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

Marc Martí-Sabaté, Dani Torrent

In this work, we derive some analytical properties of Berry's phase in
one-dimensional quantum and classical crystals, also named Zak's phase. It is
commonly assumed in the literature that this phase can only take the values 0
or $\pi$ for a centrosymmetric crystal, however we have found that this
assumption is inaccurate and it has its origin in a wrong assumption on Zak's
original paper. We provide a general demonstration that Zak's 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
also demonstrate that Zak's phase is independent of the origin of coordinates
selected to compute it. 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 material
presents a trivial phase in all the bands of the dispersion diagram. This work
shows that Zak's phase and its connection to edge states in one-dimensional
crystals should be carefully revisited, since the assumption about its
quantization has been widely used in the literature.