Hexagonal Warping Effects in the Surface States of Topological Insulator Bi${}_2$Te${}_3$

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 the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral structures. We show that this hexagonal warping term naturally explains the observed hexagonal snow-flake Fermi surface. The strength of hexagonal warping is characterized by a single parameter, which is extracted from the size of the Fermi surface. We predict a number of testable signatures of hexagonal warping in spectroscopy experiments on Bi${}_2$Te${}_3$. We also explore the possibility of a spin-density wave due to strong nesting of the Fermi surface.