The nontrivial topological phases of a one-dimensional non-Hermitian dimerized lattice with spin-orbit coupling and Zeeman field

We investigate in details the topological properties of a one-dimensional non-Hermitian dimerized lattice with spin-orbit coupling and Zeeman field. Based on the Bloch band theory, the winding number and the chiral index are found to be the topological invariants due to the effective chiral symmetry. We give a geometrical interpretation of the topological number and use it to distinguish the different topological phases. According to the geometrical meaning of the topological number, the spin-orbit coupling can change the hopping terms in the unit cell and therefore change the winding vectors. However, the Zeeman field cannot modify the trajectory shape of the winding vectors, but shift them along the x-axis. The topologically different phases of different spin-orbit coupling are studied and characterized by the different topological indexes. Furthermore, we demonstrate that the relationship between topological phases and the topological indexes by checking whether the zero-mode solution exists by diagonalizing the Hermitian operator H†H under the open boundary condition.