Berry phase oscillations in a simple model

While the Berry phase is often introduced in a variety of contexts, it is generally neglected in solid state courses due to a lack of simple models and the difficulty of the required mathematical techniques. In this paper, we calculate the Berry phase for a simple model, a one-dimensional Dirac comb with a secondary peak treated as a perturbation, using relatively straightforward techniques. Interestingly, we find that for this specific model the phase exhibits a band-dependent beat-like phenomenon as the secondary peak sweeps across the unit cell. Using perturbation theory we obtain an explicit expression for these oscillations, in excellent agreement with numerical results.