Effect of electron-electron interaction on transport in non-Galilean invariant Fermi systems: Application to the Bi_2Te_3 family of 3D Topological Insulators

We study the effect of electron-electron interaction on the resistivity of a metal where umklapp scattering is either not effective or suppressed. This can happen in cases such as in a metal near a Pomeranchuk quantum phase transition or in a system with low density of carriers, e.g., the surface states of three-dimensional (3D) topological insulators. In such cases, one must consider both interactions and disorder to obtain a finite and T dependent resistivity. The existence of the Fermi-liquid (T^2) term in resistivity of a two-dimensional (2D) metal, as we show, then depends on 1) dimension (2D vs 3D), 2) geometry (concave vs convex), and 3) topology (simply vs multiply connected) of the Fermi surface. In the case of 3D topological insulators of the Bi_2Te_3 family, upon doping the Fermi surface of 2D metallic surface states changes its shape from convex to concave due to hexagonal warping, while still being too small to allow for umklapp scattering. We show that the T^2 term in the resistivity is present only in the concave regime and demonstrate that the resistivity obeys a universal scaling form valid for an arbitrary 2D Fermi surface near a convex/concave transition.