across $V_1$, $V_2$, and the field $h$. They identify a novel charge-ordered insulator (COI) phase stabilized by $V_2$, observe an intermediate Luttinger-liquid (LL) region when $V_1$ is large, and map phase diagrams in the $(V_2,h)$ plane; $S(k)$ distinguishes MI from LL/COI while $\Delta$ is decisive for LL vs insulating phases; the entanglement entropy provides complementary but finite-size-limited support. The study demonstrates how extended interactions modify confinement physics in 1D gauge theories and provides signatures for experimental platforms such as quantum simulators.

Abstract

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.

Figures (12)

• Figure 1: Finite-size scaling extrapolation of the charge gap $\Delta L/2, L)$ for $V_1 = 1.0$ (and different values of $h$ and $V_2$ as labeled). The linear fit is performed as a function of $1/L$, and the intercept at $1/L \to 0$ yields $\Delta$.
• Figure 2: The critical charge gap $\Delta$ for $V_1=1.0$ presented as a function of $h$ (upper panel, for fixed $V_2$ as labeled) and of $V_2$ (lower panel, for fixed $h$ as labeled). Linear extrapolation of $\Delta$ to zero identifies critical points, which define the LL-COI phase transition boundary (cf. also Sec. \ref{['Phase-Diagram-S']}).
• Figure 3: The critical charge gap $\Delta$ for $V_1=4.0$ as a function of $h$ (upper panel, for fixed $V_2$ as labeled) and of $V_2$ (lower panel, for fixed $h$ as labeled). The points, from which phase transitions occur are identified by linearly extrapolating $\Delta$ to zero, it constitutes the basis for constructing the MI-LL and the LL-COI phase boundaries (cf. also Sec. \ref{['Phase-Diagram-S']}).
• Figure 4: Static structure factor $S(k)$ for $V_{1}=4.0$ and $h=0.5$ for different system sizes: $L=60$ (upper panel), $L=80$ (middle panel), and $L=104$ (lower panel). The peak at $k=\pi$ indicates the MI phase, while the peaks at $k=\pi/2$ and $3\pi/2$ signal the LL and the COI phases. The comparison illustrates the finite-size convergence of $S(k)$ toward the thermodynamic limit.
• Figure 5: Structure factor $S(k)$ across the LL to the COI phase transition for $V_1 = 1.0$ and $L=104$. The upper panel shows the transition driven by increasing the magnetic field $h$ (as labeled, at fixed $V_2=2.0$). The lower panel shows the transition driven by increasing the long-range interaction $V_2$ (as labeled, at fixed $h=0.0$). In both cases, the $S(k)$ features transition from the LL behavior to the two pronounced peaks characteristic of the COI phase.