Non-trivial Berry phase for an asymmetric one-dimensional potential in the free electron limit

We find a non-trivial Berry phase for a finite potential, even in the delocalized-electron limit. We obtain this result for a one-dimensional, periodic sawtooth potential, though our analysis is applicable to other systems. We show that this result is due to both the periodicity of the time-independent wave function and the effects of interband degeneracies. We also find, for the sawtooth potential, that each band is uniquely characterized by its Berry phase’s functional dependence on the potential’s parameters. Thus, each band may be identified by the behavior of its Berry phase.