Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory

With the advent of quantum simulators, exploring exotic collective phenomena in lattice models with local symmetries and unconventional geometries is at reach of near-term experiments. Motivated by recent progress in this direction, we study a \mathbb{Z}_2 ℤ 2 lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell hosting the \mathbb{Z}_2 ℤ 2 gauge fields. Elementary Wilson loops along pairs of these bonds allow to identify a dynamical gauge-invariant flux, responsible for Aharonov-Bohm-like interference effects in the tunneling dynamics of charged matter residing on the vertices. Focusing on an odd number of links, we show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability. We find inhomogeneous phases in which an ordered pattern of the gauge fluxes spontaneously breaks translational invariance, and intertwines with a bond order wave for the gauge-invariant kinetic matter operators. Long-range order is shown to coexist with symmetry protected topological order, which survives the quantum fluctuations of the gauge flux induced by an external electric field. Doping the system above half filling leads to the formation of topological soliton/anti-soliton pairs interpolating between different inhomogeneous orderings of the gauge fluxes. By performing a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization. Quasiparticles carrying fractional charge and bound at the soliton centers can be arbitrarily separated without feeling a confining force, in spite of the long-range attractive interactions set by the small electric field on the individual integer charges.