Dynamical Aharonov-Bohm cages and tight meson confinement in a $\mathbb{Z}_2$-loop gauge theory

TL;DR

We investigate a 1D $\mathbb{Z}_2$ lattice gauge theory with loop-based gauge fields and dynamical matter, where $\pi$-flux visons induce dynamical Aharonov-Bohm cages that constrain charge motion. By mapping to a spin-1 gauge description and employing a novel DMRG encoding, we map out finite-density phases: at $h=0$ the system self-assembles AB cages of varying length, confining charges into mesons; at finite $h$ vison mobility yields a Luttinger-liquid of mesons described by an effective constrained Bose-Hubbard model, with robust LL behavior across parameter ranges. For certain fillings, notably $\nu=2/3$, the system also forms incompressible Mott-like phases, including a $\mathbb{Z}_3$ Mott insulator of mesons locked into AB trimers, accompanied by translational-symmetry breaking and a finite charge gap. The results reveal a new confinement mechanism driven by dynamical AB caging, with clear experimental relevance for trapped-ion and Rydberg quantum simulators and potential extensions to higher dimensions and string-net frameworks.

Abstract

We study the finite-density phases of a $\mathbb{Z}_2$ lattice gauge theory (LGT) of interconnected loops and dynamical $\mathbb{Z}_2$ charges. The gauge-invariant Wilson terms, accounting for the magnetic flux threading each loop, correspond to simple two-body Ising interactions in this setting. Such terms control the interference of charges tunneling around the loops, leading to dynamical Aharonov-Bohm (AB) cages that are delimited by loops threaded by a $π$-flux. The latter can be understood as $\mathbb{Z}_2$ vortices, the analog of visons in two dimensional LGTs, which become mobile by adding quantum fluctuations through an external electric field. In contrast to a semi-classical regime of static and homogeneous AB cages, the mobile visons can self-assemble leading to AB cages of different lengths depending on the density of $\mathbb{Z}_2$ charges and the interplay of magnetic and electric terms. Inside these cages, the individual charges get confined into tightly-bound charge-neutral pairs, the $\mathbb{Z}_2$ analogue of mesons. Depending on the region of parameter space, these tightly-bound mesons can propagate within dilute AB-dimers that virtually expand and contract, or else move by virtually stretching and compressing an electric field string. Both limits lead to a Luttinger liquid described by a constrained integrable model. This phase is separated from an incompressible Mott insulator where mesons belong to closely-packed AB-trimers. In light of recent trapped-ion experiments for a single $\mathbb{Z}_2$ loop, these phases could be explored in future experiments.

Figures (16)

• Figure 1: $\mathbb{Z}_2$ Loop-chain. (a) Hardcore bosons live in the sites of a one-dimensional lattice. Two links depart from each site in a loop geometry and host spin-$\frac{1}{2}$ gauge-field variables $\sigma^\alpha_1$ and $\sigma^\alpha_2$. Each loop encloses a gauge flux $\sigma^x_1 \sigma^x_2$ that can take values in $\{0,\pi\}$. The hopping amplitude between sites connected by a loop in a $\pi$-flux state vanishes due to destructive Aharonov-Bohm interference. (b) Gauge invariant configurations in the spin-$1$ reduction of the loop-chain model, where $|\Phi^\pm\rangle$ and $|\Psi^+\rangle$ are common eigenstates of the spin-$1$ operators $(S^x)^2$ and $(S^z)^2$. Occupied sites are filled in dark-blue.
• Figure 2: Schematics of the encoding for DMRG. The bonds of the spin-$1$ LGT in Eq. \ref{['eq:spin1red']} are broken in half. Super-sites (dashed circles) enclose the original matter sites of the chain and two "half-links", except at the edges, where one "half-link" is dropped as the original chain terminates with sites. In each super-site, the left "half-links" host spin-$\frac{1}{2}$ degrees of freedom, while spin-$1$ operators live in those on the right.
• Figure 3: ED-DMRG comparison. The results obtained by using DMRG with the devised encoding are compared to ED applied on Eq. \ref{['eq:spin1red']}, for a chain of $L=9$ sites and $N=6$ particles. The groundstate energies obtained with the two methods with varying $h/t$ are plot for $J/t = 0.0,0.5,1.0$.
• Figure 4: Phase-diagram at $h=0$. Contour plot of the filling fraction $\nu=N/L$ as a function of $J/t$ and $\mu/t$, for a chain of $L=60$ sites. Red lines delimit areas of constant average gauge flux $\overline{W} = \sum_i \langle W_{\bigcirc_{i_\ell}}\rangle/(L-1)$ in which the chain is divided into clusters of size $\ell_{\rm AB}$ indicated by the red numbers inside each region. For instance, the region on the left corresponds to the un-partitioned chain of $\ell_{\rm AB}=L$ connected sites and no visons, whereas that in the bottom right corner is broken at every site $\ell_{\rm AB}=1$ by completely filling the loops with visions. Except for the small area with cages of size $\ell_{\rm AB}=5$, which host four particles each, the remaining arrangement of AB cages with $\ell_{\rm AB}\in\{1,2,3,4\}$ contain two particles each. Notice that the reference values $J^\star/t = {3\sqrt{3}}/{\pi} - \sqrt{2}$, $J'/t = \sqrt{5}- \sqrt{2}$ and $J/t = \sqrt{2}$ can be calculated analytically in the thermodynamic limit- see Supplemental Material for details.
• Figure 5: Chain-partitioning. We plot the averaged expectation value of the magnetic Wilson operators $\overline{W} = \sum_i \langle W_{\bigcirc_{i_\ell}}\rangle/(L-1)$, which provides a sense on how the number of vison changes with the parameters and, thus, on the formation of AB cages at $h=0$ and filling $\nu=\frac{2}{3}$. The fragmented chains contain AB cages separated from one another by $\pi$-flux loops (colored in red). By increasing $J/t$, we observe the transition from a fully-connected chain to an AB-trimer covering and, finally, to an AB-dimer configuration. In the thermodynamic limit, the transition points are given by $J^\star/t = {3 \sqrt{3}}/{\pi} - \sqrt{2}$ and $J/t = \sqrt{2}$.