As we know, due to the strong spin-orbit coupling in TIs, the spins are locked in the perpendicular direction of momentum; as a matter of fact, electrical conduction is robust against the back scattering at the edges or on the surfaces on the TIs. Majorana fermions, topological superconductivity, the absence of backscattering from nonmagnetic impurities.
I want to know that what is Berry phase and what is it's physical significance in TIs.
When you transform in time slowly enough an Hamiltonian into another Hamiltonian without close the gap in the middle of the transformation you can transit adiabatically from the ground-state of the initial Hamiltonian to the ground state of the final Hamiltonian. When you do that you end up accumulating a phase, which can be divided in two contributions: the first one called 'dynamical' phase is simply related to the complex oscillation of the wave-function associated to the parameter dependent ground-state energy, the remaining contribution is called the Berry phase or 'geometric phase'. The dynamical phase depends on how much time you took to do the transformation. The Berry phase, which can be written as an integral over the trajectory in parameter space of the so-called Berry connection, only depends on the shape and topology of the trajectory in the parameter space. While the Berry connection is not a gauge-invariant quantity, as it depends on your convention in defining the phases of the various ground-state, the Berry curvature, which is a generalized vector product derivative of the Berry connection in the parameter space, is gauge-invariant.
These quantities play a major role in defining the topology of a material, as the Hall conductivity is in fact proportional to the integral of the Berry curvature of all occupied single-particle electronic states over the Brillouin zone: as a consequence only materials with a non-zero Berry curvature can potentially present a non-trivial topology and edge physics.
For more information I recommend the wonderful lecture notes of David Tong: http://www.damtp.cam.ac.uk/user/tong/qhe.html
The main physical significance of the Berry phase in the topological insulators is given by what is called the TKNN invariant. This invariant allows to define properly the electric polarization P as a topological quantity within a topological insulator.
A very good book for this subject is:
https://poboiko.bitbucket.io/qm/fall16/seminar-6-adiabaticheskoe-priblizhenie/topological_insulators.pdf
José Lebreuilly Thank you very much for for giving precious time to answer this question. It helped me to understand the concept clearly. @
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