Field-theoretical approach to Anderson localization in 2D disordered fermionic systems of chiral symmetry classes (BDI, AIII, CII) is developed. Important representatives of these symmetry classes are random hopping models on bipartite lattices at the band center. As was found by Gade and Wegner two decades ago within the sigma-model formalism, quantum interference effects in these classes are absent to all orders of perturbation theory. We demonstrate that the quantum localization effects emerge when the theory is treated nonperturbatively. Specifically, they are controlled by topological vortexlike excitations of the sigma models by a mechanism similar to the Berezinskii-Kosterlitz-Thouless transition. We derive renormalization-group equations including these nonperturbative contributions. Analyzing them, we find that the 2D disordered systems of chiral classes undergo a metal-insulator transition driven by topologically induced Anderson localization. We also show that the topological terms on surfaces of 3D topological insulators of chiral symmetry (in classes AIII and CII) overpower the vortex-induced localization.

Similar vortex excitations also emerge in systems with strong spin-orbit interaction (symplectic symmetry class AII). Such systems may exhibit topological insulator state both in three and two dimensions. Interplay of nontrivial topology and Coulomb repulsion induces a novel critical state on the surface of a 3D topological insulator. Remarkably, this interaction-induced criticality, characterized by a universal value of conductivity, emerges without any adjustable parameters. Interaction also leads to a direct transition between trivial insulator and topological insulator in 2D (quantum-spin-Hall transition) via a similar critical state. The nature of this latter critical state is closely related to the effects of vortices within the Finkelstein sigma model.