Fig. 3: Phase-diagram at h = 0.

Contour plot of the filling fraction ν = N/L as a function of J/t and μ/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 ℓ_AB indicated by the red numbers inside each region. For instance, the region on the left corresponds to the un-partitioned chain of ℓ_AB = L connected sites and no visons, whereas that in the bottom right corner is broken at every site ℓ_AB = 1 by completely filling the loops with visions. A schematic representation of the flux configurations is depicted in the larger regions, in which the periodic arrangement of visons (red π-flux circles) and 0-flux loops (green circles) with periodicity ℓ_AB is manifest. Except for the small unlabeled closed area with cages of size ℓ_AB = 5, which host four particles each, the remaining arrangement of Aharonov-Bohm cages with ℓ_AB ∈ {2, 3, 4, 5} contain two particles each. Notice that the reference values \({J}^{\star }/t=3\sqrt{3}/\pi -\sqrt{2}\), \(J^{\prime} /t=\sqrt{5}-\sqrt{2}\) and \(J/t=\sqrt{2}\) can be calculated analytically in the thermodynamic limit - see Supplementary Note 3 for details.