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In this work, we derive some analytical properties of Berry's phase inone-dimensional quantum and classical crystals, also named Zak's phase, whencomputed with a Fourier basis. We show that Zak's phase can be divided in twoterms: a global phase required to make the Bloch wave periodic in the Brillouinzone and an internal phase which measures the relative delay of the differentFourier terms within the Brillouin zone. While the former phase is dependent onthe origin of coordinates of the unit cell, the latter is independent of it, sothat it can be interpreted as an internal property of the band itself. We showthat this internal phase is always zero for a symmetric crystal while it cantake any value when this symmetry is broken, showing therefore that it can beinterpreted as a measure of the assymetry of the band. Since for a symmetriccrystal Zak's phase is entirely determined by the global part, we show thatthis can be easily calculated by means of the parity of the Fourier terms atthe center and edge of the Brillouin zone, being therefore unnecessary theintegration of the modes through the unit cell and the entire Brillouin zone.We provide numerical examples analyzing the internal part for both electronicand classical waves (acoustic or photonic). We analyze the weakest electronicpotential capable of presenting asymmetry, as well as the double-Dirac deltapotential, and in both examples it is found that the internal phase variescontinuously as a function of a symmetry-control parameter, but it is zero whenthe crystal is symmetric. For classical waves, the layered material isanalyzed. Although Zak's phase has been mainly studied in connection with theexistence of edge states in finite crystals, we consider that the study of theinternal phase can be more relevant to understand bulk properties of quantumand classical crystals.

Zak's Phase in Non-Symmetric One-Dimensional Crystals