Appearance and disappearance of edge states in topological insulators

We investigate the relationship between spin Chern numbers and edge state properties in general situations, where the time-reversal symmetry may be broken. As an example, we consider a thin film of three-dimensional topological insulators sandwiched between two ferromagnetic insulators with an antiparallel magnetization configuration. A topological quantum spin Hall phase with quantized spin Chern numbers C-+/- = +/- 1, and a trivial insulator with C-+/- = 0 are found in different parameter regions. With tuning parameters, the quantum phase transition between the two phases can occur through closing of the spin spectrum gap rather than energy gap. It is further shown that for a junction between samples with different parameters, appearance of edge states at the interface is always related to the mismatch of spin Chern numbers, independent of symmetries.