Casimir Torque between Topological Insulators: a Physical Implication of the Surface State Hexagonal Warping Effect

We use a variation of the Lifshitz formula to calculate the anisotropic Casimir energy density between two topological insulators in the vacuum. We find that the hexagonal warping effect can induce a Casimir torque between the two topological insulators, $T_c\propto\sin(6\theta)$ with twisted angle $\theta$. The maximal Casimir torque at $\theta=\pi/12$ is estimated to be $\sim 10^{-19} N\cdot{m}/rad$ for Bi$_2$Te$_3$ on the [111] surface when the distance between the two topological insulators is about 20 $nm$ and the surface areas are taken to be $\sim 1 cm^2$.