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Topological order, the hallmark of fractional quantum Hall states, isprimarily defined in terms of ground-state degeneracy on higher-genusmanifolds, e.g. the torus. We investigate analytically and numerically thesmooth crossover between this topological regime and the Tao-Thouless thintorus quasi-1D limit. Using the wire-construction approach, we analyze anemergent charge density wave (CDW) signifying the break-down of topologicalorder, and relate its phase shifts to Wilson loop operators. The CDW amplitudedecreases exponentially with the torus circumference once it exceeds thetransverse correlation length controllable by the inter-wire coupling. By meansof numerical simulations based on the matrix product states (MPS) formalism, weexplore the extreme quasi-1D limit in a two-leg flux ladder and present asimple recipe for probing fractional charge excitations in the $\nu=1/2$Laughlin-like state of hard-core bosons. We discuss the possibility ofrealizing this construction in cold-atom experiments. We also address theimplications of our findings to the possibility of producing non-Abelian zeromodes. As known from rigorous no-go theorems, topological protection for exoticzero modes such as parafermions cannot exist in 1D fermionic systems and theassociated degeneracy cannot be robust. Our theory of the 1D-2D crossoverallows to calculate the splitting of the degeneracy, which vanishesexponentially with the number of wires, similarly to the CDW amplitude.

Pretopological fractional excitations in the two-leg flux ladder