Long-range Coulomb Interaction effects on Topological Phase Transitions between Semi-metals and Insulators

Topological Dirac / Weyl semi-metals is generically protected by a chiral symmetry, which refers to a lattice symmetry that protects energy gapless-ness of the semi-metals. A mirror symmetry in the Dirac semi-metal of the distorted spinel, BiZnSiO₄, is one prime example, and thus a chiral symmetry breaking transition is intrinsically tied to topological phase transitions. We find that a chiral symmetry order-parameter and the instantaneous long range Coulomb interaction in topological semi-metals reveal characteristic quantum critical behaviors. We show that a topological transition associated with a chiral symmetry is stable under the presence of the Coulomb interaction. Furthermore, the electron velocity always becomes faster than the order parameter velocity in three spatial dimensions. Thus, the transition is not relativistic. This implies a supersymmetry is forbidden by the long range Coulomb interaction in some chiral symmetry breaking transitions. Exact universal ratios of physical quantities such as energy gap are obtained, and further implications of our calculations with experimental consequences are also discussed.