Robustness of the Spin-Chern number

The Spin-Chern ($C_s$) was originally introduced on finite samples by imposing spin boundary conditions at the edges. This definition lead to confusing and contradictory statements. On one hand the original paper by Sheng and collaborators revealed robust properties of $C_s$ against disorder and certain deformations of the model and, on the other hand, several people pointed out that $C_s$ can change sign under special deformations that keep the bulk Hamiltonian gap open. Because of the later findings, the Spin-Chern number was dismissed as a true bulk topological invariant and now is viewed as something that describes the edge where the spin boundary conditions are imposed. In this paper, we define the Spin-Chern number directly in the thermodynamic limit, without using any boundary conditions. We demonstrate its quantization in the presence of strong disorder and we argue that $C_s$ is a true bulk topological invariant whose robustness against disorder and smooth deformations of the Hamiltonian have important physical consequences. The properties of the Spin-Chern number remain valid even when the time reversal invariance is broken.