Topological phases are quantum phases of matter that are characterized by robust ground state degeneracy and quantized geometric phases. Transitions between different topological phases are usually called topological phase transitions, which are characterized by discrete jumps of the topological invariant C . Upon tuning one or multiple system parameters M = ( M 1 , M 2 , … … ) , C jumps abruptly from one integer to another at the critical point M c . Typically, the topological invariant C takes the form of an integration of a curvature function F ( k ) in momentum space :