Critical phenomena and quantum phase transitions are paradigmatic concepts in modern condensed matter physics. A central example in the field of mesoscopic physics is the localization-delocalization (metal-insulator) quantum phase transition driven by disorder -- the Anderson transition. Although the notion of localization has appeared half a century ago, this field is still full of surprising new developments. In this talk, I will discuss novel materials---graphene and topological insulators---where the charge carriers have a character of 2D Dirac fermions. A non-trivial topological nature of the corresponding theories leads to remarkable physical consequences. In graphene, the charge carriers turn out to be topologically protected from Anderson localization, as long as the intervalley scattering can be neglected. A similar phenomenon takes place for surface modes of a 3D topological insulator. Furthermore, we show that, in the latter system, the Coulomb interaction leads to emergence of a quantum critical state. Remarkably, the interaction-induced critical state emerges on the surface of a three-dimensional topological insulator without any adjustable parameters. This ``self-organized quantum criticality'' is a novel concept in the field of interacting disordered systems. Finally, we predict a quantum spin-Hall transition between the normal and topological insulator phases in 2D that occurs via a similar (or identical) quantum critical point.
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