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Mass classification and manipulation of zero modes in one-dimensional Dirac systems

We present a detailed mass classification of all possible zero-energy modes in one-dimensional Dirac systems. By introducing a linear mass term into the Dirac Hamiltonian, we find that the topologically protected zero-energy modes have the mass-momentum duality. Based on the duality, we classify three fundamental zero-energy modes in 2 * 2 subspaces respectively: solitons in sublattice subspace, Majorana zero modes in Nambu subspace, and magnetic zero-energy modes in spin subspace. Within the mechanism of mass competition, isolated zero-energy modes emerge by lifting Kramers degeneracy in the combined 8 * 8 inner space and its 4 * 4 subspaces. We also propose experimental methods of manipulating possible masses in Dirac Hamiltonians. Our classification scheme could easily be extended to 2D or 3D systems and applied to investigate topologically protected states in other fields.