Invariant points amidst gauge sensitivity in cylindrical and toroidal lattices

We examine the spectrum for lattices with cylindrical and toroidal topology subject to Abelian gauge potentials. Gauges that are equivalent in planar lattices with trivial topology develop differences due to the periodic boundary conditions. But some residual gauge equivalency, evident in the spectrum, is found to remain under specific conditions. This is associated with the gauge structure being commensurate with the lattice periodicity, and the behavior of the associated field along the symmetry axes. The interplay of the gauge choice and the periodic boundary conditions leads to a class of persistent degeneracies that are found to be robust against vast changes in system parameters and even under change of topology from cylinder to torus.