Topological superconductors from one-dimensional periodically modulated Majorana chains

By analogy to the topological models of fermions in one-dimensional periodically modulated lattices, we provide a systematic method to construct topological superconductors in BDI class. We then create superlattices of Majorana fermions to interpolate several Majorana chains, and realize topological superconductors with arbitrary winding numbers. Two kinds of chiral symmetries are identified in the systems with multiple chains. Of the two winding numbers associated to the chiral symmetries, one counts the number of zero-energy modes, while the other counts the difference of the numbers of α- and β-type Majorana zero states. We also show that one α- and one β-type Majorana zero modes collapse into fractional charged zero states when they are spatially intertwined. In the systems with odd number of chains, it induces topological superconductors with coexistence of fractional charged zero states and Majorana zero states. Finally by introducing symmetry breaking term, we present an intuitive explanation of the Z(2) nature of the topological invariant in the D class.