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New two dimensional systems like surface of topological insulator andgraphene offer a possibility to experimentally investigate situationsconsidered "exotic" just a decade ago. One of those is the quantum phasetransition of the "chiral" type in electronic systems with relativisticspectrum. Phonon mediated ("conventional") pairing in the Dirac semimetalappearing on the surface of topological insulator leads to transition into achiral superconducting state, while exciton condensation in these gaplesssystems has been envisioned long time ago in the physics of the narrow bandsemiconductors. Starting from the microscopic Dirac Hamiltonian with localattraction or repulsion, the BCS type gaussian approximation is developed inthe framework of functional integrals. It is shown that due to an"ultra-relativistic" dispersion relation there is a quantum critical pointgoverning the zero temperature transition to a superconducting or the excitoncondensed state. The quantum transitions that have critical exponents verydifferent from the conventional ones. They belong to the chiral universalityclass. We discuss the application of these results to recent experiments inwhich surface superconductivity was found in topological insulators andestimate feasibility of the phonon pairing.

Chiral universality class of the normal-superconducting and the exciton condensation transition on the surface of topological insulator