Stability of the symmetry-protected topological phase and Ising transitions in a disordered U(1) quantum link model on a ladder

We revisit the U(1) quantum link model in a ladder geometry, confirming the presence of symmetry protected topological phase at zero fermion mass and finding, by finite-size scaling, that the critical exponent and the central charge are consistent with the Ising universality class for all phase transitions observed. A blind application of the Harris criterion would suggest that this criticality is lost in the presence of the disorder, however, it turns out not to be the case. We have found that the transitions survive the disorder affecting ladder's rung hoppings only, disappearing only for quite strong disorder amplitude, and they remain of Ising universality class. Only the disorder affecting ladder's legs destroys the nonzero mass phase criticality. The symmetry protected topological phase for zero mass survives a small disorder of any kind. The observed robustness against disorder is explained qualitatively using field-theoretic arguments.