Hexagonal Warping Effects on the Surface of Topological Insulators*

A single two-dimensinoal Dirac fermion state has been recently observed on the surface of topological insulator Bi$2$Te$3$ by angle-resolved photoemission spectroscopy(ARPES). We study the surface band structure using $k \cdot p$ theory and find an unconventional hexagonal warping term, which is a counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral latices. We find hexagonal warping naturally explains the observed snowflake shape of the Fermi surface, and extract its magnitude. We predict a number of testable signatures of hexagonal warping in ARPES and STM experiments on Bi$_2$Te$_3$. We also explore the possibility of a spin-density wave due to strong nesting of the Fermi surface. Ref: L. Fu, arXiv:0908.1418, PRL in press