Undergrad Berry phase in the Brillouin zone

[hokhani]

As mentioned in the literatures, the definition of the Berry phase is meaningful only for non-orthogonal states. However, in the topological insulators it is defined for quantum states of a matter which are orthogonal. How to justify this inconsistency?

 

[Demystifier]

Please provide references for both claims!

 

[hokhani]

Of course, in the book “A Short Course on Topological Insulators” by János K. Asbóth et. al. chapter 2, introduces the Berry phase based on the relative phase of two non-orthogonal quantum states. Then, in chapter 3 (Eq. 14), introduces the bulk electric polarization as the Berry phase of the occupied band across the Brillouin zone. I appreciate any help.

 

[DrDu]

If I remember correctly, you are comparing orbitals within one lattice cell for different k values. These orbitals are not orthogonal. This is not in conflict with the global many electron states being orthogonal. Let $0< |\langle \phi_1| \phi_2\rangle| <1$. Then the two local functions $\phi_{1/2}$ are not orthogonal. But the overlap of the total function of N cells goes as $|\langle \phi_1| \phi_2\rangle|^N \to 0$ if N goes to infinity.