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Topologically-ordered phases of matter, although stable against localperturbations, are usually restricted to relatively small regions in phasediagrams. Their preparation requires thus a precise fine tunning of thesystem's parameters, a very challenging task in most experimental setups. Inthis work, we investigate a model of spinless fermions interacting withdynamical $\mathbb{Z}_2$ gauge fields on a cross-linked ladder, and showevidence of topological order throughout the full parameter space. Inparticular, we show how a magnetic flux is spontaneously generated through theladder due to an Aharonov-Bohm instability, giving rise to topological ordereven in the absence of a plaquette term. Moreover, the latter coexists herewith a symmetry-protected topological phase in the matter sector, that displaysfractionalised gauge-matter edge states, and intertwines with it by aflux-threading phenomenon. Finally, we unveil the robustness of these featuresthrough a gauge frustration mechanism, akin to geometric frustration in spinliquids, allowing topological order to survive to arbitrarily large quantumfluctuations. In particular, we show how, at finite chemical potential,topological solitons are created in the gauge field configuration, which boundto fermions forming $\mathbb{Z}_2$ deconfined quasi-particles. The simplicityof the model makes it an ideal candidate where 2D gauge theory phenomena, aswell as exotic topological effects, can be investigated using cold-atom quantumsimulators.

Robust topological order in fermionic $\mathbb{Z}_2$ gauge theories: from Aharonov-Bohm instability to soliton-induced deconfinement