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We present a detailed mass classification of all possible zero-energy modesin one-dimensional Dirac systems. By introducing a linear mass term into theDirac Hamiltonian, we find that the topologically protected zero-energy modeshave the mass-momentum duality. Based on the duality, we classify threefundamental zero-energy modes in 2 * 2 subspaces respectively: solitons insublattice subspace, Majorana zero modes in Nambu subspace, and magneticzero-energy modes in spin subspace. Within the mechanism of mass competition,isolated zero-energy modes emerge by lifting Kramers degeneracy in the combined8 * 8 inner space and its 4 * 4 subspaces. We also propose experimental methodsof manipulating possible masses in Dirac Hamiltonians. Our classificationscheme could easily be extended to 2D or 3D systems and applied to investigatetopologically protected states in other fields.

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