  We quantum-simulated the 2D Harper-Hofstadter (HH) lattice model in a highly
elongated tube geometry -- three sites in circumference -- using an atomic
Bose-Einstein condensate. In addition to the usual transverse (out-of-plane)
magnetic flux, piercing the surface of the tube, we threaded a longitudinal
flux $\Phi_{\rm L}$ down the axis of the tube This geometry evokes an
Aharonov-Bohm interferometer, where noise in $\Phi_{\rm L}$ would readily
decohere the interference present in trajectories encircling the tube. We
observe this behavior only when transverse flux is a rational fraction of the
flux-quantum, and remarkably find that for irrational fractions the decoherence
is absent. Furthermore, at rational values of transverse flux, we show that the
time evolution averaged over the noisy longitudinal flux matches the time
evolution at nearby irrational fluxes. Thus, the appealing intuitive picture of
an Aharonov-Bohm interferometer is insufficient. Instead, we quantitatively
explain our observations by transforming the HH model into a collection of
momentum-space Aubry-Andr\'{e} models.
