Average spin Chern number

In this work we propose the average spin Chern number (ASCN) as an indicator of the topological significance of the spin degree of freedom within insulating materials. Whenever this number is a nonzero even integer, it distinguishes the material as a spin Chern insulator, and the number is a topological invariant whenever there is a symmetry that commutes with the spin and protects Chern numbers. If this number is not zero, it indicates that the material has nontrivial spin transport properties, and it lies close to the value of the spin Hall conductivity (SHC) within the band gap. For systems where the spin commutes with the Hamiltonian, the ASCN matches the SHC. When the noncommutativity of the spin with the Hamiltonian cannot be neglected, both values are nonzero simultaneously. The ASCN is therefore a good complement for the intrinsic contribution of the SHC, and permits us to detect topological information of the material, which is not possible alone from the value of the SHC.