Coexistence of strong and weak Majorana zero modes in an anisotropic XY spin chain with second-neighbor interactions

We theoretically investigate Majorana zero modes emerging in an anisotropic XY spin chain with second-neighbor interactions. The spin chain is mathematically equivalent to the Kitaev chain comprised of spinless fermions if only nearest-neighbor interactions are considered. Recent studies on the Kitaev chain with long-ranged interactions, have presented coexistence of several Majorana zero modes. Investigating the topological phase diagram of the anisotropic XY spin chain with second-neighbor interactions, we find coexistence of several Majorana zero modes similar to the Kitaev chain. However, we confirm that one of the zero modes is restricted into a Hilbert subspace. The mode is regarded as a so-called weak zero mode that should exhibit thermodynamical properties different from a (strong) zero mode appearing in the whole Hilbert space. Since the quantum statistics of spin is the same as a hard-core boson, we expect an experimental realization of the weak mode in not only quantum spin materials but also optical lattices for cold atom systems.