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The $\mathbf{Z}_{2}$ topological invariants in 2D and 3D topological superconductors without time reversal symmetry

In theory of topological classification, the 2D topological superconductors without time reversal symmetry are characterized by Chern numbers. However, in reality, we find the Chern numbers can not reveal the whole properties of the boundary states of the topological superconductors. We figure out some particle-hole symmetry related $\mathbf{Z}_{2}$ invariants, which provide more additional information of the topological superconductors than the Chern numbers provide. With the $\mathbf{Z}_{2}$ invariant, we define weak and strong topological superconductors in 2D systems. Moreover, we explain the causes of mismatch between the Chern numbers and the numbers of boundary states in topological superconductors, and claim that the robust Majorana zero modes are characterized by the $\mathbf{Z}_{2}$ invariant rather than the Chern numbers. We also extend the $\mathbf{Z}_{2}$ invariants to 3D non-time-reversal symmetry superconductor systems including gapful and gapless situations.