Topological or Thermodynamic phase transitions ?

Topological phases are usually described using the tenfold classification of insulators and superconductors. We offer an alternative approach in the spirit of defect classification in thermodynamic phase transitions for ordered phases. We unveil the deep analogy between these two topics where symbol and order parameter, a non vanishing quantity in the ordered phase, play similar roles. The general expression of the symbol for a given topological symmetry class, is completely analogous to the Landau expansion for the free energy. This new approach allows for a quantitative description of topological phase transitions.