The analytical properties of Zak's phase in one-dimensional quantum and classical crystals

Absence of Quantization of Zak's Phase in One-Dimensional Crystals

We derive some analytical properties of the quantum and classical phase of one-dimensional quantum and classical crystals, also named zak s phase.We provide a general demonstration that this phase can take any value for a non-symmetric crystal but it is strictly zero when it is possible to find a unit cell where the periodic modulation is symmetric.We provide numerical examples verifying this behaviour for both electronic and classical waves (acoustic or photonic).We analyze the weakest electronic potential capable of presenting asymmetry, as well as the double-dirac delta potential, and in both examples it is found that zak s phase varies continuously as a function of a symmetry-control parameter, but it is zero when the crystal is symmetric.For classical waves, the layered material is analyzed, and we demonstrate that we need at least three components to have a non-trivial zak s phase, showing therefore that the binary layered materialpresents a trivial phase in all the bands of the dispersion diagram.