We consider chiral electrons moving along the 1D helical edge of a 2D topological insulator and interacting with a disordered chain of Kondo impurities. Assuming the electron-spin couplings of random anisotropies, we map this system to the problem of the pinning of the charge density wave by the disordered potential. This mapping proves that arbitrary weak anisotropic disorder in coupling of chiral electrons with spin impurities leads to the Anderson localization of the edge states. Introduction. -Recent interest to the topological insulators (TI) is inspired by remarkable properties of their boundaries [1][2][3][4]. While the charged excitations in the bulk of TI are gapped as they are in conventional band insulators, the boundary can host gapless excitations. In the presence of a potential disorder there appear bulk electronic states in the gap. However such states are localized and thus cannot support any DC current. At the same time the boundary states of TI remain extended making the system conductive. The prediction is most dramatic for the one-dimensional (1D) edge of a two-dimensional (2D) TI where right and left moving electrons carry opposite spins: the conductance remains perfect (e 2 /h) because the potential disorder cannot flip spins of the edge electrons and thus cannot back-scatter them. As a result the usual 1D Anderson localization does not occur. On the other hand, surface imperfections in real crystals are by no means limited by the potential disorder. Since the existing experiments [2, 4] do not show perfect conductance even for short TI edges, it is important to understand the possible sources of the backscattering.

We analyze the effect of the electron-electron interaction on the resistivity of a metal near a Pomeranchuk quantum phase transition (QPT). We show that umklapp processes are not effective near a QPT, and one must consider both interactions and disorder to obtain a finite and T dependent resistivity. By power counting, the correction to the residual resistivity at low T scales as AT((D+2)/3) near a Z=3 QPT. We show, however, that A=0 for a simply connected, convex Fermi surface in 2D, due to the hidden integrability of the electron motion. We argue that A>0 in a two-band (s-d) model and propose this model as an explanation for the observed T((D+2)/3) behavior.

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