Description 
Topological crystalline insulators (TCIs) have lowenergy surface states in certain high symmetry directions, protected by crystalline symmetry instead of the usual timereversal symmetry. The nontrivial topology is mathematically characterized by a mirror Chern number. The IVVI semiconductors SnTe and related semiconducting alloys Pb_{1x}Sn_{x}(Te,Se) were recently predicted to belong to the TCI class, and topologically protected surface states with novel electronic dispersions were found to be present on certain surfaces invariant under reflection symmetry.
The four topologically protected electron bands on the (001) surface of the crystalline topological insulator Pb(x)Sn(1x)Te include two containing TypeII van Hove singularities, accessible at relatively small values of doping. The diverging density of states associated with the twodimensional Van Hove singularities enhances the possibility of Fermisurface instabilities on the TCI surface, brought about by weak repulsive interparticle interactions.
Wavefunctions corresponding to electrons in these bands have nontrivial geometric phases that effectively impart a momentum dependence to the interparticle interactions in a given band, that in turn, is reflected in the symmetries of the order parameters. Using a “multipatch” parquet renormalization group scheme, we study the effect of repulsive electron interactions on the competition of different electronic phases on the (001) surface when the chemical potential is tuned to the vicinity of the Van Hove singularities. Over a wide region of parameter space of repulsive interactions, we show that a chiral pwave superconducting phase is favoured. Implications for experiment are discussed.
