Dynamics of impurity in the environment of Dirac fermions

We study the dynamics of a nonmagnetic impurity interacting with the surface states of a 3D and 2D topological insulator (TI). Employing the linked cluster technique we develop a formalism for obtaining the Green's function of the mobile impurity interacting with the low-energy Dirac fermions. We show that for the non-recoil case in 2D, the Green's function in the long-time limit has a power-law decay in time implying the breakdown of the quasiparticle description of the impurity. The spectral function in turn exhibits a weak power-law singularity. In the recoil case, however, the reduced phase-space for scattering processes implies a non-zero quasiparticle weight and the presence of a coherent part in the spectral function. Performing a weak coupling analysis we find that the mobility of the impurity reveals a T−3/2 divergence at low temperatures. In addition, we show that the Green's function of an impurity interacting with the helical edge modes (surface states of 2D TI) exhibit power-law decay in the long-time limit for both the non-recoil and recoil case (with low impurity momentum), indicating the break down of the quasiparticle picture. However, for impurity with high momentum, the quasiparticle picture is restored. The mobility of the heavy impurity interacting with the helical edge modes exhibits unusual behaviour. It has an exponential divergence at low temperatures which can be tuned to a power-law divergence, T−4, by the application of the magnetic field.