  Three-dimensional topological insulators are characterized by the presence of
protected gapless spin helical surface states. In realistic samples these
surface states are extended from one surface to another, covering the entire
sample. Generally, on a curved surface of a topological insulator an electron
in a surface state acquires a spin Berry phase as an expression of the
constraint that the effective surface spin must follow the tangential surface
of real space geometry. Such a Berry phase adds up to pi when the electron
encircles, e.g., once around a cylinder. Realistic topological insulators
compounds are also often layered, i.e., are anisotropic. We demonstrate
explicitly the existence of such a pi Berry phase in the presence and absence
(due to crystal anisotropy) of cylindrical symmetry, that is, regardless of
fulfilling the spin-to-surface locking condition. The robustness of the spin
Berry phase pi against cylindrical symmetry breaking is confirmed numerically
using a tight-binding model implementation of a topological insulator nanowire
penetrated by a pi-flux tube.
