Topological phase and Chern number

Question

the relation between topological phase and Chern numbers is unclear to me.

For Haldane model if the Chern numbers of its two bands go from (+1,-1) to (0,0), we say that it goes from topological phase to trivial phase.

Now if (as an example) the Chern numbers go from (+1,-1) to (-1,+1), does it also count as topological phase transition? If I choose the number of edge states as topological order, it seems to me that both are in the same phase. If it's the case, what is the physical sense of having a Chern number changing from +C to -C?

Thank you very much.

Answer 1

When the Chern numbers of the two bands go from $(1,-1)$ to $(-1,1)$, the Chern number of the occupied band changes sign. Even though the number of chiral edge modes is the same, they have opposite chirality (left moving vs right moving), and also the Hall conductance changes sign. So they are different topological phases. You can't just "choose the number of edge states as topological order".