Gauge Sensitive Features of Quantum States on a Cylindrical Lattice

We examine the spectrum and the quantum states on small lattices with cylindrical topology subject to gauge potentials, natural or synthetic, that introduce position-dependent phase in the hopping terms. General gauge invariance assumed in open or infinite lattices is lost due to the periodic boundary condition and the spectrum and eigenstates can become sensitive to specific gauge assumed. This can have significant effects on the behavior of the system, such as those relevant in phenomena associated with the quantum Hall effect. We examine several distinctive features in the spectrum including avoid crossings and persistent intersections, and their dependence on the choice of gauge and other system parameters. The origins of those features are traced by comparison with systems with different boundary conditions and configurations. We correlate the impact of those spectral features with the behavior of the quantum states of the system. We determine under what conditions invariance under specific gauge choices can be restored.