Majorana Edge States and Braiding in an Exactly Solvable One-dimensional Spin Model

We derive an exactly solvable one-dimensional (1D) spin model from the three-band Hubbard model with a strong spin-orbit coupling by introducing U(1) gauge fields to the isospin states. We find that it has a topological nontrivial phase characterized by Majorana end modes which are protected by a new Z_2 topological invariant related to the parity of the lattice sites (odd or even number of sites) in the spin chain. With the protection of this Z_2 topological invariant, a novel braiding of two Majorana edge states in this strictly geometric 1D chain is realized. We also discuss the possible realization of the gauge fields.