Quasi-periodicity enforces Berry phase discontinuity in crystals

The Berry phase of an isolated electronic state is evaluated for different integration paths in reciprocal space, thanks to Riemann-Bloch states providing a consistent description in the whole Brillouin zone. The relationship with the bounded position operator is investigated for the open-path Berry phase and with different form factors of the constituent Gaussian type atomic orbitals (GTOs). When the Berry phase is evaluated over a closed loop in reciprocal space, a regularisation procedure in the complex plane is adopted and additional \textit{quasi}-symmetries are exploited here for the first time in order to rationalise the occurrence of non-analytical jumps as a function of the winding number or of the GTO broadening.