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The topological protected electronic states on the surface of a topologicalinsulator can progressively change their Fermi cross-section from circular to asnowflake shape as the chemical potential is increased above the Dirac pointbecause of an hexagonal warping term in the Hamiltonian. Another effect ofwarping is to change the spin texture which exists when a finite gap isincluded by magnetic doping, although the in-plane spin component remainslocked perpendicular to momentum. It also changes the orbital magnetic moment,the matrix element for optical absorption and the circular dichroism. We findthat the Fermi surface average of z-component of spin is closely related to thevalue of the Berry phase. This holds even when the Hamiltonian includes asubdominant non-relativistic quadratic in momentum term (which providesparticle-hole asymmetry) in addition to the dominant relativistic Dirac term.There is also a qualitative correlation between $\left\langle S_z\right\rangle$ and the dichroism. For the case when the chemical potentialfalls inside the gap between valence and conduction band, the Hall conductivityremains quantized and unaffected in value by the hexagonal warping term.

Hexagonal warping on spin texture, Hall conductivity and circular dichroism of Topological Insulator